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Slow Light, Stopped Light and Guided Light in Hot
Rubidium Vapor Using Off-resonant Interactions
by
Praveen Kumar Vudya Setu
Submitted in Partial Fulfillment
of the
Requirements for the Degree
Doctor of Philosophy
Supervised by
Professor John C. Howell
Department of Physics and Astronomy
Arts, Sciences and EngineeringSchool of Arts and Sciences
University of RochesterRochester, New York
2011
ii
Dedicated to my parents.
iii
Curriculum Vitae
The author was born in Kurnool, Andhra Pradesh, India on 4 January, 1983.
He attended the Indian Institute of Technology, Kharagpur from 2000 to 2005
and obtained his Bachelor of Science and Master of Science degrees in Physics.
He came to the University of Rochester in the Fall of 2005 for graduate studies
in Physics and received a Master of Arts degree in 2007. He pursued his doctoral
research in atomic physics and quantum optics under the supervision of Professor
John C. Howell.
iv
Publications
“A double Lorentzian atomic prism”, P. K. Vudyasetu, S. M. Bloch, D. J.
Starling, J. S. Choi and J. C. Howell, Physical Review Letters (Submitted).
“Rapidly reconfigurable slow-light system based on off-resonant Raman ab-
sorption”, P. K. Vudyasetu, R. M. Camacho and J. C. Howell, Physical Review A
82, 053807 (2010).
“Interferometric weak value deflections: Quantum and classical treatments”,
J. C. Howell, D. J. Starling, P. Ben Dixon, P. K. Vudyasetu and A. N. Jordan,
Physical Review A 81, 033813 (2010).
“All Optical Waveguiding in a Coherent Atomic Rubidium Vapor”, P. K.
Vudyasetu, D. J. Starling and J. C. Howell, Physical Review Letters 102, 123602
(2009).
“Four-wave-mixing stopped light in hot atomic rubidium vapour”, R. M. Ca-
macho, P. K. Vudyasetu and J. C. Howell, Nature Photonics 3, 103 (2009).
“Storage and Retrieval of Multimode Transverse Images in Hot Atomic Rubid-
ium Vapor”, P. K. Vudyasetu, R. M. Camacho and J. C. Howell, Physical Review
Letters 100, 123903 (2008).
“Slow-Light Fourier Transform Interferometer”, Z. Shi, R. W. Boyd, R. M.
Camacho, P. K. Vudyasetu and J. C. Howell. Physical Review Letters 99, 240801
(2007).
v
Conference Presentations
“Four Wave Mixing (FWM) and Electromagnetically Induced Transparency
(EIT) Based Coherent Image Storage in Hot Atomic Vapors”, P. K. Vudyasetu,
R. M. Camacho, and J. C. Howell, CLEO/QELS 2008 paper: QThB3 (oral pre-
sentation).
“Storage and Retrieval of Images in Hot Atomic Rubidium Vapor”, P. K.
Vudyasetu, R. M. Camacho, and J. C. Howell, Slow and Fast Light (SL) 2008
paper: SWD4 (oral presentation).
“Storing and Manipulating Multimode Transverse Images in Hot Atomic Va-
pors”, P. K. Vudyasetu, D. J. Starling, R. M. Camacho, and J. C. Howell, Laser
Science (LS) 2008 paper: LWD3 (oral presentation).
“Fast Reconfigurable Slow Light System based on Off-resonant Raman Ab-
sorption Scheme”, P. K. Vudyasetu, R. M. Camacho, and J. C. Howell, Frontiers
in Optics (FiO) 2010 paper: FThN5 (oral presentation).
vi
Acknowledgments
This thesis would not have been possible without the help of many individuals.
First and foremost, I would like to thank my thesis supervisor, Prof. John C.
Howell, for his invaluable guidance through the highs and lows of research over
past 5 years. His enthusiasm for research is infectious and made working in the
laboratory more enjoyable. I cherish his patience and his commitment towards
the progress of my graduate education, which has culminated in this thesis.
I wish to acknowledge my mentors who helped me acquire key research skills.
I would like to thank Michael V. Pack for teaching me optics experimental skills.
He was the man with answers for whatever questions I had, and I learned a great
deal from him about EIT experiments. He was very generous in helping me with
MATLAB and in lending me his books. I enjoyed working with Ryan M. Camacho
on several projects. He taught me the value of good presentations of research ideas
and I have learned a lot about scientific writing from him. He was always there
with help and kind words of wisdom whenever I needed it.
I would like to thank my lab-mates for all their help. I was very fortunate
to work with Curtis J. Broadbent, David J. Starling, Gregory Armstrong, Steven
M. Bloch and Joseph Choi on various projects, who were all fun to work with
and I have learned something or the other from each of them. I, also, had an
vii
opportunity to work with Zhimin Shi from Prof. Boyd’s group and I thank him
for the discussions on slow light devices.
I would like to extend my sincere gratitude to the university community for
all the warmth and terrific hospitality without which traveling from a far away
country for studies would not have been so wonderful. In particular, I would like
to thank Barbara Warren, Shirley Brignall, Connie Hendricks, Michie Brown and
Ali DeLeon of the Physics department and the staff at the International Services
Office, with whom I have interacted on numerous occasions and they were always
helpful.
I, also, would like to acknowledge the support from the funding agencies
DARPA DSO Slow Light and NSF.
Finally, I would like to thank my family and friends for being supportive and
encouraging. I thank my parents for all the sacrifices they made to ensure my
success.
viii
Abstract
This thesis presents the applications of some of the coherent processes in a
three-level atomