Generation and Characterization of New Quantum States of the Light Field

GGeenneerraattiioonn aanndd CChhaarraacctteerriizzaattiioonn ooff NNeeww QQuuaannttuumm SSttaatteess ooff tthhee LLiigghhtt FFiieelldd

Dissertation zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.), eingereicht am Fachbereich Physik der Universität Konstanz

Submitted by Hauke Hansen April 2000

Abstract

This thesis is concerned with the generation and characterization of new quantum states of the light field employing the nonlinear optical processes of 2- and 3-photon down-conversion. A strong focus is placed on the generation and measure-ment of pulsed optical single-photon Fock states.

The single photon Fock state is one of the most fundamental states of the light field. It is highly non-classical and reveals the particle aspect of the quantized light field most strikingly. We have performed quantum state reconstructions of pulsed single-photon Fock states with measurement efficiencies of up to 33% using the method of pulsed optical homodyne tomography. In this way non-Gaussian mar-ginal distributions of optical quantum states are measured for the first time. Non-Gaussian Wigner functions have been reconstructed from the measured data exhibiting a central dip. This experiment combines the techniques of photon counting and homodyne detection for the first time in a single experiment.

The method of pulsed optical homodyne tomography has been applied to recon-struct the Wigner function and density matrices of the pulsed vacuum and pulsed coherent states. The vacuum state has been reconstructed up to an error of 0.25%. Coherent states with average photon numbers of only a few photons per pulse have been characterized with a fidelity of 99.5%.

A crucial prerequisite for pulsed optical homodyne tomography at the single -photon level is an efficient homodyne system. A pulsed homodyne system with an ultra-low electronic noise of 565 electrons/pulse and a very efficient direct subtrac-tion has been designed, implemented and characterized. Shot noise limited behav-ior up to 2.3· 108 photons per local oscillator pulse corresponding to a maximum subtraction of 83 dB was demonstrated.

A detailed theoretical description of the experiment on the single -photon Fock state tomography is presented including the transverse and spectral structure of the gen-erated photon twins, state preparation by conditional measurements and the homo-dyne detection process. The state preparation fidelity and coincidence count rate is estimated taking into account arbitrary spatial and spectral detection and pump modes.

A theoretical treatment of non-ideal homodyne systems is provided. The homodyne detection efficiency and the optimum parameter settings are calculated under very general conditions. Various inefficiencies such as imperfect mode-matching, beam splitter imbalance or different detector efficiencies are considered.

A proposal is put forth to extend the method of conditional state preparation using repeated 2-photon down-conversions to produce higher n-Fock states as well as arbitrary truncated quantum states of the light field.

The process of 3-photon down-conversion splits a photon into three strongly corre-lated child photons. This process is a generalization of the down-conversion proc-ess to the next order nonlinear process and has so far not been demonstrated ex-perimentally. This new physical process makes accessible optical three particle correlations and a new class of quantum states of the light field – the star states. Theoretical investigations on the experimental feasibility of different schemes in-cluding χ(3)-OPOs, star state generation in single -pass or resonant configurations and GHZ-state generation employing the process of χ(3)-parametric fluorescence in bulk media or fibers are presented.

Hauke Hansen LS Prof. Dr. J. Mlynek Fachber. Physik Universität Konstanz 78457 Konstanz Printed April 28, 2000 File: DissS8.doc, 56914 words, 195 pages

Table of Contents

1 INTRODUCTION: QUANTUM TECHNOLOGY 1

2 THEORY 5

2.1 Quantum States of the Light Field 5 The Wigner Function 8 Classical and Non-classical States of the Light Field 11 Examples of Quantum States 14 What is a Photon? 17

2.2 Optical Quantum Tomography 19 Homodyne Measurements 20 The Inverse Radon Transformation 22 The Abel Transformation 22 Quantum State Sampling 25 Compensation of Experimental Losses 27

2.3 Single -photon Fock State Tomography 28 2.3.1 Introduction 28 2.3.2 Second Harmonic Generation 30

Pulsed Second Harmonic Generation 31 Group Velocity Dispersion 32

2.3.3 2-Photon Down-conversion 33 Pulsed Parametric Fluorescence 34 Expected Count Rate 43 Parametric Amplification with Focused Beams 44

2.3.4 Single-Photon Fock State Preparation 47 Projection onto the Trigger State 48 State Preparation Fidelity 50

2.3.5 Homodyne Measurements 57 Pulsed Homodyne Detection 57 Non-Ideal Homodyne Detectors 60 Theoretical Treatment of the Employed Homodyne Detector 66

2.3.6 Homodyne Measurement of a Single -Photon Fock State 67 2.3.7 Total Measurement Efficiency 70

2.4 Conditional Quantum State Pre paration 72 2.4.1 Higher Fock State Generation 72

Higher Order Parametric Fluorescence 73 Repeated Parametric Fluorescence 74 The Improvement 80

2.4.2 Preparation of Arbitrary Pure Truncated Quantum States 80

3 EXPERIMENT 85

3.1 Outline 85

3.2 The Optical Setup 89

3.3 Light Sources 91 3.3.1 Pump Laser 91 3.3.2 Primary Laser 92 3.3.3 Pulse Picker 95

3.4 Nonlinear Optical Processes 98 3.4.1 Second Harmonic Generation 98 3.4.2 2-Photon Down-conversion 101 3.4.3 Parametric Amplification 103

3.5 Photon Counting 107 3.5.1 Photon Number Distributions 109 3.5.2 Time Interval Distributions 111 3.5.3 State Preparation 114

3.6 Longitudinal and Transverse Mode Matching 117 Pulse Shaping 117 Transverse Mode Matching 119

3.7 Fast Pulsed Homodyne Detection 122 Homodyne Detector Electronics 123 Homodyne System Performance 126 Homodyne Detector Efficiency 129 Gated Homodyne Measurements 130

3.8 Data Acquisition and Monitoring 133 Monitoring of Experimental Parameters 133 Data Acquisition and Analysis 134

3.9 Measurement Efficiency 137

3.10 Quantum State Reconstruction 139 3.10.1 Vacuum State 139 3.10.2 Coherent State 141 3.10.3 Single-photon Fock State 144

4 3-PHOTON DOWN-CONVERSION 148

4.1 Introduction 148

4.2 Phase Matching 150

4.3 Materials 153

4.4 Parametric De-/Amplification 155

4 . 54 . 5 χχ (3)-OPOs 156 Pump Resonant OPO 157 Non-Pump Resonant OPO 158 Star State Threshold 159 Application: Light Controlled OPOs 160

4.6 Cascaded χχ (3) 161

4.7 Star State Generation 163

4.8 Parametric Fluorescence 164

4.9 3-PDC in Fibers 165

5 SUMMARY AND OUTLOOK 169

5.1 Summary 169

5.2 Perspectives 171 5.2.1 Possible Further Projects 171 5.2.1.1 Quantum state preparation in 2-photon down-conversion 171

Photon Added States 171 Displaced Fock States 172

Repeated 2-Photon Down-Conversion 172 Squeezed Fock states 173 2-Photon Down-Conversion with periodically poled materials 173

5.2.1.2 3-Photon Down-Conversion 174 5.2.1.3 Energy Resolving Photodetection 174 5.2.1.4 CW-Quantum Information Processing 174

6 APPENDICES 176

6.1 Optical Double Slit Wigner Function 176

6.2 Bibliography 181

6.3 Acknowledgements 186

11 IInnttrroodduuccttiioonn:: QQuuaannttuumm TTeecchhnnoollooggyy

The gap between the quantum and the classical world is closing. It seems merely a matter of time until human technology will cross the quantum boundary. What is new about these quantum features? Why do we care whether something is “quan-tum” or “classical”? – One reason is that the world according to quantum mechan-ics is very different from the world of our every day experience. In the classical world we do not expect light waves to behave like Ping-Pong balls nor a soccer ball to show an interference pattern when it hits the goal. Even less so do we expect the ball to change its direction in flight just because the audience is watching it nor our seat neighbor to sit next to us and next to the old lady a few seats further down the row at the same time. But phenomena like these are common in the world of quantum mechanics.

We also care, because quantum systems might be able to perform certain tasks much better than their classical counterparts. For example, a quantum computer1,2 – a computer operating according to the laws of quantum mechanics – could crack the security code of cash cards much faster than any classical computer. Another example of the astounding features of quantum mechanics is the communication protocol called quantum cryptography which ensures the safety of a data transmis-sions on the basis of physical laws instead of numerical complexity.

These examples are not just imaginative ideas but have experimentally been con-firmed to work – at least in principle. Quantum cryptography has been demon-strated using a 23 km long standard communication fiber under lake Geneva3. First implementations of “toy” quantum computers have successfully been realized us-ing NMR techniques4.

Quantum technology in general – quantum mechanics applied to build devices – relies on our ability to a) create, b) manipulate and c) characterize quantum states. For each of the three pillars of quantum technology great progress has been achieved in recent years: the controlled design of almost arbitrary quantum states has been investigated in a number of papers for atomic and optical quantum sys-tems5,6,7,8. Quantum state transformations on single quantum systems and the im-plementation of conditional dynamics for at least two coupled quantum systems are currently being pursued by a number of groups using very different systems like NMR, ion traps, quantum wells and quantum dots, Josephson junctions, and cavity 1 for a recent review see H.-J. Briegel, I. Cirac, P. Zoller, „Quantencomputer“, Phys. Blätt. 55, 37 (1999) 2 Cirac, Zoller; B.E. Kane, Nature 393, 133 (1998) 3 A. Muller, H. Zbinden, N. Gisin “Quantum cryptography over 23 km in installed under-lake tele-com fibre”, Europhys. Lett. 33, 335 (1996) 4 I.L. Chuang, L.M.K. Vandersypen, X. Zhou, D.W. Leung, S. Lloyd, Nature 393, 143 (1998) 5 C.K. Law, and J.H. Eberly, “Arbitrary Control of a Quantum Electromagnetic Field”, Phys. Rev. Lett. 76, 1055 (1996) 6 K. Vogel, V.M. Akulin, and W.P. Schleich, „Quantum State Engineering of the Radiation Field“, Phys. Rev. Lett. 71, 1816 (1993) 7 A. Luis, L.L. Sánchez-Soto, “Conditional generation of field states in parametric down-conversion”, Phys. Lett. A 244, 211 (1998) 8 T. C. Weinacht, J. Ahn, and P.H. Buckbaum, „Controlling the shape of a quantum wavefunction“, Nature 397, 233 (1999)

Quantum computer

2 1 Introduction: Quantum Technology

QED systems. The characterization of quantum states has evolved from the meas-urements of single parameters to quantum state reconstruction techniques9 that map out the full information about a quantum state and to minimal invasion techniques like quantum non-demolition (QND) measurements that try to minimize the back action of a measurement on the time evolution of the system under investiga-tion10,11,12. This thesis is part of these ongoing efforts and focuses on the creation and characterization of new quantum states of the light field.

Non-classical features of the light field were first confirmed in 1957 with meas-urements of anti-bunched light13, but more than 25 years passed before other non-classical features such as sub-Poissonian photon statistics14 and squeezing15,16,17,18 followed. Since these initial observations the field of quantum optics has grown rapidly and a wealth of quantum phenomena has been observed in optics and found applications e.g. in QND19,20,21 measurements, measurements on the violation of Bell´s inequalities22,23,24, Quantum Cryptography, Quantum Teleportation25,26 and GHZ27 measurements. Optics has become a successful testing ground for more general quantum phenomena.

Quantum state reconstruction in the optical domain has largely relied on the tech-nique of optical homodyne tomography, which employs similar numerical tools as the generation of 3D-images in medical tomography. Homodyne tomography has been established as a reliable technique to obtain the full information about optical quantum states in recent years28,29,30, but has also successfully been applied to the

9 for an overview see Journal of Mod. Opt. 44, Nr. 11/12 (1997), Spec. Issue “Quantum State Prepa-ration and Measurement” 10 V.B. Braginsky, “Classical and quantum restrictions on the detection of weak disturbances of a macroscopic oscillator”, Sov. Phys. JETP 26, 831 (1968) 11 C.M. Caves K.S. Thorne, R.W.P. Drever, V.D. Sandberg, M.Zimmermann, Rev. Mod. Phys 52, 341 (1980) 12 P. Grangier, J.-F. Roch, G. Roger, Phys. Rev. Lett. 66, 1418 (1991) 13 R.Q. Twiss, A.G. Little, R. Hanbury Brown, Nature 180, 324 (1957) 14 R. Short, L. Mandel, “Observation of sub-Poissonian photon statistics”, Phys. Rev. Lett. 57, 691 (1983) 15 R.E. Slusher, L.W. Hollberg, B. Yurke, J.C. Mertz, J.F. Valley, “Observation squeezing by four wave mixing in a cavity”, Phys. Rev. Lett. 55, 2409 (1985) 16 R.E. Slusher, J. Kimble et al., Spec. Issue on Squeezed Light JOSA B 4, 1465 (1987), first result in JOSA B 1986 17 R. Paschotta, M. Collett, P. Kürz, K. Fiedler, H.A. Bachor, J. Mlynek, “Bright Squeezed Light from a Singly Resonant Frequency Doubler”, Phys. Rev. Lett. 72, 214 (1994) 18 G. Breitenbach, T. Müller, S. Pereira, F.-P. Poizat, S. Schiller, J. Mlynek, „Squeezed Vacuum from a Monolithic Optical Parametric Oscillator“, J. Opt. Soc. Am. B 12, 2304 (1995) 19 S.F. Pereira, Z.Y. Ou, H.J. Kimble, “Back action evading measurements for quantum non-demolition measurements and optical tapping”, Phys. Rev. Lett. 72, 214 (1994) 20 K. Bencheikh, J.A. Levenson, “Quantum nondemolition Demonstration via Repeated Backaction Evading Measurements“, Phys. Rev. Lett. 19, 3422 (1995) 21 R. Bruckmeier, H. Hansen, S. Schiller, “Repeated Quantum Nondemolition Measurements of con-tinuous Optical Waves”, Phys. Rev. Lett. 79, 1463 (1997) 22 Z.Y. Ou, L. Mandel, Phys. Rev. Lett. 61, 50 (1988) 23 Y.H. Shih, C.O. Alley, Phys. Rev. Lett. 61, 2921 (1988) 24 P. R. Tapster, J.G. Rarity, P.C.M. Owens, Violation of Bell´s Inequality over 4 km of Optical Fiber, Phys. Rev. Lett. 73, 1923 (1994) 25 D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Winfurter, A. Zeilinger, „Experimental quan-tum teleportation“, Nature 390, 575 (1997) 26 A. Furusawa, J.L. Srensen, S.L. Braunstein, C.A. Fuchs, H.J. Kimble, E.S. Polzik, „Unconditional quantum teleportation“, Science (1998) 27 D. Bouwmeester, J.-W. Pan, M. Daniell, H. Weinfurter, A. Zeilinger, „Observation of three-photon Greenberger-Horne-Zeilinger entanglement“, Phys. Rev. Lett. 82, 1345 (1999) 28 D.T. Smithey, M. Beck, and M.G. Raymer, “Measurement of the Wigner Distribution and the Density Matrix of a Light Mode using Optical Homodyne Tomography: Application to Squeezed States and the Vacuum”, Phys. Rev. Lett. 70, 1244 (1993) 29 M. Munroe, D. Boggarvarapu, M.E. Anderson, M.G. Raymer, “Photon statistics from the phase-averaged quadrature field distribution: theory and ultrafast measurement”, Phys. Rev. A 52, R924 (1995)

Quantum optics

Homodyne tomography

1 Introduction: Quantum Technology 3

vibrational state of molecules31 and trapped ions32 as well as to transverse motional state of atoms in an atomic beam33.

Based on prior work in our group on tomographic reconstructions of cw-field states such as the vacuum, coherent, squeezed thermal and phase averaged coherent states34,35, pulsed states of the light field are investigated in this work in order to access strongly non-classical states of the light field. Pulsed optical state tomogra-phy has formerly been applied by the group of M.G. Raymer to measure the Wigner function and photon number distributions of coherent and squeezed states28 as well as to investigate the amplitude and phase structure of optical pulses36.

One of the most fundamental quantum states of the light field is the state of a single light particle, a “photon”. The major part of this thesis will be devoted to the quan-tum state reconstruction of a “photon” state – more technically called the single-photon Fock state. This state is highly non-classical and reveals the particle aspect of the quantized light field most strikingly. Its marginal distributions are of non-Gaussian shape and its Wigner function exhibits a strong negativity around the origin of phase space (Figure 1-1). Although the quantum state reconstruction of a Fock state has already been performed for the motional state of trapped Beryllium-ions37 and single-photon Fock states of the electromagnetic field have already been prepared in the optical and in the microwave range38, a reconstruction of a single-photon Fock state has so far not been achieved in the optical domain.

The single-photon Fock state reconstruction combines the techniques of homodyne measurements and single -photon counting in a single experiment for the first time. It has stimulated further developments to advance the technique of pulsed homo-dyne tomography to the single-photon level. This allows not only the reconstruc-tion of a single-photon Fock state but also of other non-classical states of the light

30 G. Breitenbach, S. Schiller, „Homodyne Tomography of classical and non-classical light”, J. Mod. Opt. 44, 2207 (1997) 31 Rochester 1995 32 D. Leibfried, D.M. Meekhof, B.E. King, C. Monroe, W.M. Itano, and D.J. Wineland, „Experimen-tal Determination of the Motional Quantum State of a Trapped Atom”, Phys. Rev. Lett. 77, 4281 (1996) 33 Ch. Kurtsiefer, T. Pfau, J. Mlynek, “Measurement of the Wigner function of an ensemble of helium atoms“, Nature 386, 150 (1997) 34 G. Breitenbach, S. Schiller, J. Mlynek, “Measurement of the quantum states of squeezed light“ , Nature 387, 471 (1997) 35 G. Breitenbach, F. Illuminati, S. Schiller, and J. Mlynek, “Broadband detection of squeezed vac-uum: A spectrum of quantum states“, Europhys. Lett. 44, 192 (1998) 36 M. Beck, M.G. Raymer, I.A. Walmsley, V. Wong, Opt. Lett. 18, 2041 (1993); M. Munroe, D. Boggavarapu, M.E. Anderson, M.G. Raymer, Phys. Rev. A 52, 52 (1995) 37 D. Leibfried, D.M. Meekhof, B.E. King, C. Monroe, W.M. Itano, and D.J. Wineland, „Experimen-tal Determination of the Motional Quantum State of a Trapped Atom”, Phys. Rev. Lett. 77, 4281 (1996) 38 B. T. H. Varcoe, S. Brattke, M. Weidinger, H. Walther, “Preparing Pure Photon Fock States of the Radiation Field”, Nature 403, 743 (2000)

Figure 1-1 Wigner function of the single-photon Fock state.

The single-photon Fock state

4 1 Introduction: Quantum Technology

field such as photon-added states39, displaced Fock states40 and higher Fock states. Experimental avenues to generate and characterize these states are discussed in this thesis.

We use entangled photon pairs created in the process of 2-photon down conversion to generate single -photon Fock states by conditional measurements41. In the proc-ess of 2-photon down conversion strongly correlated photon pairs are produced from a single pump photon. These strong quantum correlations have successfully been exploited in a number of experiments to demonstrate quantum mechanical features such as non-locality42, delayed choice43 experiments and erasing44 in a number of research groups around the world. In this work a new experimental scheme based on repeated 2-photon down conversion is discussed that – in princi-ple – allows to generate arbitrary states of the light field.

A natural extension of 2-photon down conversion is the process of 3-photon down conversion where a single pump photon is split into three child photons exhibiting strong 3-particle quantum correlations employing higher order nonlinearities in optical materials. Possible schemes to produce new quantum states of the light field based on these χ(3)-nonlinearities and the limitations of presently available materi-als are discussed in this dissertation.

This thesis is organized as follows :

Part 2 provides an introduction to quantum states of the light field and presents basic concepts like the Wigner function and the method of optical homodyne to-mography. An in depth theoretical treatment of the experiment on single -photon Fock state generation is presented including a detailed discussion of realistic (inef-ficient) pulsed homodyne detection. Perspectives for conditional quantum state preparation in 2-photon down conversion will conclude part 2 of this thesis.

Part 3 is devoted to the description of the experimental methods used for the pulsed quantum state reconstruction. Results are presented on photon count meas-urements and tomographic reconstructions of pulsed vacuum, coherent and single -photon Fock states.

Part 4 is dedicated to a summary of the investigations on 3-photon down conver-sion. Theoretical results on mode matching considerations, χ(3)-OPOs, star states and 3-photon down conversion in different optical materials are given.

A summary of the work presented in the preceding chapter together with a discus-sion of possible future perspectives in Part 5 concludes this thesis.

39 G.S. Agarwal, K. Tara, Phys. Rev. A 43, 492 (1990) 40 K.E. Cahill , R. J. Glauber, Phys. Rev. 177, 1857 (1969); 41 A. Luis and L.L. Sánchez-Soto, “Conditional generation of field states in parametric down-conversion”, Phys. Lett. A 244, 211 (1998) 42 W. Tittel, J. Brendel, T. Herzog, H. Zbinden, N. Gisin, Non-local two-photon correlations using interferometers physically separated by 35 meters“, Europhys. Lett. 40, 595 (1997); G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, A. Zeilinger, Violation of Bell's inequality under strict Einstein locality conditions“, Phys. Rev. Lett. 81, 5039 (1998) 43 T. Herzog, P. Kwiat, H. Weinfurter, A. Zeilinger, “Delayed-choice quantum eraser: theory and experiment”, QELS `95. Summaries of Papers 16. 5 (1995); Z.Y. Ou, “Nonlocal correlation in the realization of a quantum eraser”, Phys. Lett. A 226, 323 (1997); K. Yoon-Ho, Y. Rong, S.P. Kulik, Yanhua Shih, M.O. Scully, “Delayed "choice" quantum eraser”, Phys. Rev. Lett. 84, 1 (2000) 44 T. Herzog, P.G. Kwiat, H. Weinfurter, A. Zeilinger, “Complementarity and the Quantum Eraser”, Phys. Rev. Lett. 75, 3034 (1995)

Photon pairs

3-photon down conversion

22 TThheeoorryy

This part of the thesis provides the theoretical background for the experimental work described in part 3 and presents calculations and theoretical results on pulsed optical homodyne tomography and quantum state preparation based on the process of 2-photon down-conversion. Theoretical investigations of quantum state prepara-tion employing χ(3)-nonlinear optical materials are deferred to part 4.

The first chapter of this part introduces quantum states of the light field and ex-plains basic concepts underlying our work like the Wigner function and the notion of classical and non-classical states of the light field.

The second chapter will be devoted to the experimental method of optical homo-dyne tomography, its use and usefulness as well as some remarks on its different implementations.

The third chapter provides a theoretical description of our experiment on the sin-gle -photon Fock state tomography. Major issues discussed in this chapter include a calculation of the twin photon state produced in spontaneous 2-photon down-conversion, the method of conditional measurements as a means to prepare single-photon Fock states, a treatment of realistic pulsed homodyne detection – i.e. sub-ject to experimental inefficiencies such as imperfect detector efficiencies and opti-cal losses –, and discuss the effect of the measurement efficiency on the expected measurement results.

The fourth and last chapter will give a theoretical outlook on further perspectives of conditional quantum state generation based on 2-photon parametric fluores-cence.

The cursory reader may want to skip the more formal parts of this section and con-centrate on the main issues and discussions. The formal treatments can easily be distinguished from introductions and discussions since wider line spacing is used for these parts due to the greater density of formulas found there.

2.1 Quantum States of the Light Field In the realm of classical physics the state of a system is defined by specifying a value for every degree of freedom in the system’s phase space. Thus for a free point particle moving in a 3D-space, six numbers are required to completely char-acterize the state of the particle – three to pinpoint its position in space and three to fix its momentum. In the same fashion the state of a classical byte register is speci-fied by a sequence of 8 zeros and ones. A suitable measurement performed on a classical free particle does not change its state, a repeated readout of the classical byte register is possible without altering its state.

This allows classical observers to obtain information about physical systems with-out disturbing them: different observers watching e.g. the trajectory of a volleyball will observe the ball at the same position at the same instance in time as their fel-

Classical versus quantum states

6 2 Theory

low observers. The ability to observe physical objects without influencing them lies at the very heart of what we call reality in every day life.

Strangely enough this preconception about reality has been found to be incorrect for the description of quantum objects. According to the laws of quantum mechan-ics, a free particle cannot be pinpointed to a given point in space without changing its momentum as a consequence of the attempt to obtain position information. In the same way the value of a quantum byte register cannot be specified by stating a sequence of zeros and ones. Instead, every quantum bit might exist in a superposi-tion of the two distinct states 0 and 1 with different probability amplitudes.

To cope with this quantum behavior a new description of the state of a system is required. In the framework of quantum mechanics this description is based on the notion of a state vector Ψ , which characterizes the state of a simple quantum mechanical system in phase space – the Hilbert-space. If the system is subject to a Hamiltonian H , then its time evolution is determined by the Schrödinger equation:

)(ˆ)( tHtt

i Ψ=Ψ∂∂

h (2.1.1)

States that can be characterized by a wave vector are calle d pure states. To be able to describe a wider class of physical systems which also covers statistical mixtures of pure states, we employ the concept of the density matrix, which may be defined as:

∑ ΨΨ=i

iiipρ (2.1.2)

where pi denotes the probability to find the system in state iΨ and ∑ =i ip 1 .

The density operator contains all information about the state under investigation and is the most general description of a quantum state. Its knowledge allows to calculate the statistical answer to all possible measurements on the system. The

expectation value of any observable O is given by

OˆTrˆ ρ=O (2.1.3)

Thus “Knowing the state means knowing the maximally available statistical infor-mation about all physical quantities of a physical object”45, as Ulf Leonhardt puts it.

What is a state of the light field? Since classical theories cannot describe the parti-cle aspect of the light field – and the main focus of this thesis lies on measurements of light particles, photons – we will make extensive use of a quantum mechanical treatment of the light field in this thesis. A quantum description of the electromag-netic field is a crucial theoretical tool for the experiments pursued in the course of this thesis.

Starting from the classical Hamiltonian of the electromagnetic field

∫ += 3 )1

(21 223

Lo

o BErdHrr

µε (2.1.4)

we introduce the field decomposition in reciprocal, kr

-space for the vector poten-

tial Ar

, the electric field Er

and magnetic field Br

45 U. Leonhardt, “Measuring the Quantum State of Light” (Cambridge Studies in Modern Optics, 1997)

The state vector

States of the light field – field quantization

2.1 Quantum States of the Light Field 7

..)(),(

..)(),(

..)(),(

)(,0

3

)(,0

3

)(,0

3

ccerukaBkditrB

cceruaEkditrE

cceruaEkdtrA

s

trkiksk

s

trkiksk

s

trkiksk

−×=

−=

+=

∑ ∫

∑ ∫

∑∫

−

−

−

ω

ω

ω

ε

ε

ε

rrr

rrr

rrr

rv)rr

rvrr

rvrr

(2.1.5)

where 3

00

)2(2 πε

ωh=E , 3

00

)2(2 π

ωµh=B , ska ,

r describes the field amplitudes, krvε

the polarization vector, ||/ kkkrr)

= the unit k-vector and )(rur

the normalized trans-

verse Gaussian mode of the light field. ∑s

denotes the summation of the two

orthogonal polarizations. Introducing the canonical variables

)(2

);(21 *

,,,*

,,, sksksksksksk aai

PaaX rrrrrr −=+= (2.1.6)

we can rewrite the Hamiltonian in the form

∑ ∫ +=s

skskk

XPkdH )()2(

1 2,

2,

33

rrr

hωπ

. (2.1.7)

This Hamiltonian is equivalent to the Hamiltonian of a system of independent one-

dimensional harmonic oscillators, one for each field mode and polarization. Fol-

lowing the standard quantization procedure we replace the field amplitudes ska ,r

and *,ska r by the harmonic oscillator annihilation and creation operators ska ,ˆ r and

+ska ,ˆ r . Making use of the commutation relationship 1]ˆ,ˆ[ ,, =+

sksk aa rr we obtain the

well known Hamiltonian of the free field modes

∑ ∫ += +

s ksksk aakdH

vrrh )2/1ˆˆ(

)2(

1ˆ,,

33

ωπ

(2.1.8)

Thus the electromagnetic field is regarded as consisting of modes associated with a

specific k-vector and polarization, where each mode may be considered as a nor-

malized harmonic oscillator. In the quantum theory of the electromagnetic field the

field energy in every mode is quantized, each energy quantum may be interpreted

as an elementary particle of the light field.

In analogy to the harmonic oscillator we may also define the phase dependent and

Hermitian quadrature operator: )ˆˆ()(ˆ21 ϕϕϕ ii eaeaX −++= ♣, where we have

♣ Different conventions are being used to define the quadrature amplitudes– connected to different

normalizations – : )ˆˆ()(ˆ ϕϕγϕ ii eaeaX −++= , where γ = 1, 21 , ½. Throughout this thesis I will

employ the convention γ = ½, which corresponds to normalized phase space variables, i.e.

Harmonic oscillator analogy

8 2 Theory

dropped the mode subscripts sk ,r

. For ϕ = 0 this operator reduces to the canonical

position operator )ˆˆ()(ˆ21 ++= aaX ϕ , for ϕ = π/2 to the canonical momentum op-

erator, )ˆˆ(ˆˆ21 +−=≡ aaYP i .

This mode decomposition of the electromagnetic field allows us

to single out a few relevant optical modes from the rest of the

field. A pure state of a single mode of the field can be described

by expanding the field mode wave function in terms of suitable

basis functions. For a mixed state of the field we may again

resort to a density matrix description.

What is a Photon?” in this chapter).

Since these concepts are of a rather abstract nature, the question

arises, whether we can give a more intuitive picture for quan-

tum states of the light field. One possible approach to tackle this

issue is to find a phase space representation of quantum states.

The Wigner Function Since a quantum system cannot be described by a point in phase space, a phase space description of a quantum system requires some sort of distribution function – very much like a probability density distribution for a classical system (see Figure 2-1). A very powerful and widely used phase space representation is the Wigner function, which was first introduced by E. Wigner in a famous paper46 dating back to 1932.

Wigner’s original definition of this distribution was that of a classical-like phase space distribution obtained as the Fourier transform of the displaced density matrix in the quadrature rep-resentation:

∫∞∞−

+−= dqeqxqxpxW ipq2/2ˆ2/21

),( ρπ

(2.1.9)

E. Wigner chose this function from the set of possible quasi-probability distribu-tions, since from all possible phase space distributions “it seemed to be the sim-plest”46. Its popularity most likely derives from the fact that its marginal distribu-tions – obtained by integrating along a particular direction in phase space – yield the measurable wave functions of the field.

22 ˆˆˆˆˆ PXaan +== + . This leads to a commutator of 2]ˆ,ˆ[ iPX = and an uncertainty relation of

4122 ≥∆∆=⋅ YXyx σσ

46 E.P. Wigner, Phys. Rev. 40, 749 (1932)

Energy eigenstates

A particularly popular choice of basis func-tions are the energy eigen-states of the Hamiltonian (2.1.8) or Fock states

)ˆ(!/1 nann +=

, where 0 de-notes the quan-tum state when no excitation is present – called the “vacuum” state. The first excitation, the single-photon Fock state, will play a central role in this thesis and is often called a “pho-ton” (for a fur-ther discussion see the section “Negativities as a result of interferences in phase space


J. Bertrand and P. Bertrand47 have used this property to give an alternative, more intuitive definition of the Wigner function, which emphasizes its connection to measurable quantities, the marginal distributions:

∫

∞∞−

+

+−=

≡

dpPXPXW

XUUXXpr

)cossin,sincos(:

ˆˆˆ),(

θθθθ

ρθ θθ, (2.1.10)

where θ denotes the phase angle and )ˆˆexp(ˆ aaiU +−= θθ the unitary rotation op-erator in phase space.

The Wigner function has proven to be a very helpful visualization of a quantum mechanical system in phase space. It is normalized and its marginal distributions pr(X,θ) – the density shadows of the Wigner function – denote the quadrature am-

47 J. Bertrand, and P. Bertrand, Found. Phys. 17, 397 (1987)

Figure 2-1 The state of a classical system – e.g. a harmonic oscillator – can be described by a point in phase space at any instance of time. For a system of many classical oscillators a probability density distribution is used to give an overall description of the system without specifying the position and momentum of every single oscillator. In quantum mechanics even the state of a single quantum particle has to be described by a distribution function in phase space – due to Heisenbergs uncertainty law. The Wigner function is the depiction of a quantum mechanical system in phase space which most closely resembles a classical probability distri-bution. Spatial and momentum information is presented simultaneously.

10 2 Theory

plitude probability density of the system – or equivalently its wave function in coordinate- or momentum-space.

Nevertheless the Wigner function is not a true probability density distribution, since it is not positive definite for all states of the light field. In particular it may take on negative values for non-classical quantum states of the light field. A few examples of states with negative Wigner functions are depicted in the following chapter. In fact, it can be shown that the only pure states, for which the Wigner function is non-negative, are states with Gaussian marginal distributions48,49.

The Wigner function contains the full information about a quantum system: for pure states the knowledge of the Wigner function is equivalent to knowing the wave function, for more general pure or mixed states equivalent to knowing the density matrix of the system. The Wigner function can be used to calculate expec-tation values for arbitrary operators according to

∫∫== ),(),(4ˆˆtrO pxWpxWdpdxO Oπρ (2.1.11)

where WO(x,p) is the Wigner function associated with the operator O . Thus it is possible to obtain any quantum mechanical prediction of measurement outcomes as filtered projections of the Wigner function. Since our observations are limited to the measurement of Hermitian operators, “all that we can see are shadows of the states, very much in the sense of Plato’s famous parable.” as Ulf Leonhardt af-firms50.

According to Heisenberg’s uncertainty law, position and momentum cannot be measured simultaneously to arbitrary precision. This has certain implications for the Wigner function: firstly, the positive regions of the Wigner function may not be confined to a phase space region smaller than 4/1<∆⋅∆ px . Secondly, due to this necessary spread, the absolute value of the Wigner function is bound by

π/2|),(| ≤pxW .

Other commonly used quasi-probability distributions are the P-function, also called Glauber-Sudarshan function51,52, which corresponds to normal ordering of the an-nihilation and creation operators, and the Q-function corresponding to anti-normally operator ordering. The set of all possible phase space distributions can be mapped out by a parameter s ranging from –1 to 1. The relation between the quasi-probability distributions for different s is given by the formula

ss

ppxx

espxWpdxdss

spxW −′−′+−′

−′′′′′

−′= ∫∫

22 )()(

),,()(

1),,(

π (2.1.12)

where s = 1 corresponds to the P-function, s = 0 to the Wigner function and s = -1 to the Q-function. From equation (2.1.12) it can easily be seen that a smoothing hierarchy exists between the different quasi-probability distributions, the P-function being the most singular, the Q-function being most smoothed out. In fact the P-function becomes singular for coherent states and is ill behaved for all non-classical states. The Q-function on the contrary remains positive definite for all possible quantum states. The Wigner function offers a sort of compromise: it takes on negative values for all states more non-classical than squeezed states but does not exhibit any singularities.

48 N. Lütkenhaus, S.M. Barnett, Phys. Rev. A 51, 3340 (1995) 49 R.L. Hudson, Rep. Math. Phys. 6, 249 (1974) 50 U. Leonhardt, “Measuring the Quantum State of Light” , (Cambridge University Press, 1997) 51 R.J. Glauber, Phys. Rev. Lett. 10, 84 (1963) 52 E.C.G. Sudarshan, Phys. Rev. Lett 10, 277 (1963)

s-parame-trized quasi-probability distributions


These properties have led to a definition of classicality and non-classicality of states based on the quasi-probability functions. But what are classical and non-classical states of the light field and why? This will be the topic of the following section.

Classical and Non-classical States of the Light Field In quantum optics the term “quantum” often bears the connotation of “interesting”, the term “classical” carries an undertone of “uninteresting”. This is not surprising, since the phenomena of classical optics are well understood, new and fascinating physics arises from our steadily increasing ability to control and measure non-classical phenomena of the light field. At the same time this highlights the need for a rational and formal definition of what we consider as quantum properties of opti-cal systems – apart from the notion that quantum features cannot be explained us-ing the classical wave theory of electromagnetic fields.

Non-Classicality Measures

The question on how to define non-classicality has received widespread interest over the years53. The most commonly accepted, necessary non-classicality criterion is based on the Glauber-Sudarshan P-distribution. As U.M. Titulaer and R.J. Glauber put it in 1965: “Fields with positive definite P-functions … are, in fact, precisely the quantum fields which may be described in a natural way as possess-ing classical analogues” 54. In a similar way L. Mandel states in 1986: “If P is not a probability density, then the state is nonclassical” 55. Unfortunately, since the P-function cannot be measured, this criterion cannot operationally be verified di-rectly.

The study of non-classical effects of the light field has also been a rapidly growing field of experimental investigations during the last two decades – starting with the pioneering work on photon anti-bunching by J. Kimble et al.56 in 1977. Since these early days a number of non-classicality criteria have been put forth and partially been investigated experimentally:

1. If “two photo-detection processes are less likely to appear close together than further apart, we speak of antibunching”57, in the opposite case of bunching. Anti-bunching is a sufficient criterion for an optical state to be non-classical. This criterion can be framed in a more formal fashion in stat-ing that the normalized second-order correlation function has to be smaller than 1: g(2) < 1

2. According to Heisenberg’s uncertainty principle, the product of the mo-mentum and position uncertainty cannot get smaller than a specified level. If the quantum fluctuations associated with a quantum state are equally dis-tributed among two conjugate quadrature components, then Heisenberg’s

principle leads to the condition 412 ≥∆ θX . This level is called the

53 L. Mandel and E. Wolf, Ref. Mod. Phys. 37, 231 (1965); L. Mandel , Opt. Lett 4, 205 (1979); D.F. Walls, Nature 324, 210 (1986); M. Hillery, Phys. Rev. A 35, 725 (1987); C.T. Lee, Phys. Rev. A 44, R2775 (1991); W. Vogel and D.-G. Welsch, Lectures on Quantum Optics (Akademie-Verlag, Berlin, 1994); D.F. Walls, and G.J. Milburn, Quantum Optics (Cambridge University Press, Cambridge, 1995); J. Janszky, M.G. Kim, and M.S. Kim, Phys. Rev. A 53, 502 (1996); G.M. D’Ariano, M.R. Sacchi, and P. Kumar, Phys. Rev. A 59, 826 (1999) 54 U.M. Titulaer, and R.J. Glauber, Phys. Rev. 140, B 676 (1965) 55 L. Mandel, Phys. Scr. T 12, 34 (1986) 56 H.J. Kimble, M Dagenais, and L. Mandel, Phys. Rev. Lett. 39, 691 (1977) 57 L. Mandel, E. Wolf, “Optical coherence and quantum optics“, p. 712 et seq. (Cambridge Univer-sity Press, 1995, New York)

12 2 Theory

standard quantum level. If for any state 412 <∆ θX , which is possible

for states which deviate from the equal distribution of quantum noise, then these states are necessarily non-classical and called squeezed states. The degree of squeezing S is defined as the ratio of the quantum fluctuations of the state under consideration and the standard quantum level (commonly stated in dB).

3. A criterion often cited as a sufficient condition for non-classicality is a sub-Poissonian photon count statistics. Coherent states exhibit a Pois-sonian count statistics, amplitude squeezed states a clear sub-Poissonian statistics. This condition is often rephrased in terms of Mandel’s Q-parameter, which is defined as

n

nnQ

ˆ

ˆ)ˆ( 2 −∆≡ (2.1.13)

The Q-parameter takes on negative values whenever the photon count sta-tistics is sub-Poissonian58.

4. Yet another criterion for non-classicality sets a condition for the probabil-ity to detect n photons: “Except for the vacuum state, any state for which p(n) = 0 has no classical analog and is purely quantum mechanical. This is a reflection of the fact that the coherent state v contains contributions from Fock states with all possible occupation numbers”59. Although useful as a theoretical distinction tool, this criterion is hardly useful for the ex-perimentalist, since experimental uncertainties will make the distinction between a vanishingly small p(n)à 0 and an exact p(n) = 0 very difficult.

5. If Po denotes the probability for a state of the light field to have an odd number of photons and Pe that for an even number, than the state is non-classical if Po > ½ or Pe < ½ . Again Po might get very close to ½ making it hard to test this criterion experimentally.

6. Photon number oscillations , which are sometimes considered to be a sig-nature of non-classical states, are not by themselves a sufficient condition for non-classicality, since they can also be achieved for a statistical mixture of coherent states (see Figure 2-2). Only if the distance between adjacent

peaks is smaller than n2 – the width of a Poissonian distribution for a photon number expectation value n – can photon number oscillations be regarded as a true condition for non-classical states60.

7. Other criteria based on homodyne tomographic measurements have been put forth61,62, which are not discussed here, since up to this point the method of homodyne tomography has not been explained.

8. There are also criteria that are based more directly on the quasi-probability distributions. One of these criteria assures that a state is non-classical, if the characteristic function associated with its quasiprobability distri-bution decays slower than the characteristic function of the vacuum state: “There exist values of the phase ϕ for which the quadrature distribu-

58 L. Mandel, E. Wolf, “Optical coherence and quantum optics“, p. 627 (Cambridge University Press, 1995, New York) 59 L. Mandel, E. Wolf, “Optical coherence and quantum optics“, p. 543 (Cambridge University Press, 1995, New York) 60 C. Zhu and C.M. Caves, “Photocount distributions for continuous-wave squeezed light” , Phys. Rev. A 41, 475 (1990) 61 Vogel, Risken, Phys. Rev. A 40, 2847 (1989) 62 Smithey, Phys. Rev. A 70, 1244 (1993)


tions p(x,ϕ) exhibits structures that are narrower than the distribution of the ground state. Correspondingly, for those ϕ values the characteristic function G(k,ϕ) decays more slowly than the characteristic function of the

ground state”63, i.e. 42

),( kekG −>ϕ

9. Another more commonly employed criterion based on the quasi-probability distributions is the negativity of any s-parameterized qua-siprobability distribution. For example, in the case of a squeezed state all quasi-probability distributions with s > 0 will exhibit negative values in certain phase space regions. Again, as for the P-distribution criterion, to my knowledge, no proposal has been put forth on how this condition can experimentally be accessed for states where the negativities occur for s-values bigger than 0.

All of these criteria only relate to a single mode of the electromagnetic field and criteria 1.-6. are linked to particle aspects of the light field. Thus particle -like behavior of the light field is clearly considered to be a sign of non-classicality.

If we consider more than one mode of the electromagnetic field, a different reason for non-classical behavior emerges: if the two or more modes are entangled, then no classical interpretation of the state is possible . Entanglement measures as a for-mal basis for the notion of entanglement have been proposed by V. Vedral and M.B. Plenio65 and others.

Is it possible to perform an ordering of quantum states of the light field according to their degree of non-classicality? Some of the quantitative criteria mentioned above might in fact be able to provide an ordering, like e.g. according to their de-gree of squeezing or according to their Mandel Q-parameter. Unfortunately, since all criteria only pose sufficient conditions for non-classicality – apart from 8. and 9. –, there will always exist non-classical states which cannot be ordered due to any single criterion 1. – 7.. Realizing this problem C.T. Lee has proposed a non-classicality measure – a “non-classical depth” – based on criterion 9.66,67: the non-classical depth is the maximum number d = (1-s)/2, for which the s-parameterized

63 W. Vogel, „Nonclassical states: an ovservable criterion“, Phys. Rev. Lett. 84, (2000), in press 64 idea due to Kumar, private communication. 65 V. Vedral, and M.C. Plenio, “Entanglement Measures and Purification Procedures”, Phys. Rev. A 57 , 1619 (1998) 66 C.T. Lee, “Measure of the nonclassicality of nonclassical states“, Phys. Rev. Lett. 44, 2775 (1991) 67 C.T. Lee, “Theorem on nonclassical states“, Phys. Rev. Lett. 52, 3374 (1995)

p(n)

n

p(n)=0.208 p(0,n)+.978 p(1.57,n)

0 1 2 3 4 5 6 7 8 9 10

0.05

0.1

0.15

0.2

0.25

0.3

Figure 2-2 Statistical mixture of a coherent state with α = 1.57 and the vacuum state exhibiting photon number oscillations64.

14 2 Theory

quasi-probability distribution remains positive definite. The non-classical depth of e.g. a coherent state is 0, of a squeezed state 0.5, and of a Fock state 1.

The next section will introduce a few quantum states of the light field relevant for this work and discuss some of their properties.

Examples of Quantum States § Fock states have already been introduced earlier in this chapter and will

play an important role throughout this thesis. They represent the particle aspect of the quantized light field and form a particularly popular basis for the expansion of the electromagnetic field, since they are the energy eigen-functions of the free field Hamiltonian and follow a rather convenient al-gebra. The Wigner function of the n-photon Fock state is of the form

)(2222 22

))(4()1(),( pxn

n epxLpxW +−+−= π (2.1.14)

where Ln denotes the nth Laguerre polynomial. In the special case of the single-photon Fock state this simplifies to

)(2222 22

)1)(4(),( pxepxpxW +−−+= π (2.1.15)

Higher Fock states are proportional to higher order Hermite polynomials with a Gaussian roll-off to infinity. The higher the occupation number n, the more ridges and valleys the Wigner function exhibits, the total number of ridges and valleys – 1 being equal to n. All Fock states are rotational symmetric since the number operator n and the quantum mechanical phase

Figure 2-3 Non-classical quantum states of the light field.


are conjugate operators. Single -photon Fock states preparation by condi-tional measurements on a photon twin produced in 2-photon down-conversion will be discussed in detail in chapter 2.3, options to produce higher Fock states will be described in chapter 2.4.1.

§ Star states are non-classical states which occur in parametric χ(3)-deamplification in a process analogous to the squeezing interaction but employing the nonlinear coefficient of the next higher order. The states have a tri-fold symmetry in phase space and are difficult to realize experi-mentally due to the small magnitude of the χ(3)-nonlinearity in available optical materials. Perspectives of employing χ(3)-interactions in the process of 3-photon down-conversion will be the topic of chapter 4.

§ Schrödinger cat states are strongly non-classical states of the light field which arise from a coherent superposition of two coherent states with dif-ferent excitations α. They relate to the famous Schrödinger Gedanken-experiment of generating a coherent superposition of two macroscopically distinguishable states (the cat in the box is alive – or dead), and elucidate the quantum mechanical superposition principle. The Wigner function of an optical Schrödinger cat is of the form

))4cos((),( 0)(2))((2))((22 2222

022

0 xxeeepxW pxpxxpxx +−++−+−− ++= π

(2.1.16)

These states have been observed in the microwave domain, but not in the optical domain, since they are extremely vulnerable to losses and decoher-ence. An analog of the Schrödinger cat state in classical wave electrody-namics is the double slit. A reconstruction of the state of the transverse field components from a simple optical experiment will be described in the appendix (chapter 6.1)

Figure 2-4 Density matrix of a one photon added coherent state. Due to the addi-tion of a photon the population of density matrix elements has been shifted up along the diagonal leaving behind an empty row at n = 0 and an empty column at m = 0.

16 2 Theory

§ If photons are added to an arbitrary state of the light field new quantum states are generated which are labeled photon added states68:

Ψ≡Ψ + mam )(, . These states are non-classical due to a vanishing probability of finding no photon and have so far – apart from the trivial case where they reduce to one of the states mentioned above – not been generated69.

Photon added coherent states have the additional interesting property to provide a link between the Fock states and the coherent states. For vanish-ing α they approach Fock states, for large α coherent states. Therefore pho-ton added coherent states can be interpreted as a bridge between the parti-cle and the wave aspect of the electromagnetic field.

§ Displaced Fock states are Fock states with a midpoint displaced from the origin of phase space. They have received widespread interest in recent years70. Mentioned as early as in 1969 in a paper by K.E. Cahill and R.J. Glauber71 these states are defined as

∑ −−−

==

k

knk

kn kLk

nenDn )(

!

!)(ˆ, 2

2/122

αααα α (2.1.17)

and exhibit a number of interesting and useful properties, such as e.g. strong photon number oscillations. Displaced Fock states have proven to be particularly useful to find simplified series representations of quasiprob-ability distributions of the quantum field (such as the Wigner function).

Most of the states described in this section have a non-classical depth of greater than 0 and their Wigner function therefore exhibits negative regions in phase space. These negativities can be interpreted as arising from an interference of regions in

68 G.S. Agarwal, K. Tara, Phys. Rev. A 43, 492 (1990) 69 C.T. Lee, Phys. Rev. A 52, 3374 (1995) 70 M. Venkata Satyanarayana, Phys. Rev. D 32, 400 (1985); A. Wunsche, Quantum Optics 3, 359 (1991); N. Moya-Cessa, P.L. Knight, Phys. Rev. A 48, 2479 (1993); F.A.M. De Oliveira, M. Kim, P.L. Knight, V. Puzek, Phys. Rev. A 41, 2645 (1990); A.-S. F. Obada, G.M. Abd Al-Kader, Journal of Mod. Opt. 46, 263 (1999) 71 K.E. Cahill , R. J. Glauber, Phys. Rev. 177, 1857 (1969)

Wave particle dualism

Figure 2-5 Wigner function of the single-photon added coherent state. The Wigner function bears resemblance to the Fock state (the dip in the center) and to a coher-ent state (the hill on the side).


phase space where the Wigner function takes on positive values separated by a distance greater than ½ – the width of the vacuum state. These regions are areas of enhanced probability density and, if they are coherently superposed, then interfer-ences arise between them leading to negativities located around the midpoint of the connecting line in phase space. The frequency of the interference effects increases proportional to the separation between the positive regions. This behavior is rather obvious for a Schrödinger cat state, but also explains the strong negativity of the Wigner function for the single-photon Fock state.

What is a Photon? The term “photon” was coined in 1927 by G.N. Lewis72 in an attempt to develop a theory of chemical valence. He used the word “photon” to denote exchange parti-cles of radiation between atoms (which he thought could be bound to atoms). The concept of light particles bound to atoms turned out to be flawed, but the term “photon” – used for elementary light particles in analogy to protons, neutrons and electrons as the elementary constituents of matter – has caught on.

The modern concept of light particles as the smallest unit of light energy – or other electromagnetic radiation – originated from Einstein´s works on the photoelectric effect. However, in 1926 G.Wentzel and G. Beck were able to provide a complete theoretical description of this effect quantizing only the matter part of the problem and using a purely classical description of the optical field.

Even today there still is a certain amount of discussion about the concept of the “photon”: in a 1995 paper entitled “Anti-Photon”73 W.E. Lamb gives a concise history of the concept of a photon and criticizes its thoughtless use: “Talking about radiation in terms of particles is like using such ubiquitous phrases as “You know” or “I mean” …”. At the First Rochester Coherence Conference he suggested “that a license be required for use of the word ‘photon’”, and offered to issue such a license to people he thought to be properly qualified. In a preprint T.W. Marshall and E. Santos even talk about the “Myth of a photon”74.

Despite these discussions the term “photon” is widely used in present day physics and the concept has proved to be a useful and intuitive way to understand particle aspects in the behavior of light. Throughout this thesis I will adopt the most com-monly accepted “definition” of a “photon” as given by L. Mandel and E. Wolf75 based on the field quantization of the electromagnetic field:

“The discrete excitations or quanta of the electromagnetic field, corre-

sponding to the occupatin numbers n, are usually known as photons.

Thus a state ,...0,0,1,0,0... ,skr is described as a state with one photon of

wave vector kr

and polarization s.”

The extensive use of the “photon” concept in elementary particle physics is due to its role as the intermediary particle of the electromagnetic force. As the Encarta puts it:

“The photon is an elementary particle, or a particle that cannot be split into anything smaller. It carries the electromagnetic force, one of the four

72 J.H. Hildebrand: „Gilbert Newton Lewis“, Biogr. Mem. Nat´l Acad. Sci. 31, 210 (Columbia Univ. Press, New York, 1958) 73 W.E. Lamb, Anti-Photon, Appl. Phys. B 60, 77 (1995) 74 T.W. Marshall, E. Santos, “Myth of a photon”, Preprint quant-ph/9711046 75 L. Mandel, and E. Wolf, “Optical coherence and quantum optics”, (Cambridge University Press, USA, 1995)

Negativities as a result of interferences in phase space

18 2 Theory

fundamental forces of nature, between particles. The electromagnetic force occurs between charged particles or between magnetic materials and charged particles. Electrically charged particles attract or repel each other by exchanging photons back and forth.”

The electromagnetic modes introduced in standard textbooks on field quantiza-tion76 are characterized by a specified k-vector and polarization, corresponding to a plane wave expansion of the mode spectrum. The corresponding field excitations or photons accordingly have a well defined frequency. This is an idealized point of view that introduces certain restrictions: if the frequency spread of the field modes is assumed to be of delta-shape in frequency space, then no localization of the pho-ton in space is possible. The energy wave-packet is completely delocalized.

This assumption may be justified for cavity field modes, where the cavity mode spectrum replaces the continuous mode spectrum of free space, and for the descrip-tion of cw-lasers, where the first order coherence length typically is well in excess of the length of the experimental interaction region. But even in these cases the single frequency assumption is a simplified view since the cavity mode as well as laser mode has a certain – narrow – spread in frequency associated with it.

If we attempt to describe localized single-photon states traveling in space, the sin-gle frequency description no longer offers a suitable approximation. In this case we have to resort to spatio-temporal modes77, which are delocalized to a certain extent in the frequency domain, but localized in space.

If we define a wave-packet photon creation operator in terms of the familiar crea-tion operator as

∫ +−∆+ ∆∆≡ ωωωω aefdA tti ˆ)(ˆ )(* 0 (2.1.18)

then this new creation operator obeys the same commutation relations and the same algebra as the single mode operators. The corresponding Fock-states form a com-plete, orthonormal basis for all possible states of an electromagnetic wave packet field mode:

basis lorthonorma complete,

1,11

1)(for1],[ 2

N

NNNANNNA

fdAA

−=++=

=∆∆=+

+ ∫ ωω

(2.1.19)

Thus this new operators can be considered as equivalent to the commonly used single frequency creation operators and provide an alternative mode decomposition of the electromagnetic field in terms of wave-packets.

These spatio-temporal modes are much better suited to describe the single -photon states prepared by conditional measurements on photon twins produced in 2-photon parametric fluorescence that we are using in the experiments described in this the-sis. Hence we will frequently refer to the concept of spatio-temporal modes throughout this thesis.

76 C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, “Atom-Photon interactions”, (Wiley & Sons, USA, 1988) 77 A. Joobeur, B.E.A. Saleh, and M.C. Teich, "Spatiotemporal coherence properties of entangled light beams generated by parametric down-conversion" , Phys. Rev. A 50, 3349 (1994)

Spatio- temporal modes

2.2 Optical Quantum Tomography 19

2.2 Optical Quantum Tomography The word 'tomography' is derived from the Greek words ‘tomos’ meaning ‘slice’ and ‘graphae’ meaning ‘image’. Today the term “tomography” is employed in a number of circumstances to denote methods to reconstruct a higher-dimensional picture of an object from a set of lower-dimensional images taken from different angles. The object might be as plastic as the human head or as abstract as a simul-taneous picture of the time and frequency behavior of electronic circuits. Tomogra-phy has developed into a powerful visualization tool to help physicians, electronics engineers and physicists to represent relevant information about complex systems in a single picture.

In the optical domain the tomographic technique is employed to reconstruct the

78 J. Radon, Berichte der sächsischen Akademie der Wissenschaften, Leipzig, Math.-Phys. Kl. 69, 262 (1917) ♣ For a good impression see "http://neurosurgery.mgh.harvard.edu/pet-hp.htm" 79 PET Scanning, Nucleonics 11, 40 (1953) ♦ http://www.tomography.umist.ac.uk/ ♥ http://www.aracor.com/cti/Prop.html ♠ http://home.cern.ch/t/tomograp/www/

Tomography: Brief History and Applications in Different Fields

The concept of tomography was first published as early as 1826, by Abel, a Nor-wegian physicist. His method was limited to objects with rotational symmetry. In 1917, an Austrian mathematician, Radon, extended Abel’s idea to objects with arbitrary shapes78. The Abel and inverse Radon transformations are still used today to compute tomographic images.

The tomographic method has received widespread interest mainly due to its appli-cation in medical image processing. Godfrey Hounsfield and Allen Cormack were jointly awarded the Nobel prize for their pioneering work on X-ray Tomography in 1979.

Modern Siemens computer tomograph

Currently, there are a number of tomographic techniques available for studying complex biological systems and multiphase phenomena. These include, for exam-ple, infrared, optical, X-ray and Gamma-ray tomographic systems♣, positron emis-sion tomography (PET)79, magnetic resonance imaging (MRI), and sonic or ultra-sonic tomographic systems.

Less known applications of tomographic methods apart from medical purposes include process tomography in production systems♦ , reverse engineering♥, ocean acoustical tomography, analysis of scattering events in elementary particle phys-ics♠, and electrical impedance tomography.

20 2 Theory

Wigner function from experimental data. Even though it is not possible to com-pletely determine the state of a quantum system in a single measurement, it is pos-sible to infer the state – up to a certain precision – from a number of measurements performed on identically prepared systems80.

A concise and very readable treatment on optical homodyne tomography can be found in 81. A good general view on state reconstruction in quantum optics is pro-vided in 82.

The first experimental quantum state reconstruction was performed on a hydrogen in the n = 3 excited state by Havener et al83 in 1986. Pulsed optical homodyne to-mography of coherent and squeezed states was first used by M.G. Raymer et al. at the University of Oregon84 in 1992. They have also applied this technique to meas-ure photon number distributions of coherent and squeezed states as well as to in-vestigate the amplitude and phase structure of optical pulses.

The following two sections will try to answer the question: how does the method of tomographic reconstruction work – in the general case and in the special case of a phase-independent state?

Homodyne Measurements Let us assume we have an unknown optical quantum state as depicted in Figure 2-6, where the unknown state is represented by its Wigner function. This unknown quantum state – called the signal beam – is coupled to a strong local oscillator beam by overlapping the two beams at a precise 50/50 beam splitter. The two beams emerging from the outputs of the beam splitter are the sum and the differ-ence of the signal and local oscillator fields. Both beams are detected with a PIN-photodiode yielding two photocurrents

222

2

222

1

)cos(2

)cos(2

LOLOssi

LOs

LOLOssi

LOs

EEEEeEEi

EEEEeEEi

+−=−∝

++=+∝

θ

θ

θ

θ

(2.2.1)

The local oscillator beam usually is in a coherent state with a coherent amplitude α high enough, so that the quantum fluctuations in the local oscillator beam may be neglected, and the local oscillator field may be treated as a classical wave. Sub-tracting the currents of two detectors yields a signal

θθ XEi s ∝∝∆ )cos(2 (2.2.2)

which is proportional to the quadrature amplitude of the signal field. This meas-urement scheme is called a homodyne detector and used to obtain a measurement signal which is proportional to the field instead of being proportional to the field intensity as for standard photodetectors. θ denotes the relative phase angle between the local oscillator and the seed beam. From a large number of quadrature ampli-tude measurements the probability distribution of the quadrature amplitude eigen-

values XUUXXP += θθθ ρ ˆˆˆ)( may be obtained (Figure 2-6 shows an exam-

80 A. Royer, “Measurement of quantum states and the Wigner function”, Found. Phys. 19, 3 (1989) 81 U. Leonhardt, “Measuring the Quantum State of Light” , (Cambridge University Press, 1997) 82 Special issue Quantum State Preparation and Measurement, J. of Mod. Opt. 44, Nr. 11/12 (1997) 83 C.C. Havener, N. Rouze, W.B. Westerfeld, and J.S. Risley, “Experimental determination of the density matrix describing collisionally produced H(n=3) atoms”, Phys. Rev. A 33, 376 (1986) 84 D.T. Smithey, M. Beck, M.G. Raymer, and A. Faridani, Phys. Rev. Lett. 69, 2650 (1992) 70, 1244 (1993); M. Beck, M.G. Raymer, I.A. Walmsley, and V. Wong, Opt. Lett. 18, 2041 (1993); M. Munroe, D. Boggavarapu, M.E. Anderson, and M.G. Raymer, Phys. Rev. A 52, 52 (1995)


ple of a time trace of measurement values and the corresponding statistical distribu-tion – the marginal distribution).

If a set of these distributions θθθ )( XP for a sufficiently high number of angles θ is measured by varying the phase angle θ, employing e.g. a piezo-mounted mirror as a phase shifter in the local oscillator beam path, complete information about the quantum state of the signal beam is obtained. From this set of distributions either the Wigner function or the density matrix of the optical quantum state in the signal beam (or in principle the probability distribution of possible measurement out-comes for any observable) may be obtained employing suitable inversion tech-niques. In the general case of a phase-dependent Wigner function the inverse-Radon-transformation may be employed to obtain the Wigner function of the state. In the special case of a phase-independent quantum state we only need to register a single (phase-averaged) marginal distribution and use the simpler one-dimensional Abel transformation to perform the same task. These two mathematical methods are described in the following two sections.

Figure 2-6 Homodyne tomography: an unknown quantum state (signal) is over-lapped with a strong local oscillator at a beam splitter. The statistics of the differ-ence current of two detectors monitoring the outgoing beam reproduce a marginal distribution of the unknown quantum state. If the relative phase angle between the signal and the local oscillator beam is varied, a set of marginal distributions is obtained from which the Wigner function or the density matrix of the unknown quantum state may be reconstructed.

22 2 Theory

The Inverse Radon Transformation To obtain the Wigner function from the set of marginal distribution in the general case of a phase-dependent quantum state we have to invert equation (2.1.10) yield-ing

)x-sinpcosK(x),xpr(1

),(02

′+′′= ∫∫∞+∞−

θθθθπ

πxddpxW (2.2.3)

where the kernel is given by

∫∞

∞−= xiedxK ξξξ

21

)( (2.2.4)

This (back-)transformation is called the inverse Radon transformation. It is a mathematical transformation with a kernel which is only defined in the sense of a generalized function85 (like e.g. the delta-functional). Therefore a regularization procedure is required for a numerical implementation of the inverse Radon trans-formation. This regularization may be achieved by choosing an appropriate cutoff frequency k c in the integral of equation (2.2.4). This results in an approximate ker-nel

)1)sin()(cos(1

)(2

−+≈ xkxkxkx

xK ccc (2.2.5)

Unfortunately this approximate kernel function is still ill defined at x=0. Therefore we need an additional expansion of equation (2.2.4) around the origin, which we obtain by expanding equation (2.2.5) around the origin to 4. order, yielding

...)724

1(2

)(4422

−+−≈xkxkk

xK ccc (2.2.6)

To numerically calculate the kernel for a reconstruction equation (2.2.6) is used to complement the kernel function of equation (2.2.5) close to the origin. A conven-ient point to switch between equation (2.2.5) and (2.2.6) is the point |k c x| = 0.1.

This regularization procedure leads to a low-pass cutoff filtering of the experimen-tal details and will wash out details smaller than 1/k c in the reconstructed Wigner function. Since the marginal distributions obtained in the experiment will exhibit statistical noise of high frequency a suitable choice of k c is critical for a faithful reconstruction of the measured state. Thus k c has to be adapted to the experimental situation.

Short but helpful notes on the numerical implementation of the inverse Radon transformation can be found in81.

The Abel Transformation In the case of phase independent states with a rotationally symmetric Wigner func-tion a simplified form of the inverse Radon transformation may be used to infer the Wigner function from marginal data. Since phase and photon number are conjugate observables, all Fock states – being eigenstates of the number operator – are exam-ples of phase independent states.

85 I.M. Gel’fand, G.E. Shilov, Generalized Functions, (Academic Press, San Diego, 1965)


In the special case of rotationally invariant states equation (2.2.3) simplifies to

∫∞

−

′−=

rdx

rx

xrW

22

)(rp1)(

π (2.2.7)

This transformation is called Abel transform and relates the marginal distribution of a rotationally symmetric quantum state to its Wigner function. In this case we do not have to register a set of marginal distributions for different phase angles, since – due to the symmetry – all marginal distributions will be the same.

Although the kernel can be expressed in closed form in the case of the Abel trans-form it is still singular. To remove this singularity we perform the variable trans-

formation 22 rxu −= leading to the modified formula

∫∞

+

′−=

0 22

))((rp1)( du

ru

uxrW

π (2.2.8)

which is singular only for u = r = 0 and allows a straightforward numerical imple-mentation.

Figure 2-7 Effect of Gaussian filtering in the Abel transform: the upper graph shows simulated data of a Fock state marginal for an efficiency of 63% and the filtered data. High frequency statistical noise is efficiently removed and allows a faithful reconstruction of the Wigner function with an efficiency loss of 2% due to filtering.

24 2 Theory

Another complication that arises in connection with equation (2.2.7) and (2.2.8) is that the marginal distribution itself does not enter the back-projection formula, but its first derivative. The data obtained in experiments will almost inevitably suffer from statistical noise which will dominate the first derivative – in particular for low sampling rates like in our experiment.

Thus filtering of experimental marginal data is required to obtain useful reconstruc-tions. Two different low pass filters have been applied in the experiment to strongly reduce the impact of statistical noise on the reconstructions: convolution of the marginal distribution with a Gaussian of suitable width or cut-off filtering. The advantage of convoluting the marginal data with a Gaussian is that the effect of this filtering procedure can easily be interpreted in physical terms: it corresponds to an additional loss mechanism which introduces Gaussian vacuum noise into the measurement. Thus filtering further reduces the measurement efficiency, but does not introduce any other numerical artifacts. The contribution of the filter noise to the experimental result can in this case easily be stated as an efficiency factor:

4/1

12filter

filterσ

η+

= (2.2.9)

where σfilter is the bandwidth of the filter in frequency space in normalized units. Figure 2-7 exemplifies the effect of filtering for simulated measurement data corre-sponding to a measurement efficiency of 63% with additional statistical noise. The filter efficiently removes the high frequency statistical noise and reduces the effi-ciency slightly by 2%.

It is worth noting that the implementation of the inverse Radon transformation also involves a cut-off filtering of the data due to the introduction of the cut-off fre-quency k c in the integration kernel.

Without going into details, let me draw a rough sketch of the algorithm employed to obtain the reconstructions:

§ Read in the marginal data and normalize it.

§ Symmetrize the data to obtain values for the marginal distribution in radial coordinates (pr(X)).

§ Filter the marginal data employing one of the filters discussed above.

§ Calculate the Abel transformation according to

∑=

−−++

−≈max

0

))5.0()5.0(()(1

)(x

x

xuxurx

xdprrW

π (2.2.10)

where xscalexprxprxdpr /))()1(()( −+= , dpr(0) = 0 and xscale is the quadrature normalization factor obtained in the normalization procedure.

22)( rxxu −= as above and xmax is the maximum value of the quadra-ture component x measured in the experiment.

§ Check the normalization of the Wigner function.

Due to the fact that normalization, symmetrization and filtering are linear opera-tions on the marginal data their order may be chosen at convenience.

As an example for the application of the Abel transform Figure 2-8 shows a recon-struction of a Fock state from simulated homodyne data with a total measurement efficiency of η=0.63%.

Filtering


Quantum State Sampling Since the set of marginal distributions pr(X,θ)θ contains the full information about the investigated quantum state, it is also possible to infer the density matrix from this data employing a technique called quantum state sampling. Direct sam-pling was first proposed by G.M. D’Ariano et al.86,87 and D.-G. Welsch et al.88 and mathematically and numerically refined by a number of authors89,90,91,92.

As we can see from equation (2.1.11) choosing nmO nm == ,ˆ ρ the density ma-

trix in the Fock basis can be obtained from the Wigner function via an integral transform. As the inverse Radon transformation, which connects the marginal data to the Wigner function also is a linear integral transform, a linear transformation between the set of marginal distributions and the density has to exist. This linear transformation may be written in the form

∫∫∞∞−

== ),(),(pr ,0, θθθρρ πXFXdXdnm nmnm

) (2.2.11)

where Fm,n is a specific set of functions called pattern functions. The density matrix elements ρn,m are obtained from statistically sampling the pattern functions Fn,m

86 G.M. D’Ariano, M.B.A. Paris, Phys. Rev. A 49, 3022 (1993) 87 G.M. D’Ariano, C. Macchiavello, M.B.A. Paris, Phys. Rev. A 195, 31 (1994) 88 H. Kühn, D.-G. Welsch, W. Vogel, J. Mod. Opt. 41, 1607 (1994) 89 G.M. D’Ariano, U. Leonhardt, H.Paul, Phys. Rev. A 52, R1801 (1995) 90 U. Leonhardt, H. Paul, G.M. D’Ariano, Phys. Rev. A 52, 4899 (1995) 91 U. Leonhardt, M.G. Raymer, Phys. Rev. Lett. 76, 1989 (1996) 92 Th. Richter, Phys. Rev. A 211, 327 (1996)

Figure 2-8 Simulated Fock state reconstruction: the upper left section shows simu-lated marginal data generated for a total detection efficiency of η = 63%. The upper right graph shows the marginal distribution corresponding to the generated data. The associated Wigner function is depicted in the lower graph.

26 2 Theory

with the obtained marginal data. The main difficulty of this method therefore con-sists in finding the right pattern function for this sampling procedure. As deduced in 93 the pattern functions can be expressed as

θθ )()(),( nmimnmn eXfXF −= (2.2.12)

where the amplitude pattern functions fmn can be represented as the first derivative of the product of the regular and irregular Fock state wave functions in the quadra-ture representation ψn(X) and ϕn(X)

( ))()(1

)( XXx

Xf nnmn ϕψπ ∂

∂= (2.2.13)

The normalizable regular wave functions can be stated in terms of Hermite poly-nomials

2

!2

)2()(

2/1

X

n

nn e

n

XHX −

−=

πψ (2.2.14)

The irregular wave functions, which correspond to the not normalizable solutions of the time-independent harmonic oscillator Schrödinger equation, can be ap-proximately calculated using the backward recursion

( ))(2)(21

1)( 21 XnXX

nX nnn ++ +−

+= ϕϕϕ (2.2.15)

To obtain the irregular wave function up to a maximum photon number of N, it is recommended93 to start from initial values n = 4N, 4N-1 and to employ the semi-classical approximation

93 U.Leonhardt, “Measuring the Quantum State of Light“, ch. 5.2 (Cambridge Univ. Press, USA, 1997)

Figure 2-9 Examples of amplitude pattern functions for quantum state sampling.

X-values have been scaled by 1/ 2 due to a different normalization.


( )

+−

++

≈4

2)2sin(2

12sin

sin12

ππϕ nn

nn tt

n

tn (2.2.16)

with )12/arccos( += nXtn . This approximation may be trusted up to quadrature

values of )182/118(2|| 3 −−−= NNX .

To avoid the derivative in equation (2.2.13) the amplitude pattern functions fmn for mn ≥ may be calculated according to

))()(12

)()(12)()(2(1

)(

1

12/3

XXn

XXmXXXXf

nm

nmnmmn

+

+

+−

+−=

ϕψ

ϕψϕψπ (2.2.17)

This formula also yields the amplitude pattern functions for mn < since nmmn ff = .

The procedure sketched here allows an efficient numerical implementation of the direct sampling formula from equation (2.2.11).

Compensation of Experimental Losses Any experimental realization of optical homodyne tomography will inevitably suffer from experimental losses. If these losses are known they may be compen-sated for, if the generalized inverse Bernoulli transformation94,95 is applied to the density matrix lossρ obtained from the experimental data by quantum state sampling. This becomes possible since the effect of losses – modeled by a beam splitter with transmission η (= 1 – losses) – for any Fock state is known and de-scribed by the Bernoulli transformation. Inverting this transformation yields the inversion formula

∑∞

=

+− −

+

+++=

0

2/)( )1

1(ˆˆk

kloss

nm

n

kn

m

kmknkmnm

ηρηρ (2.2.18)

where ρ denotes the density matrix of the quantum state without losses. Since the

factor k)/11( η− increases exponentially with k and changes sign, this method exhibits a very high sensitivity to small deviations in the density matrix. To avoid a blow-up of small errors the k-summation can be truncated for values of m and n above which numerical and statistical errors dominate the value of the density ma-trix elements, if these elements do not carry any significant information about the quantum state.

94 T. Kiss, U. Herzog, U. Leonhardt, Phys. Rev. A 52, 2433 (1995) 95 Ch.T. Lee, Phys. Rev. A 48, 2285 (1994)

28 2 Theory

2.3 Single-photon Fock State Tomography The single-photon Fock state is one of the most fundamental states of the light field. It is highly non-classical and reveals the particle nature of light most strik-ingly. Its Wigner function exhibits a strong negativity around the origin of phase space and its marginal distribution are of non-Gaussian shape. Even though a quan-tum state reconstruction of a Fock state has already successfully been performed on the vibrational state of Beryllium-ions in a trap, no full quantum characterization of a single-photon has been achieved.

Our main motivations to embark on the project of single-photon Fock state tomo-graphy were

§ to improve our understanding of a “single-photon” by attempting to obtain a characterization of its wave function and Wigner function,

§ to measure non-Gaussian marginal distributions and negativities of the recon-structed Wigner function for the first time in the optical domain,

§ to combine the methods of photon counting and measurements of quantum field fluctuations in a single experiment,

§ and to advance the technique of homodyne tomography to the single -photon level.

2.3.1 Introduction The idea to perform a single -photon Fock state preparation by conditional meas-urements on a photon pair goes back a long way: already as early as 1985 C.K. Hong and L. Mandel prepared single -photon Fock-states by performing conditional measurements on a photon-pair produced in parametric 2-photon down-conversion.

Probably the first authors to point out the non-classical behavior of an n-photon Fock state prepared by conditional measurements and measured with a homodyne detector were B. Yurke and D. Stoler96 in 1987. They also calculated the resulting quadrature distributions as a function of the efficiency of the detector. It is these quadrature distributions that we have measured in the experiment described in this thesis.

In 1989 A. Aspect and P. Grangier97 demonstrated strongly non-classical features and interference effects of a single -photon wave packet prepared by conditional measurements on a photon-pair.

Z.Y. Ou pointed out that a photon – incident on a beam splitter – might drastically change the photon count statistic at the two output beams98. He also gave a theo-retical treatment of a homodyne measurement performed on a parametrically gen-erated photon pair using a pulsed pump source99.

96 B. Yurke, and D. Stoler, “Measurement of amplitude probability distributions for photon-number-operator eigenstates”, Phys. Rev. A 36, 1955 (1987) 97 A. Aspect , P. Grangier, G. Roger, “Wave particle duality for a single-photon”, J. of Opt. 20, 119 (1989) 98 Z.Y. Ou, “Quantum multi-particle interference due to a single-photon state” , Qu. Semiclass. Opt. 8, 315 (1996) 99 Z.Y. Ou, “Parametric down-conversion with coherent pulse pumping and quantum interference between independent fields”, Qu. Semiclass. Opt. 9, 599 (1997)

2.3 Single-photon Fock State Tomography 29

Outline of the theoretical treatment

The primary laser source of our experiment is a mode-locked, pulse-picked laser which provides transform limited ps-laser pulses. The dominant fraction of these pulses is single-pass frequency doubled. Chapter 2.3.2 describes the process of pulsed second harmonic generation to derive the expected power and pulse shape of the doubled beam and to estimate the effect of group velocity dispersion on the generated pulses.

The second harmonic pulses provide the pump power for the process of pulsed 2-photon down-conversion which produces strongly correlated photon pairs. In a simplified notation the generated quantum state may be written as

( )∫∫ Φ+=Ψ isisisis kkkdkdN 1,1),(0,0 33out

rr (2.3.1)

where N represents a normalization constant. The details of this state are derived in chapter 2.3.3 where also an estimation of the expected photon pair creation rate is given and a treatment of the induced process of parametric amplification with fo-cussed beams is presented.

A single-photon Fock state is prepared from the state in equation (2.3.1) by project-ing it onto a photon count event in the trigger beam path t1 :

sts N 11 out =Ψ′=Ψ (2.3.2)

The Fock state preparation by conditional measurement is treated in chapter 2.3.4. The fidelity of the state preparation in the presence of filters and irises is also con-sidered in this chapter.

The state sΨ is overlapped with a local oscillator mode and detected at a homo-dyne detector. This detector performs a quadrature amplitude measurement on the state sΨ resulting in marginal distributions

22

,meas, 0)1(1)(pr θθθθθ ηηρ XXXXX LOLO −+== (2.3.3)

with the measurement efficiency 2

LOs=η where LO and s symbolize the temporal and spatial mode of the local oscillator and the signal beam. From this marginal distribution the single-photon Fock state can be reconstructed employing the method of homodyne tomography described in the last chapter. Pulsed homo-dyne detection is discussed in chapter 2.3.5 where also a general method for the calculation of the efficiencies and optimum parameter settings of non-ideal homo-dyne systems is developed and applied to various inefficiencies. Chapter 0 provides a more detailed treatment of the homodyne detection of a single-photon Fock state, and finally chapter 0 discusses the effects of a non-ideal efficiency on the measured marginal distribution and the reconstructed Wigner function.

Second harmonic generation

Photon pair creation

Fock state preparation

Pulsed homodyne detection

30 2 Theory

2.3.2 Second Harmonic Generation In the process of second harmonic generation light of the fundamental frequency ω is converted to light of frequency 2ω. For a single pass configuration, focused Gaussian beams and an undepleted fundamental pump beam the generated second harmonic power may be calculated according to100,101

22 ωω PP SHGΓ= (2.3.4)

where

2

22

20

3

2 132

ωπλεπ

wLhd

cn effSHG =Γ (2.3.5)

and n denotes the refractive index, λ the wavelength of the pump beam, deff the effective nonlinearity, L the length of the doubling crystal wω the waist of the fun-damental beam when entering the crystal and h the Boyd-Kleinman-factor.

The generated second harmonic power therefore is proportional to the square of the fundamental power, to the square of the effective nonlinearity and the square of the crystal length. It is inversely proportional to the effective area of the fundamental beam assumed to be in a 00-Gaussian mode. For an elliptical beam profile we need to replace

yx www ππ ω

112 → (2.3.6)

where wx/y♣ denote the Gaussian beam waist in the x- and y-direction. Thus, in

general, focussing into the crystal more tightly will increase the intensity of the fundamental pump beam in the crystal and lead to a higher conversion efficiency. However, if the crystal length exceeds the Rayleigh-range of the pump beam then the beam will only be focused efficiently in the middle of the doubling crystal and spread considerably towards the edges of the crystal. For tightly focused beams we therefore have to account for the beam spread which leads to a correction, called the Boyd-Kleinman-factor,

22/

2/ /11 ∫−

∆

+=

L

LR

zki

dzzzi

eL

h (2.3.7)

This factor is unity in the case of wide beams, but is reduced considerably, if focus-ing becomes too tight. The two effects of increasing intensity and beam spreading counteract each other leading to an optimum conversion efficiency, if the crystal length L = 5.68 zR. In this optimum case we obtain

Ldcn

effSHG2

032

2642.1

ελ

π=Γ . (2.3.8)

Thus, if optimum focusing is chosen, the generated second harmonic power de-pends only linearly on the length of the doubling crystal.

Due to the fact that the wave fronts of the generated second harmonic beam have to fit to the square of the pump beam wave fronts the beam waist radius of the second

harmonic beam w2ω will be smaller than that of pump beam by a factor of 2 and

100 A. Yariv, Quantum Electronics – 3rd ed., p. 400 (John Wiley & Sons, 1989, USA) 101 R.W. Boyd, Nonlinear Optics, p. 93 (Academic Press, 1992, USA) ♣ Here and in the following “/” is used a for alternative subscripts thus wx/y should be read as wx or wy.


the far field half-width diffraction angle of the second harmonic beam )2/( 2ωπλθ w= will also be smaller than that of the incident beam by a factor of

2 .

Pulsed Second Harmonic Generation In the case of a pulsed pump wave equation (2.3.4) is valid only for the peak pow-ers of the second harmonic and fundamental wave. The average powers of the two beams are related by

22

2ln

1ωω

τπ

PT

P SHG ∆Γ= , (2.3.9)

where we have made use of equation (3.3.1) to replace the peak powers with the average powers.

Using the experimental parameters for BBO (λω = 790 nm, deff = 1.36 pV/m, n = 1.661, L = 3 mm, wω(L/2) = 56 µm (focusing with a 200 mm lens), τ = 1/81.2 MHz, ∆T = 1.8 ps, Pω = 1.65 W) we obtain an expected second harmonic power of 47 mW corresponding to a conversion efficiency of 2.6%. In the case of LBO (deff = 0.851 pV/m, n = 1.611) an expected power of 21 mW corresponding to a conversion efficiency of 1.3%. Both values agree well with the second harmonic powers observed in the experiment (chapter 2.3.2).

In the case of pulsed second harmonic generation we can no longer assume the fields to be monochromatic, but we have to allow for a certain spread in the fre-quency domain. In the case of transform-limited light pulses we can decompose the fields of the fundamental and the second harmonic beam into their spectral compo-nents

∫ −⋅=⋅= ωω ω degAtgAtA tipLpLpL )(~)()( 0,/0,// (2.3.10)

where the subscript p is used to denote the second harmonic wave (which is later used as the pump wave for the down converter) and the subscript L for the funda-mental pulses from the pump laser. g(t) represents the temporal shape function and

))(()(~ tgFg =ω its Fourier transform, the spectral shape function, which is nor-malized according to

∫ = 1)(~ 2 ωω dg . (2.3.11)

The nonlinear susceptibility couples the pump and the second harmonic waves

)()( 2 tAitA Lp κ−= (2.3.12)

κ describes the nonlinear coupling.

Equation (2.3.12) may be used to derive the spectral shape of the generated second harmonic pulses:

∫∫∫

∫∫ ∫

−−=

′−−′′−=

′′−= ′−−

)(~)(~

)()(~)(~

)(~)(~)(~

20,

20,

0,0,0,

LpLLLLL

LLpLLLLLLL

tiLLLL

tiLLLL

tippp

ggdAi

ggddAi

egAdegAdedtidgA LL

ωωωωκ

ωωωδωωωωκ

ωωωωκωω ωωω

(2.3.13)

32 2 Theory

Consequently the spectral shape of the generated second harmonic beam results from a convolution of the spectral shape function of the fundamental beam with itself

LLp ggg ~~~ ⊗= (2.3.14)

We therefore expect the generated second harmonic pulses to be wider in the fre-

quency domain by a factor of 2 and shorter in the time domain by the same fac-tor.

Group Velocity Dispersion Another effect that deserves attention in the case of pulsed second harmonic gen-eration is group velocity dispersion: Phase-matching assures that the wave fronts of the fundamental and second harmonic wave will maintain a fixed phase throughout the crystal. However, this does not imply that the pulses traveling through the crys-tal also possess the same group velocity. As a consequence, if the up-conversion occurs at the beginning of the crystal, the wave packet will propagate through the crystal with a different velocity than if the conversion occurs at the end of the crys-tal.

This group velocity mismatch D will lead to a maximum time delay of

lD ⋅=∆τ , 21

11

gg vvD −= , (2.3.15)

where vg1 denotes the group velocity of the harmonic and vg2 the group velocity of the sub-harmonic pulses.

Hence in the general case where the two group velocities do not coincide, the gen-erated pulses will acquire a certain delay depending on the position in the crystal where the up-conversion occurs. Integrating all the contributions for the different positions along the crystal we obtain

∫∆ ′−′⋅=∆ ττ00, )(),( ttgtdAtA ppp (2.3.16)

Assuming a Gaussian shape for g(t) we may carry out the integration

))2

(erf)2

(erf(),( 0, στ

στ ∆−−=∆ tt

NAtA pp (2.3.17)

where N is a normalization constant and erf() denotes the error function.

The pulse shape of the generated second harmonic pulses with and without disper-sive delay using the experimental parameters for frequency doubling in a BBO crystal is depicted in Figure 2-10. Group velocity dispersion leads to a significant shift of the pulse center by ∆τ/2 = 0.3 ps, but the pulse width increases only slightly from 2.355 ps to 2.366 ps. Thus the transform-limited character of the pulses is maintained in the process of second harmonic generation.

If transform-limited character of the pulses would be lost, then these pulses would not correspond to pure states but to statistical mixtures and could therefore not be employed for the generation of transform-limited single-photon Fock states. Ex-perimentally this would be the case, if femto-second laser pulses were used instead of pico-second pulses.


2.3.3 2-Photon Down-conversion In the process of 2-photon down-conversion energy is transferred from a higher frequency pump-wave to two lower frequency waves, usually labeled signal and idler. In the fundamental process associated with the interaction Hamiltonian

++−= ispI aaaiH ˆˆˆ/ˆ κh a pump photon is annihilated creating a signal and an idler photon. The possible emission wavelengths and directions are governed by the laws of energy and momentum conservation. The term parametric fluorescence is employed, if the down-conversion process occurs spontaneously as a first order scattering event.

The process of 2-photon down-conversion has been the backbone of many experi-ments in the field of quantum optics in our group as well as in other research groups around the world. In our group the interest has started with experiments on squeezed light generation102,103,104 and optical QND-measurements105,106,107. These efforts – inspired by the urge to inquire fundamental questions in quantum optics – have led to two different avenues of research: The first avenue focused on an appli-

102 S. Schiller, S. Kohler, R. Paschotta, and J. Mlynek, “Squeezing and quantum nondemolition meas-urements with an optical parametric amplifier”, Appl. Phys. B 60, S77-88 (1995). 103 G. Breitenbach, T. Müller, S.F. Pereira, J.-Ph. Poizat, S. Schiller and J. Mlynek, “Squeezed Vac-uum from a Monolithic Parametric Amplifier”, J. of Opt. Soc. Am. B 12, 2304-9 (1995). 104 K. Schneider, R. Bruckmeier, H. Hansen, S. Schiller, and J. Mlynek, “Bright squeezed light gen-eration by a continuous-wave semi-monolithic parametric amplifier”, Optics Letters 21, 1396 (1996) 105 R. Bruckmeier, K. Schneider, S. Schiller, and J. Mlynek „Quantum nondemolition measurements improved by a squeezed meter input” , Phys. Rev. Lett. 78, 1243 (1997) 106 R. Bruckmeier, H. Hansen, S. Schiller, and J. Mlynek “Realization of a paradigm of quantum measuremetns: the squeezed light beam splitter”, Phys. Rev. Lett. 79, 43 (1997) 107 R. Bruckmeier, H. Hansen, S. Schiller, and J. Mlynek “Repeated continuous quantum nondemoli-tion measurements”, Phys. Rev. Lett. 79, 1463 (1997)

- 3 - 2 - 1 0 1 2 3 4

0.1

0.2

0.3

0.4

0.5

|g(t

)|2

t

dispersive delayincluded

without dispersivedelay

Figure 2-10 Second harmonic pulses with and without dispersive delay. Even though the center of the pulses is shifted by ∆τ/2, their width does not change considerably.

34 2 Theory

cation oriented approach, which built on the expertise acquired in this field to de-velop new devices108,109,110 (such as singly, doubly and pump resonant OPOs, with the paramount aim of an optical synthesizer) and explore their applications in dif-ferent areas111,112. The second avenue continued the investigation of fundamental quantum optical issues with the work on optical homodyne tomography of classical and non-classical states of the light field (chapter 2.2). The work presented here is aimed at extending this efforts to the generation and characterization of highly non-classical states of the light field.

To gain access to highly non-classical states of the light field we decided to exploit the high degree of quantum correlation in parametrically produced photon pairs, which in turn also required the development of new tools for the characterization of these pulsed quantum states (chapter 2.3.5). In this chapter I will develop the theo-retical background for the description of the single-photon Fock state generation by pulsed parametric fluorescence.

Pulsed Parametric Fluorescence The process of parametric fluorescence produces a complex photon pair emission pattern with rich and intriguing properties. Momentum conservation enforces that photons of equal frequency are emitted into a cone (Figure 2-11). The strong quan-tum correlations between the two photons of a pair have led to a number of quan-tum optical experiments:

C.K. Hong and L. Mandel were the first ones to employ the correlation properties of the twin photons to prepare single-photon Fock states113. Coherence properties of the parametric fluorescence process have been studied in detail in several pa-pers. R. Ghosh et al.114 predicted quantum interference effects in coincidence count measurements (fourth-order interference) on correlated photons.

A number of groups have exploited polarization entanglement in type II parametric fluorescence, for quantum optical experiments. In the type II parametric fluores-cence process signal and idler photons are emitted with orthogonal polarizations in two cones. At the intersection points of the two cones it is not possible to distin-guish which photon belongs to which cone leading to polarization entanglement115, and a quantum (polarization) state of the type ),,(

21 hvvh +=ψ , where v and

h denote a vertical or horizontal polarization. This state has been used for experi-

108 K.Schneider, P. Kramper, S .Schiller, and J. Mlynek, “Toward an optical synthesizer: a single-frequency parametric oscillator using periodically poled LiNbO3”, Opt. Lett. 22, 1293 (1997) 109 K. Schneider, S. Schiller, “Narrow-linewidth, pump-enhanced singly-resonant parametric oscilla-tior pumped at 532 nm”, Appl. Phys. B 65, 775 (1997) 110 U. Strößner, A. Peters, J. Mlynek, S. Schiller, J.-P. Meyn, R. Wallenstein, “Single-frequency con-tinuous-wave radiation from 0.77 to 1.73 µm generated by a green-pumped optical parametric oscil-lator with periodically poled LiTaO3”, Opt. Lett. 24, 1602 (1999) 111 F. Kühnemann, K. Schneider, A. Hecker, A.A.E. Martis, W. Urban, S. Schiller, J. Mlynek, “Photoacoustic trace gas detection with a single-frequency continuous-wave optical parametric oscillator”, Appl. Phys. B 112 K. Bencheikh, R. Storz, K. Schneider, K. Jäck, M. Lang, J. Mlynek, and S. Schiller, „Absolute frequency stabilization of a continuous-wave optical parametric oscillator to the sub-kHz level” , OSA Tops 19, 392 (1998) 113 C.K. Hong and L. Mandel, "Experimental realization of a localized one-photon state" , Phys. Rev. Lett. 56, 58 (1986) 114 R. Ghosh, C.K. Hong, Z.Y. Ou, and L. Mandel, "Interference of two photons in parametric down-conversion" , Phys. Rev. A 34, 3962 (1986) 115 P.G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A.V. Sergienko, and Y. Shih: "New high-intensity source of polarization-entangled photons pairs" , Phys. Rev. Lett. 75, 4337 (1995)


ments to test the non-locality of quantum mechanics116, interaction free measure-ments117, quantum teleportation118,119, and entanglement swapping120.

Employing the second order scattering process in which two entangled photon pairs are created simultaneously in the same mode three-photon entanglement can be realized121, leading to Greenberger-Horne-Zeilinger states122. These states allow for tests of non-locality in a single experiment without the need for statistical viola-tions of Bell-type inequalities.

More recent work on parametric fluorescence has focused on the complications that are introduced, if pulsed laser sources are employed. Short pulses lead to com-plicated multimode entanglement123,124,125

.

116 G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, "Violation of Bell's inequality under strict Einstein locality conditions", Phys. Rev. Lett. 81, 5039 (1998) 117 P.G. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, and M.A. Kasevich: "Interaction-free meas-urements" , Phys. Rev. Lett. 74, 4763 (1995) 118 D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger: "Experimental quantum teleportation" , Nature 390, 575 (1997) 119 A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, „Un-conditional quantum teleportation“, Science 282, 706 (1998) 120 J.-W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, "Experimental entanglement swap-ping: Entangling photons that never interacted" , Phys. Rev. Lett. 80, 3891 (1998) 121 A. Zeilinger, M.A. Horne, H. Weinfurter, and M. Zukowski, "Three-particle entanglements from two entangled pairs" , Phys. Rev. Lett. 78, 3031 (1997) 122 D. Bouwmeester, J.-W. Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, "Observation of three-photon Greenberger-Horne-Zeilinger entanglement", Phys. Rev. Lett. 82, 1345 (1999) 123 Z.Y. Ou, "Parametric down-conversion with coherent pulse pumping and quantum interference between independent fields", Quantum Semiclass. Opt. 9, 599 (1997) 124 A. Joobeur, B.E.A. Saleh, and M.C. Teich, "Spatiotemporal coherence properties of entangled light beams generated by parametric down-conversion" , Phys. Rev. A 50, 3349 (1994) 125 A. Joobeur, B.E.A. Saleh, T.S. Larchuk, and M.C. Teich, "Coherence properties of entangled light beams generated by parametric down-conversion: Theory and experiment" , Phys. Rev. A 53, 4360 (1996)

Figure 2-11 In the process of (type I phase-matched) parametric χ(2)-fluorescence highly correlated signal and idler photon pairs are generated. Photons of a given frequency are emitted in cones centered on the pump beam. Photon pairs are emit-ted into opposite directions on their respective cones. Only in the frequency degen-erate case both twin photons lie on the same cone.

36 2 Theory

The interaction Hamiltonian for the χ(2)-parametric -fluorescence process in the case

of a pulsed pump beam can be written as:

..ˆˆ),,(

)();,()(ˆ

)(

volumecrystal

2/

2/

0)(33

chaaezyxudzdydx

eAkdkddtH

isrkkki

pl

l

tipppiseffispI

sip

pis

+

⋅=

++−−−

∞∞−

∞∞−

∞ −+

∫∫∫

∫ ∫∫rrrr

444 3444 21

ωωωωωωωχω

(2.3.18)

where we have taken the pump to be a classical wave. up(x,y,z) describes the trans-

verse field mode of the pump wave, Ap(ωp) is the spectral shape of the (pulsed)

pump beam♦ , and the effective nonlinearity χeff (ωs, ωi,;ωp) is given by

)(ˆ)(ˆˆ)()(2

)()();,( )2(

32

2/1322/5

iksjlljk

ioso

ispep

pixeff kokoennc

ni rrhχ

ωωπ

ωωωωωωωχ −= (2.3.19)

in the case of Type I (eoo) phase-matching. Furthermore the extension of the crys-

tal in the transverse xy-plane is assumed to be much larger than the extent of the

optical pump beam, so that the integration boundaries of the transverse integrals

may be taken to infinity (in the experiment the down-conversion crystal has a size

of 8(x) · 8(y) · 3(z) mm3, the pump beam diameter is < 3 mm (FWHM at 1/e2-

points), thus the expected error is smaller than 6.6· 10-7).

We choose our coordinate system so that the pump beam travels along the z-

direction. The pump beam focus coincides with the center of the down-conversion

crystal. If the pump beam is weakly focused, then the dependence of the transverse

pump mode up(x,y,z) on z may be neglected, so that the z-integration just yields the

crystal length♣.

After carrying out the x,y-integration the interaction Hamiltonian takes on the form

..ˆˆ))(2

(sinc

)(~)();,()(ˆ

)(,,

)()(

,,33

,,,,

chaaekkkl

l

kkuAkdkddtH

isti

k

pzizs

kkkk

isppppiseffispI

pis

z

yiysxixs

+−+⋅

⋅+=

++−+

∆

++≈

⊥⊥∫∫∫

ωωω

δδ

ωωωωχω

44 344 21

44 344 21

rr

(2.3.20)

♦ At this point we choose our time origin to coincide with the center of the pump pulse thus omitting a phase factor )exp( pp tiω− . ♣ This is a good approximation, as long as the Rayleigh length of the pump beam is much bigger than

the z-extension of the crystal, i.e. ZR,p >> l/2. This leads to the condition

p

p

n

lw

π

λ

20 >>

Inserting the experimental parameters (λ = 0.79 µm, l = 3 mm, np = 1.62) we obtain the condition w0 >> 15 µm, which is well satisfied in the experiment.

Interaction Hamiltonian


Here )(~,, ⊥⊥ + isp kku

rris the Fourier transform of the transverse optical mode of the

pump beam and approaches a delta function for unfocussed beams – with a spread

that is governed by the k-vector spread of the pump beam.

In the interaction picture the time evolution of the initial state )0(Ψ passing

through the down-conversion crystal can be written as:

)0()()(ˆ

Ψ=Ψ ∫ ′′− tHtdi

Iet h (2.3.21)

For a small perturbation the exponential time evolution operator may be expanded

to first order, yielding

)0())(ˆ1()( Ψ−=Ψ ∫ tHdti

Nt Ih

(2.3.22)

where N is a normalization constant. Higher order terms may be neglected if

1)(2

<<=Γ ∫ tHdt Iih

. In the context of our experiment Γ acquires the physical

meaning of the chance per pulse to generate a photon pair and is of the order 10-4

(full laser power) or 10-5 (pulse picked operation). Thus the perturbative approach

is well justified.

The next higher order process produces photon twin states with two photons in

each of the signal/idler channels and occurs with a rate proportional to Γ 2. Since

the trigger detector only distinguishes the cases 0=n and 0>n these events

will not be filtered out by our gating procedure and contribute to our final results.

This contribution is of the order %002.0102/2 52 =⋅≈Γ − and will therefore be

neglected throughout this thesis. However – these higher order processes may be

successfully employed for the generation of n > 2-particle correlations126 and for

the generation of higher Fock states (chapter 2.4.1).

Inserting the interaction Hamiltonian for the parametric fluorescence (2.3.20) into

equation (2.3.22) and taking the input state to be the vacuum state for both the sig-

nal and idler beam leads to

126 D. Bouwmeester, J.-W. Pan, M. Daniell, H. Weinfurter, A. Zeilinger, „Observation of three-photon Greenberger-Horne-Zeilinger entanglement“, Phys. Rev. Lett. 82, 1345 (1999)

Time evolution

38 2 Theory

),)2

(sinc)(~)();,(

0(

),)2

(sinc)(~)();,(

)(0(

),(

,,

33

,,

nintegratiotimetodue

33out

is

kk

zispispisiseff

is

isz

isppppiseff

pisisp

kkkl

kkuA

kdkdN

kkkl

lkkuA

kdkddN

is

rr

444444444444 3444444444444 21

rr

rrrr

444 8444 76

rrΦ≡

⊥⊥

⊥⊥

∆⋅+++′

+=

∆⋅⋅+′

−++=Ψ

∫∫

∫∫∫

ωωωωωωχ

ωωωωχ

ωωωδω

(2.3.23)

where isk /

r denotes a single-photon excitation in the mode associated with the

wave vector isk /

r, )/( cleffeff hχχ =′ and where we have introduced a function

Φ that carries the information about the transverse and longitudinal structure of the

photon twin. This function is made up of four major components:

1. a term governing the strength of the nonlinear interaction (χ’eff) which is

slowly varying as a function of the optical frequencies involved,

2. a term that describes the longitudinal bandwidth of the pump wave

(A(ωs+ωi)), and which reflects the pulsed nature of the pump beam,

3. the function )(~,, ⊥⊥ + isp kku

vv, which derives from the transverse k-vector-

spread of the pump beam,

4. and the sinc-function, which accounts for the spread of the phase-matching

curve due to the finite length of the crystal.

The non-vacuum part of the state in equation (2.3.23) exhibits a non-local character

and strong quantum correlations between the two photons generated in the para-

metric process. Since the state cannot be factored into a wave function describing

the signal and one describing the idler beam, it is an entangled state. Due to their

entangled nature the two photons are often regarded as a single quantum object –

called bi-photon.

We will now proceed by examining the spread of the function Φ , which will allow

us to specify the transverse and longitudinal shape of the bi-photon.

Let zkk zpp ˆ0,

0 ⋅=r

, 0sk

r, and 0

ikr

denote one specific set of wave-vectors for which

the phase-matching condition is fulfilled. Let us also take the wave vectors 0sk

r and

0ik

r to lie in the xz-plane. We also introduce the deviations from the optimum con-

figuration

Bi-photon


0

0///

0///

0///

=

−=∆

−=∆

−=∆

isisis

isisis

isisis

ϕϕϕ

θθθ

ωωω

(2.3.24)

which we assume to be small. Figure 2-12 clarifies the sign conventions and the

geometry of the setup.

First we realize that the frequency dependence of the effective nonlinearity is much

weaker than that of the other three terms. We may therefore neglect its contribution

and write

)2

(sinc)(~)(),,,,,( ,,z

ispispisisiskl

kkuAN∆

⋅−+′≈∆∆∆∆∆∆Φ ⊥⊥rr

ωωϕϕθθωω

(2.3.25)

where N’ is a normalization constant.

For a cw-pump A(ωs+ωi) approximates a delta-function. The same holds for

)(~,, ⊥⊥ + isp kku

vv, if the transverse mode function of the pump beam is that of a plane

wave. In this case Φ reduces to the well known sinc-shaped phase-matching func-

tion.

These approximation – however – are not fulfilled in our experiment, so we pro-

ceed by expanding ∆k z and yxisyx kkk /,,/ )( ⊥⊥ +=∆rr

as functions of

isisis ϕϕθθωω ∆∆∆∆∆∆ ,,,,, . We obtain

ss

s

n

si

i

i

n

i

ss

N

si

i

N

ix

si

si

nn

cN

cN

k

θλ

θλπθ

λθλπ

ωθλ

ωθλ

∆−∆+

∆−∆=∆

≡≡

≡≡

00

00

00

00

cos)(2cos)(2

sin)(sin)(

876876

48476876

(2.3.26)

Figure 2-12 Sign conventions and geometry in calculating the bi-photon band-width

40 2 Theory

)(sin2 0

iss

ssy

nk ϕϕ

λθπ

∆−∆=∆ (2.3.27)

)(sin2

coscos

0

00

iss

ss

spss

ipii

z

n

c

NN

c

NNk

θθλ

θπ

ωθ

ωθ

∆+∆+

∆−

−∆−

=∆ (2.3.28)

where Ns/i denotes the group velocity refractive index for the signal and idler beam

respectively ( )()()( /,,,/ isoisoisisois nnnN λλλλ ≈′−= ) and where we have em-

ployed the condition 0,

0, xixs kk = .

For the frequency-degenerate case ( isiosois nn θθλλλλ =⇒== )()(, ) these

expansions simplify to

ps

ssp

pssz

sis

ssy

sis

sssi

ssx

nc

NNk

nk

nc

Nk

θλ

θπω

θ

ϕλ

θπ

θλ

θπω

θ

∆+∆−

=∆

∆=∆

∆−∆=∆

00

0

00

sin2cos

sin2

cos2sin

(2.3.29)

where we have changed variables to the more intuitive and convenient quantities

sisi

sisiisp

sisiisp

ϕϕϕ

θθθθθθ

ωωωωωω

∆−∆=∆

∆−∆=∆∆+∆=∆

∆−∆=∆∆+∆=∆

,,

,,

(2.3.30)

As a further simplification we approximate the constituents of Φ in terms of Gaus-sians:

22

222

2

2

)(4

4

)(

40

)2

(sinc

2)(~

)(

lkz

kkw

pp

z

yx

p

p

elk

ew

ku

eAA

∆−

∆+∆−

⊥

∆−

≈∆

≈∆

⋅≈∆

α

σ

ω

π

ω

r (2.3.31)

where the pump pulse spread is given by 2)(

2ln21 FWHM

ppωσ ∆= and corresponds

to 0.43 nm in the experiment, the pump beam waist at the position of the down-

conversion crystal is w = 1.5 mm and α = 0.322 is a numerical factor chosen to

match the 1/e-width of the approximating Gaussian to that of the sinc-function.


Substituting equation (2.3.29) into equation (2.3.25) and making use of the Gaus-

sian approximation we obtain

))(41

exp(

)4

exp(2

),,,,(

2,

2

2,

2

2,

2

2,

2

2,

2

0

sisisipppsi

si

p

p

si

si

p

p

si

sipsisipsip

wAN

θωκθωκσ

θ

σ

θ

σ

ω

σ

ω

σ

ϕπ

ϕθθωω

θθωω

ϕ

∆∆+∆∆+∆

+∆

+∆

+∆

−

⋅∆

−⋅⋅⋅′=∆∆∆∆∆Φ

(2.3.32)

where

s

sssssi

s

sspssp

ss

ssi

ss

ssi

ss

sp

sssi

pssp

pp

clnN

c

lnNN

nw

nw

nl

Nw

c

NNlc

c

λαθπθ

κ

λ

αθπθκ

θπ

λσ

θπ

λσ

θαπ

λσ

θσ

θσα

σσ

ϕ

θ

θ

ω

ω

2200

2200

0,

0,

0,

0,

202,

cos2sin

sin2)cos(

sin2

cos2

sin2

sin

))cos((

=

−=

=

=

=

=

−+=

(2.3.33)

Using the parameters of the actual experiment in equation (2.3.33) we obtain the

phase-matching bandwidth of the photon pairs used in the experiment:

σω,p = 0.262 nm, σω,si = 0.555 nm, σθ,p = 0.065°, σθ,si = 0.00297°, σφ,si = 0.042°

or for the FWHM-spreads:

∆ω,p = 0.618 nm, ∆ω,si = 1.307 nm, ∆θ,p = 0.153°, ∆θ,si = 0.0070°, ∆φ,si = 0.099°

From equation (2.3.32) we observe that azimuthal deviations decouple from polar

and longitudinal deviations, whereas there is a correlation between polar and longi-

tudinal deviations. These correlations only exist between the differential deviations

(those indexed “si”) and between the center deviations (indexed “p”).

The physical origin of these correlations may easily be explained: If we decide to

choose ∆ωp and ∆θp in a way that we walk along the cone of frequency degenerate

pair emission, ∆k z will vanish and Φ will remain maximal. This accounts for the

center correlations. Also we may chose ∆ωsi and ∆θsi in a way that we are still

42 2 Theory

looking at photon twins, but non-degenerate ones, where we will again observe a

high degree of correlations. This accounts for the differential correlations.

Let us consider these correlations more closely. From equation (2.3.31) we learn

that, if the conditions

sis

sssi

ss n

c

Nθ

λθπ

ωθ

∆=∆00 cos2sin

(2.3.34)

ps

ssp

pss n

c

NNθ

λθπω

θ∆−=∆

− 00 sin2cos (2.3.35)

hold, then Φ does not limit the width of the generated photon pairs to first order.

To obtain the corresponding phase matching bandwidth in this case, we have to

expand the wave vector mismatch as a function of the frequency displacements

∆ωp and ∆ωsi up to second order yielding additional contributions in equations

(2.3.28) and (2.3.29)

2,

2,

)2(

21

)21

(21

sizspzspz kkkk ωω ∆′′−∆′′−′′=∆ (2.3.36)

where 0

|/ 22, ppzp kk ωω∂∂=′′ and 0222

,2

, cos|/|/00 sszszs ss

kkk θωω ωω ⋅∂∂≈∂∂=′′ .

We utilize the condition β!

)2( =∆ zk (β = 1.3916 is the HWHM of the sinc2-

function) to obtain the second order FWHM-bandwidths

)2/1(4

2

42

,

2

,

,

2

,

zsp

ppp

zs

sssi

kklc

n

klcn

′′−′′=∆ ′′

′′=∆ ′′

βπ

λ

βπλ

ω

ω

(2.3.37)

Using

∂∂

≈∂∂

+∂∂

=′′2

2

2

3

2

2

2

2

λπ

λωω

ω n

c

nc

nc

k (2.3.38)

and inserting the parameters for the BBO-down-converter employed in the experi-

ment yields si,ω∆ ′′ = 133 nm and p,ω∆ ′′ = 21.5 nm. As expected sisi ,, ωω ∆>>∆ ′′ and

sipsip ,, ∆>>∆ ′′ .

This concludes the discussion of the quantum state and the wave function of the

photon twins produced in the process of 2-photon down-conversion. The following

section gives an estimate of the number of photon pairs available for the measure-

ments. The theoretical description of the single-photon Fock state tomography is


continued with a treatment of the conditional state preparation on page in chapter

2.3.4 and concluded with a description of the homodyne measurements performed

on the generated state in chapter 0.

Expected Count Rate

The expected count rate in parametric fluorescence may be calculated according to

Fermi’s Golden Rule

fI iHfW ρπ 2ˆ2h

= (2.3.39)

where IH denotes the interaction Hamiltonian for the 2-photon down-conversion

(equation (2.3.18)) which connects the initial state isi 0,0= to the final state

is kkfrr

,= , and ρf is the density of states. For paraxial wave vectors the number

of states in the infinitesimal k-space volume is kdkdrr

33 may be approximated by

ii

iss

sis dEc

nkdd

cn

kdV

kdkdV

h

rr,

2,

26

233

6

2

)2()2(⊥⊥≈ ω

ππ. (2.3.40)

where dEi is the infinitesimal change in the idler energy. Hence the density of final

states per unit energy is given by

sisis

f dkdkdc

nnVω

πρ ,

2,

226

2

)2(⊥⊥=

h. (2.3.41)

K. Koch et al.127 have carried out the necessary calculation for Gaussian beams and

arrive at the result

spisp

peffs P

n

LdcP λ

λλε

λπ ∆=

2520

24 2

)2(h

(2.3.42)

for the power generated in the signal beam. As expected the generated signal power

is proportional to the pump power Pp and the spectral width selected for the detec-

tion ∆λs. deff denotes the effective nonlinearity, L the length of the crystal, λp/s/i the

wavelength of the pump, signal and idler wave and np the effective refractive index

for the pump wave. The number of photons emitted into a signal wavelength inter-

val of the size ∆λs is therefore given by )/( ωhss PN = .

K. Koch et al. also find an enhanced emission rate for a configuration where the

signal and pump waves copropagate along the same direction in the down-

conversion crystal due to walk-off effects in the crystal. They call this effect a “hot

44 2 Theory

spot” in parametric fluorescence. We make use of such a hot spot in our experi-

mental setup to assure minimum mode distortion for the signal mode.

Using the experimental parameters for the BBO crystal used in our experiment

(deff = 2.14· 10-12 V/m, np = 1.661, λs = λi = 2λp = 790 nm, L = 3 mm, Pp = 40 mW)

and assuming a narrow bandwidth filter with a bandwidth of ∆λs = 2 nm the ex-

pected power emitted with a wavelength of 790 ± 1 nm into a cone centered around

the pump beam is Ps = 9.1· 10-11 W corresponding to a photon rate of Ns = 36· 106

photons/s.

Only a small fraction of this cone which is transmitted through an iris with a di-

ameter of d = 1 mm placed at a distance of 70 cm from the crystal is selected for

detection. At this distance the cone has acquired a radius of r = 8.3 cm at an open-

ing angle of 6.85°. Accounting for the optical losses (ηopt = 0.99), the limited sin-

gle-photon detector efficiency (ηdet = 0.62) and the effective filter transmission

(Tf = 0.60) we obtain an expected single -photon count rate of

r

dTR foptis π

πηη24det/ = = 2.0· 104 photons/s. (2.3.43)

The experimentally observed rate coincides with this expectation to within the

measurement precision.

Parametric Amplification with Focused Beams If an additional seed beam is fed into the incoming signal mode of the down-

converter the spontaneous process of parametric amplification is transformed into

an induced process, called parametric amplification, where the signal beam is pa-

rametrically amplified and a macroscopic amount of light is generated in the idler

beam due to the energy conversion from the pump beam to the subharmonic signal

and idler beams. In the experiment this process allows us to align the down conver-

sion and the trigger detection beam path as well as to achieve mode matching be-

tween the signal and the local oscillator beam with macroscopic amounts of light.

If the seed beam is strongly attenuated it can also be utilized for the quantum state

reconstruction of coherent states and serve to prepare photon added states.

127 K. Koch, E. C. Cheung, G.T. Moore, S.H. Chakmakjian, J.M. Liu, “Hot Spots in Parametric Fluo-rescence with a Pump Beam of Finite Cross Section”, IEEE J. of Quant. El. 31, 769 (1995)


Generated Idler Power

The coupled-wave equations for the process of parametric amplification may be

written as

zkiispsi

si eAAiAz

A ∆−⊥ −=∇+

∂∂ *

//2/ κ (2.3.44)

where the normalized field amplitudes pispispispis EnA //////// /ω= for the

signal/idler/pump beam were introduced, zizsp kkkk ,, −−≡∆ is the wave vector

mismatch, pis

piseff

nnnc

d ωωωκ = the nonlinear coupling and where we have ne-

glected propagation losses (αs/i/p = 0).

For an undepleted pump wave, neglecting transverse effects and taking 0)0( =iA

the solution of the coupled differential equations (2.3.44) is given by128,129

))0(cosh()0()(

))0(sinh()0()(

zAALA

zAAiLA

pss

psi

κ

κ

=

−= (2.3.45)

We obtain the expected power produced in the idler beam by expanding Ai(L) to

first order in z♣ and using 2

////21

//////0

0/ disdisdisdisdis AISP ωµε== where

Ss/i/d denotes the effective area of the respective beam. This results in

isPAi PPP Γ= (2.3.46)

where

22

222

20

3

28

ps

ieff

iPA

ww

wLd

cn πλε

π=Γ (2.3.47)

In the pulsed case equation (2.3.46) relates the peak powers. For the average pow-

ers are connected by

ps

ipsPAi TT

TPPP

∆∆∆

Γ=2lnπ

τ (2.3.48)

Inserting the parameters for the BBO down conversion crystal employed in the

experiment (λi = 790 nm, deff = 2.14 pV/m, n = 1.661, L = 3 mm, wp = 0.68 mm,

ws = wi, τ = 1/81.2 MHz, ∆Ts = ∆TL = 1.8 ps = pT∆2 , Li TT ∆⋅≈∆ 3/1 ,

Ps = 120 mW, Pp = 40 mW,) into equation (2.3.48) we obtain an expected power in

128 R.W. Boyd, Nonlinear Optics, p. 77 (Academic Press, 1992, USA) 129 A. Yariv, Quantum Electronics – 3rd ed., p. 409 (John Wiley & Sons, 1989, USA)

46 2 Theory

the idler beam of 8.0 µW which exceeds the experimentally observed idler power

of 5.5 µW due to a limited mode-matching between the pump beam and the seed

beam. The limited mode-match mainly results from the deteriorating effect of the

narrow acceptance angle in the doubler on the pump beam mode, which has been

reduced considerably by using an LBO- instead of a BBO-crystal for the second

harmonic generation.

Beam Distortion due to Walk-Off

As already pointed out above, the down converter employed in the experiment

operates in a “hot spot” configuration, such that the pump and signal beam

copropagate within the crystal. However, this is not the case for the idler beam.

Owing to the opening angle α between the signal and idler beam we expect the

idler beam to be distorted along the x-direction.

We assume the pump and seed beam to be in a Gaussian TEM-00 mode

)/1(

/,

//

/,2

/

2

/1

)()( psRps zziw

ri

psR

psps e

zzi

zAzA

+

+=

(

(2.3.49)

with beam waists of w0,s/p, Rayleigh ranges pspspsR wnz /2

/,0/, / λπ= and a ampli-

tude of )(/ zA ps

(. In addition we presume that the z-dependence of the electric field

within the crystal will be dominated by the nonlinear interaction (weak focusing)

and neglect the z-dependence of the fields and the contribution of the transverse

Laplacian to obtain

zkiwwzx

wwy

spi eeeAAi

z

A

iw

sp

iw

sp ∆−+−−+−

−=∂∂

4847648476

((

21

222

21

222 )

11()2sin()

11(

*α

κ (2.3.50)

where the z-dependent x-shift accounts for the walk-off between the copropagating

pump and signal beam and the idler wave propagating at an angle α with respect to

the other two beams.

Integrating equation (2.3.50) across the crystal length L for an undepleted pump

and seed beam and perfect phase matching (∆k = 0) yields

⋅−−

⋅+−=

−

ii

iw

y

spiw

Lx

w

LxweAAiLA i

ααα

πκ

2sin2/Erf

2sin2/Erf

2sin2)(

2

2

((

♣ The linear expansion of Ai corresponds to the approximation of an undepleted seed/signal beam. This is well justified for our experimental setup, since we generate idler powers in the µW range with seed beam powers on the scale of 100 mW and pump powers of around 40 mW.


(2.3.51)

Figure 2-13 illustrates the beam distortion according to equation (2.3.51) for a

BBO crystal with an internal opening angle of 8.12° for various crystal lengths L.

If L/2· sin 2α becomes comparable with wi, then the resulting idler beam shape

deviates from that of a symmetric TEM-00 mode and becomes elongated along the

x-direction. The bold contour line for a crystal length of L = 3 mm corresponds to

the experimental situation with unfocused pump and seed beam.

2.3.4 Single-Photon Fock State Preparation In the experimental scheme described here, single -photon Fock states are prepared

by triggering the homodyne detection in the signal channel by a photon count event

in the idler channel. This chapter describes the state preparation process by condi-

tional measurements and derives the state preparation fidelity as first measured by

C.K. Hong and L. Mandel130. Conditional state preparation in a wider context is

discussed in chapter 2.4.

130 C.K. Hong, L. Mandel, “Experimental Realization of a Localized One-Photon State”, Phys. Rev. Lett. 56, 58 (1985)

- 1.5 - 1 - 0.5 0 0.5 1 1.5 2

- 1.5

- 1

- 0.5

0

0.5

1

1.5

2

L = 12 mm

6 mm

3 mm

0.1 mm

crystal length

BBO = 8.12°α

x

y

beam distortion due to walk-off

Figure 2-13 Beam distortion in parametric amplification due to walk -off. A BBO crystal cut as in the experiment with an internal opening angle between signal and idler beam of α = 8.12° and different length L has been assumed for the calcula-tion.

48 2 Theory

Projection onto the Trigger State We assume that the trigger detector registers photons in the spatio-temporal mode

tti

ttttt kekufdc

kd ttrr ωωω −

⊥⊥ ∫∫=Ψ )(~)(~1

,,2 (2.3.52)

where the spectral shape function )(~

tf ω is defined by the spectral filters employed

and )(~,⊥tt ku

r denotes transverse mode function of the trigger beam determined by

the irises and pinholes placed in the beam path. The phase factor tt tie ω− accounts

for the possible time delay between the pump pulse and the detection of an idler

photon at the trigger detector.

The photon count event in the trigger detector projects the twin-photon state of

equation (2.3.23) onto a single-photon state in the signal channel

( )

stitti

ititisptt

zisptpeff

istt

sittiti

istttistt

istis

keekkkkuku

/ÄklgfAc

kdkddkd

kkkee

kkkufc

kdkddkd

pstpt

pptt

rrrrrr

rrr

rrr

ωω

ωω

ωωδδ

ωωωχ

ω

ωω

−−−⊥⊥⊥⊥⊥

⊥

−

⊥⊥

−−+

+′

≈

Φ=

ΨΨ=Ψ

∫ ∫∫∫

∫ ∫∫∫

)(,,,,,

*

*0

33,

2

,**33

,2

,out

)()()(~)(~

2sinc)(~)(~

),()(~)(~1

(2.3.53)

where 0/)()(~ppspisp AAg ωωωω +=+ denotes the normalized spectral shape

function of the pump beam and we have explicitly included the phases factor of the

pump pulse which was so far hidden due to the choice of the time origin (compare

footnote on page 47). Carrying out the d3k i-integration and collecting the transverse

and longitudinal components leads to

∫−

⊥=Ψ sti

ssssss kekugGkdN tprr ωω )(~)(~

,3 (2.3.54)

where N is a normalization constant and the nonlinear gain G, the signal spectral

shape function )(~ssg ω and the signal spatial mode function )(~

,⊥ss kur

are given by


)(~)(~)(~

)coscos(2

sinc)(~)(~

)(~

,,,*

,2

,

)(*

02

⊥⊥⊥⊥⊥

−−

+=

−−+=

=

∫

∫

tsptttss

ttiitssp

stspttss

peff

kkukukdku

ec

nlgfdg

Ac

lG

tpt

rrrr

h

ωθωθωωωωωωω

χ

(2.3.55)

Thus the spectral shape of the generated single -photon Fock states is given by the

convolution of the filter function of the trigger mode with the product of the phase-

matching bandwidth and the spectral shape of the pump mode

PMps ggfg ~~~~ * ⋅⊗= , where )2/(sinc~zPM klg ∆≡ . The spatial mode of the gener-

ated photons results from a convolution of the spatial mode of the trigger and the

pump pts uuu ~~~ * ⊗= .

The normalization constant N may be determined employing the condition

1=ΨΨ ss which leads to

||1

)(~1)(~

)(~)(~)(~)(~

22

1

2

1

2,,

222

)(

,,**2332

Gc

Nc

GN

gdc

kukdGN

kkkukuggGkdkdN

ss

ssssss

kk

ssssssssssssss

ss

=⇒==ΨΨ

=

=ΨΨ

=

′

=

⊥′⊥

−

′⊥′⊥′′

∫∫

∫∫′

44 344 21444 3444 21

r

43421

rrrr

rr

ωω

ωω

δ

(2.3.56)

To ensure that we really have prepared a single -photon Fock state we have to show

that the state of equation (2.3.54) corresponds to a transform limited state. Since

the spectral shape function )(~ssg ω depends on the detection time tt this will not in

general be the case. To clarify the conditions under which a transform limited pho-

ton wave packet may be achieved I will rephrase an argument given in131.

If a filter with a bandwidth much greater than the spectral width of the photon pair

mode function Φ is employed in the trigger channel, then we may change the inte-

gration variable in equation (2.3.55) to ∆ωp = ωs+ωt and rewrite )(~ssg ω as

44444444 344444444 21)(

* ),(~)(~)(~

),(~

t

tpts

tg

tisipPMppps

tits eggdfetg

Φ

∆∫ ∆∆∆∆−≈ ωω ωωωωωω (2.3.57)

The phase factor ts tie ω accounts for a time localization of the signal photon wave

packet depending on the detection time of the idler photon in the trigger detector.

131 Z.Y. Ou, “Parametric down-conversion with coherent pulse pumping and quantum interference between independent fields”, Qu. Semiclass. Opt. 9, 599 (1997)

Transform limited single-photon states

50 2 Theory

The photon wave packets will have a temporal width determined either by the in-

verse of the differential phase-matching bandwidth 1/ si∆ ′′ or by the inverse width

of the spectral filter 1/∆t and arrive with a time distribution governed by the func-

tion gΦ(tt). In this case the trigger detection will localize the photon wave packets

to a time interval shorter than the pump pulse width making a precise synchroniza-

tion between the local oscillator pulses and the photon wave packets impossible.

If however the filter in the trigger channel has a bandwidth smaller than the spec-

tral width of the photon pair mode function Φ then we may approximate

)(~ssg ω by

444 3444 21

)(

*

*

)(~

)(~)(~)(~

t

tt

tf

tittsPMspss efdggg ∫≈ ωωωωωω (2.3.58)

In this case the resulting photon pulse in the signal channel will be transform lim-

ited and arrive with a time delay independent of the trigger detection event. This

allows a synchronization of the photon wave packets with the local oscillator

pulses. The temporal width of the photon wave packets will be governed by the

inverse of the spectral width of Φ .

In the experiment the spectral width of the photon pair mode function Φ and that of

the spectral filter are comparable. A good approximation to the transform limited

behavior is only expected, if the 0.3 nm narrow bandwidth filter is employed in the

trigger beam path.

This completes the theoretical description of the conditional Fock state preparation

process. To obtain a quantum state reconstruction of the state thus prepared pulsed

homodyne measurements have to be performed. The homodyne detection of the

single-photon Fock state is treated in chapter 0 starting on page 67.

The following section provides the theoretical background for the state preparation

fidelity in coincidence count measurements performed with single-photon counters

placed in the signal and idler channel.

State Preparation Fidelity

This section provides a theoretical treatment of the coincidence count measure-

ments between signal and idler photons registered with single -photon counters.

These measurements allow an estimation of the state preparation fidelity in 2-

photon down-conversion. A simplified theory of the Fock state preparation in 2-

photon down-conversion including multiple count events was published together

with experimental results stating a preparation fidelity of 1.06 ± 10% by C.K. Hong


and L. Mandel in 1986132. Since the corrections due to n > 1 Fock states in our case

are estimated to be of the order 10-6, we will neglect multiple photon counts in the

further treatment. A calculation of the effect of filtering and irises in the detection

process of coincidence measurements on photon twins produced in the process of

2-photon down-conversion has been given by Joobeur et al. 133,134. Since in our case

we need to be able to describe the case of general spatial and spectral detection

modes, we extend the treatment to include arbitrary modes.

The quantum state produced in pulsed 2-photon down-conversion according to

equation (2.3.23) may be written as:

),)2

(sinc)(~)(

0,0(

,,

33

isz

ispispeff

isis

kkkl

kkuA

kdkdNrrrr ∆

⋅++′

+=Ψ

⊥⊥

∫∫

ωωχ (2.3.59)

Since the vacuum contribution will not lead to photon count events we neglect this

part of the two-photon wavefunction and rewrite the state as

⊥⊥⊥⊥

⊥⊥

+∆−+

=Ψ′ ∫∫∫∫,,,,

,,

,,)(~),( isisispzpisp

isis

kkkkukg

kdkdddrrrr

rr

ωωωωωγ

ωω (2.3.60)

132 C.K. Hong, L.Mandel, “Experimental Realization of a Localized One-Photon State”, Phys. Rev. Lett. 56, 58 (1986) 133 A. Joobeur, B. Saleh, M. Teich, “Spatiotemporal coherence properties of entangled light beams generated by parametric down-conversion”, Phys. Rev. A 50, 3349 (1994) 134 A. Joobeur, B. Saleh, T.S. Larchuk, M. Teich, Coherence properties of entangled light beams generated by parametric down-conversion: Theory and experiment”, Phys. Rev. A 53, 4360 (1996)

Figure 2-14 Schematic setup employed to measure the state preparation fidelity

52 2 Theory

where the longitudinal part of the wavefunction 2

)()( ωω pp gG ≡ and the trans-

verse mode function 2

)(~)(~

⊥⊥ ≡ kukU pp

rr are normalized according to

∫ =1)(ωω pGd and 1)(~

=∫ ⊥⊥ kUkd p

rr. 2γ≡Γ denotes the photon pair generation

probability per pulse, unit k-space area and unit frequency interval.

The spectral and transverse integration in equation (2.3.60) can be factorized for

small emission cone opening angles 2/0 πθ <<s

⊥⊥⊥⊥⊥⊥ +

⋅∆−+≈Ψ′

∫∫∫∫

,,,,,, ,)(~

,),(

isispis

iszpispis

kkkkukdkd

kgddrrrrrr

ωωωωωωωγ (2.3.61)

A count event in the signal or idler beam path corresponds to a projection onto the

signal and idler wavefunctions

⊥−⋅

⊥⊥

⊥

−−

∆=Ψ ∫∫,//

)(0,///,0/,//

///,///

0/0/)()( isistrki

isisisisisis

isisisisisiDs

ketkk

Akdd

isisrrr

v

rr

ωωωθ

ηω

ω

(2.3.62)

where ηs/i denotes the optical propagation efficiency in the signal and idler beam

path and is used to account for propagation as well as filter losses:

isisoptis T /,0/,/ ⋅=ηη . As/i is the effective area of the detected mode (e.g.

As/i= π/2 w2 for a Gaussian mode with waist w) and ∆s/i the effective spectral widths

of the signal and idler filters ( isis /,/ 2 ωσπ=∆ for a filter with Gaussian spectral

shape). 2

,0/,//,/ )()( ⊥⊥⊥ −≡Θ isisisis kkkrrr

θ and 2

0,///// )()( isisisisis tT ωωω −≡

represent the normalized transverse and longitudinal detected modes and include

the effect of irises, spatial and spectral filters. ωs/i,0 is the center frequency of the

optical filters and ⊥,0/ iskr

a k-space offset of the transverse mode from the direc-

tions of correlated photon emissions. The exponential phase factor accounts for the

detection space and time offsets.

Using these mode function the coincidence rate C can be calculated according to

∫∫∫∫

∫∫∫∫ΨΨΨ′=

Ψ′Ψ′=

⊥⊥

⊥⊥

2,0,000

2

,0,0002

,

:ˆˆ:

DiDsisisPD

DiDsisisPD

rdrddtdt

IIrdrddtdtC

rr

rr

η

η (2.3.63)

Coincidence rate


ηPD designates the quantum efficiency of the single -photon detectors and we have

averaged over the random detection times ts/i,0 and the detector surfaces ⊥,0/ isrr

.

We insert equations (2.3.61) and (2.3.62) into equation (2.3.63) and make use of

),( //// isisisis ωωδωω ′=′ , ),( ,/,/,/,/ ⊥⊥⊥⊥ ′=′ isisisis kkkkrrrr

δ to reduce the num-

ber of integrations. Carrying out the time and space averaging

∫∫ ⊥⊥−

⊥ ′′=⊥ ),(),( ,/,///)(

,0/0///,/

isisisistrki

isis kkerddt isisisrrr rr

δωωδωwe obtain

⊥ΩΩΓ∆∆= ωηηη isisisPD AAC (2.3.64)

where we have introduced the quantities Ωω and ⊥Ω which denote the spectral and

transverse mode overlaps and will now be discussed in detail.

The spectral mode overlap Ωω in equation (2.3.64) is given by

)(

)()(

)()()(

)()()(

00

00

00

00

isisp

ispisppp

ipissppp

iiissspispis

TTG

TTGd

TTdGd

TTGdd

ωω

ωωωωω

ωωωωωωωω

ωωωωωωωωωω

∆+∆⊗⊗=

∆−∆−∆⊗∆∆=

∆−∆−∆∆−∆∆∆∆=

−−−+=Ω

∫∫∫

∫∫

(2.3.65)

where we have transformed the integration values according to

)2/()2/( pipsisp ωωωωωωω −+−=∆+∆=∆

2/)( pis ωωωω ∆+∆−∆=∆ (2.3.66)

and where 2/0/0/ pisis ωωω −=∆ .

If we approximate Gp and Ts/i by Gaussians with standard deviations of σω,p and

σω,s/i then we obtain

)(2

)(

2,

2,

2,

2,

2,

2,

200

2

1 isp

is

eisp

ωωω σσσωω

ωωωω

σσσπ

++∆+∆

−

++≈Ω (2.3.67)

Another simplification arises, if Gp approaches a delta function

)()( ppisPG ωδωωω ∆→−+ . This is the case for a cw-pump beam, or if the

signal and idler filter bandwidths are much larger than the pump bandwidth. In this

situation equation (2.3.65) simplifies to

)()( 00 iiss TTd ωωωωωω ∆−∆∆−∆∆=Ω ∫ (2.3.68)

The components of this overlap integral are visualized in Figure 2-15: The overlap

takes on its maximum value at the point of degeneracy (∆ωs/i0 = 0) and decreases

with growing detunings ∆ωs/i0.

Spectral mode overlap

54 2 Theory

Similarly we find for the transverse mode overlap

)(~

)()(~

)()()(~

,0,0

,0,0,,,

,0,,0,,

⊥⊥

⊥⊥⊥⊥⊥

⊥⊥⊥⊥⊥⊥⊥⊥⊥

+Θ⊗Θ⊗=

−−∆Θ⊗Θ∆∆=

−∆−∆Θ−∆Θ∆∆∆=Ω

∫∫∫∫∫∫

isisp

ispisppp

ipissppp

kkU

kkkkUkd

kkkkkkdkUkd

rr

rrrrr

rrrrrrrr

(2.3.69)

where an analogue variable transformation as in equation (2.3.62) has been used.

In the Gaussian approximation for the transverse mode functions pU~

, Θs and Θs

with standard deviations isp //,⊥σ = 1/wp/s/i we obtain a transverse mode overlap of

( ))(2

2,

2,

2,

2,

2,

2,

2,0,0

2

1 isp

is kk

isp

e ⊥⊥⊥

⊥⊥

++

+−

⊥⊥⊥⊥

++≈Ω σσσ

σσσπ

rr

(2.3.70)

where we have assumed equal beam profiles in the k x- and k y-direction.

If an unfocused pump beam is used, we can approximate the transverse pump mode

function by a delta-function )()(~

ppp kkUrr

∆→∆ δ . Equation (2.3.69) then simpli-

fies to

)()( ,0,,0 ⊥⊥⊥⊥⊥⊥⊥ −∆−∆Θ−∆Θ∆=Ω ∫∫ ipiss kkkkkkdrrrrrr

(2.3.71)

This approximation is valid, if the angular pump mode wave-vector spread is much

smaller than the wave-vector spread of the detected modes. If irises are employed

to define the shape of the detected modes, this is the case for

Figure 2-15 Spectral overlap Ωω in the case of a narrow pump bandwidth for iden-tical filters in the signal and idler beam. A deviation of the filter midpoints from the degeneracy point ωp/2 leads to a reduced overlap.

Transverse mode overlap


)/()2/( / pis wzd πλ>> where d is the diameter of the iris, z the distance between

the crystal and the iris and wp the waist of the pump beam.

In this case the mode overlap ⊥Ω is given by the geometric overlap of the two

irises with radii K1/2 and a midpoint displacement ∆k0

−+−=∆Ω⊥ )2sin2(

2)2sin2(

21

)( 22

22

11

21

22

21

20 θθθθπ

KK

KKk (2.3.72)

where ( ))/()(arccos 02/120

21/2

22/12/1 kKkKK ∆∆−−=θ . The normalized mode

overlap for equal radii K1/2 is depicted as a function of the midpoint displacement

between the irises in Figure 2-16.

If the two irises are of different size and are adjusted so that their transverse mode

functions completely overlap, then the transverse overlap integral yields the ratio of

the smaller iris area to the larger ( ),max(/),min( isis AAAA=Ω ⊥ ). Hence the

coincidence count rate will not be proportional to the product of the two iris areas

but to the square of the smaller area only.

Obviously a close analogy exists between the spectral and transverse mode over-

laps.

To be able to calculate the preparation fidelity we also derive the signal and idler

count rates. For the signal count rate we find

Figure 2-16 Spatial overlap ⊥Ω in the case where two irises of identical diameter define the transverse mode function of the detected signal and idler photons. A relative midpoint displacement between the irises leads to a reduced overlap.

Signal and idler count rates

56 2 Theory

⊥

⊥⊥

⊥⊥

ΩΩΓ∆=

ΨΨ′=

Ψ′Ψ′=

∫∫∫∫

∫∫∫∫

ssssssPD

DsisisPD

sisisPDcalcs

A

rdrddtdt

IrdrddtdtN

ωηη

η

η2

,0,000

,0,000 ::

rr

rr

(2.3.73)

with the spectral overlap

1

)()(

)()(

1

0

0

=

∆∆∆−∆∆=

∆−∆∆+∆∆∆=Ω

=

∫∫∫∫

444 3444 21 pppssss

sssispiss

GdTd

TGdd

ωωωωω

ωωωωωωω

(2.3.74)

and the transverse overlap also yielding ⊥Ω ,s = 1. The idler count rate can be calcu-

lated accordingly. Hence we have

Γ∆=

Γ∆=

iiiPDcalci

sssPDcalcs

AN

AN

ηη

ηη (2.3.75)

Introducing the normalized coincidence rate

calci

calcs

calc

is

iscalc

NN

C

II

IIC =≡

:ˆ::ˆ:

:ˆˆ:~ (2.3.76)

we define the preparation fidelity as the ratio between the measured and the calcu-

lated normalized coincidence rate

434214434421⊥

⊥Ω⋅

Ω∆∆⋅⋅

−−−=

⋅−−

−==

M

is

M

isisPDiiss

random

calc

calci

calcs

iiss

random

calc

AADCNDCN

CC

C

NN

DCNDCN

CC

C

CF

111

)()(

)()(~

~exp

ω

ωηηη

(2.3.77)

Here C and Ns/i denote the experimentally observed coincidence, signal and idler

count rates, Crandom the random coincidence rate and DCs/i the signal and idler dark

count rates. The mode matchings Mω and M⊥ are unitless, 1≤ and describe the de-

gree of overlap between the correlated photon modes emitted by the down-

converter and the detected signal and idler modes.

To cover the case of conditional state preparation in either the signal or the idler

beam path we also define a signal and idler preparation fidelity as

4342143421⊥

⊥Ω⋅

Ω∆⋅⋅

−−

=

,/,/

//////

111)(

sisi M

si

M

sisiPDisis

random

is ADCNCC

F

ω

ωηη (2.3.78)

State prepara-tion fidelity


The signal (idler) preparation fidelity provides a measure for the conditional state

preparation quality in the signal (idler) beam, if a trigger detection has been regis-

tered in the idler (signal) beam.

2.3.5 Homodyne Measurements The quantum theory of the photodetection process has been treated in a number of

textbooks on quantum optics135,136,137. The signal produced by a photodiode is –

under certain not too restrictive approximations – found to be proportional to the

normally ordered expectation value of the photon number operator :ˆˆ: aa+ . A

number of schemes to measure quadrature amplitudes instead of the photon num-

ber have been discussed in the literature138,139,140,141,142, but the most commonly

employed scheme is that of balanced homodyne detection143,144, which has already

been introduced in chapter 2.2. A signal beam is combined with a strong local os-

cillator beam – in a coherent state with a high excitation 1|| 2>>α – at a precise

50% beam splitter. The resulting two beams are measured at two photodiodes and

the two photocurrents are subtracted to yield a signal which is proportional to the

quadrature amplitude Xθ (Figure 2-17)

Pulsed Homodyne Detection

The time and frequency signal of a homodyne detector operated with a pulsed local

oscillator deviates from the cw-case. What kind of time and frequency signals do

we expect in the pulsed case?

We take the pulsed local oscillator field to be of the form

θθ στστ i

nLOLO

i

nLOLO eEEntgeEntgtE ∑∑ ∆−−=−∝ )ˆ(),(ˆ),()( (2.3.79)

where g(t,σ) describes the temporal pulse shape of the local oscillator pulses, τ

denotes the time separation between two consecutive pulses and θ represents the

relative phase between the local oscillator and the signal beam. In the case of a

135 D.F. Walls and G.J. Milburn, "Quantum Optics" (Springer-Verlag, Berlin Heidelberg, 1994) 136 L. Mandel and E. Wolf, "Optical Coherence and Quantum Optics" (Cambridge University Press, 1995) 137 M.O. Scully and M.S. Zubairy: "Quantum Optics" (Cambridge University Press, 1997) 138 H.P. Yuen, J.H. Shapiro, IEEE Trans. Inf. Theory IT-24, 657 (1978) 139 J.H. Shapiro, H.P. Yuen, J.A. Machado Mata, IEEE Trans. Inf. Theory IT25, 179 (1979) 140 J.H. Shapiro, H.P. Yuen, IEEE Trans. Inf. Theory IT-26, 78 (1980) 141 H.P. Yuen, J.H. Shapiro, “Quantum Statistics of Homodyne and Heterodyne Detection”, in Coher-ence and Quantum Optics IV, ed. L. Mandel, and E. Wolf (Plenum, New York) 142 L. Mandel, Phys. Rev. Lett. 49, 136 (1982) 143 H.P. Yuen, V.W.S. Chan, Opt. Lett. 8, 177 (1983) 144 G.L. Abbas, V.W.S. Chan, and T.K. Yee, Opt. Lett. 8, 419 (1983)

58 2 Theory

strong local oscillator in a coherent state we may neglect the quantum fluctuations

of the local oscillator LOE∆ and replace the field operator LOE with the classical

field LOE .

Local oscillator and signal beam are combined at a beam splitter with a transmis-

sion of 50%. The two fields emerging from this beam splitter, which are propor-

tional to the sum and the difference of the two incident fields

)(2/12/1 LOs EEE m⋅∝ , are detected with two photodiodes. The currents pro-

duced in the photodiodes are proportional to the field intensities

∑ ∑

∑

−+ +−−≈

−=∝

n nX

iS

iSLOLO

nS

iLO

eEeEEntgEntg

EeEntgEi

444 3444 21m

m

θ

θθ

θ

στστ

στ

)ˆˆ(),(),(

ˆ),(

2122

21

2

212

2/12/1

(2.3.80)

To proceed from the first to the second line in equation (2.3.80) we have made use

of the fact that the individual pulses are well separated from each other, so that the

pulses shape functions for different n do not overlap.

Thus the sum and difference currents are given by

∑

∑

−∝−=

−∝+=

−

+

nLO

nLO

XNntgiii

Nntgiii

θστ

στ

),(2

),(

21

221

(2.3.81)

where the number of photons per pulse in the local oscillator beam is related to the

light field by integration across the photodetector surface

)/(22

002 ωµε h∫∫= LOLO EnSdN . We find that the obtained difference sig-

nal i- is indeed proportional to the quadrature amplitude θX of the signal beam.

∫∞∞− +−= dqeqxqxpxW ipq2/22/2

1),( ρπ

Figure 2-17 Pulsed homodyne detection scheme

Time domain


The local oscillator amplifies the quantum noise of the signal beam by a factor of

LONntg ),( στ− whenever local oscillator pulses fall onto the detector, giving

rise to pulsed shot noise observation. As expected, the quantum fluctuation signal

scales with the square-root of the local oscillator photon number NLO.

Only the amplification of the quantum noise of the signal due to the mixing with

the local oscillator makes it possible to achieve a shot noise level above the elec-

tronic noise floor of the photodiodes. Even for ultra-low noise detectors as the ones

used in our experiment local oscillator powers exceeding 106 photons/pulse are

required to accomplish a satisfactory signal-to-noise-ratio (SNR).

Since the ratio between the sum and difference powers

LOLOLO NXNNii ∝=−+ ):ˆ:/(/ 2222θ increases linearly with the photon num-

ber per pulse in the local oscillator, pulsed homodyne detection also calls for a

particularly efficient current subtraction♣.

If the sum and difference currents are fed into a spectrum analyzer they give rise to

a spectrum of the shape

( ):ˆ:)/1,(~

)(

)/1,(~

)(

21/1

2

221/1

2

θτω

τω

σν

σν

XFNeGiFSA

NeGiFSA

LOn

nibw

bw

LOn

nibw

bw

∑

∑

−−−

−++

∝∝

∝∝ (2.3.82)

where F(…) represents a Fourier transform and bw/1

... symbolizes a time averaging

with a temporal width of 1/bw and ( )2),()/2,(~ σσν tgFG = . For mode-locked

lasers )/2,(~ σνG corresponds to the laser gain profile and has a typical band-

width in the THz range. ∑∑ −≈−nn

ni ne )/2(|| τπωδτω produces a frequency

comb of regular spaced frequency peaks at frequencies which correspond to multi-

ples of the inverse of the laser cavity round trip time. The width of these peaks is

proportional to the inverse of the number of terms N contributing to the summation

in ||∑ −n

nie τω , which is limited either by the spectral resolution of the spectrum

analyzer or by the inverse of the first order coherence time of the mode-locked

laser. At multiples of the mode-locking frequency a sum signal proportional to the

number of photons per pulse NLO is observed while the difference signal is propor-

tional to LON . Away from multiples of the mode-locking frequency the shot

♣ For a cw and a pulsed optical beam of the same average power the subtraction has to be more effi-

cient in the pulsed case by a factor of the power enhancement PP /ˆ .

Frequency domain

60 2 Theory

noise signal is reduced by a factor NLO, but is still present due to the white noise

character of the spectrum of the quantum fluctuations ( ):ˆ: 2θXF .

Non-Ideal Homodyne Detectors

For a realistic modeling of homodyne detectors employed in our experiments ex-

perimental inefficiencies in homodyne detection have to be accounted for. Such

inefficiencies may arise due to a number of reasons like:

different detector efficiencies of the two photodetectors,

imperfect mode-matching,

losses,

deviations from an ideal 50% beam splitter.

The following calculations are based on methods described in145 and aim at calcu-

lating quantum efficiencies and optimum parameter settings for non-ideal homo-

dyne detectors.

Figure 2-18 depicts a general scheme for the description of such homodyne sys-

tems: an input-output matrix Mt

connects the input beams Ao to the output beams

Bp. We assume the local oscillator input beam LLL AAA ∆+= to be the only beam

with a non-vanishing field amplitude much greater than the quantum fluctuations

AAL ∆>> . We could at this point assume the local oscillator beam to be a classical

wave (i.e. neglect the quantum fluctuations ∆AL), but we will carry this contribu-

tion through the calculation to show that it does contribute to the final results.

The signal beam As and the other input beams A3…An are assumed not to carry an

amplitude significantly exceeding the level of quantum fluctuations ( ss AA ∆= ,

nn AA ...3...3 ∆= ). Not all of these fields have to be freely accessible or to correspond

to real optical beams. Some of them may be used to account e.g. for distributed

losses or non-ideal photodetector efficiencies. Three types of output beams are

distinguished: the beams B1…Bh which hit the photodetector labeled D1, the beams

Bh+1…Bi impinging on photodetector D2 and the remaining beams Bi+1…Bj. Thus

145 R. Bruckmeier, and S. Schiller, “Properties of the linearized quantum optical bus“, Phys. Rev. A 59, 750 (1999)

General scheme


∆

∆∆

∆+

=

+

+

n

s

LL

j

i

i

h

h

A

A

A

AA

M

B

B

B

B

B

B

M

t

M

M

M

3

1

1

1

or ∑ ⋅=p

pqpq AMB (2.3.83)

The photodetector currents i1/2 may then be approximated as

( )

( )∑ ∑

∑ ∑

∑ ∑ ∑

∈

∈

∈ ∈ =

+∆+∝

+∆+≈

=∝

2

**22

1

**2

1 1

2

...1

21

..

..

Dj ppjpLjLLjL

Dk qqkqLkLLkL

Dk Dk nqqkqk

ccAMAMAMi

ccAMAMAM

AMBi

(2.3.84)

where terms of the order O( )2A∆ have been neglected and the beams impinging

on the detectors are added incoherently since they are in orthogonal modes. The

photocurrent i2 is amplified by a factor g before both photocurrents i1 and i2 are

subtracted to yield the homodyne signal 21 igii −=∆ .

A

A

D1

D2

vacuuminputs…

A A

M…

…

……

localoscillator

signal

B

B

B B

B … B

-

i

i ∆ig

L

3 n

s

1

h

i+1 j

h+1 i

2

1

Figure 2-18 General scheme for the description of inefficient homodyne systems.

62 2 Theory

To be able to observe the weak signal resulting from the quantum fluctuations

∆Αο we need to suppress the terms proportional to 2

LA . Their contribution to ∆i

vanishes, if

∑ ∑∈ ∈

⋅=1 2

22

Dk DjjLkL MgM (2.3.85)

Therefore we have to provide an experimentally accessible free parameter such as

the electronic gain g or the reflectivity of the central beam splitter R that may be

adjusted to fulfill condition (2.3.85).

If equation (2.3.85) holds then the difference current ∆i can be written as

( ) ( )

( ) ( )

( )∑

∑ ∑∑

∑∑

∑ ∑∑ ∑

≠

+=

−−

≠

Γ

∈∈

Γ

∈∈

∈∈

+∆Γ+

∆−∆Γ+∆+∆Γ=

+∆

⋅−+

+∆

⋅−=

+∆−+∆∝∆

sqqLq

XY

is

isLs

X

is

isLs

sqqL

DjjqjL

DkkqkL

isL

DjjsjL

DkkskL

Dj ppjpLjL

Dk qqkqLkL

ccAA

eAeAAieAeAA

ccAAMMgMM

cceAAMMgMM

ccAMAMgccAMAMi

sss

o

s

..

)Im()Re(

..

..

....

*

)2/(2)(2

*

)(2

*

*

2

*

1

*

*

2

*

1

*

2

**

1

**

444 3444 21444 3444 21

444444 3444444 21

444444 3444444 21

πθθ

θθ

θ

θθ

θ

(2.3.86)

where we have introduced the relative phase θ between the local oscillator beam

and the signal beam ( θiss eAA −→ ) and separated the contribution of the signal

beam from the remaining contributions. Only the first contribution of the signal

beam – proportional to Re(Γs) – corresponds to the desired measurement of the

field quadrature Xs(θ), all other contributions deteriorate the quality of the meas-

urement. The contribution proportional to Im(Γs) represents an admixture of the

orthogonal field quadrature component. This contribution disappears, if the input-

output-matrix Mt

is real (no phase shifts).

For the variance of the difference current we obtain

∑≠

∆Γ+Γ+Γ=∆sq

qLqsLssLs AAYAXAi2*2222222 )()Im(4)()Re(4 θθ

(2.3.87)

Power balancing


We can now identify the quantities 2

qΓ as the traction of (unwanted) quantum

noise from the qth beam. Using this we may calculate the efficiency of the homo-

dyne detector, which is given by

∑ ∆Γ

Γ=

qqq

s

A2*

2)Re(η (2.3.88)

Please note that in equation (2.3.88) the summation has been extended to include

the signal beam again. If the auxiliary beams ( sq ≠ ) are assumed to be in the vac-

uum state ( 4/12 =∆ qX ) then equation (2.3.88) may be simplified further to

∑

=Γ

Γ=

nqq

s

...1

2

2)Re(η (2.3.89)

with

∑∑∈∈

⋅−=Γ1

*

1

*

DjjqjL

DkkqkLq MMgMM (2.3.90)

as defined in equation (2.3.86).

ΓL vanishes, if condition (2.3.85) is fulfilled. The quantum noise of the local oscil-

lator beam therefore does not contribute to the difference current or the efficiency

of the homodyne system. Equation (2.3.88) implies that if one or more of the auxil-

iary vacuum inputs are replaced by squeezed beams an enhanced homodyne effi-

ciency becomes possible.

Equations (2.3.89) and (2.3.85) are the central results of this discussion and will

now be used to calculate the efficiencies and parameter settings for homodyne de-

tectors with various inefficiencies.

Optical Losses or Different Detector Efficiencies

As a first example we consider the homodyne setup sketched in the left diagram of

Figure 2-19 depicting a homodyne system with optical losses or equivalently non-

ideal photodetector efficiencies, which we model by beam splitters with transmis-

sions η1/2 placed in the arms of the homodyne detector. Since the beam splitter

matrix for a beam splitter with transmission T (intensity) is given by

)),(),,(( TRRTM BS −= , we obtain for the input-output-matrix of this setup

−−

−=

2

1

22

11

10

01

ηη

ηηηηRT

TRM (2.3.91)

Homodyne Efficiency

64 2 Theory

To suppress the current contributions due to 2

LA we have to fulfill condition

(2.3.85) which for the system considered here is of the form

21 ηη TgR = (2.3.92)

We may either choose the beam splitter reflectivity R or the electronic gain g as the

adjustable parameter to satisfy (2.3.92). Choosing the beam splitter reflectivity

leads to

21

2

21

1 ,ηη

ηηη

η+

=+

= RT (2.3.93)

Using the input-output-matrix we may also calculate the contributions to the differ-

ence current for all the beams

)1(

)1(

0

)(

224

113

21*22

*11

21*22

*11

ηη

ηη

ηη

ηη

−=Γ

−=Γ

=−=−=Γ

+=−=Γ

T

R

TRMMMM

TRMMMM

LLLLL

sLsLs

(2.3.94)

corresponding to a homodyne efficiency of

)1()1(

)(

)1()1()(

)(

122221

21

2121

22112

21

221

−−+−−+

=

−+−++

+=

ηηηηηη

ηηηηηηηηηη

ηηη

TRTR

TR

(2.3.95)

To first order in the losses 2/12/1 1 η−=l we obtain

22

1 2121 ηηη

+=

+−≈

ll (2.3.96)

S

LO

R

D1

D2

vacuum

vacuum

η

η

3

4

1

2

S

LO

R

MM

D1

D2

vacuum 3

Figure 2-19 Homodyne detector inefficiencies: left diagram: imperfect detectors with different efficiencies; right diagram: imperfect mode-matching


For fixed η1, and η2 varying from 0 to 1 we find optimum operation conditions for

η2 = 1 and η2 = 0 (corresponding to an unbalanced homodyne scheme as described

in146). For η1 = 1 the optimized efficiency η as a function of η2 never drops below

86%.

Using an electronic gain g = η1/ η2 to meet condition (2.3.92) we obtain a slightly

different expression for the homodyne efficiency, but the same expansion for low

losses (equation (2.3.96)).

For 21 ηη = equation (2.3.95) simplifies to 2/1ηη = . In this case the non-ideal

efficiencies η1/2 have the same effect as optical losses in the signal beam before it

reaches the beam splitter.

Imperfect Mode-Matching

The effect of imperfect mode-matching on homodyne operation may be calculated

starting from the right diagram in Figure 2-19 and carrying out the same analysis as

in the last section.

The mode mismatch is described by a beam splitter with a transmission of MM that

does NOT introduce additional quantum noise into the system, but distributes a

mode Φ into a part mode-matched to the reference mode Ψ and an orthogonal,

non-mode-matched part Ψ¬ according to Ψ¬−+Ψ=Φ MMMM 1 .

Condition (2.3.85) is satisfied for g = 1 and R = T = ½ resulting in a homodyne

efficiency of η = MM.

Imbalanced Beam Splitter

In the case of an imbalanced beam splitter ( %50≠R ) no efficiency loss occurs, if

the electronic gain is adjusted to g = R/T. If a homodyne system is employed where

the electronic gain cannot be adjusted, a beam splitter imbalance of 5.0−=∆ R

leads to a reduced homodyne efficiency ∆−≈ 21η .

Lossy Beam Splitter

The result for optical losses from the next to last section may not be applied, if the

beam splitter itself exhibits internal losses, which are different for the reflected and

transmitted beams. This is the case e.g. for a polarizing beam splitter cube where

the reflectivity typically reaches values up to 99.7% whereas the transmission is

limited to below 98%.

146 L. Mandel, E. Wolf, „Optical coherence and quantum optics“, p. 1052 (Cambridge, New York, 1995)

66 2 Theory

Assuming that the output fields incident on the two detectors are given by

642

531

)1()1(

)1()1(

ALTALRALTALRB

ALRALTALRALTB

TRLTsR

RTLRsT

++−+−−=

++−+−= (2.3.97)

where LT and LR denote transmission and reflection losses, we obtain a homodyne

efficiency of

RTRT

RT

LRLTLLTR

LLTR

22)1()1(

)1()1(

++−−−−

=η (2.3.98)

for TLLR RT )1/()1( −−= . Assuming a beam splitter reflectivity of R = 50% the

efficiency may be approximated by RT LL 441 −−=η .

Theoretical Treatment of the Employed Homodyne Detector

To perform pulsed homodyne tomography a homodyne detector has been used in

the experiment, in which the charges produced in the two photodiodes are directly

subtracted on a capacitor before the amplification takes place. Therefore no elec-

tronic gain adjustment is possible to fulfill condition (2.3.85) in our case. Power

splitters consisting of a polarizing beam splitter cube and a λ/2-plate where intro-

duced into the two arms of the homodyne detector to make a precise power balanc-

ing possible to compensate for e.g. photodiode efficiency differences. The com-

plete experimental scheme for the homodyne detector employed in the measure-

ments is depicted in Figure 2-20. Following through the same steps as for the more

simple configurations we obtain optimum power balancing, if

TLRL

RT

R

R

−−

=11

)(

)(

22

11

θθ

(2.3.99)

where )( 2/12/1 θR denote the reflectivities of the power splitters, T = 1-R the re-

flectivity of the dielectric beam splitter, which is adjusted for a reflectivity close to

50% by tuning the angle of the beam splitter, and L the losses at the antireflection

coatings of the dielectric beam splitter. For the choice of reflectivities as in equa-

tion (2.3.99) a homodyne efficiency of

2

)21()1()(21 21

11ηη

θη λ+

∆−−≈ RLRTTT lensHD (2.3.100)

is obtained. Here Tλ represents the transmission of the λ/2-plates, 2/1−=∆ RR

and η1/2 denotes the photodiode efficiencies and Tlens the transmission of the focus-

sing lenses (not shown in Figure 2-20).

A second option that was considered for the design of a pulsed homodyne system

was based on combining two polarizing beam splitter cubes with a λ/2-plate sand-


wiched in between, which act as a beam splitter with a variable transmission. As

pointed out before, optimum operation for such a device requires to adjust the elec-

tronic gain for both photocurrents individually, which is not possible with the

homodyne system employed in the experiment. Additionally one of the beams di-

rected onto the photodiodes would be transmitted through a polarization beam

splitter cube. This would introduce additional losses since the maximum transmis-

sion for beam splitter cubes typically ranges below 98%, whereas reflectivities

exceeding 99.5% are achieved. For these reasons the setup depicted in Figure 2-20

was preferred.

2.3.6 Homodyne Measurement of a Single-Photon Fock State

As we have shown in the preceding section a homodyne measurement corresponds

to a measurement of the quadrature amplitude Xθ,LO where the subscript LO repre-

sents the longitudinal and transverse mode of the local oscillator beam. We assume

a local oscillator quadrature amplitude eigenstate to be of the form

∫ −−⊥= θ

ωθ ω XekugkdcX LOLOLO ttrki

LOLOLOLOLOLO))((

,3

, )(~)(~ rrr (2.3.101)

where we have explicitly written out the LOkr

-dependence and introduced a possi-

ble time delay between the local oscillator pulses and the pump pulses tLO.

S

LO

R

D1

D2

-

TT R ( )θ

R ( )θ

i

i∆i

1

2

3

4

5

6

7

8

9

10 11 12 13

1415

16

17

18

19 20

21 22

23

24

25 26

27

28

a

b

a b

LL

λ 1

2

1

2

λ

Figure 2-20 Scheme of the homodyne detector setup used in the experiments.

68 2 Theory

Projecting the state prepared by triggering the homodyne detection by a photon

count event in the signal beam (equation (2.3.54)) onto the local oscillator state we

obtain

1)(~)(~)(~)(~

1)(~)(~

)(~)(~)(

1)(~)(~

)(~)(~

,,*

,2*

)(,,

*

*33

)()(,,

*

*333,

||

θω

θωω

ω

θωω

ωθ

ωωω

ωωδ

ωω

Xkukukdeggd

Xekuku

eggkkkdkdc

Xeekuku

eggkdkdrdcX

mm

sssLOs

mm

tisssLOs

tissLOLO

tissLOLOLOssLO

tirkkissLOLO

tissLOLOsLOsLO

LOs

LOs

LOLO

LOsLOs

LOLO

44444 344444 21

rr

444444 3444444 21

rr

rr

rr rrr

⊥=

⊥⊥⊥

=

−

−−⊥⊥

−

−−−⊥⊥

−

∫∫

∫∫

∫∫∫

=

−=

=Ψ

(2.3.102)

1θX is nothing else then the harmonic oscillator wave function for the rotated

quadrature amplitude Xθ (equation (2.2.14)) which for rotationally symmetric states

like the Fock states does not depend on the angle θ and – in the case of the single

photon Fock state – reduces to the analytic expression147

2

22

1)( 4 XeXXX −==Ψπ

(2.3.103)

We identify the terms ⊥mm and mm|| as the square roots of the transverse and

longitudinal mode matchings ⊥MM and MM||

2*2*2

||||

2*2,,

*,

22

)(~)()(~)(~||

),(),()(~)(~||

∫∫

∫∫∫

−===

=== ⊥⊥⊥⊥⊥

tgttgdteggdmmMM

yxuyxudydxkukukdmmMM

sLOLOti

sssLOs

sLOsssLOs

LOsωωωω

rr

(2.3.104)

Based on these results we may now separate the local oscillator mode into a part

mode-matched to the single-photon wave packet in the signal channel (s) and a

non-mode-matched part ( s¬ ). As we have seen in the last section these two con-

tributions add up incoherently since they arise from orthogonal modes of the light

field and give rise to a density matrix for the optical mode measured with the

homodyne detector of

ssMMssMMmeas ¬¬ ΨΨ−+ΨΨ= )1(ˆ ηηρ (2.3.105)

where ηΜΜ = ⊥MM · MM||. The expected marginal distributions can now be calcu-

lated according to equation (2.1.10):

147 see e.g.: U. Leonhardt, “Measuring the Quantum State of Light”, p. 23 (Cambridge Studies in Modern Optics, 1997)

Transverse and longitudinal mode-matching

Marginal distribution


( )

22

22

0)1(1

)1(

)1(

ˆˆ)(pr

θθ

θθ

θθ

θθθ

ηη

ηη

ηη

ρ

XX

XX

XX

XUUXX

tottot

stotstot

sstotsstot

meas

−+=

Ψ−+Ψ=

ΨΨ−+ΨΨ=

=

¬

¬¬

+

(2.3.106)

where we have replaced ηMM with ηtot – the total measurement efficiency – for

reasons discussed in the next section, and we have assumed that no excitation is

present in the non-mode-matched part. Thus the imperfect mode-matching will

lead to an admixture of the vacuum state to the measured state as expected from the

discussion in the previous section.

Using the wave function of the vacuum state in the quadrature representation

2

40

2)( XeX −=Ψ

π (2.3.107)

and the single-photon Fock state wave function from equation (2.3.103) we arrive

at the result

222 ))1(4(/2)pr(X, X

tottot eX −−+= ηηπη (2.3.108)

The effect of a limited efficiency on the Wigner function of the measured state is a

smoothing according to148

),Gauss),(

),(2

),,(

))()((2 22

tot

PPXX

tot

(X, PPXW

ePXWPdXdPXW tot

ηπ

η η

⊗=

⋅′′= ∫∫′−+′−

−

(2.3.109)

In the case of a single -photon Fock state the convolution in equation (2.3.109) re-

duces to a linear combination of the Fock and Vacuum Wigner functions weighted

by the total experimental efficiency ηtot and 1-ηtot respectively. Thus we obtain

222 )1)14((

2),( X

tottottot eXXW −−+−= ηηπ

η (2.3.110)

for the Wigner function of a single-photon Fock state detected with a total effi-ciency ηtot.

We can also interpret this result by stating that we are not able to measure the

Wigner function in the presence of losses. Instead we are measuring an s-

parameterized quasi-probability distribution with an s-parameter ranging from

s = 0 (Wigner function) in the ideal case of ηtot = 1 to s = –1 (Q function) in the

case of a total measurement efficiency of 50%.

148 U. Leonhardt, “Measuring the Quantum State of Light”, p. 80 (Cambridge Studies in Modern Optics, 1997)

Wigner function

70 2 Theory

Figure 2-21 depicts the Wigner function and marginal distribution of the detected

Fock state for various values of the efficiency. A negativity in the Wigner function

is achieved for measurement efficiencies exceeding 1/2. A dip in the marginal dis-

tribution already shows up if the efficiency surpasses 1/3.

2.3.7 Total Measurement Efficiency As we have seen in chapter 2.3.5 any losses and inefficiencies decrease the meas-urement efficiency of the homodyne system. Since all of these effects lead to an incoherent admixture of Gaussian noise to the measured quadrature value, we summarize the effect of all the experimental non-idealities in a single quantity – the total measurement efficiency. For this reason we have replaced the mode-matching efficiency with the total measurement efficiency in the last section.

Factors contributing to the total measurement efficiency are:

Figure 2-21 Influence of the measurement efficiency on expected marginal distri-butions and the Wigner function. An efficiency of 33% needs to be exceeded to see a dip in the marginal distribution. To be able to experimentally demonstrate nega-tive values of the Wigner function the efficiency needs to surpass 50%.


1. The transverse mode-matching ⊥MM and

2. the longitudinal mode-matching MM|| as discussed in the last section and specified in equation (2.3.104).

3. The optical efficiency of the signal channel ηopt,sig, including power losses induced by irises and a mirror as well as distributed losses.

4. The homodyne detector efficiency ηHD according to equation (2.3.100).

5. False measurements due to trigger dark count events: if the trigger detec-tor fires spontaneously a homodyne measurement is read out even though no single-photon Fock state was present in the signal channel – the homo-dyne detector measures vacuum fluctuations. Since the accessible quadra-ture amplitudes for the vacuum and the single -photon Fock state overlap, for any single homodyne read-out we cannot distinguish between wanted and unwanted measurements. Thus, if we denote the count rate of the trig-ger detector with N and the dark count rate with NDC , we are registering vacuum fluctuations with a probability NDC /N. Since the measured prob-ability distribution arises from statistically sampling the quadrature ampli-tude distribution obtained from triggered measurements of the homodyne detector, these false measurements give rise to an efficiency contribution of

NN DCDCprep /1, −=η (2.3.111)

6. Additional technical noise terms arise due to the imperfect pulse selection of the pulse picker cpulsepick (chapter 3.3.3) and due to

7. the electronic noise of the homodyne detector and the data acquisition electronics ηel (chapter 3.7). These contributions to the total measurement efficiency are discussed in more detail in the experimental part of this the-sis.

Collecting these contributions we obtain the total measurement efficiency

elpulsepickHDprepHDsigopttot cpMMMM ηηηη ⋅⋅⋅⋅⋅⋅= ⊥ ,,|| (2.3.112)

72 2 Theory

2.4 Conditional Quantum State Preparation in Repeated Parametric Fluorescence

Conditional measurements provide a promising method to prepare quantum states of the light field. The basic idea is to entangle the quantum system under consideration with an auxiliary system and to prepare it in the desired state owing to the state reduction associated with an appropriate measurement on the auxiliary system. In the scheme described in this chapter repeated 2-photon down-conversion is used to entangle two traveling optical modes – labeled signal and idler. A single-photon counter is placed in the output idler beam to achieve a state preparation in the signal mode.

Quantum state manipulation and engineering according to von Neumann´s projec-tion principle has been discussed in 149. Employing conditional measurements a number of new quantum states of the light field can be prepared including Fock states150, displaced Fock states, photon added states154 and Schrödinger-cat-like states153. Physical systems considered in the literature include Jaynes-Cummings type cavity QED151, parametric amplifiers152, the output beams of a beam splitter with different classical and non-classical input states153,154, four-wave mixers155, and trapped ions156. Conditional state generation in 2-photon down-conversion has been discussed in 150. The results presented in this chapter were obtained in a col-laboration with Jens Clausen and Prof. Welsch at the University of Jena157

2.4.1 Higher Fock State Generation Theoretical proposals to produce Fock states have been investigated in a wealth of papers by a number of authors. Proposed systems include high-Q cavities158, quan-tum jumps in ion traps159, the application of certain nonlinear optical Hamilto-nians160, excited single molecules in a microcavity trap161, and micromasers162. The

149 J. von Neumann, „Mathematical Foundations of Quantum Mechanics“, (Princeton University Press, Princeton, NJ, 1955) 150 A. Luis, L.L. Sánchez-Soto, “Conditional generation of field states in parametric down-conversion”, Phys. Lett. A 244, 211 (1998) 151 M. Brune, S. Haroche, S. Lefèvre, J.M. Raimond, and N. Zagury, Phys. Rev. Lett 65, 976 (1990); M. Brune, S. Haroche, S.J.M. Raimond, L. Davidovich, and N. Zagury, Phys. Rev. 45, 5193 (1992), K. Vogel, V.M. Akulin, and W.P. Schleich, Phys. Rev. Lett. 71, 1816 (1993) 152 K. Watanabe, and Y. Yamamoto, Phys. Rev. A 38, 3556 (1988), S. Song, C.M. Caves, and B. Yurke, Phys. Rev. A 41, 5261 (1990); B. Yurke, W. Schleich, and D.F. Walls, Phys. Rev. A 42, 1703 (1990), M. Ban, Phys. Rev. A 49, 5078 (1994) 153 M. Dakna, T. Anhut, T. Opatrný, L. Knöll, and D.-G. Welsch, “Generating Schrödinger cat-like states by means of conditional measurements on a beam-splitter”, Phys. Rev. A 55, 3184 (1997) 154 M. Dakna, L. Knöll, D.-G. Welsch, „Photon-added state preparation via conditional measurement on a beam splitter“, Opt. Comm 145, 309 (1998) 155 M. Ban, Opt. Comm. 130, 1281 (1996) 156 J.I. Cirac, R. Blatt, A.S. Parkins, and P. Zoller, Phys. Rev. Lett. 70, 762 (1993); R.L. De Matos Filho, and W. vogel, Phys. Rev. Lett. 76, 4520 (1996) 157 J. Clausen, H. Hansen, L. Knöll, J. Mlynek, D.-G. Welsch, “Conditional quantum state engineer-ing in repeated 2-photon down-conversion“, in preparation 158 J.R. Kuklinski, , „Generation of Fock States of the Electromagnetic Field in a High-Q Cavity through the Anderson-localization”, Phys. Rev. Lett. 64, 2507 (1990) 159 J.I. Cirac, R. Blatt, A.S. Parkins, P. Zoller, “Preparation of Fock States by Observatin of Quan-tum Jumps in an Ion Trap”, Phys. Rev. Lett. 70, 762 (1993) 160 S.Y. Kilin, D.B. Horoschko, “Fock State Generation by the Methods of Nonlinear Optics”, Phys. Rev. Lett. 74, 5206 (1995) 161 F. De Martini, G. Di Guiseppe, and M. Marrocco, “Single-Mode Generation of Quantum Photon States by Excited Single Molecules in a Microcavity Trap”, Phys. Rev. Lett. 76, 900 (1996) 162 F. De Martini, D. Boschi, M. Marroco, Proc. of the 3. Int. Aalborg Summer School (Nonlinear Optics) (1996)

2.4 Conditional Quantum State Preparation 73

generation of n = 1 Fock states has successfully been demonstrated in nonlinear optics, in ion traps and in microwave cavities163. A scheme to produce higher order Fock states in a beam splitter array with single-photon Fock states at the input is discussed in 164.

Higher Order Parametric Fluorescence The scheme of single -photon Fock state generation employing the process of pulsed 2-photon down-conversion can be extended to higher order processes with the aim to generate Fock states with n > 1. In these higher order processes n > 1 photon pairs are created and emitted into the signal and idler emission channels. A symmetric N-multiport is placed in the idler channel, which distributes an input channel to N-output channels with equal probability, and the output channels are monitored with nN ≥ single-photon counters. Conditioning the read-out of the signal channel on n coincident clicks projects the generated state onto an approxi-mate n-photon Fock state. Only an approximate state preparation is achieved, since it is possible that m > n photon pairs are emitted from the down-converter, of which only n are counted at one of the photodetectors (due to the limited detector efficiency and the possibility that two or more photons hit the same detector). Figure 2-22 depicts the envisaged experimental setup.

In a simplified notation the state produced in parametric fluorescence including higher order processes can be written as

is

nnisisis

isinaa

out

nn

e ispi

γγγ

ακ

!12

21

ˆ

221100

00)2(

++++=

=Ψ=Ψ++−

K

h

(2.4.1)

The probability per pulse that n photons are produced in parametric fluorescence, impinge on a symmetric N multiport and result in n clicks is found to be

),,(),( !1 nnNDnN nnn ηΓ=Π (2.4.2)

where Γ∼ γ2 denotes the chance to generate a photon pair in a single pump pulse, η the detection efficiency of the trigger detectors. D(N,n,n) represents a statistical factor that accounts for the probability to register m clicks employing N photodetectors if n photons impinge on a symmetric N multiport.

163 C.K. Hong, L. Mandel, Phys. Rev. Lett. 56, 58 (1985); D. Leibfried, D.J. Wineland, Phys. Rev. Lett. 77, 4281 (1997); X. Maitre, E. Hagley, G. Nogues, C. Wunderlich, J.M. Raimond, S. Haroche, Phys. Rev. Lett. 79, 769 (1997); 164 O. Steuernagel, “Synthesis of Fock states via beam splitters”, Opt. Comm. 138, 71 (1997)

Figure 2-22 Setup for the preparation of n>1 Fock states in parametric fluores-cence higher order processes. A symmetric multiport distributes the n photons at its input to N output channels with equal probabilities.

74 2 Theory

∑=

− ⋅−

=m

i

niNn

ii

N

m

N

NnmND

0)1(

1),,( (2.4.3)

The fidelity of the state preparation - the probability to find the optical system in the desired state - can now be evaluated to be:

)1(

),,()1,,(

)2(1

12 ++−+

≈n

nnNDnnND

Fηγ

(2.4.4)

As a specific example consider the n=4 Fock state:

In the case of N = 6 trigger detectors and assuming Γ = 0.1% and η = 65% results in a preparation fidelity of 89.9% but a production rate decreased by Γ3= 10-9 in comparison with the single pair creation rate.

Repeated Parametric Fluorescence The main drawback of the straightforward approach to generate n > 1 Fock states in higher order parametric fluorescence is the low generation rate which scales as Γn where Γ is typically on the scale of 10-4-10-2. This quickly renders the produc-tion of higher Fock states impractical for increasing n. To achieve a higher genera-tion probability a scheme based on repeated monitored 2-photon down-conversion according to the schematic setup illustrated in Figure 2-24 may be employed.

The idea

A pulsed laser pumps a down-conversion crystal. The idler channel of the down-conversion process is monitored by a trigger detector while the signal beam is cir-culated in a resonator with an optical path length synchronized to the pump pulse separation. After an initial count event has been registered in the trigger detector the photon pulse is repeatedly passing the down-conversion crystal. Further pair generation events are monitored by the trigger detector. After n trigger events have been detected within a specified maximum number of round trips N, the state is coupled out of the cavity by a cavity dumper.

To obtain a mathematical description of this process we need an evolution formula for the idler mode after one roundtrip. Three possible cases have to be considered:

Figure 2-23 Combinatorial factor for the detection of n photons, if n photons im-pinge on a symmetrical N-multiport


A1 The photon wavepacket circulates undergoing roundtrip losses T. No pa-rametric process occurs.

A2 The photon wavepacket suffers from roundtrip losses. A parametric proc-ess occurs but the trigger detector does not click.

B The photon wave packet suffers from roundtrip losses. A parametric proc-ess occurs and the idler photon is registered by the trigger detector.

Since the system suffers from losses we cannot describe the resonated signal mode by a state vector, but we need to employ a density matrix formalism. For the gen-eration of Fock states a reduced density matrix description including only the di-agonal elements of the density matrix suffices to describe the process for four rea-sons:

1. A photon addition only couples density matrix elements diagonally, i.e.

1,1ˆ),1(ˆ)1,(ˆˆˆˆ −−+ ⋅=−−==′ ∑∑ ji

ijkl

ijljklikij jijljkiiaa ρδρδρρ (2.4.5)

Diagonal density matrices are therefore transformed to diagonal matrices by a photon addition.

2. Photons impinging on one input port of a beam splitter behave as inde-pendent classical particles, if a vacuum state is incident on the other input port. Thus a decaying Fock state may be described in the diagonal basis.

3. Since we do not know whether a parametric events has occurred, if the trigger detector does not register a photon, we cannot distinguish between the cases A1 and A2. In an experiment we will perform many measure-ments and therefore produce an incoherent superposition of all possible re-alizations of sequences of cases A (A1 or A2) and B. Incoherent superposi-

Figure 2-24 Setup for the generation of n>1 Fock states in repeated parametric fluorescence.

76 2 Theory

tions of states with diagonal density matrices in the Fock basis will again result in diagonal density matrices.

If we denote the reduced density matrix by dr

where nndn ρ= , the effect of

any operator ô acting on ρ) may be described by a reduced operator Ô = ô+ô since

∑ ∑ ∑==′

=j k j

jjjikijkjiii ooo

jkjj

ρρρδρ

2* || (2.4.6)

The reduced density matrix dr

is nothing else but the photon number distribution:

ndnp =)( .

Roundtrip losses may be accounted for by a beam splitter with a transmission T = 1 – losses. The effect of a beam splitter in the Fock basis is described by the operator165

ijij

ijjiij TR

j

iBTR

j

ib

=⇒

−= +−+ ˆ)1(ˆ (2.4.7)

According to equation (2.4.5) the reduced diagonal operator corresponding to the creation operator â+ is given by

)1,(ˆ +⋅=+ jijAij δ (2.4.8)

After this preliminary remarks we may now proceed to find a quantitative descrip-tion of the effect of a round trip.

Let Γ denote the probability per pulse to produce a photon pair, η the detection

efficiency of the trigger detector and +B account for the round trip losses. Then we obtain the following equation for the two experimentally distinguishable cases A (no photodetection occurs) and B (photodetection occurs) up to second order in the parametric process:

)(222

41)1(

)(2224122

41)1(

^

^

ˆ]ˆ))1(1(ˆ[ˆ

ˆ]ˆ)1(ˆ)1(1))1()1(1[(ˆ

n

RT

n

n

RT

n

dBAANd

dBAANd

B

A

r

4444444 34444444 21

r

r

4444444444444 34444444444444 21

r

++++

++++

−−Γ+Γ=

−Γ+−Γ+−Γ−−Γ−=

ηη

ηηηη

(2.4.9)

where N is a normalization factor.

Neglecting the nonlinear terms due to higher order parametric processes the system of coupled differential equations describing the evolution along path A becomes linear and may be solved by a Laplace transformation yielding:

)0(1^

1)( ])1)1[((ˆ dRTsLNd

ART

An

r

4444 34444 21

r⋅−+=

↔

−− (2.4.10)

where L-1 denotes the inverse Laplace transform of the matrix argument inside the angled brackets. s is the argument in Laplace space. Please note that the (...)-1 refers

to a matrix inversion. )0(dr

denotes the reduced density matrix at the beginning of

165 Campos, Saleh, Teich, Phys. Rev. A 40, 1371 (1989)

Round trip equations

Evolution without photon-detections


the evolution. If the preparation procedure described above is employed, then the

initial state will be prepared in a single-photon Fock state ( ,...)0,1,0()0( =dr

).

Figure 2-25 depicts the time evolution of the elements of )(ndv

according to the solution found above as well as for a numerical computation using the parameter values T = 0.1%, Γ = 0.3%, η = 70%. The single-photon diagonal matrix element

)(1

ndr

decays slowly as a function of the number of round trips distributing popula-tion to the vacuum element as well as to the higher photon number elements.

Any single state preparation conditioned on n detections in the trigger detector may now be written as:

rL

rd RT RT RT RT RT d N n Nn B A

NA

NB A

Ni

i

n

n= ⋅ ⋅ ⋅ + =↔ ↔ ↔

=

−− ∑

^ ^

, ( )1 2 10

1

1

(2.4.11)

where Ni denotes the number of round trips between the ith and (i+1)th photon de-tection event. The expected prepared state according to the procedure stated in the beginning of this section is then given as an incoherent superposition of all possibe single preparations weighted by their respective probabilities:

rK L

rd p N N RT RT RT RT RT dn

N n

N

n n B AN

AN

B AN

N N n

n

i

= ⋅ ⋅ ⋅∑=

−

↔ ↔ ↔

= −∑ ∑ −

max

( , , )^ ^

1 1 01 2 1 (2.4.12)

where the probability for this preparation to occur is given by:

p N N p d p d p d p d p dn n B n AN

n AN

N B N ANn( , , ) ( ) ( ) ( ) ( ) ( )1 1 1 2 1 0

1 2

1 1

1K L− − − += ⋅ ⋅ ⋅− (2.4.13)

Since the probabilities depend on the di, a general formula is difficult to obtain. However using

ndjdAAdpppj

jji

jijiBBA ˆˆ)(10,

ηηηη Γ=⋅Γ=Γ=Γ=∧−= ∑∑∞

=

++ (2.4.14)

approximate solutions may be calculated, if an approximated closed expression for the expectation value of the photon number n may be found.

Figure 2-25 Time evolution of the reduced density matrix, if no photons are regis-tered by the trigger detector (T = 0.1%, Γ = 0.3%, η = 70%).

The prepared state

78 2 Theory

Let dBdrr

+≡′ ˆ be the photon number distribution after the initial state has under-gone round trip losses and before it reenters the down-conversion crystal. Then the photon number expectation value at this point is given by

nRdjRTR

i

jid

dBidndn

jj

jR

i

iji

jj

jijij

==

=

=′′=′

∑∑∑

∑

=

−

+

44 344 21

,

ˆ

(2.4.15)

According to equation (2.4.9) we may now calculate the photon number expecta-tion value n ′′ after the down-converter in the case where no photon is registered by the trigger detector to first order in Γ

]1)3(()1())1(1[(

])1()1())1(1[(

222221

21

+−++∆−Γ+−Γ−=

+′−Γ+′−Γ−≈′′

−

−

nRRnRnRnR

nnn

ηην

ηην

(2.4.16)

where we have replaced the normalization operator N with the normalization con-stant ν –1. With the help of equation (2.4.9) this normalization constant may be approximated using

Evolution of the average photon number

Figure 2-26 Evolution of a n = 1 Fock state in the setup depicted in Figure 2-24 with no photons being registered at the trigger detector. The diffusion in n-space can clearly be seen. Purple and red colors correspond to high, blue and white to low occupation probability.


n

n

djd

djijjid

n

jj

jj

jij

Ai

i

)1(

)1(

)1()1())1(1(

])1,()1()1(),())1(1[(

1

)normalized is state (initial 1

, ˆ1

!

ηννν

ην

ηη

δηδην

ν

−Γ=−′′=

−Γ+=

+−Γ+−Γ−=

++−Γ+−Γ−=′′=′′

+=

∑∑

∑∑+

&

43421321

44 344 21321

(2.4.17)

yielding

∑−

=−Γ+=

1

0

)()( )1(1m

i

im nην (2.4.18)

Employing 1)0( =n and 0)0( =∆n as the initial condition we obtain the initial growth or loss rate G:

)0()0()1()1(

)1(

))1(2(

)1(21)1(1

)1(31

nTnnn

TT

n

G44 344 21

& −−Γ≈−=⇒

−−Γ+≈−Γ+

−−Γ+=

η

ηη

η

(2.4.19)

Therefore the evolution of the average photon number may be approximated by

mTm en ))1(2()( −−Γ≈ η (2.4.20)

where m is the number of round trips elapsed. Figure 2-26 shows a comparison between the exact time evolution of the average photon number and the approxi-mate solution of equation (2.4.20).

Using the approximate solution for the average photon number we may also give an approximation for the normalization factor ν (m) according to equation (2.4.18)

Figure 2-27 One possible realization of a n = 3 Fock state preparation. The preparation fidelity after 100 round trips is 66%.

80 2 Theory

1

1)1(1

)1(2

)1())1(2()(

−−

−Γ+≈−−Γ

−−−Γ

T

mTm

e

eη

ηην (2.4.21)

The Improvement Employing repeated 2-photon down-conversion the higher Fock state generation rate is increased approximately by a factor of N in comparison to the simple scheme since

p N p d NBN

2 01 1( ) ( ( ))≈ − − ≈ Γη (2.4.22)

where N is the maximum number of round trips you allow for. N is limited by the preparation error (=1-fidelity) which can be approximated to

ε η≈ + −N T R( ( ))Γ 1 (2.4.23)

Assuming ε = 1/3 to be acceptable and setting the experimental parameters to T = 0.1%, Γ = 0.3%, and η = 70% leads to an improvement of the generation rate by a factor of 213. For the n=2 Fock state this improves the rate to 2/3 of the single pair production rate.

The improvement of the rate raises with decreasing T and Γ and with increasing η.

2.4.2 Preparation of Arbitrary Pure Truncated Quantum States

SU(1,1) Transformation at a Parametric Down-Converter

We consider the interaction Hamiltonian of the non-degenerate parametric χ(2)-

interaction ( )+++ −−= 3213212

2 ˆˆˆˆˆˆ/ˆ aaaaaaiH PDCPDC κh and use the parametric ap-

proximation 33ˆ aia −→ to obtain the Hamiltonian of the parametric amplifier

( ) ( ) 12*

2122112211

22211

ˆˆˆˆˆˆˆˆˆˆˆˆ

ˆ21

ˆˆ21

ˆˆˆ

aazaazaaaaaaaa

HaaaaH PDCPA

++++−

+++

++

++−++≡

+

++

+=

φφ

hh (2.4.24)

where ( ) 2/21 ωωφ hh +=± and *3

2 az PDCκh= . Introducing a energy shift

( )2211 ˆˆˆˆˆˆ aaaaHH PAPA++

− −−=′ φ this Hamiltonian can be expressed as a linear

combination of the SU(1,1) generators 21ˆˆˆ KiKK ±=± and 3K with

( ) 2/ˆˆˆˆˆ21121 aaaaK += ++ , ( ) ( )iaaaaK 2/ˆˆˆˆˆ

21121 −= ++ and ( ) 2/ˆˆˆˆˆ22111 aaaaK ++ += .

Applying the disentanglement theorem166 we obtain the non-unitary time evolution

operator

2221*

1211

323

ˆˆ*ˆˆˆˆˆˆ*

ˆ)(ˆ2ˆ)(

ˆˆ

aaaaRaaRaa

KiKiKi

HiPA

TeeT

eee

eU

RTRT

PA

++++ −−−−

−+

′

=

=

=ϕϕϑϕϕ (2.4.25)

where

166 K. Wodkiewicz, J.H. Eberly, J. Opt. Soc. Am. B 2, 458 (1985)


ββ

ϑ

ββ

φβϑ

ϕ

ϕ

sinhsinh

sinhcoshcosh

*zieR

ieT

T

T

i

i

==

+== +

(2.4.26)

and 22++= φβ z . PAU relates the two-mode input state ρ to the two-mode

output state +=′ PAPA UU ˆˆˆˆ ρρ . In the Heisenberg-picture the field operators are

transformed as

−

−=

′′

++2

1**

2

1

ˆ

ˆ

ˆ

ˆ

a

a

TR

RT

a

a (2.4.27)

in close analogy to the SU(2) transformation describing a beam splitter, but pre-

serving the photon number difference instead of the photon number sum:

22112211 ˆˆˆˆˆˆˆˆ aaaaaaaa ++++ −=′′−′′ . Operator functions accordingly transform as

)ˆˆˆ,ˆˆ(ˆ)ˆ,ˆ(ˆˆ2

*1

*2121

++++ +−−= aTaRaRaTFUaaFU PAPA (2.4.28)

Conditional Measurements

We consider the case where an arbitrary input signal mode sρ is coupled to a pure

input idler mode iΨ at a parametric amplifier. The output idler mode is moni-

tored with a trigger detector. If this trigger detector has registered an output idler

state tΨ , then the signal mode is left in state

( )

44 344 2144 344 21Y

tPAis

Y

iPAt

ttis

UUN

TrN

ˆˆ

1

1

ˆˆ ΨΨΨΨ=

ΨΨ′=′+−

−

ρ

ρρ (2.4.29)

where N is a normalization constant. The non-unitary conditional operator Y does

not depend on the input signal state and acts on the signal mode as a single mode

operator. Hence the trigger detection in the idler path leads to a conditional state

preparation in the signal beam.

Expressing the input idler state as ( )∑∞

=

++ ≡=Ψ0

22 0ˆˆ0ˆm

mmi aFaf and the de-

tected trigger state as ( )∑∞

=

++ ≡=Ψ0

22 0ˆˆ0ˆn

nnt aGag we may specify the condi-

tional operator Y in a closed form

+−

−= 11ˆˆ*

1

*

1* ˆˆ)ˆ(ˆˆ aa

s

PA TaTR

GaRFY (2.4.30)

82 2 Theory

where the subscript 1)||1(/)||1( 22 ≥−+−= TTs indicates s-ordering167.

Making use of equation (2.4.28) we note that the parametric time evolution opera-

tor transforms the coherent displacements )(ˆ2/12/1 αD according to

)(ˆ)(ˆˆ)(ˆ)(ˆˆ2

*12

*2112211 αααααα TRDRTDUDDU PAPA +−−=+ . Coherent dis-

placements in the idler mode are coupled to coherent displacements in the signal

mode. Thus in the special case of coherently displaced idler mode states

0)ˆ(ˆ)(ˆ2+=Ψ aFDi α and 0)ˆ(ˆ)(ˆ

2+=Ψ aGDt β we obtain the expression

−==

−=

*

***

*

**ˆ)0,0(ˆˆˆ

R

TDY

R

TDY PAPA

αββα

βα (2.4.31)

As a consequence of equation (2.4.31) no s-ordering is required in equation

(2.4.30), if the input or output idler beam is in a coherent state ( 1ˆ =F or 1ˆ =G ).

State Preparation

Suppose we intend to prepare a signal mode quantum state that can be expressed –

or at least approximated – by a truncated Fock state expansion

∑=

=ΨM

mnc

0 (2.4.32)

For this state there exist M solutions of the equation 0=Ψ kβ (k = 1…M) corre-

sponding to M zeros of the Q-function. Therefore the state Ψ may be generated

according to

∏∏=

++

==−=Ψ

M

kkk

MM

kk

M DaDM

ca

M

c

111 0)(ˆˆ)(ˆ0)ˆ( βββ (2.4.33)

by applying a series of coherently displaced photon additions to the vacuum state.

The idea

If the parametric amplifier considered above is fed with coherent pulses, then it

performs coherently displaced photon additions whenever a parametric process

takes place in the down-conversion crystal. Hence it is possible to prepare arbitrary

quantum states with a truncated Fock state expansion, if the idler beam input of the

parametric amplifier is fed with a suitable series of coherent states, the output idler

beam is monitored with a trigger detector and the signal beam is repeatedly fed

back into the input signal mode of the parametric amplifier in a setup similar to that

167 K.E. Cahill, R.J. Glauber, Phys. Rev. 177, 1857 (1959)


depicted in Figure 2-24 (with an additional seed beam inserted into the input idler

mode).

A similar scheme employing a beam splitter array with coherent states and Fock

states as the input states has been proposed by Jens Clausen et al. in 168,169.

Let us consider a single parametric process first and assume that a coherent state is

fed into the idler mode of the parametric amplifier ki α=Ψ and that the trigger

detector registers a single -photon 1=Ψt . In this case we find

−

−= +−− +

*

**

11ˆˆ*

*

*)( ˆˆˆˆ 11

R

TDaT

RDRY kaakk

PAαα

(2.4.34)

with the help of equation (2.4.31). The operator )(ˆ kPAY describes a displaced photon

addition as expected. For a vanishing coherent amplitude αk = 0 we recover a sim-

ple photon addition as used in the calculation in chapter 2.4.1. If the resulting sig-

nal state is fed back into the input mode of the parametric amplifier and the process

(2.4.34) is repeated M times with different αk, then we obtain a total evolution op-

erator )1()2()( ˆˆ...ˆˆPAPA

MPAPA YYYY = and a resulting prepared state

0ˆ0ˆ1 *

1*

1

||||2

||1)3(

2 12

2

∏ ∑= =

++−

+−

−−⋅=

∑=

M

k

M

kll

llM

R

TM

MMi

PAT

T

RT

aeTReY

M

kk ααα

ξ

(2.4.35)

where ξie is an irrelevant phase offset, αM+1 = 0 and we have used

)(ˆ)ˆ(ˆ)(ˆ * ααα DaaD −= ++ and aaaa

TaTaTˆˆ**ˆˆ* ˆˆ

++++ = .

To obtain the desired state Ψ we equate (2.4.35) with (2.4.33) which yields the

required size of the coherent amplitudes

∑∑=

+=

+ −=⇔−

=M

klll

lMk

k

M

kll

llM

k TTT

R

T

T

R

T)( *

1*2

*

1*

*ββα

ααβ (2.4.36)

For the generation rate we obtain

∑=

−+−=

M

kk

R

T

MMM

M

eTRc

MN 1

22

2

||||

||1

)3(22

||||||

! α (2.4.37)

where we have made use of the fact that the truncated state Ψ is normalized.

Since Γ+=+= 1||1|| 22 RT we find MMcMN Γ≤ 2||/! . Thus for Γ typically

168 J. Clausen, M. Dakna, L. Knöll, D.-G. Welsch, Phys. Rev. A 59, 1658 (1999)

84 2 Theory

much smaller than unity the generation rate rapidly decreases with increasing M.

This was to be expected as we have assumed that a parametric process occurs every

round trip.

In a realistic setup a problem might arise due to the fact that single-photon counters

do not distinguish between 1 and many photons. Thus the state projection assumed

in the calculation above is only achieved, if 1|| 2 <<kα which is expected only for

intended states with low average photon number. To overcome this problem, the

coherent displacement can spatially be separated from the parametric process by

placing an additional beam splitter in the signal round trip path. A coherent dis-

placement at this beam splitter may be achieved, if its reflectivity is close to unity

so that the insertion of a strong coherent state in its seed beam port will add a co-

herent excitation α τ , where τ is the reflectivity of the beam splitter, without sig-

nificantly changing the quantum fluctuations of the signal beam. In any case an

experimental realization of the scheme discussed above has to provide coherent

idler pulses mode matched to the circulating signal state and a pulse-to-pulse co-

herent excitation adjustment.

A higher preparation rate but lower preparation fidelity is expected, if the condition

of consecutive parametric processes is relaxed. In this case the coherent displace-

ments have to be triggered on a photon count event in the idler path. Unregistered

photon additions in the down-converter will be a more serious problem for the

general scheme than for the higher Fock state generation, since already the addition

of a single photon will completely wash out the quantum mechanical phase of the

prepared state.

169 J. Clausen, M. Dakna, L. Knöll, D.-G. Welsch, Phys. Rev. A 60, 726(E) (1999)

3.1 Outline 85

33 EExxppeerriimmeenntt

3.1 Outline

The single-photon Fock state 1 is one of the most fundamental states of the light field. Absorption and emission of photons from the electromagnetic field by atoms or molecules are among the most basic matter-light interactions. Photons are the fundamental exchange particles of the electromagnetic force that mediate electro-magnetic interactions. Due to its fundamental relevance the single -photon Fock state has attracted considerable theoretical and experimental attention not only in the field of quantum optics, but also in solid state physics and elementary particle physics.

First steps towards the generation of optical Fock-states were connected to the ef-forts to employ photon emissions by single-quantum systems to obtain non-classical photon count statistics. Since an optical transition in a single-quantum system only emits one photon at a time, photon emissions from such systems are less likely to occur close to each other than further apart. This effect is called anti-bunching. Observations of anti-bunched light by Kimble et al.170 in 1977 have been among the pioneering experiments in the field of quantum optics.

The idea to use single quantum systems to generate single -photon states of the light field has further been pursued by a number of groups in the last decade employing optically active molecules: De Martini et al. have presented evidence of strong anti-bunching of single Oxazin molecules excited by optical pulses in an active microtrap171,172, Orrit et al. have used rapid adiabatic passages in DBATT mole-cules at cryogenic temperatures around 1.8 K to produce single photons173 and Moerner et al. have produced single photons at room temperature using Terrylene molecules excited with optical pulses. The last two experiments suffered from the drawback that the photons were emitted into a 4π-solid angle and therefore difficult to transfer into a single-optical mode.

Already in the late 80’s P. Grangier and A. Aspect 174 as well as C.K. Hong and L. Mandel175 have used conditional state preparation by measurements on correlated photon pairs generated in an two-photon emission cascade or in the process of 2-photon down-conversion to prepare conditional single-photon Fock state as de-scribed in chapter 2.3.4..

170 H.J. Kimble, M. Dagenais, L. Mandel, Phys. Rev. Lett. 39, 691 (1977) 171 F. De Martini, G. Di Giuseppe, M. Marrocco, “Single-Mode Generation of Quantum Photon States by Excited Single Molecules in a Microcavity Trap”, Phys. Rev. Lett. 76, 900 (1996) 172 F. De Martini, O. Jedrikiewicz, P. Mataloni, “Generation of quantum photon states in an active microcavity trap”, J. of Mod. Opt. 44, 2053 (1997) 173 Orrit et al., Phys. Rev. Lett. 83, 2722 (1999) 174 P. Grangier, G. Roger, A. Aspect, “Experimental evidence for a photon anticorrelation effect on a beam splitter : a new light on single-photon interferences” , Europhys. Lett. 1, 173 (1986) 175 C.K. Hong, L. Mandel, “Experimental realization of Localized One-Photon States”, Phys. Rev. Lett. 56, 58 (1986)

Anti-bunching

Single-molecule emissions

Conditional measure-ments

86 3 Experiment

C.K. Law and J. Kimble have proposed to generate a “bit stream of single-photons single-photon pulses” – also called a photon pistol, where the emission of a single photon may be controlled by pulling a classical trigger – in an Λ-type three-level atomic system strongly coupled to a cavity in 1997176. A related scheme to generate single-photon states in a cavity QED setup was proposed by A. Kuhn et al.177 in 1999. Experimental efforts for an experimental realization of these proposals are under way.

The group of S. Haroche et al. have been able to store, read-out and manipulate Fock states in the microwave regime inside a high-finesse, superconducting cavity employing highly excited circular Rydberg states178. Microwave Fock state genera-tion is also possible in micromasers, where single atoms are strongly coupled to a microwave cavity179. Recently Fock state generation in a micromaser has experi-mentally been demonstrated by B.T.H. Vascoe et al.180 in Munich.

The NTT research labs, Hamamatsu Photonics and the group of Prof. Yamamoto at the University of Stanford have recently succeeded to produce a “single-photon turnstile” based on electronically controlled single-photon emissions by a GaAs/AlGaAs quantum well structure181. In a recent extension of work they em-ploy optically pumped micro-pillar structures to obtain up to 78% coupling effi-ciency into a single optical mode.

The great number of theoretical and experimental efforts to produce optical single photon states highlights the relevance of these states in the field of quantum optics. Despite all these efforts no full experimental quantum characterization of a single-photon Fock state has been achieved in the optical domain. However, the Fock state density matrix has already been determined for a different physical system: the motional state of atoms in an ion trap182.

Our aim is to efficiently prepare a single -photon Fock state in a well defined mode and to obtain the full quantum information about this very fundamental state of the light field employing the method of pulsed optical homodyne tomography. In this way we want to contribute to the understanding of light particles, since the tomo-graphic measurement of a single -photon Fock state requires an intimate under-standing of the Fock state preparation and characterization.

Furthermore this experiment is – to our knowledge – the first experiment that com-bines the methods of single-photon counting with homodyne measurements of the optical field quadrature amplitudes in a single experimental setup. Thus light inten-sity (photon number) and optical field (quadrature amplitudes) are measured simul-taneously. J.G. Rarity et al. demonstrated non-classical interference between inde-pendent sources183 in an experimental setup similar to ours but exclusively employ-ing single-photon counters instead of a homodyne system.

The single-photon Fock state tomography experiment also intends to further de-velop the method of pulsed homodyne tomography and to enhance our understand-ing of quantum state preparation by conditional measurements with this experi- 176 C.K. Law, H.J. Kimble, “Deterministic generation of a bit-stream of single-photon pulses”, J. of Mod. Opt. 44, 2067 (1997) 177 A. Kuhn et al., Appl. Phys. B 96, 373 (1999) 178 S. Haroche et al., e.g. : Nature 400, 239 (1999) 179 A. Napoli, A. Messina, “Conditional generation of non-classical states in a non-degenerate two-photon micromaser: single-mode Fock state preparation. II”, J. of Mod. Opt. 44, 2093 (1997) 180 B. T. H. Varcoe, S. Brattke, M. Weidinger, H. Walther, “Preparing Pure Photon Fock States of the Radiation Field”, Nature 403, 743 (2000) 181 Yamamoto et al., Nature 397, 500 (1999) 182 D. Leibfried, D. M. Meekhof, B.E. King, C. Monroe, W.M. Itano, and D.J. Wineland, “Experi-mental Determination of the Motional Quantum State of a Trapped Atom”, Phys. Rev. Lett. 77, 4281 (1997) 183 J.G. Rarity, P.R. Tapster, R. Loudon, “Non-classical interference between independent sources”, preprint quant-ph/9702032, (1997)

Photon pistol

Single-photon turnstile

Single-photon Fock state tomography

3.1 Outline 87

ment. In this way we also hope to make a number of other strongly non-classical states of the light field accessible (compare chapter 5.2.1.1).

Figure 3-1 provides a schematic general view of the experimental setup (a more detailed version of the setup is shown in chapter Fehler! Verweisquelle konnte nicht gefunden werden.): We employ a mode-locked Ti:sapphire laser in combi-nation with a pulse picker to obtain transform-limited pulses at 790 nm with a repetition rate between 204 and 816 kHz and a pulse duration of 1.8 ps. The prop-erties and specifications of the light sources will be described in chapter 3.3.

Most of the radiation is single-pass frequency doubled in a BBO-crystal yielding 173 µW at 395 nm and then passed on to a BBO-crystal cut for Type I down-conversion. The down-converter is operated in a frequency degenerate, but spa-tially non-degenerate configuration yielding up to 25000 photon pairs per second into a 2 nm wide filter. The number of detected photon pairs strongly depends on the spatial and spectral filters employed. A description of the nonlinear optical processes can be found in chapter 3.4.

A single-photon counter is placed in one of the emission channels – labeled trigger – to detect photon pair creation events and to trigger the readout of a homo-dyne system placed in the other emission channel - labeled signal. In this way only those pulses are selected for homodyne measurements where a photon has been emitted into the signal channel, thus preparing single-photon Fock states by condi-tional measurements. Photon counting measurements and the fidelity of state preparation by conditional measurements are discussed in chapter 3.5.

We use a small fraction of the original optical pulses from the pulse picker - split off before the frequency-doubler - as the local oscillator for the homodyne system. These pulses have to be temporally and spatially mode-matched to the photons in the signal channel. Mode matching considerations are explained in chapter 3.6. In order to be able to detect the quantum noise of single-photons at the pulse picking

Figure 3-1 Strongly simplified experimental setup.

Experimental setup

88 3 Experiment

rate, we have developed a homodyne design with ultra-low electronic noise (565 electrons per pulse) and high subtraction efficiency (> 83 dB). A detailed descrip-tion of the pulsed homodyne system is given in chapter 3.7.

Chapter 3.8 deals with the topic of data acquisition and monitoring of the ex-perimental parameters. Chapter 3.9 links together the experimental parameters discussed in earlier chapters to obtain an estimation of the total measurement efficiency. Finally in chapter 3.10 experimental results on the reconstruction of the vacuum state, coherent states and single -photon Fock states are presented.

Figure 3-2 Main part of the optical setup. The down-converter crystal is mounted in the aluminum oven in the center of the picture. The trigger and signal beam single-photon-counters can be made out left and right behind the down-converter. The homodyne system is located at the back of the right side.

3.2 The Optical Setup 89

3.2 The Optical Setup Figure 3-3 offers a more detailed version of the optical setup. The table is divided into four compartments according to the level of optical power present. The com-partments are separated by light shields.

The first compartment in the lower left corner of the setup depicted in Figure 3-3 houses the Ti:sapphire laser and the frequency doubler, the optical power is around 1.7 W. The second compartment in the middle left section of the diagram accom-modates greater parts of the local oscillator and seed beam paths. Power levels in this compartment are in the mW-range. The third compartment in the upper left corner of the experimental setup contains the main experimental components: the down converter, the single-photon counters and the homodyne system. It can opti-cally be sealed from the rest of the table by light tight separators to assure a mini-mum amount of stray light in this compartment since both, the single-photon counters as well as the homodyne detector, are very sensitive to stray and back-ground light at 790 nm.

A fourth compartment to the right of the three compartments described above, was added to house the pulse picker and its monitor detectors as well as the second harmonic pump light dump and monitor detector. Two kinematic mirrors M2 and M8 serve to either redirect the laser beam through the pulse picker or – if removed – to have the full laser power available for higher light intensities and photon count

Figure 3-3 Setup of the optical table; all major components have been labeled to allow an unambigu-ous identification.

90 3 Experiment

rates in order to ease the alignment for the down-conversion process.

There are three main beam paths in the optical setup:

1. The pump beam path, which receives the major part of the optical power and contains the frequency doubler to convert the infrared laser radiation to the blue. The remaining laser power is separated from the blue light employing a spectral filter and redirected into a beam dump. Optical components in this beam path have been labeled “P”. The lenses LP1 and LP2, which are used to focus the laser beam into the crystal and to collimate it again, also serve to enlarge the pump beam waist by a factor of 1.5. The intention of enlarging the beam is to make mode matching between the optical beams easier.

About 10% of the laser power is transmitted through the dielectric beam splitter M9. The transmitted beam propagates to two power splitters, each consisting of a λ/2-retardation plate and a polarizing beam splitter cube, which select an adjustable amount of power to be used either for the seed or the local oscillator beam.

2. The seed beam is used only for alignment purposes and for the reconstruction of coherent quantum states. During all other measurements it is blocked. The first beam splitter cube PBS1 forms the beginning of the seed beam path. This beam then passes through a 3:2-optical telescope for beam expansion and is sent towards the down-converter. The telescope assures a compatible beam waist of the pump beam and the seed beam at the down-conversion crystal and also allows to mode match the waist size and waist position of the seed and pump beam. The optical path length difference between the two beams can be adjusted precisely without changing the beam position using an optical trombone to obtain an optimum pulse overlap in the down-conversion crystal. Optical components in the seed beam path are labeled “S”.

3. The local oscillator beam path starting at PBS2 is also equipped with a beam expanding telescope and an optical trombone to enable a path length adjustment and spatial mode match of the local oscillator beam to the seed beam and the mode of the single-photons. Components in this beam path are labeled “LO”.

The down-conversion process produces 790 nm radiation being emitted into two channels – labeled signal and idler channel. The idler beam is directed towards a single-photon counter which serves as the trigger detector. At a later stage of the experiment a spatial filter, which is not shown in Figure 3-3, has been installed in the signal channel to improve the transverse mode selection.

The signal radiation propagates to the homodyne detector where it is mode matched to the local oscillator beam. Alternatively, the signal beam may be redi-rected towards a second single -photon counter for coincidence measurements using the cinematic mirror MS7.

Beam paths

3.3 Light Sources 91

3.3 Light Sources For the single-photon Fock state reconstruction photon wave packets have to be generated in a well controlled fashion. This requires a careful choice of the light sources used in the experimental implementation. Requirements to be considered include:

§ The need for short, but not too short laser pulses at a frequency that can ef-ficiently be detected with standard photo diodes and single -photon count-ers.

§ A pulse repetition rate that does not exceed the time resolution of the de-tection system of ca. 1 MHz.

§ A moderately high power level to drive the inefficient process of paramet-ric fluorescence.

§ Good transverse mode quality to obtain good mode-matching.

To meet this requirements we use a pulse picked, mode-locked Ti:sapphire laser which provides ps-pulses at a repetition rate just below 1 MHz.

3.3.1 Pump Laser An Argon ion laser (Spectra Physics) is employed as the pump laser for the pri-mary Ti:sapphire mode-locked laser. Important technical data of this laser are col-lected in the table below. The laser is not equipped with a beam stabilization which became a standard equipment only for later models of the same type. As a conse-quence of this frequent realignment of its output beam proved to be necessary for optimum laser operation.

The pump laser exhibits power oscillations at multiples of the supply net AC-frequency, with a strong peak at 300 Hz. These oscillations are transferred to the Ti:sapphire laser.

The laser tube has been in operation for more than 4300 hrs showing only a slow increase in current consumption with time and requiring only minor cavity read-justments from time to time. This remarkable longevity may probably be explained by the almost optimum operation conditions: high supply current and stable output power.

Technical data of the pump laser

Type Ar-ion laser model 2035-35 from Spectra Physics, manufactured Febr. 1988

Output power 0...15 W multiline @ 457.9 – 514.5 nm

Specifications noise: 0.5 % rms (10 Hz – 2 MHz) , power stability: 3.0 % in a 30 minutes period after 2 hrs warmup, beam diameter (1/e2): 1.8 mm, beam divergence: 0.45 mrad, cavity length 1.71 m standing wave (87 MHz mode spacing)

Operational parameters 7.8 - 10 W output power at 41-45 A current consumption tube pressure: 560 V at 50 A

Power supply Spectra Physics, Model 270, 0…60 A current @ 480 V 3-phase (power consumption < 38 kW), water cooled ( ≥ 13.3 l/min)

92 3 Experiment

3.3.2 Primary Laser The primary laser in our experiment is a longitudinally pumped, actively mode-locked Tsunami Ti:sapphire laser from Spectra Physics. Active mode-locking is achieved by means of an acousto-optic modulator (AOM), which allows the laser to operate without mode-locking drop-outs for extended periods of time. The cavity length of 3.7 m leads to a repetition rate of 81.2 MHz corresponding to a time sepa-ration of the pulses of τ = 12.3 ns.

Dispersion compensation in ps-operation relies on a Gires-Tournois interferometer (GTI) – a Fabry-Perot-interferometer with a highly reflecting end mirror resulting in almost unit reflectivity, but a frequency dependent phase shift – which enables pulse widths of ∆T = 1.4 – 1.8 ps. This temporal width corresponds to a pulse bandwidth of 0.54 mm. The pulse duration is strongly dependant on the length of the GTI cavity which may be controlled by an external knob and the power level of the laser (see the inset in Figure 3-4). Optimum pulsing is achieved at output pow-ers between P = 1.5 and 1.8 W average power. According to

PT

P ˆ2lnτ

π ∆= (3.3.1)

this corresponds to a peak power of P = 14.1 – 18.2 kW and a power enhancement

of PP /ˆ = 4636.

The temporal shape of the laser pulses is expected to be of sech2-shape and has been measured employing an auto-correlator (Figure 3-4). Results of the auto-correlation measurement are shown in Figure 3-4. Auto-correlation width of 2.0 – 2.5 ps have been recorded, consistent with pulse length of 1.4 – 1.8 ps. Assuming transform-limied behavior we may also infer the spectral width of the pulses using ∆Τ ∆ν = 0.315 (0.441 for a Gaussian) yielding ∆ν = 3.5 1011 Hz or ∆λ = 0.79 nm (FWHM @ 1.8 ps pulse length).

Aut

ocor

rela

tion

sign

al [

a.u.

]

1/cosh(t) fit2 γGaussian fit

-6 -4 -2 0 2 4 6

0

2

4

6

8

10

Time [ps]

400 600 800 1000 1200 1400 1600 1800 20001,6

1,8

2,0

2,2

2,4

2,6

2,8

3,0

Puls

lengt

h[p

s]

P(Ti:Sa) [mW]

Laser Pulses

Figure 3-4 Laser pulses from the Ti:sapphire laser measured with an auto-correlator. The inset shows the dependence of the pulselength on the pump power.


Wavelength tuning in the ps-operation mode is accomplished using a birefringent filter. This tunability allows us to select an emission frequency which coincides with the center frequency of the narrow bandwidth filter in the trigger channel (790.6 nm).

Intensity RMS noise of the Tsunami laser has been observed to be 1.06% for a measurement interval of 1 hour and 0.36% in a 30 s interval. Frequency noise is mainly induced by pump laser noise and shows a strong peak at 300 Hz (see Figure 3-5). Figure 3-6 depicts a long term measurement of the pump laser power, the Ti:sapphire laser power and the power of the frequency doubled light.

The laser beam leaving the Tsunami laser exhibits a substantial ellipticity with a waist size of w = 0.6 mm along the vertical direction and w = 0.9 mm along the horizontal direction. This ellipticity is due to the folded cavity design, where the laser mode hits the cavity mirrors at angles other than normal incidence, thus giving rise to an astigmatism. This astigmatism is compensated with the Brewster-angle cut Ti:sapphire-rod to a large extent.

Figure 3-5 Low frequency laser intensity fluctuations of the Ti:sapphire laser ob-tained form a numerical Fourier transform of laser power time traces.

94 3 Experiment

Technical data of the primary laser

Type Tsunami mode-locked Ti:sapphire laser from Spectra Physics

Power output Operated at 1.6 – 1.8 W average output power corresponding to a peak power of 11.6 kW

Repetition rate 81.2 MHz (2 x 1.846 m = 3.692 m cavity length in a folded standing wave resonator, 1 pulse every 12.3 ns

Pulses ∆Τ = 1.4 - 1.8 ps (typ. 1.8) @ pump powers >8 W, corresponding to ∆λ = 0.78 nm spectral pulse width (FWHM @ 1.8 ps pulse length, sech2-pulse), ∆ν = 3.5 1011 Hz, and a pulse length of 0.54 mm

Beam parameters Collimated, strongly elliptical beam: 1/e2-width: vertical 0.6 mm, horizontal 0.9 mm, beam divergence < 0.6 mrad (0.034°), polarization > 500:1 vertical

Figure 3-6 Long term stability of the pump laser power, Ti:sapphire power, and SHG power.


3.3.3 Pulse Picker To reduce the measurement bandwidth, which is imposed by the pulse repetition rate, we employ a pulse picking system (NEOS, N17389) to select single pulses from the pulse-train of the mode-locked Ti:sapphire laser. The pulse picker is based on a TeO2-acousto optic modulator (AOM), which is fed with 7W RF pulses with a duration of 10 ns and a carrier frequency of 389.5 kHz provided by an HF-driver (NEOS, N64389-SYN). Despite the fact that this pulse picker has been spe-cifically designed for pulse picking of mode-locked Tsunami lasers, it failed to achieve the specifications originally quoted by NEOS.

The driver has two modes of operation: pulse picking and cw-outcoupling. The latter is used mainly for adjustments where a higher power level in the outcoupled beam is advisable. In pulse picking operation the RF-power level may be set with a control knob. For optimum performance an adjustment of the relative timing of the acoustic pulses with respect to the laser pulses as well as a fine-tuning of the phase of the driving frequency with respect to the RF pulses is required.

The pulse rate division factor can be selected manually and was set to be 400 (203 kHz pulse repetition rate), 200 (406 kHz) or 100 (812 kHz). Higher repetition rates result in higher count rates in the experiment, but the time resolution of detector electronics limits the maximum repetition rate to 1 MHz.

The figure of merit for the pulse picker operation is its total contrast ratio (CR) – the power ratio between the selected pulse and the sum of all neighboring pulses. An admixture of neighboring pulses, which do not correspond to photon pair emis-sions, will result in an unwanted contribution to the power deposited on the homo-dyne system photo diodes. Due to the limited bandwidth of our detection system the detector will perform an effective time averaging over the selected and the un-wanted pulses leading to a contribution of vacuum noise to the final measurement

150 160 170 180 1900.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7RF Pulse Envelopes

pow

erkn

obtu

rns

0

10

9

8

7

6

5

4

3

2

1

Vol

tage

(arb

.uni

ts)

Time (ns)

Figure 3-7 RF pulses from the pulse picker driver. Optimum operation is achieved for a power knob setting of 6-7 turns.

96 3 Experiment

or equivalently an efficiency factor of CRpp /11−=η .

The contrast ratio of the pulse picker was quoted to be 200:1. In fact only much lower contrast ratios are accomplished with the delivered device. To characterize the behavior of the pulse picker contrast ratio measurements employing different types of fast photo diodes have been performed. The contrast ratio achieved strongly depends on correct driver adjustments. Preceding pulses are suppressed strongly, but the 3 trailing pulses significantly contribute to the outcoupled power. Single pulse suppression ratios of 22:1 up to 51:1 for the highest trailing pulse and total contrast ratios of up to 20:1 have been obtained for typical operation condi-tions.

Figure 3-8 Contrast ratio measurements with 3 different fast photo diodes. The center graph shows oscilloscope traces for the different diodes, the upper graph an 30:1 blow-up of these traces and the lower graph the pulses of the Ti:sapphire laser for comparison.


Both, contrast ratio and outcoupling efficiency depend on the RF power level of the driver. Optimum operation is achieved as a compromise of acceptable contrast ratios and higher outcoupling efficiency (Figure 3-9).

Synchronization of the RF driver to the mode-locked pulses is attained by connect-ing the synchronization output of the Tsunami to the RF driver synchronization input. The pulse synchronization output of the driver is used to gate the trigger signal of the signal detector with the pulse picked pulses in order to reduce the dark count rate.

To achieve reasonably fast switching times and efficient outcoupling the laser beam has to be focused tightly into the acousto-optic crystal. This results in a con-siderable k-vector spread of the laser beam traversing the crystal which in turn leads to a variation of the outcoupling efficiency across the laser beam – with an optimized outcoupling in the center of the beam. Laser power is therefore trans-ferred most efficiently in the beam center and drops towards the sides, resulting in an elliptical beam shape with a reduced horizontal beam waist.

Technical data for the pulse picker Type Neos Technologies N17389 pulse picker with

in/outcoupling optics N71389, sold in Germany by ELS Driver Neos N64389-SYN HF-Driver, delivers 10 W 10 ns

pulses of RF energy, input frequency: 41 MHz, output frequency 389.5 MHz (x 9.5), peak output power: 7 W, RF pulse width: 10 ns, cw output power: 0.6 W, power adjust: > 20 dB, phase adjust > 180°, output into 50 Ω

Pulse rate selectable 1:10 … 1:1000 Operational parameters power knob setting: 6…8 (7), 204 - 816 kHz rep. rate

(81.2:400-100), pulse picking efficiency: up to 65%, typically 55%, total contrast ratio: 23:1 (4.6 %)

5 6 7 8 9 10

10

20

30

40

50

contrast ratio (total)

contrast ratio (first afterpulse)

outcoupled power

pulse height

cont

rast

ratio

rf power knob(turns)

0

10

20

30

40

50

60

70

outc

oupl

ing

effic

ienc

y(%

)

Figure 3-9 Contrast ratio and outcoupling efficiency as a function of RF power of the driver pulses. Whereas improved contrast ratios may be obtained at reduced power levels this also leads to a reduced outcoupling efficiency. For the measure-ments the pulse picker was typically operated at power knob settings between 6 and 7.

98 3 Experiment

3.4 Nonlinear Optical Processes Nonlinear processes couple optical waves of different frequencies and provide the main physical interactions in our experiment. There are three nonlinear optical processes employed in the experiment:

§ Second harmonic generation to convert the 790 nm laser pulses to 395 nm pulses which are used as a pump wave for the down-conversion process.

§ The center piece of the experiment is a 2-photon down-converter which con-verts the incident 395 nm pump pulses to photon pairs at 790 nm in a sponta-neous scattering process, called parametric fluorescence. It is these strongly correlated photon pairs which allow the preparation of single -photon Fock states by conditional measurements.

§ If the 2-photon down-conversion is induced by seeding the parametric con-version with an additional beam – mode-matched to one of the sub-harmonic beams – parametric amplification occurs, a process that converts pump wave energy to signal and idler energy. In this way the power in the seed beam is slightly enhanced and a macroscopic amount of light is produced in the trigger beam.

3.4.1 Second Harmonic Generation Frequency doubling is performed in a single pass configuration either in a type I critical phase-matched BBO crystal or in an LBO crystal from CASIX. BBO was originally chosen for frequency doubling since it exhibits a high effective nonlinear coefficient♦ (1.36 pm/V) and a still acceptable group velocity mismatch (-1.986 ps/cm). The option to use a LiIO3-crystal , which possesses an even higher effective nonlinear coefficient for frequency doubling from 790 nm to 395 nm (4.287 pm/V), was discarded due to its higher group velocity mismatch (-5.718 ps/cm) and smaller acceptance angle (0.241 mrad/nm as compared to 0.347 mrad/nm in BBO). Since even a weak focusing into the doubling crystal with a 200 mm lens still leads to a considerable narrowing of the horizontal beam size due to the small acceptance angle, the BBO crystal was later replaced with an LBO crystal, which possesses a lower effective nonlinearity (0.851 pm/V), but has the advantage of offering a re-duced group velocity mismatch (-1.281 ps/cm) and a strongly enhanced acceptance angle (1.449 mrad/nm).

Efficient second harmonic generation (SHG) requires matching of the refractive indices for the ordinary sub-harmonic wave and the extraordinary harmonic wave to fulfill the momentum conservation condition. For type I (ooe) SHG the phase-matching condition acquires the form:

0)2,/()( =− λϕθλ eo nn ♣. (3.4.1)

♦ all values calculated with the program NL04win.exe provided by CASIX ♣ The refractive index of the BBO and LBO crystals may be calculated according to the Sellmeier equation:

2/

/2

/// )4(

)3(

)2()1()( λ

λλ eo

eo

eoeoeo a

a

aan −

−+= ,

222 )/sin()()/cos()(

)()(),/(

ϕθλϕθλ

λλλϕθ

oe

oee

nn

nnn

+=

where: ao(1) = 2.7539, ao(2) = 0.01878, ao(3) = 0.01822, ao(4) = 0.01354, ae(1) = 2.3753, ae(2) = 0.01224, ae(3) = 0.01667, ae(4) = 0.01516 for BBO and ao(1) = 3.4157, ao(2) = 0.04703, ao(3) = 0.03531, ao(4) = 0.0880, ae(1) = 2.9187, ae(2) = 0.03515, ae(3) = 0.02822, ae(4) = 0.00364 for LBO.

3.4 Nonlinear Optical Processes 99

This leads to phase-matching angles of θ = 29.55°, ϕ = 0° for BBO and θ = 90°, ϕ = 32.7° for LBO, where angle tuning in the θ-direction is required for BBO and in the ϕ-direction for LBO.

The power produced in the process of second harmonic generation is proportional to the square of the effective nonlinear coefficient deff according to (compare chapter 2.3.2)

22

2ln

1ωω

τπ

PT

P SHG ∆Γ= , where

2

22

20

3

2

)2/(

116

LwhLd

cn effSHG

ωω πλεπ=Γ ,

(3.4.2)

where L is the length of the crystal, n the refractive index and h denotes the Boyd-Kleinman-factor which accounts for the effect of focusing, λω the wavelength of the fundamental beam and wω(L/2) the Gaussian beam waist of the fundamental beam at the entrance or exit point of the crystal (compare equations (2.3.4) and (2.3.5)).

To include the effect that the pulse width ∆Τ also depends on the laser power we

use a quadratic expansion of ∆Τ in terms of ωP to rewrite equation (3.4.2) in the form

2

,

2

2 )(1 optPPB

PAP

ωω

ωω −+

= , (3.4.3)

Figure 3-10 Laser power and second harmonic power as a function of pump laser power for pulsed operation. The Ti:sapphire laser threshold is fitted to be at 2.96 W pump laser power. The power dependence of the second harmonic power devi-ates from the quadratic behavior expected for a cw-laser, but is fitted nicely, if the power dependence of the Ti:sapphire pulse length is taken into account.

SHG Power

100 3 Experiment

where optP ,ω denotes the average laser power for which the shortest pulse duration is achieved. Figure 3-10 shows a measurement of the power dependence of the average second harmonic power on the pump laser power or equivalently on the Ti:sapphire laser power. Using a functional dependence as in equation (3.4.3) an almost perfect fit of the experimental data is obtained (Figure 3-10).

A maximum second harmonic power of 43 mW is obtained for the BBO crystal at 1.52 W laser power and 15.3 mW for the LBO crystal at 1.49 W. The expected second harmonic power calculated according to equation (3.4.2) is 40 mW in the case of BBO and 17 mW for LBO.

As has already been discussed in chapter 2.3.2 a group velocity mismatch D be-tween the fundamental and the second-harmonic wave will lead to a pulse spread

lD ⋅=∆τ , 21

11

gg vvD −= , (3.4.4)

where vg1 denotes the group velocity of the harmonic and vg2 the group velocity of the sub-harmonic pulses.

If this pulse spread becomes comparable to the time duration of the laser pulses, the transform-limited character of the pulses will be lost and the pulses are – from a quantum mechanical point of view – converted into a statistical mixture of pure states not suited for the preparation of single -photon Fock states (see also chapter 2.3.2). From equation (3.4.4) we obtain a dispersive pulse delay of ∆τ = 0.6 ps for the 3 mm long BBO crystal with D = -1.986 ps/cm and of ∆τ = 0.38 ps for the 3 mm long LBO crystal with D = -1.281 ps/cm. Even though this effect leads to a noticeable midpoint displacement of ∆τ/2, its contribution to the pulse width is < 1% (see chapter 2.3.2) and may be neglected. However, group velocity mismatch would be a dominant effect, if femto-second laser pulses were used. This is the main reason why we have chosen to operate our experiment with ps-pulses.

The process of parametric amplification in the down-conversion crystal has been used to obtain an experimental confirmation that dispersive pulse spread does not exert a significant influence on the length of the doubled pulses. From a measure-ment of the pulse length of the trigger beam produced due to parametric amplifica-tion we may infer a pulse length of the doubled pulses of 1.23 ps which coincides

well with the theoretical expectation of 1.8 ps / 2 = 1.27 ps (see Figure 3-15 in chapter 3.4.3).

In critical, angular phase matching the angle of the crystal with respect to the (sub-harmonic) laser beam influences the refractive index. As a function of the angle the generated second harmonic power exhibits a sinc2-power dependence. Thus effi-cient doubling will only occur within a small angular range. The width of this an-gular range is called the acceptance angle. Since the focusing of the sub-harmonic beam into the crystal leads to a certain angular spread within the beam, the limited angular range for phase-matched second harmonic generation may influence the beam profile of the generated second harmonic beam, if the angular spread of the sub-harmonic beam becomes comparable to the acceptance angle. For BBO this effect leads to a substantial narrowing of the second harmonic beam even for weak focusing. This compromises efficient mode-matching in the subsequent down-conversion process. This detrimental effect is strongly reduced – by a factor of 4.2 – for the LBO crystal, leading to an improved mode quality of the generated har-monic beam.

Due to the elliptical nature of the laser beam emitted by the Ti:sapphire laser and the effect of angular narrowing due to the limited acceptance angle, the generated second harmonic beam also exhibits an ellipticity. In the case of the LBO crystal

Group velocity dispersion

Acceptance angle


the vertical waist of the second harmonic beam was measured to be d = 2w =1.1(2) mm (FW@1/e2(intensity)), the horizontal waist to be d = 1.4(4) mm (FW@1/e2).

Technical data for the doubler

BBO LBO

Type BBO crystal 2×2×3 mm, θ = 29.35° (spec. 29.5°, ideal: 29.55°), ϕ = 0°, AR coated @ 790 & 395 nm, for type I (ooe) frequency doubling 790 nm → 395 nm from CASIX.

LBO crystal, 2×2×3 mm, cut at θ = 90°, ϕ = 32.7°, AR coated @ 790 nm & 395 nm for type I (ooe) frequency doubling 790 nm → 395 nm from CASIX

Nonlinearity dooe = 1.36 pm/V, ΓSHG = 4.1 10-4 /kW (measured 6.4 10-4 /kW, increased due to rest pulsing)

dooe = 0.851 pm/V, ΓSHG = 2.57 10-4 /kW

Max. SHG (measured)

45.3 mW 15.3 mW

Refractive index no(790 nm) = ne(θ, 395 nm) = 1.661

no(790 nm) = ne(ϕ, 395 nm) = 1.611

Acceptance angle

0.347 mrad/nm (θ) 1.449 mrad/nm (ϕ)

Group velocity mismatch

-1.986 ps/cm → 0.60 ps -1.281 ps/cm → 0.38 ps

Losses < 0.5 % < 0.3 %

3.4.2 2-Photon Down-conversion The main physical interaction in the experiment is the process of 2-photon down-conversion which annihilates a pump wave photon at 395 nm and generates two strongly correlated photons at 790 nm. If this process occurs spontaneously, it is called parametric fluorescence, if it is induced by the insertion of an additional seed beam, parametric amplification occurs. The twin photons emitted in parametric fluorescence are entangled, which allows the preparation of a single -photon Fock state in one of the emission channels – denoted signal –, if a photon is detected in the other channel – denoted idler. A theoretical treatment of the process of para-metric fluorescence has been given in chapter 2.3.3.

We employ a 3 mm long BBO crystal from CASIX, cut at θ = 35.7°, ϕ = 0°, in a single-pass configuration to mediate the 2-photon down-conversion process in a type I (ooe) critically phase-matched configuration. The crystal produces frequency degenerate, but spatially non-degenerate photon pairs. The down-conversion crys-tal is cut in a way, that the direction of the walk-off of the pump beam (produced in the frequency doubler described in the last section) coincides with the direction of the signal beam in the crystal to minimize mode distortions in the generated signal

102 3 Experiment

beam. This direction is called a “hot spot” in parametric fluorescence184 and occurs at an angle of θ = 31.6° corresponding to an internal opening and walk-off angle of 4.062°. The crystal surfaces are oriented perpendicular to the generated signal beam to minimize losses in the signal beam path. The total measurement efficiency is only degraded by losses in the signal path. Losses in the pump and idler beam may be tolerated, as long as they do not significantly decrease the pair production.

These internal angles relate to an external angle between the incident pump beam and the normal to the crystal surface of 2.9°. This corresponds to external opening angles between the outgoing pump beam and the signal beam of 6.7° and between the pump and idler beam of 7.0° (Figure 3-12). Both angles do not coincide due to the tilt of the crystal with respect to the pump beam which leads to different dif-fraction angles. Correspondingly the angle between the signal and the idler beam is 13.7°. These angles were verified to coincide those obtained experimentally.

The losses of the down-conversion crystal were assessed to be < 1% for the pump wave at 395 nm and < 0.3% for the signal beam at 790 nm. The measured nonlin-earity of ΓSHG = 2.8 10-4 1/kW clearly exceeds the theoretical value of 1.0 10-4 1/kW due to residual pulsing as in the case of the frequency doubler.

The following two sections describe measurements performed with the down-conversion crystal.

Technical data for the down-converter

Type BBO crystal 8×8×3 mm, θ = 35.0° (spec. 35.5°, ideal: 31.6°), ϕ = 0°, AR coated @ 790 & 395 nm, for type I (eoo) down-conversion 790 nm → 395 nm from CASIX.

Nonlinearity dooe = 2.14 pm/V, ΓPA = 1.0 10-4 1/kW (measured 2.8 10-4 1/kW, increased due to residual pulsing)

Refractive index no(790 nm) = ne(θ, 395 nm) = 1.657

Opening angle 6.8° external opening angle (pump-signal or pump-idler), 4.1° internal

184 K. Koch, E. C. Cheung, G.T. Moore, S.H. Chakmakjian, J.M. Liu, “Hot Spots in Parametric Fluo-rescence with a Pump Beam of Finite Cross Section”, IEEE J. of Quant. El. 31, 769 (1995)

Figure 3-11 Opening angle and walk -off angle as a function of the angle θ be-tween the incident pump beam and the optical axis. For θ = 31.6° the two angles coincide. Please note that the angles in the inset are not drawn to scale.


internal

Group velocity mismatch

-1.986 ps/cm → 0.60 ps

Losses < 1.0 % for the pump wave and < 0.3% for the signal wave at optimized angles

3.4.3 Parametric Amplification Figure 3-13 illustrates the experimental setup for measurements of parametric am-plification. A small part of the laser power is split off before the frequency doubler and inserted into the signal mode of the down converter from the back. This beam is labeled seed beam. It allows to replace the spontaneous process of parametric fluorescence by an induced process.

In addition a kinematic mount may be inserted into the seed beam path to direct the seed beam into the idler channel instead of the signal channel.

The process of parametric amplification has been used in the experiment for differ-ent purposes:

§ Alignment: Aligning the down-conversion process is much more difficult with the sporadic emissions produced in the process of parametric fluores-cence than with the macroscopic amounts of light generated in the seeded process. Parametric amplification allows for an easy alignment of the sin-

Figure 3-12 The crystal is cut to eliminate walk -off beam distortion in the signal mode and to minimize the losses in the signal beam. To achieve this the opening angle of the down-converted signal beam and the walk -off angle coincide. The signal beam is oriented perpendicular to the crystal surface.

104 3 Experiment

gle-photon counter placed in the signal and idler beam path (for a more de-tailed description of the alignment procedure confer chapter 3.6).

§ Transverse mode matching between optical beams incident on a beam splitter is commonly achieved by monitoring the interference fringes be-hind the beam splitter. This method fails for single-photon states. To facili-tate a mode matching following the traditional method, the alignment of parametric amplification is optimized and mode matching is performed be-tween the parametrically amplified seed beam and the local oscillator beam.

§ Generation of one -photon added coherent states may be achieved by in-serting a strongly attenuated seed beam – reduced to an average photon number of only a few photons – into the signal mode of the down con-verter.

§ A determination of the pump pulse length produced in the frequency doubler is obtained by measuring the pulse duration of the idler pulses generated in the process of parametric amplification.

The power generated in the idler mode is proportional to the pump beam power PSHG and to the power in the seed beam Ps (chapter 2.3.3). This allows to generate macroscopic amounts of optical power in the idler beam, which may more easily be monitored than the weak emission produced in the process of parametric fluores-cence. With collimated pump and seed beam idler beam powers of up to Ps = 6.3 µW for pump beam powers of PSHG = 45 mW and seed beam powers of Ps = 170 mW have been generated. The expected idler power according to equation (2.3.48) is 5.2 µW. Higher idler powers may be achieved with focused pump and seed beams.

Since the pump beam power itself exhibits a quadratic dependence on the laser power, a cubic dependence of the optical power in the idler path is expected. This is confirmed by the experimental data presented in Figure 3-14.

Parametric amplification is only observed, if the optical path length of the seed beam is synchronized with that of the pump beam, so that both pulses arrive in the down converter at the same time. The idler power generated in the process of pa-rametric amplification is proportional to the temporal overlap of the pump and seed pulses.

spDC

psspDC

ssppDCi

PP

tttttPtPtd

ttPttPdttP

⊗Γ=

−=∆∆−′⋅′′Γ=

−⋅−Γ=∆

∫∫

00

00

where)()(

)()()(

(3.4.5)

Figure 3-13 Schematic experimental setup to measure parametric amplification. An additional seed beam is split off before the doubler and inserted into the signal mode of the down converter from the back.

Cubic power dependence


Thus, the pulse length of the idler pulses generated in parametric amplification is determined by a convolution of the pump with the seed pulse duration.

By changing the relative optical path length between the pump and seed beam paths we obtain a measurement of the second-harmonic pump pulse duration. As-suming Gaussian shapes of the pulses the temporal width of the pump pulses ∆Τp can be determined as a function of the width of the seed beam pulses ∆Τs = ∆Τl and the measured width of the idler pulses ∆Τi:

22sip TTT ∆−∆=∆ (3.4.6)

Figure 3-15 depicts the result of a measurement of the idler power as a function of the pulse delay between the pump and seed pulses. A measured idler pump width of 2.18 ps and a seed pump width of 1.8 ps yields an inferred pump pulse duration of 1.23 ps, in good agreement with the theoretical expectation.

Figure 3-14 Power dependence of the parametric amplification. The power in the trigger beam is proportional to the seed beam power and the second harmonic power. Since the second harmonic power is itself a quadratic function of the laser power, the trigger beam power exhibits a cubic power dependence.

106 3 Experiment

Figure 3-15 Pulse width of the doubled pulses measured using parametric amplifi-cation (see chapter 3.4.3.). The graph shows the trigger beam power as a function of the time delay between the seed and the doubled pulses. From the known width of the laser pulses the pulse width of the doubled pulses can be inferred.

3.5 Photon Counting 107

3.5 Photon Counting Single-photon counting is an indispensable prerequisite for our measurements since we need to be able to observe the extremely low light levels produced in parametric fluorescence and to prepare single -photon Fock states by detecting single -photons of the photon twins produced in the down-conversion process, in this way obtain-ing information on when a photon wave packet travels towards the homodyne de-tector.

We have employed two single-photon counters – one in the trigger channel and one in the signal channel of the down converter – to check the alignment of the para-metric fluorescence and to measure the state preparation fidelity due to coincidence measurements. These measurements are described in greater detail in section 3.5.3 later in this chapter. We have also made use of the single -photon counters to meas-ure photon number distributions (section 3.5.1) and time interval distributions (sec-tion 3.5.2).

To achieve sufficiently low dark count rates we are employing commercially avail-able EG&G single-photon counting modules, which exhibit dark count rates of only a few ten counts/s owing to a special selection process of the avalanche photo diodes used in these devices. We use two SPCM-AQ-131 single-photon counters equipped with Slik Si-avalanche photodiodes, actively quenched, which exhibit a single-photon detection efficiency of η = 0.62 at 790 nm and dark count rates of τD = 61 cts/s (trigger detector HH1) and τD = 176 cts/s (signal detector HH2) respectively. The dead time of the detectors was found to correspond exactly to the specified dead time of 31 ns, the after pulsing probability is 0.2% and the timing resolution 300 ps (Figure 3-17). The detectors are very sensitive to light when switched on and have to be turned on only in dim light conditions.

The number of photons impinging on the detector per second as a function of the registered photon count rate Rmeas may be calculated according to

η

τ DCRRphotons measDmeas −⋅−

=)1/(

# , (3.5.1)

Figure 3-16 Photo of the optical table in January 1999. The three experimental compartments for high, intermediate and low light levels are separated by black wooden shields.

108 3 Experiment

where τD is the detector dead time, DC is the dark count rate and η is the detector efficiency. For experimentally observed count rates typically not exceeding a few tenthousand counts per second the loss in count rate because of the dead time of the detector may safely be neglected.

Apart from dark counts due to spontaneous diode discharges, stray background light also contributes unwanted counts. These counts are minimized in our experi-ment by carefully controlling the ambient and stray light conditions employing filters, irises and a careful optical shielding of the crucial experimental parts from the bright parts of the table. In this way stray light and background counts could be reduced to about 30 cts/s making possible measurements even at very low count rates.

Technical data of the single -photon count modules Type EG&G single-photon counting modules SPCM-AQ-131, bought

from Laser Components, Trigger detector HH1: Ser. Nr.: 3741, Signal detector HH2: Ser. Nr.: 3553

Specifications Slik Si-Avalanche photodiode, actively quenched, supply voltage 5 V (DO NOT EXCEED 5.3 V, not even when turned on!), draws 1.5 A on switch on, 0.7…0.8 A at 25°C and <= 1 Mcts/s, dead time: 31 ns, after pulsing probability: 0.2%, detector efficiency η = 0.62 @ 790 nm, η/nd = 1.12 10-2 s, timing resolution: 300 ps ALWAYS POWER ON DETECTORS IN DIM LIGHT CONDITIONS (< 1Mcts/s)!

Module HH1 dark counts: specified: 67 cts/s, measured: 61 cts/s, max. count rate: 13.3 Mcts/s, dead time: specified and measured 31 ns, pulsewidth: 8.8 ns, active area diameter (η>50% @ 650 nm):

Figure 3-17 Timing resolution of an EG&G single-photon detector measured with a SRS620 frequency counter with a time resolution of 500 ps. The measured total timing resolution of 577 ps coincides well with the theoretical expectation of 583 ps expected for a 300 ps resolution of the single-photon detector.

Darkening


200 µm

Module HH2 dark counts: specified: 180 cts/s, measured: 176 cts/s, max. count rate: 18.8 Mcts/s, dead time: 29 ns, pulsewidth: 8.4 ns, active area diameter (η>50% @ 650 nm): 170 µm, this module showed bistable behavior and increased dark count rates and was sent in for repair twice

3.5.1 Photon Number Distributions We have employed single-photon counters to verify the photon number distribu-tions of the light state produced by the laser. This state is expected to be a coherent state with a Poissonian photon number distribution185

nn

en

nnnp −=

!),( (3.5.2)

where ),( nnp denotes the probability to measure n photons for a coherent state with an average number of photons of n . For a pulsed state the question arises, whether the photon number statistics of a pulse train will also be of Poissonian nature, if the single pulses exhibit a Poissonian photon number distribution. In the case of the photon count statistics for two Poissonian pulses with equal photon number this is the case since

∑

∑

=

−−−=

−⋅=

−⋅=

n

i

ninni

n

i

inen

ien

ninpnipnnp

0

02

)!(!

),(),(),(

185 L. Mandel and E. Wolf, "Optical Coherence and Quantum Optics" (Cambridge University Press 1995)

Figure 3-18 Photon count statistics for coherent states with average photon num-bers nav = 12.01 and 6.47. Poissonian and Gaussian distribution with the same average photon numbers are shown for comparison.

110 3 Experiment

!

)2(!)!(

!!

),(2

2

0

2

2 nen

iinn

nen

nnpnnn

i

nn

n

−

=

=

−=

⋅−= ∑

43421

(3.5.3)

In the general case of N pulses we make use of the polynomial theorem

n

niii N

Nii

n

N

=

∑=+++ K K

21 1

to obtain a Poissonian distribution again with an N

times higher average photon number

nNn

N en

nNnnp −=

!

)(),( (3.5.4)

Thus for a pulsed coherent beam it does not matter whether we are observing the photon number distribution of single pulses or of a number of pulses within a given time interval. This result does not hold in the case of an average photon number n which changes in time – as in the case of a significant contribution by technical laser noise.

Figure 3-18 shows the result of a photon number distribution measurement for the strongly attenuated laser pulses with a repetition rate of 81.2 MHz measured with an 0.01 s time window and average photon numbers of 12.01 and 6.47 pho-tons/0.01 s. Poissonian distributions (bars) and Gaussian distributions (line) with corresponding average values are shown for comparison. For low average photon numbers, the data is fitted more closely with a Poissonian distribution than with a Gaussian distribution, but – even for the case n = 6.47 – the Gaussian distribution still provides a good approximation:

n

nn

en

nnp 2

)( 2

2

1),(

−−

≈π

(3.5.5)

For an 0.01 s time window we expect a contribution of 1.76 cts/s for the signal detector HH2. If we assume independent spontaneous diode discharges with an equal probability to occur within equal time intervals, we expect the dark count

Figure 3-19 The single-photon detector dark count statistics also clearly follows a Poissonian distribution.

Coherent states


distribution to also follow a Poissonian statistic 186 – an assumption which is well justified for our detectors as illustrated in Figure 3-19, where a clear deviation from a Gaussian behavior is obvious.

A characteristic property of a Poissonian distribution is that the variance of the distribution σ2 equals the mean value n . We have investigated the dependence of the variance on the mean photon number for the pulses emitted by the Ti:sapphire laser and have verified a Poissonian behavior up to an average photon number around 1000 photons (Figure 3-20). For higher values of n we observe a variance that is bigger than the mean, which may be attributed to technical laser intensity noise. Assuming independent noise contributions to the variance from the detector dark counts (DC), the Poissonian shot noise (SN) and the technical laser noise (LN), the total variance will be of the form

2222222 nnDCLNSNDCtot γσσσσσ ++=++= (3.5.6)

Thus for high average photon numbers the dominant noise contribution will arise from technical laser noise. From the data presented in Figure 3-20 we infer an rela-tive RMS intensity noise γ = 6.3 10-3 = 0.63%, which is in agreement with the in-tensity noise figures stated in chapter 3.3.2.

3.5.2 Time Interval Distributions Another quantity of interest for a quantum state that may be measured with single -photon counters is the time interval distribution P(τ). P(τ) denotes the probability distribution to measure a time interval τ between successive count events.

A formal derivation of the time interval distribution results in187

186 D.F. Walls and G.J. Milburn, "Quantum Optics" (Springer-Verlag Berlin Heidelberg 1994) 187 L. Mandel, E. Wolf, “Optical coherence and quantum optics“, p. 720 et seq. (Cambridge Univer-sity Press, 1995, New York)

Figure 3-20 Photon count statistics for coherent states with different average pho-ton numbers. The variance should equal the average photon number nav . At higher power levels technical laser noise hides this property.

112 3 Experiment

:)(ˆ)(ˆ:)(ˆ

)()(ˆ

τηττ

η+= ∫

+′′−

tIetITtI

SP

t

ttdtIS

, (3.5.7)

where η designates the quantum efficiency of the single -photon counter, S the de-tector surface, T is the time ordering operator, : : assures normal ordering and the expectation value K also includes a time averaging.

For a coherent state the time interval distribution has a decay time which is equal to the inverse of the single-photon count rate R and equation (3.5.7) strongly simpli-fies to

( ) ττ ReRP −= (3.5.8)

This expected behavior is well fulfilled for the experimental data shown in Figure 3-21, where the single-photon count rate of 131600 cts/s coincides with the decay rate of 129000 1/s up to an error of 2%.

The inset in Figure 3-21 depicts the time interval distribution for time scales down to 10-8 m where the pulsed nature of the laser light is clearly revealed in the comb like structure of the distribution. The detector dead time of 31 ns also shows up clearly.

Measuring the coincidence count time interval distribution of counts in the signal channel after a photon has been registered in the idler channel reveals a very differ-ent behavior which strikingly demonstrates the correlated nature of the photon pair

Coherent light

Figure 3-21 Time interval distribution of a pulse train of coherent laser pulses from the Ti:sapphire laser. As expected for a coherent beam the single-photon count rate coincides with the decay rate up to an error of 2%. The inset illustrates the pulsed nature of the beam and the dead time of the single-photon counters.

Photon pairs


emissions in parametric fluorescence: A strong correlation for coincidence meas-urements corresponding to two photons being registered in the two detectors within a single pulse peaks up from the shallow background of random coincidences (Figure 3-22).

The total number of pair coincidences Cexp is approximately given by

tRC lisPC ∆⋅⋅Γ= ηηη 2exp , where Rl = 81.2 MHz denotes the repetition rate of the laser pulses and Γ the chance per laser pulse to generate a photon pair, ηPC = 0.62 the quantum efficiency of the single-photon counters, ηs = 0.847· 0.65· 0.97 = 0.534 and ηs = 0.829· 0.65· 0.97 = 0.523 the total optical efficiencies in the signal and idler beam paths and ∆t = 10 s the measurement time. Random coincidences are ex-

pected to occur at a rate tRC lisPCrandom ∆⋅⋅Γ= 22 ηηη . In the data depicted in

Figure 3-22 a pair count rate of C = 92683 cts/s has been observed corresponding to Γ = 1/940. A random coincidence rate of Crandom = 96 cts/s leads to a pair genera-tion probability of Γ = Crandom/C = 1/965. The two measurements of Γ coincide up

Figure 3-22 Time interval distribution for photon pairs generated in parametric fluorescence. In contrast to the coherent state measurements two single-photon counting detectors have been employed to obtain this measurement. Count rates are given as a function of relative arrival times at both detectors. This illustrates the extremely high degree of correlation between the photons in the trigger and idler channel.

114 3 Experiment

to an error of 2.7%. The pair production rate in this measurement is much higher than in the actual measurements since only wide band pass filters have been used to filter the fluorescence photons in the signal and idler beam paths.

3.5.3 State Preparation The single-photon Fock state preparation fidelity has been tested employing a sec-ond single-photon counter in the signal channel instead of the homodyne system and performing coincidence count measurements. Using the theoretical results obtained in chapter 2.3.4 the quantum state preparation fidelity may be calculated according to

⊥

⊥

⊥

⋅⋅⋅−

−=

⋅⋅⋅−

−=

⋅⋅⋅−−

−=

,,

,,

111)(

111)(

111

)()(

sssPDii

random

i

iiiPDss

random

s

isPDiiss

random

MMDCNCC

F

MMDCNCC

F

MMDCNDCN

CCF

ω

ω

ω

ηη

ηη

ηηη

(3.5.9)

We observe signal and trigger count rates on the scale of 20 000 cts/s at the full repetition rate of 81.2· 106 transmitted through a 2 nm wide spectral filter. This corresponds to a pair creation probability per pulse of ca. 2.5 10-4. In first order perturbation theory a down-converted photon pair does not increase the pair crea-tion probability. Thus the expected random coincidences are lower than the pair coincidences by the same factor. This expectation has been confirmed experimen-tally by the data presented in the last section. We will therefore neglect the random coincidences in the further calculations (experimentally we observe no more than 0 - 1 random coincidences/s).

Optical filters are employed to narrow down the broadband emission from the down-converter and to achieve a better spectral mode overlap. Figure 3-23 depicts measured transmission curves for the main filters employed. The relevant charac-teristics of all the filters employed are summarized in the table at the end of this chapter. According to equation (2.3.35) the filters also influence the detected spa-tial mode since the direction of phase-matched emission depends on the wave-length according to

ss

psspis

NN

λλ

θ

θθθ

∆⋅

−=∆=∆

0

0

/sin

cos

21

(3.5.10)

where the experimental parameters are Ns = 1.685, Np = 1.740, 0sθ = 4.1°,

λs = 790 nm. ∆θs/i denotes the angular width of the detected emission cone of the down-converter. This angular width corresponds to a spatial spread of ∆x = ∆θs/i· z at a distance z from the down-conversion crystal. For FWHM-width of 0.3, 2 and 10 nm we obtain ∆x = 0.16, 1.1 and 5.3 mm respectively, comparable to the trans-verse width of the pump beam and possible irises in the beam paths. The observed spatial width is further smoothed out owing to the imperfect transverse correlations which arise from the finite width of the pump beam, as can be seen from equation (2.3.69) or (2.3.23). The pump beam width corresponds to a spatial spread of ∆xp = 2.2 mm (FWHM).

We now proceed to discuss the preparation fidelity for realistic experimental condi-tions.

Filters


We employ a combination of a 2 nm bandwidth filter (F2) and a 10 nm blocking filter in the signal and idler beam path. Assuming the transmission curves of the filters and the spectral pump mode to be of Gaussian shape the spectral mode over-lap may be approximated for ∆ωs/i0 = 0 as

222,/isp

isisMM

∆+∆+∆

∆∆== ωω (3.5.11)

where ∆p/s/i denotes the effective spectral width of the respective mode.

We note that the spectral mode overlap exhibits an upper bound of 1/ 2 which is achieved for ∆p = 0 and ∆s = ∆i . This upper bound arises due to the fact that a pho-ton registered away from the center wavelength in one of the beam paths will be detected with a decreased probability in comparison to a center detection, because of the lower off-center filter transmission. This upper bound only exists for coinci-dence detection with absorptive filters and is not present in the homodyne detection where an optimized mode overlap of 1 may be achieved.

Using ∆p = 1.1 nm and ∆s/i = 2 nm we obtain a spectral mode overlap of Mω = 0.66.

Irises are employed in the signal and idler paths defining the transverse mode func-tions. The irises are placed at a distance of d = 50 cm from the down-conversion crystal. The aperture of these irises is set to the minimum diameter R = 1.6 ± 0.2 mm for which the measured coincidence rate is not noticeably diminished. For equal iris diameters the geometric overlap between the two irises (equation (2.3.72)) may be approximated as

−∆

−⋅=∆Θ⊗ΘK

k

KkAA isis )1(

11

)( 020 ππ

(3.5.12)

where K = k s· R/d = 1.3· 104 1/m. Assuming the transverse pump mode to be a sym-metric Gaussian with a standard deviation of σp = 1/wp = 0.88· 104 1/m we can ap-proximate the transverse mode overlap as

−−

−== ⊥⊥p

pis

Kerf

KMM

σπ

π

σ

π)1(

)1(2

1,/ = 0.60 (3.5.13)

Figure 3-23 Transmission curves for the narrow band dielectric filters with width 2 nm and 0.3 nm. The center frequencies do not coincide.

Spectral mode overlap

Transverse mode overlap

116 3 Experiment

This estimation of the transverse mode overlap will also be used to estimate the total measurement efficiency in chapter 3.9.

For the set of filters employed and assuming optical losses of 1% we obtain optical propagation efficiencies of ηs = 0.66 and ηi = 0.68. Using ηHD = 0.63 and a meas-ured coincidence rate of C = 3190 cts/s, signal and idler count rates of Ns = 18700 and Ni = 19600 as well as dark count rates DCs = 802 cts/s and DCi = 90 cts/s yields preparation fidelities of

F = 1.03 ± 0.10, Fs = 1.05 ± 0.10, Fi = 1.00 ± 0.10

Filter Characteristics F0.3 (2x)

∆λ = 0.3 nm, λmax = 790.3 nm (specified 790.04 nm), Tmax = 50% (specified 36%), blocking range: 600 – 940 nm (Omega Optics)

F2 (2x)

Spectral width (FWHM): ∆λ = 2 nm, center wavelength: λmax = 790.6 nm (specified 790.2 nm) maximum transmission (intensity): Tmax = 84.7%, 82.9% blocking range: 600 – 940 nm manufactured by Omega Optics

F10 (2x)

∆λ = 10 nm, λmax = 790 nm, Tmax = 80%, blocking range UV – FIR (OD 6) (Laser Components)

F20 (2x)

∆λ = 20 nm, λmax = 790.3 nm, Tmax = 65%, blocking range UV – 1200 nm (OD 5) (Omega Optics)

3.6 Longitudinal and Transverse Mode Matching 117

3.6 Longitudinal and Transverse Mode Matching

To be able to measure the field of a single -photon wave packet it is not enough to direct the beam onto a detector. Instead it is necessary to match the wave function of the photon to that of the local oscillator beam. Furthermore, for photon wave packets we do not only have to achieve an optimum transverse mode matching of the local oscillator and signal beam, but also an optimum temporal overlap of the pulses, a longitudinal mode matching. Matters are further complicated due to the fact that the signal mode does not contain macroscopic amounts of light but only single photons. Yet another factor which complicates a good mode matching of the single photons to the local oscillator beam is that the process of parametric fluores-cence produces a complicated multi-mode object, which has to be projected onto a single Gaussian mode (chapter 2.3.4) to obtain a good transverse mode matching.

Longitudinal and transversal mode-matching play a crucial role for the efficiency of the single-photon Fock state reconstruction.

Pulse Shaping The primary light source of the experiment, the mode-locked Ti:sapphire laser, provides laser pulses with a spectral width of ∆λL = 0.78 nm and a spatial extent of 0.54 mm. The transform-limited behavior of these pulses has been confirmed by comparing the spatial extent of the pulses with their spectral width as measured with an auto-correlator. Ps-pulses are chosen to assure that group velocity mis-match in the process of second harmonic generation does not compromise the transform-limited character of the pulses. Group velocity dispersion in other optical elements e.g. lenses may be neglected for ps-pulses and will only become signifi-cant, if more than 1 m of material is traversed.

The spectral shape of the doubled pulses )(~ ωpumpg is given by a convolution of

the spectral shape of two laser pulses ))(()(~ tgFg laserlaser =ω :

Figure 3-24 Evolution of the spectral width of the light pulses: The spectral width of the laser pulses is increased in the process of second harmonic generation. This spectral pulse spreading may not be reversed in the process of 2-photon down-conversion. Instead, if a spectral filter of delta-width is employed in the idler detec-tion the full spectral width of the pump pulses is transferred to the signal pulses resulting in a spectral mode mismatch between the local oscillator pulses and the signal mode.

118 3 Experiment

laserpump

laserlaserpump ggg

ωω ∆≈∆⇒

⊗=

2

~~~ (3.6.1)

As has been shown in chapter 2.3.2 and confirmed by the pulse length measure-ments described in chapter 3.4.3 the spectral width of the pulses is increased by a

factor of 2 in the process of second harmonic generation, whereas their spatial extent is foreshortened by the same factor resulting in a pulse length of 0.38 mm.

It is not possible to return to the prior width in the process of 2-photon down-conversion. Instead, the spectral shape of the photons in the signal channel is de-termined as a convolution of the spectral shape of the filters employed in the idler detection with the shape of the pump pulses multiplied with the phase matching bandwidth

phasematchpumpfiltersignal gggg ~~~~ ⋅⊗= (3.6.2)

In our case the phase matching bandwidth is much bigger than the pump band-width. Therefore the spectral width of the down converted signal photon will mainly be determined by a convolution of the filter bandwidth with the pump bandwidth. The optimum case is achieved for a filter with a delta width where the pump bandwidth will just be transferred to the signal:

laserpumpsignal

filterpumpphasematch

ωωω

ωωω

∆≈∆≈∆⇒

∆>>∆>>∆

2 (3.6.3)

For an optimum state preparation it would be advantageous to use spectral filtering of the pump beam to obtain a signal bandwidth corresponding to the width of the local oscillator pulses. At the pump wavelength of 395 nm the narrowest commer-cially available dielectric filters exhibited a width of 2 nm (FWHM). Filtering the pump beam with this filter would reduce the pump bandwidth by 13% to 0.98 nm in this way increasing the measurement efficiency by 3.5%, but at the same time reducing the available pump power to Tfilter = 0.655· 0.98/1.13 = 56.8%. Due to the

Figure 3-25 Mode overlap of two Gaussians 221 |),(),(| ∫ ωσωσω dGG as a func-

tion of their relative width


low count rate this reduced pump power would decrease the efficiency by at least 8%♣.

Two types of dielectric filters with spectral width of 0.3 nm and 2 nm (FWHM) have been used to filter the parametrically produced photons in the trigger channel (the transmission functions of the filters are depicted in Figure 3-23). The 0.3 nm filter is considerably narrower than the spectral width of the pump wave and pre-pares the photons in the signal path to have a bandwidth of ∆λs=1.48 nm. This leads to a longitudinal mode matching with the local oscillator pulses

22

|| )(~)(~)()( ∫∫ ⋅=⋅= ωωω LOtriggerLOtrigger ggdtgtgdtMM (3.6.4)

of MM|| = 93.1%. Using the 2 nm filters in the trigger channel instead leads to a bandwidth of ∆λs=2.47 nm and a reduced longitudinal mode matching of MM|| = 61.6%. Owing to the higher count rate resulting in a smaller contribution by background counts higher measurement efficiencies were obtained with the 2 nm filter.

Transverse Mode Matching Transverse mode matching has to be ensured at three stages of the experiment:

1. The beam passing through the pulse picker has to be mode matched with the full power laser beam to enable switching between full power and pulse picked operation without realignment.

2. The pump beam has to be mode matched with the seed beam to achieve a good conversion efficiency in the down-conversion process and to ensure reliable state preparation.

3. The transverse mode of the signal beam in the Fock state reconstruction measurements is determined by the state preparation in the trigger channel. Thus, the local oscillator has to be mode matched to the mode of the sig-nal photon wave packets prepared by the trigger measurement. This mode matching directly affects the total measurement efficiency. Because of the fact that mode matching to the sporadic single-photon events is very difficult, an indirect procedure has been employed in the experiment: the local oscillator beam is mode matched to the seed beam and the trigger de-tection is designed so that the corresponding signal mode coincides with the seed beam mode.

1. A good mode overlap between the full power laser beam and the beam passing through the pulse picker is achieved by roughly fixing the beam position of the laser beam with two irises and threading the pulse picked beam through these irises. Fine tuning of the pulse picked beams direction is achieved by optimizing the second harmonic power. This results in a quite precise alignment of the pulse picked beam, since the generated second harmonic power critically depends on a precise angle adjustment and the distance of the second harmonic beam monitor detector from the pulse picker is larger than 2 m. Controlling whether a good mode matching is preserved between the seed and local oscillator beam when switching from the full laser power to the pulse picked operation offers another way to check the alignment of the pulse picked beam to the laser beam at full power. Perfect overlap between the pulse picked and full power beam may not be achieved due to the influence of the acousto-optic pulse selector on the beam shape (chapter 0).

♣ Filtering with a Fabry-Perot-cavity is not an option either since the cavity length would need to be much smaller than half the spatial extent of the pump photons of 190 µm and resonant filter cavities of this size are not easy to realize.

Mode matching pulse picker – laser beam

120 3 Experiment

2. To align the spontaneous process of parametric fluorescence we use the induced process of parametric amplification. In this way macroscopic amounts of light can be used for the adjustments and a Gaussian mode can be selected from the multi-mode emission produced in the parametric fluorescence process. The seed beam is fixed and the pump beam and down-conversion crystal are adjusted to obtain an optimum parametric amplification. Irises and a spatial filter are used in addition to the spectral filters in the idler channel to detect only those photons in the idler beam which correspond to signal photons emitted into the seed beam mode. We have verified that the alignment obtained in this way also results in an optimized coincidence count rate, if the signal beam is redirected to a second single -photon counter by the cinematic mirror MS7.

No absolute value can be stated for the mode matching between seed and pump beam. Mode matching between the two beams is maximized by optimizing the parametrically generated power in the idler beam. An absolute reading would re-quire sufficiently accurate knowledge of the effective nonlinearity of the down-conversion crystal, the Boyd-Kleinman-factor as well as of a number of other ex-perimental parameters. Since measured nonlinearities usually deviate from the theoretically expected values by up to a factor of 2, theoretical predictions may not be exploited to obtain absolute values of the mode matching. Interference tech-niques may not be used since the two beams have different frequencies and are only coupled due to the nonlinear interaction in the crystal.

3. Relying on the alignment procedure described above, we perform an indirect mode matching of the single-photon states prepared in the signal channel and the local oscillator beam by mode matching the local oscillator beam to the seed beam. Typical values of 96% with maximum values of 98% have been achieved for the mode matching between the local oscillator and the seed beam.

Projecting the idler radiation onto a single spatial mode is achieved by passing the idler emissions through irises and a spatial filter. Since the spatial filter consists of

- 6 - 4 - 2 0 2 4 6r

0

0.05

0.1

0.15

0.2

0.25

0.3

I(r)

GaussAiry

0 0.5 1 1.5 2 2.5 3ρ0 0/w

0

0.2

0.4

0.6

0.8

Airy intensitypattern

Figure 3-26 Mode overlap of an Airy function with a Gaussian mode. Optimum overlap of 81.5 % is achieved for ρ0 = 0.793 w0.

Mode matching seed beam – pump beam


a focusing lens, a pinhole placed in the focal plane of the focusing and collimating lens the spatial mode selected by such a filter does not have a Gaussian shape, but is given as the Fourier-transform of the transmission function of the pinhole, which in the case of a circular aperture is an Airy-function. Due to the fact that the mode overlap between an Airy function and a Gaussian is limited to 81.5%, mode match-ing to the mode selected by spatial filtering will be limited to the same value, re-sulting in according loss in total measurement efficiency. This situation may be improved by using irises in the idler and local oscillator path to select only the center part of the beam profile and cut off the outer rings of the Airy function. In this way the mode matching may approach 100%.

A problem arises due to the ellipticity and possible astigmatism of the laser beam which leads to an ellipticity and astigmatism in the seed, local oscillator and pump beam. Since we use a spatial filter with a round pinhole this ellipticity is not pre-sent in the signal mode prepared by the measurement in the trigger detector leading to an imperfect mode match.

Realignments of the optical system are necessary on an almost daily basis to pre-serve the mode matchings due to the insufficient stability of the primary laser sys-tem.

122 3 Experiment

3.7 Fast Pulsed Homodyne Detection Fast pulsed homodyne detection is employed to measure the quadrature distributions of the field states under investigation. These distributions are then used for the reconstruction of the optical quantum states. A theoretical treatment of pulsed homodyne detection including experimental inefficiencies has been given in chapter 2.3.5.

Implementing a reliable pulsed homodyne detection with good signal to noise ratio has been one of the main experimental challenges in pursuing the aim of single-photon Fock state reconstruction. Requirements to be met include:

§ High bandwidth: Detection with a sufficiently high count rate requires a high repetition rate. This in turn calls for a high bandwidth of the homodyne detection system.

§ Efficient subtraction: Even for local oscillator powers as low as N = 1 mio. photons per pulse, corresponding to a shot noise level of ∆N = 1000 photons per pulse and an average optical power of 204 nW at a repetition rate of 812 kHz, subtraction with a signal-to-noise-ratio (power) of (∆N/N)2 > 60 dB has to be achieved. The subtraction efficiency sets an upper boundary for the local oscillator power.

§ Ultra-low noise: Limited subtraction efficiency forces us to work with low power levels in the local oscillator beams. Therefore we need a homodyne system with ultra-low electronic noise. If e.g. we again assume the local oscillator pulses to contain 1 mio. photons per pulse, then the electronic noise has to be lower than 1000 electrons per pulse to be able to see the shot noise with a signal to noise ratio above 1. The electronic noise sets a lower boundary for the local oscillator power.

§ Gating: The low photon generation probability per pulse on the scale of

-homodyne detector

BSlocal oscillator

?∆n n= ≈ 1000

n 1000 000, ,

12.3 ns 20,40,...×

signalp

p

( )

( )

1 10

0 1

4≈

≈

−

-0,002 0,000 0,0020

50

100

150

200

250

P(I)

I

!

!

!

!

fast detection → →

(∆ ≈few photons amplification

n/n) 60 dB2 → →low photon generation rate

high bandwidthultra low noise

efficient subtractiongating

Figure 3-27 Requirements for a fast pulsed homodyne detection system. The num-bers given correspond to typical experimental values.

3.7 Fast Pulsed Homodyne Detection 123

10-4 – 10-5 makes a gating procedure necessary to reduce the effective dark count rate of the trigger detector.

In this chapter the homodyne detector electronics are described first, then the results obtained with the homodyne detector are discussed and finally the gating procedure is explained.

Homodyne Detector Electronics To realize a fast pulsed homodyne system we have tried to use our experience with low noise photo detectors, employed for homodyne detection of cw-light fields, to develop a suitable homodyne detector on the basis of transimpedance amplifica-tion. With an ultra-low noise preamplifier Amptek A250 we achieved a (power) signal-to-noise-ratio (SNR) of 62 dB between the amplified laser pulses and the electronic noise for local oscillator pulses around 1 mio. photons per pulse at a pulse repetition rate of 204 kHz. Since the shot noise level for pulses containing 1

Mio. photons is NNNIISNL /1)/(ˆ/ 222 ==∆= = -60 dB below the signal level we were limited to an unsatisfactory signal-to-noise-ratio of 2 dB (1.6) for shot noise detection at local oscillator powers not exceeding 51 nW.

We therefore decided to abandon the approach based on transimpedance amplifica-tion and to build on the experiences in the group of Prof. Raymer. Prof. Raymer and co-workers have used subtraction on a stripline188 – a current line impedance matched to the photodiodes employed – in connection with pulse shaping amplifi-cation to implement a pulsed homodyne detection189. In this scheme subtraction takes place before the amplification.

We have modified this scheme by connecting the two S3883 Si-PIN photodiodes (Hamamatsu) n-to-p and placing them at a distance (4 mm) much smaller than that corresponding to their bandwidth of 300 MHz (1 m). The photodiodes exhibit a quantum efficiency of 94% (specified: 91%) at the measurement wavelength of 790 nm and are chosen for their low noise-equivalent-power (NEP) of 6.7 10-15 W/Hz1/2. They are reverse biased to achieve a fast time response.

The positive and negative charges produced by the optical pulses are collected and physically subtracted at a capacitor with a capacitance of 470 pF much larger than the capacitances of the photo diodes (6 pF). In this way very high subtraction effi- 188 P. Horowitz, W. Hill, The art of electronics, P. 879, 902 (Cambridge University Press, 1989, New York) 189 D. Smithey, “Complete experimental state determination of classical and nonclassical light“, Ph.D. thesis, Univesity of Oregon (1993)

Figure 3-28 Homodyne detector electronics. Charges produced on the diodes are subtracted directly on a capacitor, preamplified and submitted to pulse shaping amplification.

Direct charge subtraction

124 3 Experiment

ciencies exceeding 83 dB are achieved making possible local oscillator powers in excess of 200 mio. photons per pulse. To avoid charge build up at the incoupling capacitor in the case of imperfect charge balancing between the two diodes the incoupling capacitor is grounded through a high-Ohm resistance.

The difference charge is than pre-amplified using a 2SK152 FET in connection with the low-noise preamplifier A250 (Amptek) and further amplified using a 5-pole pulse shaping amplifier based on two low-noise A275 amplifiers (Amptek). A schematic of the homodyne detector electronics is shown in Figure 3-28.

Detector gain and linearity have been verified by connecting a 1 pF capacitor to the incoupling capacitor of the detector electronics and inserting small controlled amounts of charges by applying voltage steps of a known size to the 1 pF capacitor. In this way the amplification of the detector electronics was determined to be 3.01 µV/electron. The electronic noise of the detector electronics with disconnected diodes has been measured to be 1.1 mV corresponding to 367 electrons. When the photodiodes are connected an electronic noise of 1.7 mV corresponding to 565 electrons is obtained. Figure 3-29 demonstrates the excellent linearity of the detec-tor electronics.

Figure 3-30 shows time traces of the pulse signal propagating through the detector electronics (using the charge inserter and square pulses of well defined heights). After preamplification a voltage step proportional to the inserted charge is ob-served which relaxes with a long decay time of Tf = 192 µs (position 1 in Figure 3-28, first graph in Figure 3-30). This decay time is replaced with a much shorter decay time of 300 ns due to pole zero differentiation190 (position 2, second graph). This signal is amplified – without any measurable decrease in signal-to-noise ratio (position 3, 3rd graph) – and low pass filtered to obtain a more symmetric signal

190 P. Horowitz, W. Hill, The art of electronics, P. 247 (Cambridge University Press, 1989, New York)

Figure 3-29 Amplifier gain and linearity of the homodyne detector measured by inserting small, controlled amounts of charges.

Detector gain and linearity


shape thus improving the effective signal-to-noise-ratio (position 4, 4th graph). The signal is amplified again (position 5, 5th graph) and transformed from a unipolar to a bipolar signal by AC-coupling it to a 50 Ω output impedance.

Technical data of the homodyne detector

Photo diodes

Type: Hamamatsu S3883 Si-PIN diodes

Specifications : qu.eff: 91 % @ 790 nm, measured: 94 %, sensitivity > 0.58 A/W(× 1.57)), band width: 300 MHz, noise

-15 1/2

Figure 3-30 Signal propagation through the homodyne detector amplifier electron-ics. Extremely low noise performance is achieved combining charge sensitive pre-amplification followed by 5-point pulse shaping amplification. Pulses peak up at positive voltage steps across the charge inserter and down at negative.

126 3 Experiment

equivalent power: NEP: 6.7 10-15 W/Hz1/2, active diameter: 1.5 mm, spectral response: 320 - 1000 nm, peak sensitivity wavelength 840 nm, dark current: 0.05 nA, power dissipation max 50 mW, reverse voltage max. 30 V, terminal capacitance: 6 pF

preamplifier Type: A250 ultra-low-noise hybrid charge sensitive preamplifier from Amptek

FET: 2SK152, Ids=3 mA

Specifications: rise time 8.7 ns at 6 pF(2.5 ns at 0 pF), output impedance: 100 Ω, decay time Tf = Rf Cf = 300 MΩ 1 pF = 300 µs (internal feedback components), voltage output: -4.6 V … 2.8 V, gain bandwidth product: 1.5 GHz, gain at 204 kHz: 74 dB

Pulse shaping am-plifiers

Type: A275 hybrid differential pulse shaping amplifier from Amptek

Specifications: power: 15 mW, slew rate 100 V/µs, input noise 4 nV/sqrt(Hz), 200 MHz gain bandwidth product, sup-ply voltage ± 8 V, input offset voltage: 2 mV, input capaci-tance: 4 pF, differential input resistance: 44 kΩ, common mode rejection : 95 dB, power supply rejection ration 60 dB, pulse risetime 9 ns, open loop output resistance: 750 Ω

Detector gain 3.01 µV/electron (corresponds to 2.83 µV/photon)

Detector noise 565 electrons (367 electrons if photo diodes are discon-nected)

Homodyne System Performance Figure 3-31 shows typical time traces of the homodyne detector difference signal for an average optical power of 8.1 µW corresponding to 79.4 mio. photons per

Figure 3-31 Difference signal of the homodyne system for a local oscillator power of 8.1 µW (79.4 mio. photons per pulse) at a repetition rate of 204 kHz. Pulsed shot noise clearly peaks up from the electronic noise background every 4.9 µs. Many single time traces are combined resulting in the depicted noise distribution.


pulse. The noise of the optical pulses peaks up from the background of electronic noise at the repetition rate of the laser pulses with a high visibility. For a coherent local oscillator and vacuum in the signal input port of the homodyne detector the pulsed noise exhibits a Gaussian statistics (chapter 3.10.1).

To prove that the pulsed noise shown in Figure 3-32 is indeed shot noise we have performed a number of tests:

§ Is the standard deviation of the pulsed noise of the right magnitude?

§ Does the standard deviation scale with the square root of the local oscilla-tor power?

§ Is the spectrum of the observed pulse noise white?

Figure 3-32 shows the standard deviation of the pulsed noise as a function of the local oscillator power. Subtracting the noise background corresponding to 730 electrons/pulse – which is higher than stated in the last section due to background lighting and imperfect current supply noise suppression (see below) – the standard deviation of the noise scales very nicely with the square root of the local oscillator

power as predicted for shot noise ( PNNI ∝=∆∝∆ 2 ) for local oscillator powers up to 230 mio. photons per pulse. This corresponds to a maximum subtrac-tion of 83.6 dB. For local oscillator powers exceeding 230 mio photons per pulse complete cancellation of the pulses from the two photo diodes fails due to slight differences in the pulse shape of the positive and negative charge pulses, even though the square root power dependence is preserved to even higher powers, if only small sections of the pulses are monitored.

The absolute value of the observed standard deviation corresponds to the expected shot noise magnitude to within the precision of the charge insertion capacitor used to determine the detector gain.

Figure 3-32 Shot noise level as a function of local oscillator power showing the expected square root power dependence up to local oscillator powers of 230 mio. photons per pulse. The open squares show the measured noise variances, the blue squares have been obtained from these values by subtraction of the noise back-ground.

Square root power scaling

128 3 Experiment

Figure 3-33 provides another confirmation that the measured noise indeed is the shot noise of the optical beam: The noise registered at a local oscillator power of 4.6 µW and a repetition rate of 812 kHz – corresponding to 22.5 mio photons per pulse – is to a very good approximation frequency independent (white) noise up to a frequency of 100 kHz. The signal-to-noise-ratio between the shot noise signal and the electronic noise of 9.0 dB corresponds well to the value of 9.2 dB expected

from the time domain measurements ( 6105.22 ⋅ photons per pulse/ 565 electrons per pulse).

The inset in Figure 3-33 illustrates the influence of background light sources: the contributions of the monitor flicker and the 100 Hz flicker of the light bulbs to-gether with their harmonics can clearly be distinguished in the cyan colored trace. 50 Hz supply noise and the laser light fluctuations also show up in the traces.

Figure 3-33 Noise spectrum of the pulse noise signal from the homodyne detector compared to the electronic noise floor. The shot noise shows the expected fre-quency independent behavior up to a frequency of 100 kHz. The inset illustrates the effect of different background light sources like a computer monitor and a light bulb.

White noise spectrum

High sensitivity


Figure 3-34 demonstrates the sensitivity of the homodyne system in the time do-main. A single light bulb (left graph) and imperfectly filtered supply net noise (right graph) result in significant contributions to the noise voltage emitted by the homodyne detector. This high sensitivity to background light sources is the price that has to be paid for the excellent noise and amplifier performance of the pulsed homodyne system. To eliminate these noise sources the homodyne system has been operated with a car battery as the power supply and in dim light conditions.

Homodyne Detector Efficiency The electric charges produced by optical pulses have to be balanced to better than 1:1000th between the two arms of the homodyne detector for shot noise detection. Since diode efficiencies will vary across the active area of both photo diodes the mechanical setup has to provide a good mechanical stability to decrease the ampli-tudes of mechanical oscillations leading to beam displacements on the photo diodes and a sensitive adjustment of the optical powers has to be possible to achieve good balancing. Polarizing beam splitters are used in combination with λ/2-plates to allow optical power adjustments in both arms of the homodyne detector.

Losses at all optical elements in the homodyne setup as well as a deviation of the dielectric beam splitter from the ideal 50/50 case decrease the homodyne detector efficiency and have to be kept as low as possible for optimum operation.

The dielectric beam splitter is tilted to obtain an optimized reflectivity of R=50.05%, corresponding to an imbalance ∆R = 0.1%, at an angle of 47°. The optical properties of the elements in the homodyne setup are stated in the table below. A total optical efficiency of ηHD,opt = 96.3% is measured, which agrees well with the sum of all loss terms (3.8%).

The total efficiency of the homodyne detector is be estimated as

904.0)1( , =⋅⋅∆−= PDoptHDHD R ηηη (3.7.1)

Figure 3-34 Sensitivity of HD system: the left graph shows the influence of a single light bulb in the room. The 100 Hz modulation of the supply net clearly shows up on the difference signal of the HD. The right graph shows electronic noise due to an imperfectly filtered net supply voltage. For the tomographic measurements all light sources are turned off in the laboratory and a car battery is used as the volt-age source for the homodyne detector.

Optical losses

130 3 Experiment

Technical data for the homodyne system (continued)

Optical properties Losses: lenses: 0.5/0.3%, λ/2: 0.5/0.2%, BS: R=50.05%, loss ~ 2%, PBS HHIIa: Tmax=99.7%, Tmin=0.82%, Rmax=99.8%, Rmin=1.32% PBS HHIIb: Tmax=99.9%, Tmin=0.11%, Rmax=99.9%, Rmin=0.92% Total transmission: T1=48.2%, T2=48.5% Total optical efficiency: η=96.3%

Photodiodes Type: Hamamatsu S3883 Si-PIN photodiodes quantum efficiency: measured 94%, specified 91%.

Homodyne detector efficiency

90.4%

Gated Homodyne Measurements State preparation of the single -photon Fock state is achieved by conditioning the read-out of the homodyne signal on a coincidence between a count event in the trigger detector and the pulse synchronization signal from the pulse picker. In this way only those pulses are selected for an actual measurement where a photon pair has been produced and the photon in the idler channel is detected by the trigger detector.

Figure 3-35 provides a schematic of the pulse electronics to achieve the conditional state measurement: The output signal of the trigger detector is connected to a first input (In1), amplified in a low noise amplifier SLC560 and discriminated with a latched ultrafast comparator AD96685 (DC1). The discriminated trigger signal serves to enable a monoflop MF1 (Monostable Multivibrator MC10198) that pro-duces a pulse of variable duration 0.5 … 10 µs whenever a transition from the logi-cal state 0 to 1 occurs at its other input during the time the trigger signal exceeds the discrimination level. This logical transition occurs, if the pulse synchronization

Figure 3-35 Schematics of the gating electronics: The homodyne detection is con-ditioned on the coincidence of a trigger event with the pulse synchronization signal from the pulse picker. Monoflops introduce variable time delays to fine tune the coincidence window and the sampling instant.

Pulse electronics


signal, connected to a second input (In2) – time delayed at an identical monoflop (MF2) and discriminated – coincides with the trigger count. The variable time de-lay of the pulse synchronization signal because of the monoflop MF2 allows to compensate for the different times the synchronization and the trigger signal re-quire to propagate to the pulse electronics.

If a coincidence event occurs, then the gating monoflop MF1 emits a pulse to the control input of a monolithic track & hold circuit AD9100. The signal input of this track & hold circuit is directly connected to the homodyne detector output (In3). For the duration of the gating monoflop pulse the track & hold circuit tracks the voltage produced by the homodyne detector. The duration of the gating monoflop pulse is chosen to track the homodyne signal up to the instance in time when the pulsed shot noise signal reaches its peak value. Then it holds the acquired voltage long enough for the fast data acquisition card to read out the value from output Out3. The additional outputs Out1 and Out4 are used to monitor the performance of the discriminated trigger and pulse synchronization signals. The output Out2 is used to control the coincidence signal of the gating monoflop. The pulse electronics has been implemented using ECL logical components to provide switching and delay times of a few nanoseconds. Outputs 1, 2 and 4 are converted to TTL signals to be able to drive the 50 Ω impedance of BNC cables.

Figure 3-36 exemplifies the timing behavior of the gating electronics for a typical count event. Figure 3-37 demonstrates the operation of the track and hold circuit.

Due to the gating with a 9-20 ns coincidence window we are able to strongly re-duce the effective dark count rate in the experiment. Since the ECL-logic requires about 4-8 ns to notice a changed logical state we are able to achieve an effective gating time window of about 5 ns at MF1. This results in a dark count reduction by a factor of

2451

ns 5kHz 816 =⋅=∆

∆=

gate

pulses

t

tα (3.7.2)

The electronic dark count rate of the EG&G single -photon counter in the trigger channel is 56 cts/s. An additional 30 cts/s result from to residual stray light in the lab resulting in a total dark count rate of 86 cts/s. Gating enables us to reduce the effective dark count rate to 0.35 cts/s (measured: 0.3 cts/s).

Figure 3-36 Gating electronics: Time trace of a gated count event. The gating time window is determined by the width of the trigger counter signal and can be ad-justed down to less than 20 ns.

Dark count reduction

132 3 Experiment

Technical data of the pulse electronics

DC1/2

(latched ultrafast comparators AD96685)

delay ∆t = 2.5 ± 0.05 ns, switching time < 1 ns, Vin = -2.5 … 5 V (never exceed supply voltage 5 V), output current Iout = 30 mA can drive 50 Ω to –2 V, Cin= 2 pF, Rin= 0.2 MΩ Logic 0: < -1.5 V, Logic 1: > -1.1 V

MF 1/2

(Monostable Multivibrator MC10198)

input: logic 1: > -0.98 V, Logic 0: < -1.63 V, output: logic 0: -1.85/-1.65 V, logic 1: -0.81/-0.96 V, minimum trigger pulse length: 2 ns, switching time: trigger input: 4 ns; fast trigger input: 2 ns, rise time: 20% à 80% typ. 2.25 ns, enable setup time: 1.0 ns, fast trigger can not be used since that mode does not allow to trigger the positive edge

TH

(monolithic track & hold AD9100)

acquisition time: 16 ns to 0.01% (10-4), jitter < 1 ps, 250 MHz tracking bandwidth, noise figure: 3.3 nV/Sqrt(Hz), integrated output noise: 45 µV, analog input resistance: 800 kHz, capacitance: 1.2 pF, logic input: 0: -1.8…-1.5 V, 1: -1.0…0.8 V, output voltage: -2…2 V, hold noise: 300 th V/s rms, sloop rate: 1 mV/µs, settling time to 1 mV: 7 ns

ECL à TTL

(MC10H125)

output: logic 0: < 0.5 V, logic 1: > 2.5 V, input: logic 0: -0.81 … -1.13 V, logic 1: -1.48…-1.95 V, prop. Delay: 0.85…3.35 ns, rise/fall time: 0.3…1.2 ns

Figure 3-37 Track & Hold: The track & hold circuit follows the homodyne signal up to the point where the pulsed shot noise reaches its maximum voltage and then holds this value for the computer data acquisition.

3.8 Data Acquisition and Monitoring 133

3.8 Data Acquisition and Monitoring Tomographic inversion methods usually involve amounts of data and data process-ing that require the use of complex numerical methods and high computational powers to generate the reconstructed pictures. Even though the amount of data is strongly reduced in the case of rotationally symmetric quantum states such as a single-photon Fock state, we have found it advantageous to rely on computer based task automatization to

§ monitor experimental parameters online,

§ acquire measurement data and control the acquisition parameters,

§ perform online data analysis and quantum state reconstruction,

§ invoke auxiliary programs for simulation, testing, data display and data manipulation.

Program user interfaces have been realized with Labview®, since this graphical programming language allows an easy front panel design and provides high level routines for data display, analysis and storage. Time critical computational routines e.g. the inverse Radon transformation have been implemented in C++ and inte-grated into the Labview® programs as external code.

Monitoring of Experimental Parameters An enhanced multifunction I/O-card PCI-MIO-16XE-50 from National Instru-ments with 16 A/D input channels and 2 D/A output channels with a 16 bit resolu-tion and a maximum sampling rate of 20 kS/s is used to monitor experimental pa-rameters. The monitoring program provides a fast and comprehensive control of the main experimental parameters:

Figure 3-38 Front panel of the monitoring program displaying the main experi-mental parameters online. Mainly dark colors have been chosen for the front pan-els to reduce background light emitted by the computer monitor. The lower left display shows a red background color as a warning signal, that the SHG power has left the allowed parameter range.

134 3 Experiment

§ the Ti:sapphire laser power

§ the second harmonic power generated – either at the full power level or in the pulse picked mode

§ the pulse picked power

§ the pump laser power

§ the local oscillator power

These powers are detected with a total of 7 monitor detectors (compare Figure 3-3) and updated at a data acquisition rate that can be set to range from 2 ms to arbitrary slow. An additional fast photo detector is employed to measure the contrast ratio, but its output is not read in to the computer.

The laser power, second harmonic power and pulse picked power are also pre-sented graphically on the front panel of the monitoring program to be able to judge the stability in time of these parameters. Acoustic and optical warnings are issued, if one of the displayed experimental parameters leaves a predefined parameter range. All acquired data may be saved to a data file. Data is written in ASCII-format to allow easy editing and importing to data analysis programs.

Figure 3-38 shows a screen shot of the front panel of the monitoring program.

Data Acquisition and Analysis Homodyne data are read out with a fast I/O board T3012 (Imtec)191. This I/O board has 2 analog input channels with a bit depth of 12 bits (16 bits in combined opera-

191 compare also G. Breitenbach, Quantum state reconstruction of classical and nonclassical light and a cryogenic opto-mechanical sensor for high-precision interferometry, dissertation (UFO Disserta-tion, Bd. 349, 1998, Konstanz)

Figure 3-39 Front panel of the data acquisition program. The upper left graph displays data acquired in a single measurement run, the lower graph displays marginal distributions generated from 40 measurement runs as a color-coded in-tensity plot. Acquisition parameters and controls are displayed on the left side of the screen.

3.8 Data Acquisition and Monitoring 135

tion) and allows sampling rates up to 30 MHz for a single channel and 60 MHz in combined operation. In addition to earlier measurements a memory segmentation software has been added to be able to acquire many measurements without a time consuming reset of the I/O-card. A minimum segmentation size of 128 samples in conjunction with an on-board memory of 512 kS allows to read in a maximum of 4096 measurements before a board reset has to be issued.

Figure 3-39 shows the main panel of the data acquisition program. One of the board’s input channels is connected to the track & hold output of the gating elec-tronics and used to acquire data within a sampling time of typically 33 ns. A suit-able selection of typically 30 – 40 values is averaged to give a single measurement result. The measured data points may be subjected to high pass filtering by sub-tracting a gliding average to remove low frequency fluctuations and a voltage off-set. For each measurement run a marginal distribution is generated. All marginal distributions are collected and can be displayed individually, together or summed up. The displayed distribution may be fit with a nonlinear least squares fit to obtain an estimation of the measurement efficiency. An online quantum state reconstruc-tion is performed on demand. Figure 3-40 depicts the front panel that pops up, if a quantum state reconstruction is performed, displaying the marginal data before and after filtering and symmetrization as well as a cut through the reconstructed Wigner function. For a discussion of the reconstruction algorithms confer chapter 2.2. To optimize the reconstruction procedure e.g. by changing the filter type or width ad-ditional windows may be opened displaying the marginal data and the filter func-tion in Fourier space, the effect of symmetrization and data or Wigner function normalization.

For non-rotationally symmetric states a different version of the measurement pro-gram is used that sorts the experimentally acquired data according to the phase angle corresponding to the measurement and invokes the inverse Radon transfor-mation (see also chapter 3.10.2).

Figure 3-40 Front panel displaying the marginal distribution (left graph) before and after filtering and symmetrization, and a cut through the reconstructed Wigner function (right graph) together with a plot of the theoretical Wigner function for the specified efficiency.

136 3 Experiment

Technical data of the data acquisition boards

Multiple I/O board

PCI-MIO-16XE-50 (National Instru-ments)

16 single-ended or 8 differential input channels with a 16 bit resolution and a maximum sampling rate of 20 kS/s, max voltage: ± 11 V, voltage error: ± 76 µV max, slew rate: 2 V/µs, noise: 40 µV rms (DC to 1 MHz), crosstalk: –85 dB max (0 … 20 kHz),

2 D/A output channels with a 12 bit resolution, providing ± 10 V output with a sampling rate of 20 kS/s, current drive ± 5 mV, 0.5 mV offset error,

8 digital TTL/CMOS input/output channels (not used)

Fast I/O board

T3012 (Imtec)

2 analog input channels, 12 … 16 bit sampling depth, data acquisition rate 30 MHz max for a single channel, 60 MHz max for 2 channels connected, max. bandwidth: 15 MHz, memory capacity: 512 kS per channel, memory segmentation with minimum memory segment size of 128 samples, input impedance: 1 MΩ / 18 pF, max voltage: 50 V, measurement range: 6 ranges available, ± 100 mV … ± 5 V, SNR 63 dB at 20 MHz data acquisition rate, sensitivity: 0.0488 mV/digit (± 100 mV) … 2.44 mV/digit (± 5 V)

3.9 Measurement Efficiency 137

3.9 Measurement Efficiency We are now in a position to discuss the figure of merit for the single-photon Fock-state reconstruction: the total measurement efficiency. The effect of this efficiency on the expected marginal distribution and on the reconstructed Wigner function has been discussed in chapter 0: A dip in the Wigner function can be observed for effi-ciencies higher than 25%, to see a dip in the marginal distribution a total measure-ment efficiency of at least 33% is required, and a negativity in the reconstructed Wigner function will only be seen, if the efficiency exceeds 50%.

A number of factors contribute to the total measurement efficiency obtained in the experiment (compare chapter 0):

elpulsepickHDprepHDsigopttot cpMMMM ηηηη ⋅⋅⋅⋅⋅⋅= ⊥ ,,|| (3.9.1)

The transverse mode -matching ⊥MM between the signal and the local oscillator modes cannot be measured directly, since it is not possible to obtain a macroscopic interference signal between the single -photons emitted into the signal mode and the local oscillator mode. The signal mode in the measurements of the single-photon Fock state is prepared by projecting the highly complex multimode object, which is produced in the process of parametric fluorescence, onto the trigger channel mode. The transverse trigger mode is defined by a spatial filter and irises in the idler beam path. Using only a spatial filter is not sufficient, since the mode selection will not exclude beams with a wider diameter but the same wave front curvature. Also the selected mode will not have a Gaussian shape, but will be an Airy-function, which limits the optimum achievable overlap to 81.5%.

To handle the problem of signal – local oscillator mode matching we have made use of the additional seed beam, which can be inserted into the signal mode from the back. The crystal tilt and pump beam is adjusted to obtain maximum parametric amplification in the idler beam as described in chapter 3.4.3. The mode selection in the idler channel is adjusted to obtain optimum transmission of the parametrically generated mode – in this way achieving an optimum overlap with the detected trig-ger mode.

The local oscillator mode is mode-matched to the seed beam mode in the standard way. A mode-matching efficiency – calculated from the visibility of the interfer-ence fringes between the local oscillator and the seed beam – of typically MMLO,s = 96% with best values around 98% have been achieved in the experiments.

An additional efficiency factor arises from the imperfectness of the transverse cor-relations of the photon twins. This factor has been estimated in chapter 3.5.3 to M⊥ = 60.0%.

Combining these contributions we obtain

⊥⊥ ⋅= MMMMM sLO, = 0.58 ± 0.05 (3.9.2)

Good longitudinal mode-matching ||MM requires careful spectral shaping of the photon wave packets produced in the down-conversion process to match its spec-tral shape to that of the local oscillator pulses. As explained in chapter 0 a perfect pulse shaping is not possible in our experimental setup, since the pulse broadening in the doubling process cannot be reversed in the process of parametric down-conversion. As described in chapter 3.6, the measured bandwidth of ∆λp=1.45 nm (FWHM) coincides well with the theoretical expectation and cannot easily be fil-tered to obtain a better longitudinal mode-matching.

138 3 Experiment

Using the results from chapter 3.6 we obtain ||MM = 0.93 ± 0.03, filtering with the

0.3 nm filter in the trigger beam path, and ||MM = 0.62 ± 0.05 for the 2 nm filter. Unfortunately the increase in efficiency due to narrower filtering is more than compensated by a higher contribution of dark count events to the measured distri-bution.

The optical efficiency in the signal path is limited by the losses in the AR-coating of the signal photon leaving the down-conversion crystal (0.5%), the reflection losses of the mirror in the signal path (0.4%), and transmission losses in passing through the irises (which will be neglected, since the irises are set to have an aper-ture wider than the beam diameter for the actual measurements). Therefore we have ηopt,sig = 0.99 ± 0.005.

The homodyne detector efficiency is mainly limited by the optical losses in the homodyne beam paths and the efficiency of the photodiodes and has been deter-mined to be ηHD = 0.904 ± 0.01 (chapter 3.7).

The state preparation efficiency is influenced by spectral filtering, the spatial overlap of the prepared signal beam, and the ratio of dark counts to twin photon count events. We have covered the first two effects already, so that a contribution of

NNp DCDCprep −=1, (3.9.3)

remains. In the case where the narrow band filter with a width of 0.3 nm has been used, a trigger count rate around 1 ct/s was observed, with the 2 nm filter we ob-tained a data acquisition rate of about 13 cts/s. Owing to the gated detection we were able to reduce the dark count rate of the trigger detector from around 86 cts/s to 0.3 cts/s (chapter 3.7). This results in a preparation efficiency of only pprep,DC = 0.7 ± 0.1 for the 0.3 nm filter and a preparation efficiency of pprep,DC = 0.98 ± 0.07 for the 2 nm filter. Since this effect more than compensates for the decreased mode overlap, the best measurements were obtained with the 2 nm filter.

The dissatisfactory total contrast ratio of the pulse picker of CR = 25:1 (chapter 0) leads to another loss in efficiency. This is due to the fact that the bandwidth of the homodyne detector electronics is limited to 1 MHz. The homodyne detector thus performs an effective time averaging with a decay time on the scale of 1 µs. Be-cause of this effect the trailing pulses following each selected pulse contribute to the signal produced by the homodyne detector electronics. Since the chance that another photon pair is emitted into the signal mode in one of the following pulses is negligibly small, this effect leads to an efficiency factor of

96.01

1 =−=CR

C pulsepick ± 0.03 (3.9.4)

The last factor that contributes to the total measurement efficiency is the electronic noise. It contains the effect of the electronic noise of the homodyne detector’s pulse shaping electronics as well as the tracking error of the track & hold-circuit (σT&H = 90 electrons/pulse (0.27 mV)), and the digitalization error of the fast IO-card (σΙ Ο = 202 electrons/pulse (0.61 mV)). For a local oscillator power of ap-proximately PLO = 8· 107 photons per pulse (ULO = 27 mV as used in the measure-ments) and an electronic noise of EN = 513 electrons/pulse we obtain an electronic efficiency of

222

& ENP

P

IOHTLOHD

LOHDel

+++⋅

⋅=

σση

ηη = 0.98 ± 0.02 (3.9.5)

3.10 Quantum State Reconstruction 139

The contribution of the electronic noise is therefore rather small – mainly due to the good noise performance of the homodyne detector electronics.

Collecting all the terms we obtain:

0.3 nm filter 2.0 nm filter

⊥MM 0.58 ± 0.05

||MM 0.93 ± 0.03 0.62 ± 0.05

ηopt,signal 0.99 ± 0.005

ηHD 0.904 ± 0.01

pprep,HD 0.7 ± 0.1 0.98 ± 0.07

Cpulsepick 0.96 ± 0.03

ηel 0.98 ± 0.02

ηtot 0.32 ± 0.12 0.30 ± 0.11

3.10 Quantum State Reconstruction

3.10.1 Vacuum State As a first test of the pulsed homodyne system we have reconstructed a pulsed vac-uum state by blocking the signal beam. Figure 3-41 shows a typical readout of the homodyne system difference signal as a color coded density plot of many single oscilloscope sweeps. The picture was taken at a pulse repetition rate of 812 kHz and a local oscillator power of 99· 106 photons per pulse. The enhanced noise due to the quantum fluctuations plotted for four consecutive pulses can clearly be seen.

From Figure 3-41 it is obvious that 812 kHz is the highest possible pulse repetition rate to operate at. For the next higher possible pulse repetition rate of 2 MHz the homodyne signals of neighboring pulses will overlap so that the homodyne system readout will be influenced by the quantum fluctuations of the neighboring pulses.

To obtain the marginal distribution of the pulsed vacuum state we have registered

Figure 3-41 Color graded oscilloscope display of the HD signal.

140 3 Experiment

983040 single quadrature measurements in 240 separate measurement runs span-ning a total of 5 s time, each single data point corresponding to a field strength measurement of the vacuum state as seen by the homodyne system. The local oscil-lator power was set to 89· 106 photons per pulse, where a good signal-to-noise-ratio of 21 dB assures a faithful measurement of the quantum fluctuations.

The inset in Figure 3-42 depicts homodyne data acquired in a single measurement run, exhibiting a Gaussian noise statistics as expected for the vacuum. The result-ing marginal distribution, obtained by binning up all data points in 128 amplitude bins is shown on the side panels together with a Gaussian fit. Because of the high number of data points the expected relative statistical fluctuations of the bin values are 1.1% assuming an even distribution of the data point across the bins. The mar-ginal data coincides with an ideal Gaussian distribution up to a χ(2)-value of 7.1· 10-

8.

The reconstruction of a single marginal distribution is sufficient in the case of a vacuum measurement, since the vacuum state is phase independent, its Wigner function therefore rotationally symmetric. If the measured state would exhibit a phase dependence this would clearly show up in the registered data due to the phase drift across the optical table (compare next section). For the measured vac-uum state no such phase dependence was observed.

In the case of a phase independent state as the vacuum we may employ the Abel transformation (compare chapter 2.2) to reconstruct the Wigner function from the marginal distribution. The Wigner function of the pulsed vacuum state, recon-structed from the measured data is shown in the center of Figure 3-42. The agree-ment of the reconstructed Wigner function with the theoretical expectation is rather good.

We may also use the acquired data to sample the density matrix in the Fock basis employing the method of quantum state sampling as discussed in chapter 2.2.. Figure 3-43 depicts the absolute values of the resulting density matrix ρ(m,n) to-gether with the photon number distribution p(n) = ρ(n,n) up to m = n = 6. The den-sity matrix element of the vacuum contribution is evaluate to be ρ(0,0) = 1.0025 very close to one. All other absolute values of the density matrix elements are 0.01

Figure 3-42 Reconstruction of a vacuum state: the inset shows a part of the mar-ginal data used to obtain the marginal distribution. The marginal distribution ex-hibits only minor deviations from a Gaussian. Employing the Abel transformation the Wigner function is reconstructed from the experimental data and shown in the center of the plot.


or smaller. The fidelity of the quantum state reconstruction is )0,0(0ˆ0 ρρ ==F = 1.0025. This is another confirmation that the measured data

describes a vacuum state to a high precision.

3.10.2 Coherent State To be able to reconstruct phase-dependent quantum states, we need to be able to perform phase sensitive measurements, where the measured quadrature phase is determined by the relative angle between the local oscillator and the signal beam. Examples of phase-dependent quantum states of the light field that we intend to measure are not limited to coherent states but also include photon-added coherent states as well as displaced and squeezed Fock states (compare chapter 5.2.1.1). Even for measurements of states with rotationally symmetric Wigner functions we need to test for possible deviations from the phase-independent behavior to make sure that the measured state does not exhibit phase-dependent quantum fluctua-tions.

In a first step to characterize the phase behavior of our optical setup we have moni-tored the passive phase drift between the signal/seed beam and the local oscillator beam in our optical setup by inserting a seed beam into the signal beam and moni-toring the interference fringes between the two beams. Figure 3-44 shows a plot of the root-Allan-variance

2

11)1(2

1)( ∑

=

∆∆+

∆−∆

−=∆

N

iiiRA N

Vττ

ϕϕτ (3.10.1)

as a function of the drift time together with a time trace of the relative phase. In

equation (3.10.1) τ

ϕ∆

∆ i denotes the average of the relative phase between the local

oscillator and the signal beam iϕ∆ averaged over a time interval ∆τ, and N repre-sents the number of time intervals in the registered time trace.

As expected the root-Allan-variance increases with the drift time for intermediate values of ∆τ. For large times the root-Allen-variance rolls off reflecting the fact that an average relative phase exists and the actual relative phase fluctuates around this value not deviating from it by more than 70°. For small values of the drift time

Figure 3-43 Absolute value of the density matrix elements up to photon numbers of n = m = 6 and photon number distribution of a vacuum state. The same data as in Figure 3-42 has been used to generate these plots.

142 3 Experiment

the root-Allan-variance increases due to flicker noise. For our optical setup we obtain a phase stability of 1° in 52 ms and 10° in 2.2 s. Hence to measure a quadra-ture distribution with a phase accuracy of 1° we cannot exceed a measurement duration of 52 ms.

To be able to change the relative phase in a controlled fashion we employ a mirror mounted on a ring-piezoelectric transducer stack in the seed beam path. This trans-ducer allows us to shift the relative phase by 4/3 π by applying an external voltage to the transducer.

The limited amplitude of this phase modulation, the sine-modulation applied to it to avoid sudden changes of the piezo stack velocity together with the phase drift renders a direct mapping of the obtained homodyne data to the corresponding phase angle according to the externally applied voltage difficult. Therefore we do not use the external transducer-voltage to obtain the phase of the data but use a somewhat different scheme for our phase dependent measurements.

Assume g(X,P) to be any – not necessarily non-negative – distribution function of the Cartesian coordinates X and P. Then the center points of the X- and P-marginal distributions are given by

∫∫∫∫

=

=

),(

),(

PXgPdPdXP

PXgXdPdXX (3.10.2)

How does the measured center point of the marginal distribution behave, if the phase changes?

Figure 3-44 Phase drift between the local oscillator and the seed beam on the op-tical table. To keep the phase drift below 1° the measurement time needs to be less than 33 ms. The inset shows an actual time trace of the phase over more than 200 sec drift time.


The rotated center point is calculated according to

θθ

θθ

θθ

θθθθθθθθθ

sincos

sin)),((cos)),((

),()sincos(

sincos where),(

PX

PXgPdPdXPXgXdPdX

PXgPXdPdX

PXXPXgXdPdXX

+=

+=

+=

+==

∫∫∫∫∫∫∫∫

(3.10.3)

Thus if we investigate any non-rotationally symmetric Wigner function, we can easily infer the phase angle θ from the center point of the marginal distribution (up to multiples of π) – without using any prior knowledge about the state we are in-vestigating.

This notion allows us to get an approximate value of the marginal center point by taking the average over the marginal data in a time interval where the angle does

not change considerably. If we denote this average value by θX~

and the maximum

average value by )~

max(ˆθθ XX = , then the angle for any given – short – time in-

terval may be inferred as

θθθ XX ˆ/~

arccosinferred = (3.10.4)

This provides us with a mapping of the experimental data to the desired phase an-gles. Unfortunately this mapping is not linear, leading to an unequal mapping den-sity to the corresponding phase angles (i.e. the θX

~-interval connected to angles

close to 0 or π will be much smaller than around π/2)

As a first example of a phase dependent quantum state we have measured a pulsed coherent state, produced by strongly attenuating the seed beam inserted into the signal mode to a level, where every seed beam pulse only contains a few photons. To do this we had to decrease the seed beam power level by 17 orders of magni-tude (from typical power level of 100 mW down to 10-18 mW) without loosing the mode matching between the signal/seed beam and the local oscillator beam. We have achieved this by inserting 5 neutral density filters in the seed beam path and compensating for the additional optical pathlength with the optical trombone. Care-ful adjustment of the neutral density filters was required to avoid a transverse beam displacement of the seed beam.

Figure 3-45 Wigner function of a coherent state with excitation α = 2.27 recon-structed from experimental data.

144 3 Experiment

Figure 3-45 depicts the result of a measurement of a coherent state with an excita-tion of α = 2.27 (inferred from the registered data). In contrast to the measurements of G. Breitenbach et al. this excitation does not reflect the average photon number within a frequency side band of a bright coherent beam192 but directly corresponds to an average photon number of =n 5.14 photons per pulse in the signal beam. 262144 data points have been registered yielding the marginal data displayed in the inset of Figure 3-45 (64 quadrature distribution with 128 bins each).

We have applied the inverse Radon transformation (compare chapter 2.2) choosing a cutoff frequency of 7.25 to this marginal data to obtain the Wigner function de-picted in the center of the plot. The reconstructed Wigner function reflects the ex-pected 2-dimensional Gaussian phase space distribution with a width equal to that of the vacuum Wigner function. The ripples at the side of the main peak are nu-merical artifacts and arise due to measurement imperfections.

Quantum state sampling again yields the density matrix elements of this state as well as its photon number distribution (Figure 3-46). A slightly lower average pho-

ton number of ∑ == )max(1 )(m

m npn = 5.01 photons per pulse corresponding to a co-

herent excitation of α = 2.24 is inferred from the measured photon number distri-bution. Even though the reconstructed Wigner function closely resembles the ex-pected theoretical Wigner function of a coherent beam with the same excitation, some differences between the experimentally determined photon number distribu-tion and the theoretical photon number distribution of a coherent state are visible in Figure 3-46. Nevertheless the comparison of the reconstructed density matrix with that of an ideal coherent state with an amplitude α = 2.24 yields a state preparation fidelity of αρα measˆ=F = 99.5%.

3.10.3 Single-photon Fock State Results of single-photon Fock state quantum reconstructions obtained using the method of pulsed optical homodyne tomography are shown in Figure 3-47 and Figure 3-48. The figures include the quadrature measurements, the resulting mar-ginal distribution and the corresponding Wigner functions. Since the Fock states are expected to be phase independent and no phase dependence due to phase drifts

192 G. Breitenbach, S. Schiller, J. Mlynek, “Measurement of the quantum states of squeezed light“ , Nature 387, 471 (1997)

Figure 3-46 Absolute value of the density matrix elements in the Fock basis ob-tained by quantum state sampling and photon number distribution corresponding to the marginal data shown in Figure 3-45. A Poissonian distribution with the same average n is shown for comparison.


was observed in the experiment, the Abel transform has been used to reconstruct the Wigner functions from the marginal data.

A 2 nm wide spectral filter and irises have been used in the trigger beam path de-termining the spectral and transverse trigger detection mode. The laser center fre-quency was adjusted to coincide with that of the spectral filter at 790.6 nm. The laser was operated at an output power of 1.58 W yielding 45 mW of 395.3 nm pump light. A pulse picking rate of 812 kHz was chosen, reducing the pump power to 330 µW and yielding a data acquisition rate of 13 cts/s. The local oscillator power was set to 8.9· 106 photons per pulse. The mode matching between the local oscillator and the seed beam was measured to be 96% for the data in Figure 3-47 and 87% for the data in Figure 3-48. All other parameter values coincided with the ones specified in chapter 3.9.

Figure 3-47 depicts the result of a measurement run where 700 single homodyne measurements were taken yielding the marginal distribution depicted at the side panels. Also shown on the side panels is a fit of the experimental data with a theo-retically generated marginal distribution. The best fit was obtained for a detection efficiency of 33%. The measurement run was interrupted after 700 measurements due to laser instabilities. Deviations from the theoretical distribution are within the statistical errors. The obtained marginal distribution clearly deviates from the vac-uum marginal – also shown in Figure 3-47 for comparison – and exhibits a fla t-tened non-Gaussian shape.

The reconstructed Wigner function corresponding to the measured data is shown in the center of Figure 3-47. It exhibits a shallow but clearly visible dip in the center. This dip arises even though the marginal distribution is not expected to show a dip at a measurement efficiency of 33% (compare Figure 2-21). No negativities of the Wigner functions are observed due to the limited measurement efficiency. A fit for the efficiency of the reconstructed Wigner function yields a slightly higher effi-ciency of 35%.

Figure 3-47 Wigner function of single-photon Fock state measured with an effi-ciency of 33%. A clear deviation of the marginal distribution from a Gaussian is observed. The Wigner function exhibits a central dip, as expected for a single-photon Fock state, even though the marginal distribution does not exhibit a dip at 33%

146 3 Experiment

The measured data presented in Figure 3-48 show the result of an extended meas-urement run taking 21 minutes and consisting of a total number of 16200 single data points. The higher number of data points accounts for decreased statistical errors in the resulting marginal distribution, which only exhibits minor deviations from the theoretical fit. Owing to the long measurement time only a reduced meas-urement efficiency of 22% was obtained. At this efficiency the deviation from the vacuum marginal are less pronounced and the corresponding Wigner function does not exhibit a central dip but is only flattened at the top in comparison to the vac-uum Wigner function.

Applying quantum state sampling to the data sets yields the density matrices de-picted in Figure 3-49. The vacuum and single-photon density matrix elements ρ(0,0) and ρ(1,1) peak up from the noise background. For the first measurement (η = 0.33) we find ρ(1,1) = 0.32, ∑n nn ),(ρ = 0.98 and | ρ(n,m)| < 0.1 for (n,m)

≠ (0,0), (1,1). For the second measurement (η = 0.22) we find ρ(1,1) = 0.22,

Figure 3-48 Wigner function of single-photon Fock state measured with an effi-ciency of 22%. 17000 data points result in a marginal distribution with small sta-tistical errors.

Figure 3-49 Density matrices of the reconstructed single-photon Fock states. Left: η = 0.33; right: η = 0.22.


∑n nn ),(ρ = 1.001 and | ρ(n,m)| < 0.007 for (n,m) ≠ (0,0), (1,1).

Using the efficiency determined from the experimental parameters experimental losses can – at least partially – be compensated for employing the inverse Bernoulli transformation (compare chapter 2.2). Due to the sensitivity to small errors the inverse Bernoulli transformation can only successfully be applied to the second data set. Even for this set the density matrix needs to be truncated beyond n = m = 2. To perform the inverse Bernoulli transformation we make use of the independ-ently determined expected measurement efficiency as discussed in chapter 3.9. Using the decreased value for the mode matching of 87% we obtain an expected total measurement efficiency of 26%. Inverting the reconstructed density matrix shown in the right graph of Figure 3-49 the density matrix depicted in Figure 3-50 is obtained. The ρ(1,1) element of the inverted density matrix clearly dwarfs all other elements and achieves a value of 0.90. The corresponding Wigner function resembles the ideal single-photon Wigner function.

Figure 3-50 Density matrices of the reconstructed single-photon Fock state after applying the inverse Bernoulli transformation. The measurement efficiency ex-pected for the experimental configuration with a value of 0.26 was used as the inversion parameter.

Figure 3-51 Wigner function of the inverse Bernoulli transformed experimental data. The inset shows a comparison between a cut through the inferred Wigner function and the ideal single-photon Wigner function.

148 4 3-Photon Down-Conversion

44 33--PPhhoottoonn DDoowwnn--CCoonnvveerrssiioonn

4.1 Introduction This part summarizes investigations on the nonlinear optical process of 3-photon down-conversion. In the process of 3-photon down-conversion a pump photon is split into three photons of lower energy as a result of a nonlinear interaction of third order according to the Hamiltonian

)ˆˆˆˆˆˆˆˆ(ˆ32103210

)3()3( ++++ −−= aaaaaaaai

H κh (4.1.1)

where the indices 0,1 and 2 denote the subharmonic waves and the index 3 the harmonic pump wave. The nonlinear coupling κ (3) arises owing to the third order nonlinear susceptibility χ(3) and – in the degenerate case 021 ωωω == – is given by

)()(16

333

3*1

3)3(

3310

21)3( xuxuxd

V

rrh∫= χ

εεε

ωκ (4.1.2)

where )(1 xuv and )(3 xu

v are the spatial mode functions of the subharmonic and harmonic wave.

If the process of 3-photon down-conversion occurs spontaneously then it is called χ(3)-parametric fluorescence in analogy to the process of χ(2)-parametric fluores-cence treated earlier. The process is induced by inserting at least one additional subharmonic seed beam, χ(3)-parametric amplification or deamplification occurs. 3-photon down-conversion may also be used to build χ(3)-nonlinear devices such as a χ(3)-optical parametric oscillator (OPO). χ(3)-parametric fluorescence is discussed in chapter 4.8, χ(3)-parametric amplification or deamplification in chapter 4.4, and

Figure 4-1 Level scheme of the 3-photon down-conversion process: A pump pho-ton of frequency ω3 is split into 3 subharmonic photons of frequencies ω0, ω1 and ω2

4.1 Introduction 149

χ(3)-optical parametric oscillators are treated in chapter 4.5.

In contrast to the second order nonlinearity the existence of a non-vanishing χ(3)-susceptibility does not require an anisotropic optical material, but is observed in all kind of materials, including optical crystals, glasses, semiconductors, organic mate-rials and gases. Unfortunately the third order nonlinear susceptibility typically is on the scale of χ(3) = 10-20…10-22 m2/V2 for nonlinear optical crystals. It is therefore much smaller than the second order susceptibility with typical values around χ(3) = 10-12 m/V. Properties of possible nonlinear materials for 3-photon down-conversion are summarized in chapter 4.3.

The χ(3)-interaction has so far found widespread applications in nonlinear optics: the Kerr-effect, phase conjugation, third-harmonic generation, 4-wave mixing, soliton generation and induced second harmonic generation in fibers are effects based on the third order nonlinearity and have successfully been used for experi-mental as well as commercial applications. A χ(3)-susceptibility may also give rise to an effective χ(2)-susceptibility, if one of the fields is applied as a static electric

field ( DCeff E)3()2( χχ = )193.

However, despite this extensive applications of the χ(3)-interaction, the process of 3-photon down-conversion has so far not been demonstrated experimentally. This is most likely due to the low values of the χ(3)-nonlinearity, which renders many experiments difficult to realize or achieves only generation rates too low to be of practical interest. If a demonstration of the process of 3-photon down-conversion with acceptable generation rates may be achieved, this could open up a new field of research in quantum optics.

In the frequency degenerate case the process of 3-photon down-conversion may give rise to a new quantum state of the light field: the star state. The star state is characterized by Wigner function with non-Gaussian marginal distributions and a threefold symmetry in phase space (Figure 4-2). It was first discussed by S.L. Braunstein and R. McLachlan in 1987194 in a treatment on generalized squeezed states. In fact the star state may be considered as the third order nonlinear analogue of a squeezed state. This aspect is supported by the fact that star states are charac-

193 R.H. Stolen, H.W.K. Tom, “Self-organized phase-matched harmonic generation in optical fibers”, Opt. Lett. 12, 585 (1987) 194 S.L. Braunstein, R. McLachlan, “Generalized Squeezing“, Phys. Rev. A 35, 1659 (1987)

Figure 4-2 Wigner function of a star state

The star state


terized by photon number distributions, where only photon numbers which are multiples of three are occupied. K. Banaszek and P. Knight have shown that for star states produced in a single -pass configuration negativities in the Wigner func-tion may arise due to a phase-space interference of the three arms of the star state195. These negativities are easily smoothed out by losses or inefficiencies in the photodetection process (compare equation (2.3.109) in chapter 0). A total detection efficiency exceeding 80% is required to preserve these negativities. However, the result of K. Banaszek and P. Knight demonstrates that star states are associated with a greater non-classical depth (compare chapter 2.1) than squeezed states. In this work we have investigated the experimental feasibility to generate star states in a single-pass configuration or employing χ(3)-OPOs, based on theoretical work performed in our group by Timo Felbinger196,197 (compare chapter 4.7).

The subharmonic photons generated in the process of 3-photon down-conversion are strongly quantum correlated photon triples, very much like the photon pairs produced in the process of 2-photon down-conversion. These pronounced three particle correlations may be exploited to directly generate GHZ198,199-states

( )↓↓↓+↑↑↑=Ψ2

1 (4.1.3)

as discussed in chapter 4.9 or to demonstrate entangled entanglement200.

Only a summarizing treatment of our investigation on 3-photon down-conversion will be given in this thesis wherever subjects have already been described in 201,197.

4.2 Phase Matching For efficient nonlinear coupling in 3-photon down-conversion phase-matching needs to be achieved, i.e. energy and momentum conservation have to be fulfilled within the nonlinear optical material:

3210

3210

ωωωω =++=++ kkkk

rrrr

(4.2.1)

where the two conservation laws are interconnected inside a nonlinear optical me-

dium by the refractive index according to cTnkk iiiiiii /),...),,,((ˆ ωϕθω=r

(i = 0,1,2,3). In the degenerate case ( 021 ωωω == ) equation (4.2.1) reduces to

)3()( 31 ωω nn = (4.2.2)

The effective refractive index ni may depend not only on the respective frequen-cies, but also on the propagation angles θi and ϕi as well as the temperature, doping concentration and other physical parameters that may possibly be adjusted to achieve phase-matching.

195 K. Banaszek, P. Knight, “Quantum Interference in three-photon down-conversion”, Phys. Rev. A 55, 2368 (1997) 196 T. Felbinger, “Simulation nichtlinearer optischer Systeme mit der Quantentrajektorienmethode”, Diploma Thesis, University of Konstanz (1996) 197 T. Felbinger, S. Schiller, J. Mlynek, “Oscillation in 3-photon down-conversion and generation of non-classical states”, Phys. Rev. Lett. ?? (1997) 198 D.M. Greenberger, M.A. Horne, A.Zeilinger, in M. Kafatos (editor) Bell’s Theorem, Quantum Theory, and Conceptions of the Universe“ , p. 73-76 (Kluwer Academic, Dordrecht, Netherlands, 1989) 199 D.M. Greenberger, M.A. Horne, A. Shimoni, A. Zeilinger, Am. J. of Phys. 58, 1131 (1990) 200 A. Zeilinger 201 C. Hettich, “Untersuchungen zur Photonendrittelung und Erzeugung und Charakterisierung von Ein-Photon-Fockzuständen”, diploma thesis, University of Konstanz, (1997)

GHZ states

4.2 Phase Matching 151

For 3-photon down-conversion in uniaxial crystals we need to distinguish 4 differ-ent types of phase-matching according to the polarizations of the participating beams:

III Type,,

,,

II Type,,

,,

I Type,,

,,

→→

→→

→→

oooe

eeeo

ooee

eeoo

oeee

eooo

(4.2.3)

where o denotes ordinary and e extraordinary polarization. These phase-matching types are the same as the ones used to characterize third-harmonic generation (THG).

However, the phase-matching situation we are faced with in 3-photon down-conversion is very different from the situation in third-harmonic generation or four-wave mixing, where three of the four participating wave vectors are pre-determined by the insertion of optical beams. For the χ(3)-Kerr effect and phase-conjugation the phase-matching conditions are intrinsically fulfilled. Phase-matching for third-harmonic generation has been discussed as early as 1965 by J.E. Midwinter et al202 and for pulsed pump radiation in 203.

In the 3-photon down-conversion process each of the four participating waves car-ries 4 degrees of freedom: the 3 components of the wave vector and the polariza-tion. Assuming the polarization to be fixed the system is characterized by 3 × 4 = 12 degrees of freedom. Since the pump wave vector is determined by the pump beam mode and energy and momentum conservation specify 4 more degrees of freedom, 5 degrees of freedom remain for the subharmonic waves. Even if we fix one of the subharmonic wave vectors in the experimental setup – either by using a cavity or by employing irises and spectral filters – we are still left with 2 degrees of freedom. Consequently in general we will not observe one phase matched process but a whole spectrum of phase matched processes.

202 J.E. Midwinter, J. Warner, “The effects of phase matching method and of crystal symmetry on the polar dependence of third-order nonlinear optical polarisation”, 16, 1667 (1965) 203 P. Qiu, N.J. Fisch, “Third-harmonic generation in β-BaB2O4”, App. Phys. B 45, 225 (1988)

Figure 4-3 Phase matching in 3-photon down-conversion: All wave vector combi-nations lying on the blue curves are phase matched. The curves are drawn to scale for a BBO crystal pumped at 355 nm, λ0 = 1064 nm and k 0 fix. Three different con-figurations for different crystal tilt angle θ are shown: (a) θ = 36.82°, (b) θ = 37.32° (degeneracy), (c) θ = 37.82°,

Types of phase-matching

Degrees of freedom


Figure 4-3 exemplifies phase-matching for 3-photon down-conversion in a BBO-crystal in the case, where waves may freely propagate along different directions. Since we are assuming type I (ooo→ e) phase-matching the three subharmonic wave vectors exhibit an ordinary polarization. Therefore their refractive indices do not depend on their propagation direction within the nonlinear medium, so that the depicted phase-matching curves may be rotated around their symmetry axis to yield a 2-dimensional surface of phase-matched processes.

A configuration for a BBO crystal cut for the degenerate process ωω hh ⋅→ 33 is depicted in Figure 4-3 (b): if one photon is emitted into the degenerate mode, all processes lying on the phase-matching surface as depicted are phase-matched and therefore efficient. The contribution of degenerate photon triples to the total emis-sion into one mode has been found to be on the scale of 1/50 = 2% for nonlinear materials such as BBO-crystals or a Rb:Xe-gas mixture.

Even if we impose the additional condition of collinear propagation of all beams – by employing optical fibers or additional subharmonic seed beams – phase-matching may still not be unique as can be seen in Figure 4-3 (b) where three col-linear phase-matched processes are possible only one of which corresponds to the degenerate process.

If the phase-matching angle is detuned from the point of degeneracy and we do not fix one of the wavelength of the subharmonic beams any more, we may again ob-tain a whole set of phase-matched wavelength combination as depicted in Figure 4-4.

Figure 4-4 Phase-matched wavelength combination for type I 3-photon down-conversion in a BBO crystal where the angle between the optical axis and the propagation direction of the pump wave θ is reduced from the point of degenerate down-conversion (355 nm → 3 · 1064 nm) by 0.15° (θ = 37.17°).

Non-collinear case

Collinear case

4.3 Materials 153

4.3 Materials Since the existence of a χ(3)-susceptibility does not require an anisotropic materia l as in the χ(2)-case, the class of materials that may be considered to demonstrate the process of 3-photon down-conversion is significantly wider. For an experimental implementation of this process a candidate material has to meet a number of crite-ria:

§ Phase-matching has to be possible.

§ The material should exhibit low losses,

§ a high damage threshold and

§ a high χ(3)-nonlinearity.

To obtain a particularly high χ(3)-susceptibility a resonance line close to 2ω1 may be advantageous, since in this way the effective nonlinearity may be enhanced by 3 to 4 orders of magnitude without introducing losses at the optical frequencies ω1 and ω3.

The table at the end of this chapter summarizes the relevant parameters for the most promising candidates for 3-photon down-conversion in crystals and other materials.

In nonlinear optical crystals phase-matching is typically ensured employing the material birefringence. Since the wavelength ratio between the harmonic and the subharmonic wave ω3/ω1 is larger than in the χ(2)-case either a more pronounced birefringence is required or subharmonic wavelengths further in the infrared have to be chosen to achieve phase-matching. Doping optical host materials with suit-able elements (e.g. rare earth ions) might provide another – less explored – option to achieve phase-matching.

The size of the nonlinear χ(3)-susceptibility in a solid state material may be esti-mated according to a generalization of Miller’s rule – which relates the size of the second order nonlinear susceptibility χ(2) to the refractive index204 – to the χ(3)-nonlinearity205,206,207, if the frequencies employed are well separated from the mate-rial resonances:

222342)3( /m 106.5)1( Vn −⋅−≈χ (4.3.1)

Typical values of the χ(3)-nonlinear susceptibility for suitable optical crystals such as BBO, LiIO3, YVO4 or Rutil range from 10-20 to 10-22 m2/V2 – roughly 10 orders of magnitude smaller than the corresponding χ(2)-susceptibility.

Organic crystals such as PDA-TS may exhibit significantly higher nonlinearities in the scale of 10-18 m2/V2, but are presently not sufficiently well characterized – values for the refractive index and the damage threshold are not yet known – and often not available with a size and optical quality suitable for quantum optical ap-plications.

204 R.C. Miller, “Optical second harmonic generation in piezoelectrical crystals”, Phys. Rev. Lett. 5, 17 (1964) 205 J.J. Wynne, “Optical third-order mixing in GaAs, Ge, Si, InAs”, Phys. Rev. 178, 1295 (1969) 206 C.C. Wang, “Empirical relation between the linear and the third-order nonlinear optical suscepti-bilities”, Phys. Rev. B 2, 2045 (1970) 207 T. Hashimoto, T. Yoko, S. Sakka, “Sol-gel preparation and third-order nonlinear optical proper-ties of TiO2 thin films”, Chem. Soc. Jap. 67, 653 (1994)

Criteria for suitable χχ ( 3 )( 3 ) -materials

Crystals


Semiconductor materials also offer high values of the χ(3)-nonlinear susceptibil-ity, but typically become transparent only in the infrared, where efficient single-photon detection proves to be difficult. Furthermore the birefringence in these ma-terials is too weak to allow phase-matching.

Gas mixtures have successfully been employed for third-harmonic generation. A mixture of Rb and Xe has proven to be suited best for the conversion of 1064 nm to 355 nm radiation208,209,210. Phase matching in gas vapors is achieved conveniently by mixing gas species with normal and anormal dispersion. Due to the low density of scattering centers as compared to solid state materials the χ(3)-susceptibility thus achieved is lower by 5 to 7 orders of magnitude. An additional advantage of using gas mixtures is the fact that the nonlinear material may not be damaged even at extremely high peak intensities, since a gas mixture will quickly recover, even if the high field strength causes an electric discharge.

Glasses exhibit a nonlinear optical χ(3)-susceptibility in the scale of 10-19 … 10-20 m2/V2, about two orders of magnitude larger than for typical optical crystals. They are easily available with good optical quality, but no easy way has been discovered to obtain phase-matching.

One possible option to achieve phase-matching might be to dope glasses with ma-terials that exhibit a narrow absorption line close to the pump frequency like rare earth elements. Assuming a Terbium doped filter glass (produced e.g by Schott) with a 30 weight percent admixture of Terbium (corresponding to density of 3.84 Terbium atoms per m3) with a refractive index of 1.591 exhibiting a single absorp-tion line at 490 nm, an oscillator strength between f =10-4…10-6 and a linewidth of γ = 300 cm-1 211 a change of the refractive index of up to ∆n = 0.09 may be ex-pected, which might just be sufficient to achieve phase-matching – at the price of high losses in the pump beam.

Quasi-phase-matched materials with a periodic inversion of the nonlinearity have successfully been used to achieve phase-matching for χ(2)-processes. Even though an inversion of the χ(3)-nonlinearity cannot be performed since the χ(3)-nonlinearity does not result from material anisotropies, quasi-phase-matching may still be realized by a periodical modulation of the nonlinearity. A first proposal for quasi-phase-matched third-harmonic generation has been put forth by J.M. Rax et al.212 in 1992 and a first demonstration employing a stack of microscope plates has been achieved by D.L. Williams et al.213 in 1996.

Material class

Material χχ ( 3 )( 3 ) [m2/V2]

χχ (2)(2) Transparency range [µm]

Phase-matching

properties

Crystals BBO 4.2· 10-22 (1064 nm)

Yes 0.2 - 3.5 Crystal angle: 0.2-3.5 µm

High damage threshold: 2· 1014 W/m2, low absorp-tion : 0.1%/cm, 214

LiIO3 9.1· 10-21 Yes 0.35 - 5.0 > 880 nm Damage threshold: 7· 1013

208 J.F. Joung, G.C. Bjorklund, A.H. Kung, R.B. Milles, S.E. Harris, “Third-harmonic generation in phase-matched Rb vapor”, Phys. Rev. Lett. 27, 1551 (1971) 209 R.B. Miles, S.E. Harris, “Optical third-harmonic generation in alkali metal vapors”, IEEE J. of Qu. El. 9, 470 (1973) 210 H. Puell, K. Spanner, W. Falkenstein, W. Kaiser, C.R. Vidal, “Third-harmonic generation of mode-locked Nd-glass laser pulses in phase-matched Rb-Xe mixtures”, Phys. Rev. A 14, 2240 (1976) 211 McFarlain, private communication 212 J.M. Rax, N.J. Fisch, “Third-harmonic generation with ultrahigh-intensity laser pulses“, Phys. Rev. Lett. 69, 772 (1992) 213 D.L. Williams, D.P. West, A. King, “Quasi-phasematched third harmonic generation”, Laser photonics group, Department of Physics and Astronomy, University of Manchester, Manchester M13 9PL, UK (1996) 214 P. Qiu and A. Penzkofer, Appl. Phys. B, 45, 225 (1988); H. Nakatani et al., Appl. Phys. Lett. 53, 2587 (1988); I.V. Tomov et al., Appl. Opt. 31, 4172 (1992)

Semiconductors

Gas mixtures

Glasses

QPM-materials

4.4 Parametric De-/Amplification 155

(1064 nm) W/m2, 215 YVO4 7.6· 10-21 No 0.4 – 3.2 > 1064 nm Rutil 5.6· 10-20

(1900 nm) No 0.45 – 6.0 > 1500 nm 216

Organic crystals

PDA-TS 5.6· 10-18 (1064 nm)

No ? ? Insufficient characteriza-tion, available only in small quantities, 217

PTS 7· 10-19 No ? ? Insufficient characteriza-tion, Low damage threshold, 218

Glas As2S3 5.6· 10-19 No ? Possibly in fibers

Absorption: 1.6· 10-3/ cm at 2.4 µm, 219

Gas Rb:Xe 3.9· 10-27 (1064 nm)

No Everywhere away from resonances

Gas mixture Low effective nonlinearity, 220

4.4 Parametric De-/Amplification Starting from the non-linear Maxwell equations and using the rotating wave ap-proximation we may derive the dynamic equations for the photon fluxes

( kkkk EnA ω= ) in the case of degenerate 3-photon down-conversion:

31*1

)13(*3

23

)33()3(31

)3(33

3

3*31

)13(*1

21

)11()3(3

2*1

)3(11

1

63

633

13

13

AAAiAAieAiAdz

dA

AAAiAAieAAiAdz

dA

zkki

zkki

κκκγ

κκκγ

++−−=

++−−=

−−

−−

(4.4.1)

where

3/10

03/1 2 εε

µσγ m= (4.4.2)

denotes the linear losses and κ (3) describes the nonlinear coupling responsible for the 3-photon down-conversion as defined in equation (4.1.2). κ (11) and κ (33) are the nonlinear couplings due to the self-Kerr effect whereas κ (13) accounts for the cross-Kerr effect between the degenerate subharmonic waves and the pump wave.

Neglecting the Kerr-terms and assuming phase-matching we obtain the simplified equations

3

1)3(

333

32

1)3(

111 3

AAdz

dA

AAAdz

dA

κγ

κγ

−−=

+−= (4.4.3)

where we have replaced A3 with iA3 to obtain real solutions. Depending on the sign of A3 – the relative phase between the pump and the subharmonic wave – in equa-

215 M. Okada, Appl. Phys. B. 18, 450 (1971) 216 T. Hashimoto, T. Yoko, S. Sakka, The chem. Soc. of Jap. 67, 653 (1994) 217 M. Sinclair et al., in J. Messier et al. (ed.), “Nonlinear Optical Effects in Organic Polymers”, p. 29 (Kluwer Academic, Dordrecht, Netherlands, 1989) 218 P.A. Chollet et al., Rev. Phys. Appl. 22, 1221 (1987) 219 H. Kobayashi et al., J. Appl. Phys. 74, 3683 (1993); K. Nassau, S.H. Wemple, Electron. Lett. 18, 450 (1982) 220 R.B. Miles, S.E. Harris, IEEE J. of Quant. El. 9, 470 (1973); H. Puell et al., Phys. Rev. A 14, 2240 (1976)


tion (4.4.3) energy is either transferred from the harmonic pump wave to the sub-harmonic wave (parametric amplification) or from the subharmonic wave to the harmonic wave (parametric deamplification).

No full analytic solution for equation (4.4.3) has been found. However in the case, where the pump wave is much stronger than the subharmonic wave and may be assumed to remain constant throughout the nonlinear interaction equation (4.4.3) may be integrated to yield

z

Az eeA

zA1

1

11)0(3

)3(1

1)1(3

)(γγγκ

γ

+−≈ (4.4.4)

Numerical example

Figure 4-5 depicts the degree of χ(3)−parametric amplification in the degenerate case for a 1 cm long BBO crystal cut for type I degenerate 3-photon down-conversion (355 nm → 3· 1064 nm, θ = 37.3°) with an effective nonlinearity of 2.52· 10-22 m2/V2. The damage threshold for BBO is 2· 1014 W/m221, material losses are 0.1%/cm at both wavelengths. Due to the small value of the third order nonlin-ear susceptibility rather high pump intensities (exceeding 2.5 kW for a spot size of 50 µm) have to be achieved. A maximum parametric amplification of 35% is achieved, if the signal and pump intensity are set to half the damage threshold. The amplification factor does converge to a value below 1 for low pump intensities due to the optical losses in the crystal.

4.5 χχ (3)-OPOs If the nonlinear material mediating the parametric interaction is placed inside an optical resonator, macroscopic amounts of light may be generated at the subhar-monic frequencies in the case, where the nonlinear gain inside the resonator ex-ceeds the resonator losses. Such devices are called optical parametric oscillators (OPO). OPOs based on the χ(2)-nonlinearity have successfully been established as

221 M.J. Weber, “Handbook of Laser Science and Technology, vol. 3”, (CRC Press, Florida, 1986)

Figure 4-5 Parametric amplification and deamplification as a function of the pump intensity in a BBO crystal in the degenerate case (355 nm → 3· 1064 nm).

4.5 (3)-OPOs 157

laser sources with an extremely wide spectral coverage, narrow bandwidth and good stability. So far OPOs have exclusively been realized employing χ(2)-nonlinear coupling. χ(3)-OPOs are of interest since they may provide three subhar-monic waves instead of two and – under certain circumstances – may be used to generate a new quantum state of the light field: the star state.

The theory for the degenerate, pump-resonant χ(3)−optical parametric oscillator (χ(3)−OPO) has been worked out by Timo Felbinger et al. in 222 at the University of Konstanz. We just state their major results here to be able to discuss the experi-mental feasibility of χ(3)−OPOs and star states preparation.

Pump Resonant OPO The complete quantum mechanical equations of motion for the interacting pump and signal mode in the Heisenberg picture may be derived employing the input output formalism

tiin

ti

in

p

p

ebaaaiaaiaaia

ebaaaiaaiaaaiaω

ω

γκκκγ

γκκκγ

ˆ2ˆˆˆˆˆ2ˆˆ)(ˆ

ˆ2ˆˆˆˆˆ2ˆˆ3ˆ)(ˆ

3311)13(2

33)33(3

1)3(

3333

31331

)13(211

)11(3

21

)3(1111

++++∆−−=

++++∆−−=++

+++

&

&

(4.5.1)

where we have chosen a reference frame rotating with a frequency ω3, inb denotes

the annihilation operator for the external pump mode, 11 3/ ωω −=∆ p and

33 3/ ωω −=∆ p are the detunings of the signal and pump resonator mode fre-quencies from the pump frequency and the cavity losses are given by

iii

ii TLn

cLn

cT

2)11( ≈−−=γ (4.5.2)

Here L represents the length of the resonator and Ti the round trip losses of the respective mode.

222 T. Felbinger, “Simulation nichtlinearer optischer Systeme mit der Quantentrajektorienmethode” , Diploma thesis, University of Konstanz (1996); T. Felbinger, S. Schiller, J. Mlynek, “Oscillation in 3-photon down-conversion and generation of non-classical states”, Phys. Rev. Lett. 80, 492 (1997)

Figure 4-6 Schematics of the degenerate χ(3)-OPO. A pump wave of frequency ω3 is coupled to a degenerate signal mode of frequency ω1. The pump wave may be resonantly enhanced or not.


These equations reflect the threefold symmetry in phase space.

The classical equations of motion may be obtained from the quantum mechanical equations by replacing the field operators with their expectation values –

3/13/1 a=α and 31333 )(2ˆ ϕωγε i

in ePb −== h – and factorizing the correla-

tions ( jijiji aaaa αα=≈ ˆˆˆˆ ).

If the detunings are chosen as 23

)13(21

)11(1 2 ακακ +=∆ and

21

)13(23

)33(3 2 ακακ +=∆ the influence of the self-Kerr- and cross-Kerr-effect

is cancelled and we obtain an OPO behavior as depicted in Figure 4-7.

Below a threshold pump power the trivial solution 01 =α is the only stationary solution of the dynamic equations. Above this critical pump power three stable and three unstable non-zero solutions with equal amplitude but phase-shifted by 120° in phase-space exist in addition to the trivial zero-solution which remains stable even above threshold. The oscillation threshold is determined to be

)3(

2/13

2/313

, 278

κ

γγωh=OPOthP (4.5.3)

Since the stable solution directly at the threshold pump power has a non-vanishing amplitude, a first order phase transition occurs at the oscillation threshold of the χ(3)-OPO. This is a pronounced contrast to the χ(2)-case, where a second order phase transition occurs at the oscillation threshold. Consequently the χ(3)−OPO does not start oscillating by itself even above threshold.

Non-Pump Resonant OPO

To treat the non-pump resonant χ(3)-OPO we solve equations (4.4.3) with the addi-tional constraint of the self-consistency condition for the signal wave

Figure 4-7 Parametric amplification and deamplification as a function of the pump intensity in a BBO crystal in the degenerate case (355 nm → 3· 1064 nm).

Oscillation threshold

4.5 (3)-OPOs 159

)0(1)( 111 ATLA −= (4.5.4)

For small round trip losses the signal amplitude may be considered as a constant. Using this approximation equations (4.4.3) may be integrated to yield a stationary solution for the pump wave in terms of the circulating signal field A1

1

)33(

41

2)33(11

136

3)2()(

AL

ALTLAAstat

κ

κγ ++= (4.5.5)

An approximate expression for the oscillation threshold in the non-pump-resonant

case may be obtained by calculating the minimum of )( 13 AA stat :

L

TLA th κ

α 2/3112

,3)(

274 +

= (4.5.6)

A comparison of this result with the threshold in the pump-resonant case yields

resonantpumpthresonantpumpnonth PTL

TP −−−

+= ,2/3

33

1,

)(

2

α (4.5.7)

Numerical example

To get a feeling for the orders of magnitudes involved in building a χ(3)-OPO we again consider the case of a 1 cm long BBO crystal cut for degenerate 3-photon down-conversion. Due to the rather large walk-off angle of 4.9° we assume a non-pump-resonant OPO setup with round trip losses T1 = 2%, yielding plane wave optical intensities of I1,th = 5.4· 1013 W/m2 and I3,th = 0.15· 1013 W/m2. This threshold intensities are already dangerously close to the damage threshold of the BBO crys-tal.

For a pump-resonant OPO – neglecting the walk-off and assuming T3 = 2% – we obtain threshold intensities of I1,th = 3.9· 1012 W/m2 and I3,th = 2.9· 1010 W/m2 re-duced by the pump enhancement 3/1 T≈ (T3 >> α L)

A quantum treatment of the dynamic equations employing a quantum trajectory method reveals, that the bright light emitted by the OPO is squeezed, and that the zero-solution produces a star state, if the unstable branch reaches the level of the quantum fluctuations. However, no negativities of the Wigner function arise for star states produced with a χ(3)-OPO.

Star State Threshold

A perturbative quantum treatment of the χ(3)-OPO yields a measure for the devia-tion of the produced star states from the vacuum state

31

3)3(

3 γγεκ

=s (4.5.8)

where γ1 and γ3 again denote the losses in the subharmonic and harmonic wave – or alternatively the inverse cavity decay times of the respective modes – and

1333 )(2 −= ωγε hP represents the pump field amplitude. The zero solution pro-

duces a star state, only if the deviation parameter 2.03 ≈s . This yields a threshold for the star state generation of

OPOthstarth PP ,)3(31

32)3(

321

, 1627

2 κ

γγω

κ

γγ== h (4.5.9)

Oscillation threshold


Hence only if )3(31 κγγ ≈ will the threshold for the star state generation and the

OPO threshold be of the same order of magnitude.

Numerical example

For the case of a χ(3)-OPO based on a BBO crystal with identical parameters as considered above we obtain a star state threshold power of 2.4· 1012 W, correspond-ing to an threshold intensity of 2.7· 1021 W/m2. Hence the star state threshold is 11 orders of magnitude larger than the OPO oscillation threshold and exceeds the damage threshold of the BBO crystal by 7 orders of magnitude.

Application: Light Controlled OPOs

Instead of two subharmonic waves, usually labeled signal and idler, the χ(3)-OPO offers a third subharmonic wave, which we label control. If this additional wave is used to control the frequency of the signal and idler wave, a light controlled OPO may be realized. Since small changes in the control wavelength may result in major changes in the signal and idler wave, this device may also be considered as an opti-cal frequency transistor: it amplifies small wavelength changes.

Figure 4-8 demonstrates this effect for type II degenerate and non-degenerate phase-matched 3-photon down-conversion in BBO cut at a crystal angle of θ = 46.91°. Tuning λ0 by 0.064 µm produces signal radiation between 1.064 and 2.5 µm and idler radiation between 1.064 µm and 0.74 µm.

Figure 4-8 Light controlled OPO using type II phase-matching: tuning of the seed beam wavelength λ0 allows to control the signal and idler wavelengths λ1 and λ2. Small differences in seed beam wavelength λ0 lead to great differences in the sig-nal and idler wavelengths λ1 and λ2.

4.6 Cascaded ((3) 161

4.6 Cascaded χχ (3) Since the χ(3)-nonlinearity has been found to be too small for an experimental realization of a χ(3)-OPO at reasonable pump power levels, the question arises whether it is possible to create an effective χ(3)-interaction combining different χ(2)-processes. We consider the χ(2)-processes of difference-frequency-generation (DFG) ωωω +→ 23 and 2-photon down-conversion ωωω +→2 inside an opti-cal resonator as depicted in Figure 4-9.

Starting from the interaction Hamiltonians for the two processes

( )

−−=

−−=

++

+++

2212

21

)2(2

321321

ˆˆˆˆ2

ˆˆˆˆˆˆ

aaaai

H

aaaaaai

H

PDC

DFGDFG

κ

κ

h

h

(4.6.1)

with the nonlinear couplings♣

∫

∫

−=

−=

22*

13)2(

3,2

1,0

2/32

2*2

*1

3)2(

3,2,1,0

2/3

),,2(

),2,3(

uurd

uuurd

rr

PDC

rrr

DFG

ωωωχεεε

ωκ

ωωωχεεεε

ωκ

h

h

(4.6.2)

and employing the quantum Languevin equation

[ ] iniiiii

PDCDFGi baaHH

ia γγ 2ˆˆ,ˆ 2 +−+=

h& (4.6.3)

we obtain the dynamical equation for the system in the case where all three waves are resonated in the cavity

1113221)2(

1

2223121

)2(

2

33333213

ˆ)(ˆˆˆˆˆ

ˆ)(ˆˆˆ2

ˆ

2ˆ)(ˆˆˆ

aiaaaaa

aiaaaa

baiaaa

DFG

DFG

inDFG

∆−−+=

∆−−+−=

+∆−−−=

++

+

γκκ

γκκ

γγκ

&

&

&

(4.6.4)

If the losses in the intermediary wave are much smaller than the losses in the pump and signal wave γ2 << γ1, γ3, then the coupling of the intermediary wave to the en-vironment will be much weaker. Hence the outcoupled subharmonic power will be dominated by emissions at the signal wavelength. If the externally applied pump

field inb3 changes only slowly in time, the intermediary wave takes on its stationary value

312

21

2

)2(

22 ˆˆˆ2

0ˆ aaaaaDFG

++−=⇒=γ

κγ

κ& (4.6.5)

where we set the detuning for the intermediary wave ∆2 = 0. We eliminate â2 in

equation (4.6.4) applying equation (4.6.5) and replace iiii eaa ϕˆˆ → for i = 1,2,3 to

♣ The couplings defined here and in equation (4.1.2) deviate from the ones used in the diploma thesis of Timo Felbinger by a factor -i. This is achieved by exploiting the freedom to choose the offset phases for the optical waves and results in real couplings. Positive real couplings yield a more appro-priate description of the physical processes.


explicitly include the choice of phase for the three optical fields and obtain the reduced equations

111211

)11(331

)13()3(3

21

)3(1

33333311)13()3(3

1)3(

3

ˆ)(ˆˆ2ˆˆˆˆˆ3ˆ

ˆ2ˆ)(ˆˆˆˆˆ

22231

2231

aieaaeaaaeaaa

baieaaaeaa

ieff

ieff

ieff

inieff

ieff

∆+−−+=

+∆+−−=

−+−+−+−+

−+−−

γκκκ

γγκκ

ϕϕϕϕϕ

ϕϕϕϕ

&

&

(4.6.6)

where the effective χ(3)-couplings are given by

2

2)2()11(

2

2)13(

2

)2()3(

2

2

γκκ

γκ

κ

γκκκ

=

=

=

eff

DFG

eff

DFG

eff

(4.6.7)

Choosing the phases 2/3 31 πϕϕ −=− and 2/2 πϕ = leads to

111

211

)11(331

)13(3

21

)3(1

33333311)13(3

1)3(

3

ˆ)(ˆˆ2ˆˆˆˆˆ3ˆ

ˆ2ˆ)(ˆˆˆˆˆ

aiaaiaaaiaaa

baiaaaiaa

effeffeff

ineffeff

∆+−+−=

+∆+−+−=

+++

+

γκκκ

γγκκ

&

&

(4.6.8)

Comparing equations (4.5.1) and equation (4.6.8) we find that the dynamical equa-tions describing the cascaded system and a simple 3-photon down-converter are identical apart from the missing self-Kerr effect for the cascaded system and an

additional minus-sign in front of the cross-Kerr-term in the equation for 1a& . If we

choose the detunings as 11)13(

3 ˆˆ aaeff+=∆ κ and 11

)11(33

)13(1 ˆˆ2ˆˆ aaaa effeff

++ +−=∆ κκ , the

Kerr terms are cancelled and we may directly transfer the results of chapter 4.5 to the cascaded system and expect the same quantum behavior of the cascaded system as for the 3-photon down-converter.

Figure 4-9 Realization of an effective χ(3)-nonlinearity using a combination of two χ(2)-processes: a first nonlinear crystal couples the pump wave of frequency ω3 = 3ω to two subharmonic waves with frequencies ω2 = 2ω and ω1 = ω employ-ing the process of difference frequency generation. A second crystal cut for 2-photon down-conversion couples the intermediary wave at ω2 = 2ω is to the signal wave at frequency ω1 = ω. If the losses of the intermediary wave are much smaller than the losses of the pump and signal wave (γ2 << γ1, γ3 ), then the device acts as an effective 3-photon down-converter.

Cascaded coupling coefficients

4.7 Star State Generation 163

The main difference between the two systems is the size of the coupling terms, which we expect to be much larger in the cascaded χ(2)-case due to the larger size of the χ(2)-nonlinearity.

Numerical example

Assuming λ1 = 1064 nm, a nonlinear coupling χ(2) = χDFG = 7· 10-12 m/V, a refrac-

tive index == 23/2/1,3/2/1 rn ε 2.23, a effective mode volume of 1.57· 10-12 m3 (waist

size w0 = 10 µm, interaction length L = 1 cm), linear losses of the intermediary wave of α2 = 2.5%/cm and a mirror reflectivity at the intermediary wavelength of

99.9% we obtain an effective coupling of =)3(effκ 47 1/s as compared to

=)3(BBOκ 5.8· 10-6 1/s in the case of direct 3-photon down-conversion in a BBO crys-

tal. This corresponds to an expected χ(3)-OPO threshold of 2 mW and a star thresh-old of 3.6 kW.

4.7 Star State Generation Star states may be generated in a single -pass configuration or employing a χ(3)-OPO.

Only degenerate photon triples contribute to the star state. If a single -pass configu-ration is chosen for the star state generation, non-unique phase-matching leads to an admixture of thermal states to the star state, which will quickly wash out the characteristic threefold phase-space symmetry. Since we only expect about 2% of all generated photons in the detected mode to result from a degenerate down-conversion (compare chapter 4.2) and there is no way to distinguish between the degenerate and non-degenerate photon emissions into a single mode, we will not be able to observe a star state in a single -pass configuration with an efficiency exceed-ing 2%. A further complication may arise in non-centro-symmetric materials such as nonlinear optical crystals due to competing χ(2)-processes. Since the χ(2)-nonlinear coupling exceeds the χ(3)-coupling by at least 7 orders of magnitude non-phase-matched χ(2)-processes may already dominate the χ(3)-photon triple genera-tion. Therefore an observation of a star state in a single -pass configuration has to rely on materials which do not exhibit a χ(2)-nonlinearity and has to strongly sup-press the non-degenerate photon triples in comparison to the degenerate emissions. This might be possible in photonic band-gap fibers223.

As discussed in chapter 4.5. in the case of star state generation employing a χ(3)-OPO, pump intensity levels required to exceed the star state threshold are on the scale of 1021 W/m2, far above the oscillation threshold and far above the damage threshold of available nonlinear optical materials.

If a cascaded process is used to obtain an effective χ(3)-coupling, the power levels may be reduced drastically for high-finesse cavities even though the star state threshold still requires power levels of a few kW even for tight focussing. To achieve these power levels in an experimental setup it will be necessary to employ pulsed lasers with a preferred pulse length exceeding 10 ns to be able to work in the quasi-cw regime for the optical resonator with a round trip path length not ex-ceeding 30 cm. Synchronous pumping with mode-locked ps-pulses might be an-

223 J.C. Knight, T.A. Birks, P.St.J. Russell, D.M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding”, Opt. Lett. 21, 1547 (1996)

Single-pass star state generation

Star state generation in a χχ (3)-OPO

Star state generation in a cascaded system


other option, but may be complicated by different group velocities of the pump, intermediary and signal wave inside the optical crystals.

In summary our investigations on the experimental feasibility of the generation of star states have shown that the single-pass generation of star states does not seem to be realistic due to the spectrum of parasitic phase-matched χ(3)-processes. χ(3)-OPOs do not offer realistic options to generate star states either since the required power levels are too high. Star state generation employing a cascaded χ(3)-process deserves some attention and may possibly provide a way to generate star states.

4.8 Parametric Fluorescence In analogy to the process of χ(2)-parametric fluorescence, which has been treated in depth for a pulsed pump in chapter 2.3.3, the process of χ(3)-parametric fluores-cence is a spontaneous quantum mechanical scattering process that generates highly correlated photon triples. As already stated in the introduction these generic three-particle correlations might be used for a number of quantum optical experi-ments such as the generation of GHZ-states to test the non-local character of quan-tum mechanics or entangled entanglement.

Quantum mechanical scattering theory leads to a differential scattering cross-section (in cgs-units) of

002103

210332

31

30

2102

31

32)3(

3

23

2

000

)(

)(162

fluxphoton pump eunit volum unit time

in scattered photons ofnumber )(

ωωωωωδ

δχω

π

ωσ

dd

kkkknnn

kkkkdkd

n

c

ddkd

Ω−−−⋅

−−−=

⋅⋅Ω

=

∫∫vvvv

rrrh

r

(4.8.1)

into the angle element dΩ0 and the frequency interval dω0.

If the angle element and the frequency interval are sufficiently small this leads to a expected power emitted into ∆Ω0 ∆ω0 of

0033

30

40

22)3(

0648

ωωχ

∆∆Ω= PMnc

LnP

h (4.8.2)

where L denotes the interaction length and M is a numerical factor which for gases and crystals takes on a value of M = 200· 1016 and all other parameters have to be given in the cgs-system.

Numerical example

For a 1 cm long BBO crystal cut for degenerate type I 3-photon down-conversion (355 nm → 3· 1064 nm) with χ(3) = 2.25· 10-22 m2/V2, a peak pump power of 56 MW, ∆λ0 = 50 nm and ∆Ω0 = 10-5 we expect a parametrically generated subhar-monic power P0 = 3.9· 10-15 W corresponding to 20000 photons/s.

An estimation of the non-phase-matched emission due to χ(2)-parametric fluores-cence into the same mode (assuming an effective crystal length of 25 µm) leads to an expected optical power of 10-7 W. Hence the χ(3)-emission will be completely hidden in a χ(2) background, if a χ(2)-susceptibility is present.

Expected power in χχ (3)-parametric fluorescence


For an average pump power of 10 W and assuming realistic single -photon counters the expected coincidence rate is calculated to be 3 coincidences per day, a value much too low to perform experiments with it.

4.9 3-PDC in Fibers

If the process of χ(3)-parametric fluorescence takes place inside an optical fiber, the problem of a too low generation rate may be overcome by a strongly enhanced interaction length. In the non-degenerate case where the three photons of a photon triple are emitted with three distinct frequencies, a prism or grating may be em-ployed to spatially separate the three photons and to send them onto three single -photon counters in an experimental setup as depicted in Figure 4-10. A triple coin-cidence count measurement of the photon counters provides evidence that the reg-istered photons have been produced in the process of 3-photon down-conversion.

The emission of photon triples produced in the process of χ(3)-parametric fluores-cence into modes with fixed directions may be calculated employing a Green’s function approach in analogy to a treatment of χ(2)-fluorescence by P.W. Milonni et al.224 in order to obtain an estimate for the photon triple production rate in an opti-cal fiber.

For the non-degenerate case we obtain a power in the signal mode of

( ) ∫

−

+++

⋅=

=

1

2

2222

3210

23210

21203210

1

330

54

2)3(2

0

00

1

1111

129621

ω

ωωωω

ωωωω

επ

χµε

cn

ddn

cwwwwwwww

nnnnd

Pc

LP WWh

(4.9.1)

where LW W denotes the effective interaction length, wi the mode radii of the respec-tive beams in the fiber and the ω1-integration extents across all phase-matched combinations yielding a photon in mode 0. ω2 is then determined by the energy

224 P.W. Milonni, H. Fearn, A. Zeilinger, “Theory of two-photon down conversion in the presence of mirrors”, Phys. Rev. A 53, 4556 (1996)

Figure 4-10 3-photon down-conversion in fibers: frequency non-degenerate pho-ton-triples generation may be verified employing the setup shown above.


conservation. If in addition ω1 is restricted to a certain frequency interval, equation (4.9.1) yields the expected triple coincidence count rate.

Numerical example

For a As2S3-fiber phase-matched for λ0 = 800 nm, λ1 = 1100 nm, λ2 = 1280nm, assuming all mode radii to be 10 µm, all refractive indices to be 2.5, an effective interaction length of 1 m and a pump power of 1 W at 340 nm we obtain an ex-pected photon triple generation rate of Rtrip = 1000/s and single-mode powers of P0/1/2 = 1.7· 10-14 W corresponding to 90000 photons/s for the fluorescence into 4 nm wide detection intervals around 800, 1100 and 1280 nm.

To be able to verify the generation of photon triples produced by 3-photon down-conversion, triple coincidence counts have to be distinguished from background coincidence counts arising from either random coincidences or coincidence counts where only two of the three registered photons belong to a photon triple, whereas the third count event has been produced by a background count.

If we label the three photon count detectors A,B and C and denote the photon triple generation rate by Rtrip, the single-photon background count rate by Rback, and the detector dark count rates by RDC, A/B/C then the expected coincidence rate is given by

2)2()1()3( ττ ∆+∆+= RRRRABC (4.9.2)

where

+++=

++=

=

¬¬¬¬¬¬

¬¬¬

,,,,,,

)2(

,,,2)1(

3)3(

1C

C

B

B

A

AcorrLCBA

C

C

B

B

A

AtripCBAL

tripCBAL

RRRRRRRR

RRRRR

RR

ηηηη

ηηηηηηη

ηηηη

(4.9.3)

Here ηL = 1-losses represents the optical propagation efficiency (assumed to be equal for all three emission channels), ηA,B,C denotes the quantum efficiencies of the three single-photon detectors, ∆τ the coincidence gating time window and

CBADCbackCBALCBA RRR //,//,// +=¬ ηη the counts registered by the A/B/C-

Figure 4-11 Contributions to the triple coincidence count rate as a function of the gating time window. Only the constant contribution directly reflects the photon triple generation rate. The dashed vertical lines mark the positions where a signal-to-noise ratios of 1 (right line) and 10 (left line) are achieved.

Triple coinci-dence counting


detector that do not result from photon triples emitted into the three monitored emission channels.

The total measured coincidence rate is made up of three individual contributions which exhibit a different dependence on the gating time window ∆τ: The first term R(3), which is independent of the gating time window, yields a direct measurement of the photon triple generation rate, if the optical losses and the detector efficien-cies are known. The second term, proportional to ∆τ, accounts for photon triple detections, where only two photons of a photon triple have been registered and the third detector clicked due to a background or dark count event occurring within the gating time window. The third contribution proportional to ∆τ 2 arises from triple counts where either only one photon of the photon triple has been detected or three background or dark count detector clicks occurred within the gating time window.

Hence a reliable detection of the photon triple generation rate either requires a very short gating time window or the individual contributions to the triple coincidence count rate have to be determined by varying the gating time window.

Numerical example

Using the same conditions as assumed in the numerical example above (Rtrip = 1000/s, Rback = 90000/s), assuming an optical transfer efficiency of 67%, a quantum efficiency of ηA = 2.6% and a dark count rate of RDC,A = 40000 cts/s of an InGaAs-single-photon counter for the 1280 nm wave, ηB = 5% and RDC,B = 8000 cts/s of a Ge-single-photon counter for the 1110 nm wave and ηC = 30% and RDC,C = 500 cts/s of a Si-single-photon counter for the 800 nm wave a gating time window of 120 ns is required for a signal-to-noise ratio R(3)/(R(2) ∆τ + R( 1) ∆τ) of 1 and a gating window of 25 ns for a signal-to-noise ratio of 10. Figure 4-11 illustrates the individual contributions as a function of the gating time win-dow.

χ(3)-parametric fluorescence in fibers may also provide an option to generate GHZ-states. GHZ-states arise, if photon triples are emitted into two quantum mechanical paths that interfere coherently. These interfering paths do not have to be spatially separated, they may also be separated in time as first noted by J.D. Franson225.

225 J.D. Franson, “Bell inequality for position and time”, Phys. Rev. Lett. 62, 2205 (1989)

Figure 4-12 GHZ-state generation employing photon triples produced in 3-photon down-conversion in a fiber and Franson type interferometers in each of the three emission channels.

GHZ state preparation


Figure 4-12 depicts an experimental setup to generate GHZ-states employing the photon triples produced in χ(3)-parametric fluorescence in fibers. A Franson inter-ferometer is placed in each of the emission channels. If a triple coincidence count is registered it may either be due to a photon triple generation where all 3 photons have taken either the long or the short path. Both possibilities interfere coherently, if the first order coherence length of the pump beam exceeds the path length differ-ence between the long and the short arm of the Franson interferometer. On the other hand the path length difference between the long and the short path should be much larger than the spatial extent of the photons emitted in the process of 3-photon down-conversion (which will be less than a 0.3 millimeter for a coherence time of less than a ps for the photon triples produced in χ(3)-parametric fluores-cence) to be able to clearly distinguish between the two paths.

5.1 Summary 169

55 SSuummmmaarryy aanndd OOuuttllooookk

5.1 Summary

Single-Photon Fock State Tomography

In the experimental project described in this thesis the method of optical homodyne tomography has been applied to pulsed states of the light field:

§ Quantum state reconstructions of a single -photon Fock states prepared by conditional measurements on photon twins produced in the process of spontaneous 2-photon down-conversion have been performed with meas-urement efficiencies up to 33%.

§ Non-Gaussian marginal distributions and non-Gaussian shaped Wigner functions were demonstrated for the first time in the optical domain.

§ The method of pulsed optical homodyne tomography has been applied to reconstruct the Wigner function and density matrices of pulsed vacuum and pulsed coherent states. The vacuum state has been reconstructed up to an error of 0.25%. Coherent states with average photon numbers of only a few photons per pulse have been characterized with a fidelity of 99.5%.

§ A pulsed homodyne system with an ultra-low electronic noise of 565 electrons/pulse and a very efficient direct subtraction was designed, im-plemented and characterized. Shot noise limited behavior up to 2.3· 108 photons per local oscillator pulse corresponding to a maximum subtraction of 83 dB was demonstrated. This allowed to advance the technique of pulsed homodyne tomography to the single-photon level.

§ Photon counting measurements and homodyne detection were combined for the first time in a single experiment.

A detailed theoretical treatment of single -photon Fock state tomography has been presented including the transverse and spectral structure of the photon twins generated, state preparation by conditional measurements and the homodyne detec-tion process. The state preparation fidelity was estimated taking into account arbi-trary spatial and spectral detection modes and the spectral and spatial spread of a pulsed pump mode.

A theory of pulsed homodyne detection and non-ideal homodyne systems has been provided, which offers a method to calculate the homodyne detection effi-ciency and the optimum parameter settings under very general conditions. The effect of various inefficiencies such as imperfect mode-matching, beam splitter imbalance, optical losses or different detector efficiencies has been considered.

Conditional State Preparation in repeated 2-Photon Down-Conversion

A proposal was put forth to extend the method of conditional state preparation employing repeated 2-photon down-conversion to produce higher n-Fock states as

170 5 Summary and Outlook

well as arbitrary truncated quantum states of the light field. The proposed scheme allows an n = 2 Fock state preparation with a rate enhanced by a factor of more than 200. An n = 3 Fock state preparation fidelity exceeding 70% and a gen-eration rate of 200 preparations per second are found for a photon pair creation probability of Γ = 3· 10-3, cavity roundtrip losses of 0.1% and a single-photon detec-tion efficiency of 70%. Only very low rates for arbitrary truncated state generation up to an n = 4 of 7· 10-3 preparations per second can be expected for the same parameters.

3-Photon Down-Conversion

Experimental perspectives based on the process of 3-photon down-conversion have been investigated:

§ Phase-matching requirements do not to provide a unique configuration for 3-photon down-conversion due to the high number of degrees of freedom in k-vector space. 5 degrees of freedom remain unspecified, if only a pump wave is inserted. Even if in addition to the pump wave one of the subhar-monic waves is completely specified or if all subharmonic waves forced to be collinear, a whole spectrum of phase-matched processes is expected.

§ The realization of a χχ (3)-OPO appears possible using nonlinear optical crystals but requires exceedingly high pump intensity close to the damage threshold of the optical material. An oscillation threshold of 43 W for tight focusing is expected in the case of a BBO crystal cut for degenerate 3-photon down-conversion (355 nm → 1064 nm). The χ(3)-OPO exhibits a phase transition of first order leading to a separation of the oscillation threshold and the threshold for star state generation

§ Direct star state generation does not appear feasible with presently avail-able χ(3)-nonlinear media. Minimum requirements for a material to render star state generation possible are a χ(3)-nonlinear susceptibility exceeding 5· 10-17 m2/V2 with a damage threshold on the scale of 1014 W/m2. Star state generation may be possible employing a cascaded χ(2)-process inside an optical resonator which yields an effective χ(3)-coupling with a coupling strength about 7 orders of magnitude stronger than for a native χ(3)-nonlinearity.

§ Only an unsatisfactory coincidence count rate of 3 photon triples per day may be expected for non-degenerate 3-photon down-conversion with com-mercially available laser sources. If phase-matching may be achieved in an optical fiber, photon triple generation rates of 1000 triples per second may be expected.

§ Parametric χχ (3)-de-/amplification yielding an expected amplification of about 35%/cm close to the damage threshold of the nonlinear material is calculated.

5.2 Perspectives 171

5.2 Perspectives

5.2.1 Possible Further Projects

5.2.1.1 Quantum state preparation in 2-photon down-conversion

Based on the work on single-photon Fock state reconstruction a number of follow-up projects are possible to further pursue the aims of quantum state preparation and characterization. These projects use the know-how in the field of photon counting and pulsed homodyne tomography that has been acquired during this dissertation.

Photon Added States Single-photon added coherent states (compare chapter 2.1) may be generated using the process of seeded 2-photon down-conversion in a setup as sketched in Figure 5-1. In this scheme a weak coherent state induces the 2-PDC in the down-conversion crystal. A trigger detector in the idler channel is used to condition measurements in the signal beam to the cases where a photon has been detected in the idler channel. In this way a single-photon added coherent state is prepared in the signal channel. In addition to the already existing experimental setup either the count rate has to be increased by a factor of 30 or more or an active phase stabiliza-tion has to be implemented to stabilize the relative phase of the seed/signal beam with respect to the local oscillator.

Photon-added states provide one out of infinitely many possible continuous paths from particle states of the light field (Fock states) to wave states (coherent states)

Figure 5-1 Schematic for the generation of single-photon added states. If the trig-ger detector fires, a photon has been added to the seed beam, thus producing a photon added state in the signal beam.


by varying the coherent amplitude α in the seed beam. This experiment would therefore allow an in depth investigation of the wave particle duality for the elec-tromagnetic field. One could for example shine this state onto a beam splitter and monitor the behavior of the photon count statistics at its output ports.

Displaced Fock States Displaced single-photon Fock states (chapter 2.1) may be produced using the experimental scheme pictured in Figure 5-2: first, a Fock state is generated in 2-PDC conditioned on the measurement of a click in a trigger detector. This state is then overlapped with a coherent state of considerable excitation (|α | >> 1) at a beam splitter with almost unit transmittivity r and thus displaced in phase space by r α.

Overlapping displaced Fock states with unknown quantum states allow direct measurements of the Wigner function of the unknown state.

Repeated 2-Photon Down-Conversion The options of using repeated 2-photon down-conversion to generate higher n Fock states and also – in an advanced setup introducing an additional coherent displace-ment into the loop – to produce arbitrary states of the light field (up to a certain maximum photon number n) have been discussed in detail in chapter 2.4.. Re-peated 2-photon down-conversion is a newly developed experimental scheme that has not been implemented so far and that – in principle – allows the generation of arbitrary states of the light field, if the photon addition in the down-converter is combined with a displacement in phase space. As for the displaced Fock states such a displacement may be achieved by inserting an additional beam splitter in the optical loop.

Figure 5-2 Schematic for the generation of displaced Fock states: A Fock state is overlapped with a coherent beam at a beam splitter with high transmittivity. In this way a coherent excitation is added while the quantum fluctuations of the Fock state are nearly preserved.


Squeezed Fock states Squeezed Fock states may be generated, if the Fock states prepared by conditional measurements are passed through a squeezer. These states are no longer eigenstates of the photon number operator and the rotational symmetry of the Wigner function is replaced by an elliptical symmetry. Similarly to the Fock states these states are no minimum uncertainty states but will exhibit sub-Poissonian photon number statistics (with a Q-parameter depending on the degree of squeezing).

2-Photon Down-Conversion with periodically poled materi-als One interesting challenge in 2-photon down-conversion is to achieve pair produc-tion probabilities exceeding 1% or even 10%, since in this case higher order proc-esses start to contribute significantly to the radiation produced. This would allow to produce and investigate higher particle correlations with acceptable count rates following the lines of the GHZ-experiment performed in the group of Prof. Zeilin-ger. Higher production rates in 2-photon down-conversion may be achieved by either using higher average pump light intensities (shorter pump pulses do not augment the production rate since this rate depends on the pump power in a linear fashion: no matter how the pump photons are distributed in time, each experiences the same scattering probability) or by increasing the efficient nonlinearity.

This could be achieved by employing periodically poled nonlinear optical materials for the down-conversion. For experiments where the photon pairs have to be emit-ted into a single mode e.g. to generate states with higher particle correlations, the maximum length of the nonlinear crystal that may be used is limited by the degree of group velocity dispersion and the filtering in the detection channels. These re-straints may be removed, if the primary aim is to produce photon pairs with a high rate.

Figure 5-3 Setup for the generation of n>1 Fock states in repeated parametric fluorescence.


5.2.1.2 3-Photon Down-Conversion

Perspectives and possible experiments employing the process of 3-photon down-conversion have been discussed in detail in part 4 of this thesis. Even though the generation of star states is not a realistic experimental option, a first demonstration of 3-photon down-conversion would spur a new field of research, exploiting the strong three particle correlations of the photon triples produced. If sufficient count rates could be achieved, a GHZ-type experiments and a demonstration of entangled entanglement could be first experiments exploiting this process.

From the present perspective the most promising candidates for 3-photon down-conversion with sufficiently high photon triple production rates are fiber based schemes either employing photonic band gap materials (transversely drilled fibers) or doped fibers. The main issue is to achieve phase matching for this process in fibers. Nonlinear optical crystals are no realistic option due to the low expected triple production rates. Organic crystals with higher χ(3)-nonlinearities are not available in sufficiently big sample with of an acceptable optical quality and – in general – exhibit rather low damage thresholds.

5.2.1.3 Energy Resolving Photodetection

Energy resolving photodetectors are photodetectors which do not only distinguish

between no photons 0=n and photons falling onto the detector 0>n , but produce a signal which is proportional to the number of photons measured. The technology for these kind of detectors exists, but has – to my knowledge – not been applied for quantum optical experiments so far. Energy resolving photodetectors are either based on avalanche solid state photodiodes operated in the proportional-ity range with a single photon resolution or on the generation and measurement of cooper pairs in supra-conducting materials.

Energy resolving photodetectors may be used to

§ Measure photon number distributions directly and apply this as a new characterization tool for optical quantum states

§ Use the ability to project an optical state onto a specific photon number

state n for conditional state preparation.

5.2.1.4 CW-Quantum Information Processing

In the past a number of experiments have been performed which have used con-cepts developed for discrete optical observables in connection with single-photon counters and transferred them to the basis of quadrature amplitudes and optical quantum noise measurements. Examples include the successful implementation of an EPR-experiment and quantum teleportation with continuous variables.

In performing a sequential QND-measurement in 1996 we have been able to dem-onstrate cw-triple beams exhibiting quantum correlations which allowed to reduce the variance measured in one beam to below the standard quantum limit, if the information obtained by detecting the other two beams were used. Unfortunately these measurements were limited to one quadrature component of the electromag-netic field modes. Realizing a sub-shot noise measurement on two orthogonal quadrature components of the electromagnetic field for cw-triple beams will allow to perform a cw-GHZ experiment.


Furthermore, quantum correlation in cw-beams may be employed to realize ex-perimental schemes for quantum error correction or purification schemes. Efforts along these lines might provide a useful complement to other experimental ap-proaches based on discrete variables. They may also be more easy to implement.

As mentioned above we are in a particularly good situation to realize cw-quantum information processing schemes, since we may build on a broad experimental ex-pertise in this field.

66 AAppppeennddiicceess

6.1 Optical Double Slit Wigner Function In this chapter a reconstruction of the Wigner function of the transverse optical field distribution behind a double slit is presented. This is intended as an example of the tomographic method and will also allow us to reflect on our preconceptions about what is quantum and what is not.

The diffraction pattern of a double slit illuminated with a plane wave calculated according to classical optical theory is given by

2

2211

221

2

)2/(Sinc

2

)2/(Sinccos),(

r

e

r

dxkb

r

e

r

dxkbxd

ri

ezxI

ikrikr

S

ikr

−+

−≈′∝ ∫ θ

λ

(6.1.1)

where the optical wave propagates along the z-direction, 222/1 )2/( dxzr −+= ,

k = 2π/λ denotes the wave vector of the optical wave, b the width of the slits, and d the distance between the two slits. and where we have reduced the 2-dimensional problem to a one-dimensional due to its translational invariance along the y direc-tion (assuming the beam diameter to be much larger than the separation of the two slits).

From this function we can obtain the Wigner function of the transverse field distribution (which should not be confused with the Wigner function in the Hilbert-space of the quadrature am-plitudes used before). To do this we realize that the time evolu-tion of the optical wave diffracted from the double slit results in a sheering motion in phase space (compare the figure at the page margin). Now, since a sheering can always be split into a

Figure 6-1 Diffraction pattern and calculated Wigner function for the double slit

Double Slit Diffraction Pattern


rotation and a stretch, the diffraction pattern at each point in space corresponds to a marginal distribution at a given angle stretched by a certain amount. In particular, the transverse field distribution right behind the screen corresponds to the marginal distribution of the spatial coordinate and the field distribution in the far field corre-sponds to the marginal distribution of the momentum coordinate.

Taking this into account we obtain the Wigner function of the double slit:

≤−−≤−−−−−

⋅=otherwise,0

x2 if)),2((Sinc)cos(2)2(

2 if)),2((Sinc)2(

),( 121 bxbppdxb

bdxdxbpdxb

pxW bπ (6.1.2)

Figure 6-2 Experimental layout and data processing for the reconstruction of the double slit Wigner function:

178 6 Appendices

With its two distinct positive regions and the interference pattern in between this Wigner function closely resembles that of a Schrödinger cat (Figure 2-3). Some differences in appearance – like the dove tails extending along the p-directions – owe their existence to the fact that the transmission function for each of the slits is a step function and not of Gaussian shape.

Of course, the main difference between the Schrödinger cat Wigner function dis-cussed in chapter 2.1 and the Wigner function of the double slit results from the fact that both functions are defined in different phase spaces: The phase space of the double slit diffraction Wigner function is that of the transverse field, whereas the phase space we have used in chapter 2.1 is the phase space of the canonical coordinates for the harmonic oscillator modes of the electromagnetic field.

In a straightforward experimental configuration we have reconstructed the Wigner function of the transverse field of a light field behind a double slit. The experimen-tal procedure is depicted in Figure 6-2: A coherent cw-laser beam at 532 nm illu-minates a double slit. A focusing lens is moved in the z-direction to image the dif-fraction pattern at different distances from the double slit onto a screen. The image is registered with a CCD-camera. Each picture is converted into a 1-dimensional marginal distribution, all marginal distributions together reproduce the diffraction pattern of the double slit (Figure 6-1, left diagram).

The inverse Radon transformation is applied to this data and yields an experimental reconstruction of the Wigner function of the transverse field produced by our dou-ble slit (Figure 6-3). As expected the Wigner function exhibits two positive regions – corresponding to the two slits – and an oscillatory behavior with strong negativ-ities along the x = 0 axis – due to the interference of the two coherent parts of the wave being transmitted through the two slits.

The diagonal stripe pattern that is visible in the reconstructed Wigner function is an artifact which can be attributed to the fact that our measurements did not cover the

Figure 6-3 Double Slit Wigner function reconstructed from experimental data

Experiment


full angular range of marginal distributions (for a more detailed discussion of the effects of limited angular ranges in tomographic reconstructions see226).

In quantum optics negativities of the quasiprobability distributions such as the Wigner function are considered to be a property of non-classical states227. The dou-ble slit Wigner function clearly exhibits such negativities along the x-axis. But here we are dealing with a purely classical experiment – or are we not?

Clearly in the case of the double slit we reconstruct the Wigner function of a clas-sical field distribution (i.e. measurable with almost arbitrary precision). The nega-tivities arise from interference of the light originating from the two coherently il-luminated slits. This is nothing more – or less – than a consequence of the wave nature of the light field.

It is only if we view this experiment in the framework of quantum electrodynamics, that things start to appear in a different light. If we accept that the light field is quantized, then we can – conceptually – arrange things in such a way that the aver-age photon number in the setup is much less than one at any given instant in time. In this case the field can no longer be measured without disturbing it, since every measurement will have considerable influence on its further time evolution. In this picture the wave function of a single light particle extends across the two slits, photons traveling through the setup interfere with themselves.

From early demonstration experiments we know that the same double slit pattern is reproduced, even if only single-photons pass through the apparatus228 – the power level does not change the physics of the experiment. The light particles do not in-teract with each other in this purely passive, linear setup. Thus it makes no differ-ence, whether the average photon number in the setup is less than one or a few 1018 photons like in our experiment.

From this point of view the optical experiment resembles an experiment performed by Christian Kurtsiefer229 in our group. He investigated the diffraction pattern pro-

226 C. Kurtsiefer, “Atomoptische Experimente zu nichtklassischen Zuständen der Bewegung von meta-stabilen Edelgasatomen und Atom-Photon-Paaren“, Dissertation, S. 128 ff. (UFO Dissertation Band 315, 1997) 227 Ching Tsung Lee, “Measure of the nonclassicality of nonclassical states”, Phys. Rev. A 44, 2775 (1991) 228 Parker S., “Single-photon double-slit interference-A demonstration.”, Am. J. of Phys. 40, 1003 (1972) 229 C. Kurtsiefer, T. Pfau, J. Mlynek, “Measurement of the Wigner function of an ensemble of helium atoms”, Nature 386, 150 (1997)

Figure 6-4 Vertical and horizontal cut through experimentally reconstructed Wigner function (along the colored lines shown in Figure 2-5). Strong oscillations and negative values can clearly be seen along the p-axis.

Discussion

180 6 Appendices

duced by a beam of excited He-atoms passing through a double slit and obtained a Wigner function of the transverse motional state of the atoms. This experiment clearly strikes us as a quantum experiment – mainly because we are used to think about atoms as particles. If we observe wave phenomena associated with matter, this is something which does not occur in the classical world. In contrast to this, light behaves as a wave in the classical world. That might be the reason why with light we are more excited in seeing aspects of particle behavior– like shot noise or bunching/anti-bunching for example – than observing wave phenomena like the double slit diffraction pattern.

In fact our optical double slit experiment can easily be explained employing only Maxwell´s classical theory of electromagnetic waves. That is why I would con-sider it a purely classical experiment. The non-classical aspects of the experiment only arise, if we employ the concept of light particles, photons with an energy ωh or light particles, photons. On the other hand we cannot understand the atom inter-ference experiment of C. Kurtsiefer et al. without employing quantum theory and Planck´s constant h to make a connection between the mass and velocity of the

atom and its wave vector ( kvmpr

hvr =⋅= ). Thus I consider the atomic double slit

experiment a non-classical experiment.

Thus, what we take for granted with light is what we find astonishing with matter and the other way around, whereas in fact, in a full quantum description of either, both have particle and wave aspects.

In concluding this detour I would like to mention that a number of non-classical double slit experiments have been performed in optics exploiting the non-local character of photon pairs230,231,232. The continued interest in the double slit might be due to the fact that “The simple double slit experiment contains the very heart of quantum mechanics”, as R. Feynman put it233.

230 E.J.S. Fonseca, P.H. Souto Ribeiro, S. Padua, Ch. Monken CH, “Quantum interference by a nonlocal double slit.”, Physical Review A 60, 1530 (1999) 231 Hong CK, Noh TG, “Two-photon double-slit interference experiment”, J. Opt. Soc. of Am. B 15, 1192 (1998) 232 Zeilinger et al., ??, 1997/8 233 R. Feynman, Lectures on physics – Part III, (Addison Wesley, USA, 1965)

Figure 6-5 Reconstructed Wigner function of a beam of He-atoms behind a double slit obtained by Ch. Kurtsiefer et al229: (a) reconstruction from experimental data, (b) reconstruction from numerically simulated data covering the same limited range of angles as the experiment.


6.2 Bibliography Citations in the alphabetic order sorted according to first author:

1. G.L. Abbas, V.W.S. Chan, and T.K. Yee, Opt. Lett. 8, 419 (1983) 2. G.S. Agarwal, K. Tara, Phys. Rev. A 43, 492 (1990) 3. G.M. D’Ariano, M.B.A. Paris, Phys. Rev. A 49, 3022 (1993) 4. G.M. D’Ariano, C. Macchiavello, M.B.A. Paris, Phys. Rev. A 195, 31 (1994) 5. G.M. D’Ariano, U. Leonhardt, H.Paul, Phys. Rev. A 52, R1801 (1995) 6. A. Aspect , P. Grangier, G. Roger, “Wave particle duality for a single-photon”, J. of Opt. 20,

119 (1989) 7. G.M. D’Ariano, M.R. Sacchi, and P. Kumar, Phys. Rev. A 59, 826 (1999) 8. M. Ban, Phys. Rev. A 49, 5078 (1994) 9. M. Ban, Opt. Comm. 130, 1281 (1996) 10. K. Banaszek, P. Knight, “Quantum Interference in three-photon down-conversion”, Phys. Rev.

A 55, 2368 (1997) 11. M. Beck, M.G. Raymer, I.A. Walmsley, V. Wong, Opt. Lett. 18, 2041 (1993) 12. K. Bencheikh, J.A. Levenson, “Quantum nondemolition Demonstration via Repeated Backac-

tion Evading Measurements“, Phys. Rev. Lett. 19, 3422 (1995) 13. K. Bencheikh, R. Storz, K. Schneider, K. Jäck, M. Lang, J. Mlynek, and S. Schiller, „Absolute

frequency stabilization of a continuous-wave optical parametric oscillator to the sub-kHz level”, OSA Tops 19, 392 (1998)

14. J. Bertrand, and P. Bertrand, Found. Phys. 17, 397 (1987) 15. D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, A. Zeilinger, „Experimental

quantum teleportation“, Nature 390, 575 (1997) 16. D. Bouwmeester, J.-W. Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, "Observation of

three-photon Greenberger-Horne-Zeilinger entanglement", Phys. Rev. Lett. 82, 1345 (1999) 17. R.W. Boyd, Nonlinear Optics, p. 93 (Academic Press, 1992, USA) 18. V.B. Braginsky, “Classical and quantum restrictions on the detection of weak disturbances of a

macroscopic oscillator”, Sov. Phys. JETP 26, 831 (1968) 19. S.L. Braunstein, R. McLachlan, “Generalized Squeezing“, Phys. Rev. A 35, 1659 (1987) 20. G. Breitenbach, “Quantum state reconstruction of classical and nonclassical light and a cryo-

genic opto-mechanical sensor for high-precision interferometry”, dissertation (UFO Disserta-tion, Bd. 349, 1998, Konstanz)

21. G. Breitenbach, T. Müller, S. Pereira, F.-P. Poizat, S. Schiller, J. Mlynek, „Squeezed Vacuum from a Monolithic Optical Parametric Oscillator“, J. Opt. Soc. Am. B 12, 2304 (1995)

22. G. Breitenbach, S. Schiller, „Homodyne Tomography of classical and non-classical light”, J. Mod. Opt. 44, 2207 (1997)

23. G. Breitenbach, S. Schiller, J. Mlynek, “Measurement of the quantum states of squeezed light“ , Nature 387, 471 (1997)

24. G. Breitenbach, F. Illuminati, S. Schiller, and J. Mlynek, “Broadband detection of squeezed vacuum: A spectrum of quantum states“, Europhys. Lett. 44, 192 (1998)

25. H.-J. Briegel, I. Cirac, P. Zoller, „Quantencomputer“, Phys. Blätt. 55, 37 (1999) 26. R. Bruckmeier, K. Schneider, S. Schiller, and J. Mlynek „Quantum nondemolition measure-

ments improved by a squeezed meter input” , Phys. Rev. Lett. 78, 1243 (1997) 27. R. Bruckmeier, H. Hansen, S. Schiller, and J. Mlynek “Realization of a paradigm of quantum

measuremetns: the squeezed light beam splitter”, Phys. Rev. Lett. 79, 43 (1997) 28. R. Bruckmeier, H. Hansen, S. Schiller, and J. Mlynek “Repeated continuous quantum nondemo-

lition measurements”, Phys. Rev. Lett. 79, 1463 (1997) 29. R. Bruckmeier, and S. Schiller, “Properties of the linearized quantum optical bus“, Phys. Rev.

A 59, 750 (1999) 30. M. Brune, S. Haroche, S. Lefèvre, J.M. Raimond, and N. Zagury, Phys. Rev. Lett 65, 976

(1990); M. Brune, S. Haroche, S.J.M. Raimond, L. Davidovich, and N. Zagury, Phys. Rev. 45, 5193 (1992), K. Vogel, V.M. Akulin, and W.P. Schleich, Phys. Rev. Lett. 71, 1816 (1993)

31. K.E. Cahill , R. J. Glauber, Phys. Rev. 177, 1857 (1969) 32. C.M. Caves K.S. Thorne, R.W.P. Drever, V.D. Sandberg, M.Zimmermann, Rev. Mod. Phys 52,

341 (1980) 33. Campos, Saleh, Teich, Phys. Rev. A 40, 1371 (1989) 34. I.L. Chuang, L.M.K. Vandersypen, X. Zhou, D.W. Leung, S. Lloyd, Nature 393, 143 (1998) 35. Cirac, Zoller 36. J.I. Cirac, R. Blatt, A.S. Parkins, P. Zoller, “Preparation of Fock States by Observatin of Quan-

tum Jumps in an Ion Trap”, Phys. Rev. Lett. 70, 762 (1993) 37. J.I. Cirac, R. Blatt, A.S. Parkins, and P. Zoller, Phys. Rev. Lett. 70, 762 (1993)

182 6 Appendices

38. C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, “Atom-Photon interactions”, (Wiley & Sons, USA, 1988)

39. P.A. Chollet et al., Rev. Phys. Appl. 22, 1221 (1987) 40. M. Dakna, T. Anhut, T. Opatrný, L. Knöll, and D.-G. Welsch, “Generating Schrödinger cat-like

states by means of conditional measurements on a beam-splitter”, Phys. Rev. A 55, 3184 (1997) 41. M. Dakna, L. Knöll, D.-G. Welsch, „Photon-added state preparation via conditional measure-

ment on a beam splitter“, Opt. Comm 145, 309 (1998) 42. A. Elitzur, L. Vaidman, Found. Phys. 23, 987 (1993) 43. T. Felbinger, “Simulation nichtlinearer optischer Systeme mit der Quantentrajektorienmetho-

de”, Diploma Thesis, University of Konstanz (1996) 44. T. Felbinger, S. Schiller, J. Mlynek, “Oscillation in 3-photon down-conversion and generation

of non-classical states”, Phys. Rev. Lett. 80, 492 (1997) 45. R. Feynman, Lectures on physics – Part III, (Addison Wesley, USA, 1965) 46. E.J.S. Fonseca, P.H. Souto Ribeiro, S. Padua, Ch. Monken CH, “Quantum interference by a

nonlocal double slit.”, Physical Review A 60, 1530 (1999) 47. J.D. Franson, “Bell inequality for position and time”, Phys. Rev. Lett. 62, 2205 (1989) 48. A. Furusawa, J.L. Srensen, S.L. Braunstein, C.A. Fuchs, H.J. Kimble, E.S. Polzik, „Uncondi-

tional quantum teleportation“, Science (1998) 49. R.J. Glauber, Phys. Rev. Lett. 10, 84 (1963) 50. I.M. Gel’fand, G.E. Shilov, Generalized Functions, (Academic Press, San Diego, 1965) 51. R. Ghosh, C.K. Hong, Z.Y. Ou, and L. Mandel, "Interference of two photons in parametric

down-conversion" , Phys. Rev. A 34, 3962 (1986) 52. P. Grangier, J.-F. Roch, G. Roger, Phys. Rev. Lett. 66, 1418 (1991) 53. P. Grangier, G. Roger, A. Aspect, “Experimental evidence for a photon anticorrelation effect on

a beam splitter : a new light on single-photon interferences” , Europhys. Lett. 1, 173 (1986) 54. D.M. Greenberger, M.A. Horne, A.Zeilinger, in M. Kafatos (editor) Bell’s Theorem, Quantum

Theory, and Conceptions of the Universe“ , p. 73-76 (Kluwer Academic, Dordrecht, Nether-lands, 1989)

55. D.M. Greenberger, M.A. Horne, A. Shimoni, A. Zeilinger, Am. J. of Phys. 58, 1131 (1990) 56. A. Zeilinger 57. S. Haroche et al., e.g. : Nature 400, 239 (1999) 58. T. Hashimoto, T. Yoko, S. Sakka, “Sol-gel preparation and third-order nonlinear optical

properties of TiO2 thin films”, Chem. Soc. Jap. 67, 653 (1994) 59. C.C. Havener, N. Rouze, W.B. Westerfeld, and J.S. Risley, “Experimental determination of the

density matrix describing collisionally produced H(n=3) atoms”, Phys. Rev. A 33, 376 (1986) 60. T. Herzog, P. Kwiat, H. Weinfurter, A. Zeilinger, “Delayed-choice quantum eraser: theory

and experiment”, QELS `95. Summaries of Papers 16. 5 (1995) 61. T. Herzog, P.G. Kwiat, H. Weinfurter, A. Zeilinger, “Complementarity and the Quantum

Eraser”, Phys. Rev. Lett. 75, 3034 (1995)M. Hillery, Phys. Rev. A 35, 725 (1987) 62. C. Hettich, “Untersuchungen zur Photonendrittelung und Erzeugung und Charakterisierung

von Ein-Photon-Fockzuständen”, diploma thesis, University of Konstanz, (1997) 63. J.H. Hildebrand: „Gilbert Newton Lewis“, Biogr. Mem. Nat´l Acad. Sci. 31, 210 (Columbia

Univ. Press, New York, 1958) 64. C.K. Hong and L. Mandel, "Experimental realization of a localized one-photon state" , Phys.

Rev. Lett. 56, 58 (1986) 65. Hong CK, Noh TG, “Two-photon double-slit interference experiment”, J. Opt. Soc. of Am. B

15, 1192 (1998) 66. P. Horowitz, W. Hill, The art of electronics, (Cambridge University Press, 1989, New York) 67. R.L. Hudson, Rep. Math. Phys. 6, 249 (1974) 68. J. Janszky, M.G. Kim, and M.S. Kim, Phys. Rev. A 53, 502 (1996) 69. Journal of Mod. Opt. 44, Nr. 11/12 (1997), Spec. Issue “Quantum State Preparation and

Measurement” 70. A. Joobeur, B.E.A. Saleh, and M.C. Teich, "Spatiotemporal coherence properties of entangled

light beams generated by parametric down-conversion", Phys. Rev. A 50, 3349 (1994) 71. A. Joobeur, B.E.A. Saleh, T.S. Larchuk, and M.C. Teich, "Coherence properties of entangled

light beams generated by parametric down-conversion: Theory and experiment", Phys. Rev. A 53, 4360 (1996)

72. J.F. Joung, G.C. Bjorklund, A.H. Kung, R.B. Milles, S.E. Harris, “Third-harmonic generation in phase-matched Rb vapor”, Phys. Rev. Lett. 27, 1551 (1971)

73. B.E. Kane, Nature 393, 133 (1998) 74. S.Y. Kilin, D.B. Horoschko, “Fock State Generation by the Methods of Nonlinear Optics”,

Phys. Rev. Lett. 74, 5206 (1995) 75. H.J. Kimble, M Dagenais, and L. Mandel, Phys. Rev. Lett. 39, 691 (1977) 76. T. Kiss, U. Herzog, U. Leonhardt, Phys. Rev. A 52, 2433 (1995) 77. J.C. Knight, T.A. Birks, P.St.J. Russell, D.M. Atkin, “All-silica single-mode optical fiber with

photonic crystal cladding”, Opt. Lett. 21, 1547 (1996) 78. H. Kobayashi et al., J. Appl. Phys. 74, 3683 (1993); K. Nassau, S.H. Wemple, Electron. Lett.

18, 450 (1982)


79. K. Koch, E. C. Cheung, G.T. Moore, S.H. Chakmakjian, J.M. Liu, “Hot Spots in Parametric Fluorescence with a Pump Beam of Finite Cross Section”, IEEE J. of Quant. El. 31, 769 (1995)

80. A. Kuhn et al., Appl. Phys. B 96, 373 (1999) 81. H. Kühn, D.-G. Welsch, W. Vogel, J. Mod. Opt. 41, 1607 (1994) 82. F. Kühnemann, K. Schneider, A. Hecker, A.A.E. Martis, W. Urban, S. Schiller, J. Mlynek,

“Photoacoustic trace gas detection with a single-frequency continuous-wave optical parametric oscillator”, Appl. Phys. B

83. J.R. Kuklinski, „Generation of Fock States of the Electromagnetic Field in a High-Q Cavity through the Anderson-localization”, Phys. Rev. Lett. 64, 2507 (1990)

84. Ch. Kurtsiefer, T. Pfau, J. Mlynek, “Measurement of the Wigner function of an ensemble of helium atoms“, Nature 386, 150 (1997)

85. C. Kurtsiefer, “Atomoptische Experimente zu nichtklassischen Zuständen der Bewegung von metastabilen Edelgasatomen und Atom-Photon-Paaren“, Dissertation, S. 128 ff. (UFO Disser-tation Band 315, 1997)

86. P.G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A.V. Sergienko, and Y. Shih: "New high-intensity source of polarization-entangled photons pairs" , Phys. Rev. Lett. 75, 4337 (1995)

87. P.G. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, and M.A. Kasevich: "Interaction-free measurements" , Phys. Rev. Lett. 74, 4763 (1995)

88. W.E. Lamb, “Anti-Photon”, Appl. Phys. B 60, 77 (1995) 89. C.K. Law, and J.H. Eberly, “Arbitrary Control of a Quantum Electromagnetic Field”, Phys.

Rev. Lett. 76, 1055 (1996) 90. C.K. Law, H.J. Kimble, “Deterministic generation of a bit-stream of single-photon pulses”, J.

of Mod. Opt. 44, 2067 (1997) 91. C.T. Lee, “Measure of the nonclassicality of nonclassical states“, Phys. Rev. Lett. 44, 2775

(1991) 92. C.T. Lee, “Theorem on nonclassical states“, Phys. Rev. Lett. 52, 3374 (1995) 93. D. Leibfried, D.M. Meekhof, B.E. King, C. Monroe, W.M. Itano, and D.J. Wineland, „Experi-

mental Determination of the Motional Quantum State of a Trapped Atom”, Phys. Rev. Lett. 77, 4281 (1996)

94. U. Leonhardt, “Measuring the Quantum State of Light” , (Cambridge University Press, 1997) 95. U. Leonhardt, H. Paul, G.M. D’Ariano, Phys. Rev. A 52, 4899 (1995) 96. U. Leonhardt, M.G. Raymer, Phys. Rev. Lett. 76, 1989 (1996) 97. U. Leonhardt, “Measuring the Quantum State of Light” (Cambridge Studies in Modern Optics,

1997) 98. A. Luis and L.L. Sánchez-Soto, “Conditional generation of field states in parametric down-

conversion”, Phys. Lett. A 244, 211 (1998) 99. N. Lütkenhaus, S.M. Barnett, Phys. Rev. A 51, 3340 (1995) 100. A. Luis and L.L. Sánchez-Soto, “Conditional generation of field states in parametric down-

conversion”, Phys. Lett. A 244, 211 (1998) 101. X. Maitre, E. Hagley, G. Nogues, C. Wunderlich, J.M. Raimond, S. Haroche, Phys. Rev. Lett.

79, 769 (1997) 102. L. Mandel and E. Wolf, Ref. Mod. Phys. 37, 231 (1965) 103. L. Mandel , Opt. Lett 4, 205 (1979) 104. L. Mandel, Phys. Scr. T 12, 34 (1986) 105. L. Mandel, E. Wolf, “Optical coherence and quantum optics“, (Cambridge University Press,

1995, New York) 106. L. Mandel, Phys. Rev. Lett. 49, 136 (1982) 107. T.W. Marshall, E. Santos, “Myth of a photon”, Preprint quant-ph/9711046 108. F. De Martini, G. Di Guiseppe, and M. Marrocco, “Single-Mode Generation of Quantum Pho-

ton States by Excited Single Molecules in a Microcavity Trap”, Phys. Rev. Lett. 76, 900 (1996) 109. F. De Martini, D. Boschi, M. Marroco, Proc. of the 3. Int. Aalborg Summer School (Nonlinear

Optics) (1996) 110. F. De Martini, O. Jedrikiewicz, P. Mataloni, “Generation of quantum photon states in an active

microcavity trap”, J. of Mod. Opt. 44, 2053 (1997) 111. R.L. De Matos Filho, and W. vogel, Phys. Rev. Lett. 76, 4520 (1996) 112. McFarlain, private communication 113. J.E. Midwinter, J. Warner, “The effects of phase matching method and of crystal symmetry on

the polar dependence of third-order nonlinear optical polarisation”, 16, 1667 (1965) 114. R.B. Miles, S.E. Harris, “Optical third-harmonic generation in alkali metal vapors”, IEEE J. of

Qu. El. 9, 470 (1973) 115. R.C. Miller, “Optical second harmonic generation in piezoelectrical crystals”, Phys. Rev. Lett.

5, 17 (1964) 116. P.W. Milonni, H. Fearn, A. Zeilinger, “Theory of two-photon down conversion in the presence

of mirrors”, Phys. Rev. A 53, 4556 (1996) 117. N. Moya-Cessa, P.L. Knight, Phys. Rev. A 48, 2479 (1993); 118. A. Muller, H. Zbinden, N. Gisin “Quantum cryptography over 23 km in installed under-lake

telecom fibre”, Europhys. Lett. 33, 335 (1996)

184 6 Appendices

119. M. Munroe, D. Boggarvarapu, M.E. Anderson, M.G. Raymer, “Photon statistics from the phase-averaged quadrature field distribution: theory and ultrafast measurement”, Phys. Rev. A 52, 52 (1995)

120. H. Nakatani et al., Appl. Phys. Lett. 53, 2587 (1988) 121. A. Napoli, A. Messina, “Conditional generation of non-classical states in a non-degenerate

two-photon micromaser: single-mode Fock state preparation. II”, J. of Mod. Opt. 44, 2093 (1997)

122. J. von Neumann, „Mathematical Foudatins of Quantum Mechanics“, (Princeton University Press, Princeton, NJ, 1955)

123. A.-S. F. Obada, G.M. Abd Al-Kader, Journal of Mod. Opt. 46, 263 (1999) 124. M. Okada, Appl. Phys. B. 18, 450 (1971) 125. F.A.M. De Oliveira, M. Kim, P.L. Knight, V. Puzek, Phys. Rev. A 41, 2645 (1990) 126. Orrit et al., Phys. Rev. Lett. 83, 2722 (1999) 127. Z.Y. Ou, “Nonlocal correlation in the realization of a quantum eraser”, Phys. Lett. A 226, 323

(1997) 128. Z.Y. Ou, “Quantum multi-particle interference due to a single-photon state” , Qu. Semiclass.

Opt. 8, 315 (1996) 129. Z.Y. Ou, “Parametric down-conversion with coherent pulse pumping and quantum interference

between independent fields”, Qu. Semiclass. Opt. 9, 599 (1997) 130. Z.Y. Ou, L. Mandel, Phys. Rev. Lett. 61, 50 (1988) 131. J.-W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, "Experimental entanglement

swapping: Entangling photons that never interacted" , Phys. Rev. Lett. 80, 3891 (1998) 132. Parker S., “Single-photon double-slit interference-A demonstration.”, Am. J. of Phys. 40, 1003

(1972) 133. R. Paschotta, M. Collett, P. Kürz, K. Fiedler, H.A. Bachor, J. Mlynek, “Bright Squeezed Light

from a Singly Resonant Frequency Doubler”, Phys. Rev. Lett. 72, 214 (1994) 134. S.F. Pereira, Z.Y. Ou, H.J. Kimble, “Back action evading measurements for quantum non-

demolition measurements and optical tapping”, Phys. Rev. Lett. 72, 214 (1994) 135. H. Puell, K. Spanner, W. Falkenstein, W. Kaiser, C.R. Vidal, “Third-harmonic generation of

mode-locked Nd-glass laser pulses in phase-matched Rb-Xe mixtures”, Phys. Rev. A 14, 2240 (1976)

136. P. Qiu, N.J. Fisch, “Third-harmonic generation in β-BaB2O4”, App. Phys. B 45, 225 (1988) 137. P. Qiu and A. Penzkofer, Appl. Phys. B, 45, 225 (1988) 138. J. Radon, Berichte der sächsischen Akademie der Wissenschaften, Leipzig, Math.-Phys. Kl. 69,

262 (1917) 139. J.G. Rarity, P.R. Tapster, R. Loudon, “Non-classical interference between independent sour-

ces”, preprint quant-ph/9702032, (1997) 140. J.M. Rax, N.J. Fisch, “Third-harmonic generation with ultrahigh-intensity laser pulses“, Phys.

Rev. Lett. 69, 772 (1992) 141. Th. Richter, Phys. Rev. A 211, 327 (1996) 142. A. Royer, “Measurement of quantum states and the Wigner function”, Found. Phys. 19, 3

(1989) 143. S. Schiller, S. Kohler, R. Paschotta, and J. Mlynek, “Squeezing and quantum nondemolition

measurements with an optical parametric amplifier”, Appl. Phys. B 60, S77-88 (1995). 144. K. Schneider, R. Bruckmeier, H. Hansen, S. Schiller, and J. Mlynek, “Bright squeezed light

generation by a continuous-wave semi-monolithic parametric amplifier”, Optics Letters 21, 1396 (1996)

145. K.Schneider, P. Kramper, S .Schiller, and J. Mlynek, “Toward an optical synthesizer: a single-frequency parametric oscillator using periodically poled LiNbO3”, Opt. Lett. 22, 1293 (1997)

146. K. Schneider, S. Schiller, “Narrow-linewidth, pump-enhanced singly-resonant parametric oscil-latior pumped at 532 nm”, Appl. Phys. B 65, 775 (1997)

147. Th. Schulze, B. Brezger, R. Mertens, M. Pivk, T. Pfau, and J. Mlynek, „Writing a superlattice 148. M.O. Scully and M.S. Zubairy: "Quantum Optics" (Cambridge University Press 1997) 149. R. Short, L. Mandel, “Observation of sub-Poissonian photon statistics”, Phys. Rev. Lett. 57,

691 (1983) 150. J.H. Shapiro, H.P. Yuen, J.A. Machado Mata, IEEE Trans. Inf. Theory IT25, 179 (1979) 151. J.H. Shapiro, H.P. Yuen, IEEE Trans. Inf. Theory IT-26, 78 (1980) 152. Y.H. Shih, C.O. Alley, Phys. Rev. Lett. 61, 2921 (1988) 153. M. Sinclair et al., in J. Messier et al. (ed.), “Nonlinear Optical Effects in Organic Polymers”, p.

29 (Kluwer Academic, Dordrecht, Netherlands, 1989) 154. R.E. Slusher, L.W. Hollberg, B. Yurke, J.C. Mertz, J.F. Valley, “Observation squeezing by four

wave mixing in a cavity”, Phys. Rev. Lett. 55, 2409 (1985) 155. R.E. Slusher, J. Kimble et al., Spec. Issue on Squeezed Light JOSA B 4, 1465 (1987), first

result in JOSA B 1986 156. D.T. Smithey, M. Beck, and M.G. Raymer, “Measurement of the Wigner Distribution and the

Density Matrix of a Light Mode using Optical Homodyne Tomography: Application to Squeezed States and the Vacuum”, Phys. Rev. Lett. 70, 1244 (1993)

157. Smithey, Phys. Rev. A 70, 1244 (1993)


158. D.T. Smithey, M. Beck, M.G. Raymer, and A. Faridani, Phys. Rev. Lett. 69, 2650 (1992) 159. D. Smithey, “Complete experimental state determination of classical and nonclassical light“,

Ph.D. thesis, Univesity of Oregon (1993) 160. S. Song, C.M. Caves, and B. Yurke, Phys. Rev. A 41, 5261 (1990) 161. O. Steuernagel, “Synthesis of Fock states via beam splitters”, Opt. Comm. 138, 71 (1997) 162. R.H. Stolen, H.W.K. Tom, “Self-organized phase-matched harmonic generation in optical

fibers”, Opt. Lett. 12, 585 (1987) 163. U. Strößner, A. Peters, J. Mlynek, S. Schiller, J.-P. Meyn, R. Wallenstein, “Single-frequency

continuous-wave radiation from 0.77 to 1.73 µm generated by a green-pumped optical para-metric oscillator with periodically poled LiTaO3”, Opt. Lett. 24, 1602 (1999)

164. E.C.G. Sudarshan, Phys. Rev. Lett 10, 277 (1963) 165. P. R. Tapster, J.G. Rarity, P.C.M. Owens, Violation of Bell´s Inequality over 4 km of Optical

Fiber, Phys. Rev. Lett. 73, 1923 (1994) 166. W. Tittel, J. Brendel, T. Herzog, H. Zbinden, N. Gisin, Non-local two-photon correlations using

interferometers physically separated by 35 meters“, Europhys. Lett. 40, 595 (1997) 167. U.M. Titulaer, and R.J. Glauber, Phys. Rev. 140, B 676 (1965) 168. I.V. Tomov et al., Appl. Opt. 31, 4172 (1992) 169. R.Q. Twiss, A.G. Little, R. Hanbury Brown, Nature 180, 324 (1957) 170. K. Vogel, V.M. Akulin, and W.P. Schleich, „Quantum State Engineering of the Radiation

Field“, Phys. Rev. Lett. 71, 1816 (1993) 171. B. T. H. Varcoe, S. Brattke, M. Weidinger, H. Walther, “Preparing Pure Photon Fock States of

the Radiation Field”, Nature 403, 743 (2000) 172. V. Vedral, and M.C. Plenio, “Entanglement Measures and Purification Procedures”, Phys.

Rev. A 57 , 1619 (1998) 173. M. Venkata Satyanarayana, Phys. Rev. D 32, 400 (1985) 174. W. Vogel and D.-G. Welsch, Lectures on Quantum Optics (Akademie-Verlag, Berlin, 1994)T.

C. Weinacht, J. Ahn, and P.H. Buckbaum, „Controlling the shape of a quantum wavefunction“, Nature 397, 233 (1999)

175. W. Vogel, Risken, Phys. Rev. A 40, 2847 (1989) 176. W. Vogel, „Nonclassical states: an ovservable criterion“, Phys. Rev. Lett. 84, (2000), in press 177. D.F. Walls, and G.J. Milburn, Quantum Optics (Cambridge University Press, Cambridge, 1995) 178. D.F. Walls, Nature 324, 210 (1986) 179. C.C. Wang, “Empirical relation between the linear and the third-order nonlinear optical sus-

ceptibilities”, Phys. Rev. B 2, 2045 (1970) 180. K. Watanabe, and Y. Yamamoto, Phys. Rev. A 38, 3556 (1988) 181. M.J. Weber, “Handbook of Laser Science and Technology, vol. 3”, (CRC Press, Florida, 1986) 182. G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, A. Zeilinger, Violation of Bell's inequality

under strict Einstein locality conditions“, Phys. Rev. Lett. 81, 5039 (1998) 183. E.P. Wigner, Phys. Rev. 40, 749 (1932) 184. D.L. Williams, D.P. West, A. King, “Quasi-phasematched third harmonic generation”, Laser

photonics group, Department of Physics and Astronomy, University of Manchester, Manchester M13 9PL, UK (1996)

185. Wineland et al., 2000 186. A. Wunsche, Quantum Optics 3, 359 (1991) 187. J.J. Wynne, “Optical third-order mixing in GaAs, Ge, Si, InAs”, Phys. Rev. 178, 1295 (1969) 188. Yamamoto et al., Nature 397, 500 (1999) 189. A. Yariv, Quantum Electronics – 3rd ed., p. 400 (John Wiley & Sons, 1989, USA) 190. K. Yoon-Ho, Y. Rong, S.P. Kulik, Yanhua Shih, M.O. Scully, “Delayed "choice" quantum

eraser”, Phys. Rev. Lett. 84, 1 (2000) 191. H.P. Yuen, J.H. Shapiro, IEEE Trans. Inf. Theory IT-24, 657 (1978) 192. H.P. Yuen, J.H. Shapiro, “Quantum Statistics of Homodyne and Heterodyne Detection”, in

Coherence and Quantum Optics IV, ed. L. Mandel, and E. Wolf (Plenum, New York) 193. H.P. Yuen, V.W.S. Chan, Opt. Lett. 8, 177 (1983) 194. B. Yurke, and D. Stoler, “Measurement of amplitude probability distributions for photon-

number-operator eigenstates”, Phys. Rev. A 36, 1955 (1987) 195. B. Yurke, W. Schleich, and D.F. Walls, Phys. Rev. A 42, 1703 (1990), 196. A. Zeilinger, “Physik der Quanteninformation”, Spektrum der Wissenschaft, A59 (Febr. 1999) 197. A. Zeilinger, M.A. Horne, H. Weinfurter, and M. Zukowski, "Three-particle entanglements

from two entangled pairs" , Phys. Rev. Lett. 78, 3031 (1997) 198. C. Zhu and C.M. Caves, “Photocount distributions for continuous-wave squeezed light”, Phys.

Rev. A 41, 475 (1990)

186 6 Appendices

6.3 Acknowledgements First of all I want to thank my wife Gesa for her support in 1001 occasions during the last 3 1/2 years, her tolerance when I spent another night working, her faith and readiness to motivate me when I was down and to hang out with me when I was stressed. She has also made a direct contribution to this thesis by pulling out nu-merous orthographic, rhetoric and layout errors. Thanks = :-).

I thank my parents, who are still taking a very vivid interest in the sometimes rather peculiar doings of their son. Without their continuous support in many ways pro-gress would have been much more difficult – and I would also very likely be a different person today.

The results presented here would not have been possible without the support and stimulation by various people around me. I want to thank:

§ my diploma students Christian Hettich and Thomas Aichele for their commit-ment to our research project. I am particularly grateful to Christian Hettich for the joint conceptual work in the beginning of my dissertation and to Thomas Aichele for his patience and for proof reading greater parts of this thesis.

§ the Danish exchange graduate student Peter Lodahl, who brought a bit of Scandinavian attitude to the south-Baden Konstanz and for his contributions to our experiment during his 9-month stay at the University of Konstanz.

§ Alex Lvovsky for sharing part of my future dreams in quantum optics and his critical attitude, which kept me hunting for a deeper physical understanding of the single-photon Fock state tomography during the work on this thesis.

§ my supervisor Stephan Schiller for his support, various constructive discus-sions and for always keeping an eye on the progress of the work. I am also grateful to Achim Peters for looking after things when Stephan was moving to Düsseldorf and to Oliver Benson for common speculations about the future of optics quantum optics and for his various suggestion on how to improve this thesis.

§ Jürgen Mlynek who did not only provide the financial, organizational and physical setting for the work presented here, but who also does a great job in looking after the people, who work together with him. He has also taught me that there is no point in “burning off fireworks in the basement” and “not to talk about it, but just do it”. I wish him all the best for his future plans.

§ Harald Schnitzler, Tobias Boley, Anabel Schmidt, Claus Braxmeier, Stefan Seel, Rafael Storz, Andreas Hecker, Dennis Weise and Ulrich Strößner for making life and physics more fun in P845. I am sincerely missing Klaus Schneider and Raid Al’Tahtamouni. I wish they could still be with us.

§ I also thank Alex Beirer, Kamel Bencheikh, Jo Bellanka, Richard Conroy, Daniel Dekorsy, Luca Haiberger, Evgeny Kowalchuk, Michael Lang, Thomas Müller, Torsten Petelski, Oliver Pradl, Pinno Ruoso, Jürgen Schoser and Ilkka Tittonen for being part of the real “Konstanz Quantum Metrology“ experi-ence.

§ Thomas Kalkbrenner, Marcus Ramstein and Vahid Sandoghdar for their good will in sharing resources.

§ Stefan Eggert for help and a heap of discussions on the electronics side of quantum optics, Stefan Hahn for supporting our work in a number of ways, Frau Heinzen and Ute Hentzen for keeping part of the paper work away from us.

6.3 Acknowledgements 187

§ Robert Bruckmeier for the joint effort in performing cw-QND measurements and for his support towards the first steps of my new professional life.

§ Wofgang and Helmut Hänsel for sharing countless physical and non-physical thoughts and a lot of enjoyable moments with me.

§ Gregor Weihs and Wolfgang Tittel for various discussions and some nice din-ners in Napoli.

§ Timo Felbinger for his work on the χ(3)-OPO and the star state as well as for his readiness to discuss a number of problems and perspectives in quantum optics with me.

For inspiring discussions I thank Alain Aspect, Hans Bachor, Konrad Banaszek, Arno Bandilla, Almut Beige, Robert Boyd, Stefan Dürr, Claudia Keller, Peter Knight, Prof. Kumar, Christian Kurtsiefer, Kim Lam, Ulf Leonhardt, Matteo Paris, Martin Plenio, Eugene Polzik, Gerd Rempe, Andreas Sizmann, Stuart Swain, Prof. Tewari, Harald Weinfurter Dirk-Gunnar Welsch, and Anton Zeilinger.

Wasting Photons Starting with pump laser powers around 9 W (2.2 1019 photons/s) we produce 1.7 W of average power at 790 nm with the primary Ti:sapphire laser. At a repeti-tion rate of 81.2 MHz this corresponds to 8.3 1010 photons per pulse. A pulse picker selects every 200th pulse with a outcoupling efficiency around 55%, corre-sponding to 8.5 mW average power, which is then frequency doubled yielding 38 µW of 395 nm light – 93 Mio. photons/pulse. This pump light generates photon pairs which are spatially and frequency selected, to obtain gated photon pairs at a rate between 1 and 20 photons per second. Thus for each photon we measure about 2 1018 photons are spent. Luckily enough photons are cheap to produce and do not leave behind a mess when disposed.

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Generation and Characterization of New Quantum States of the Light Field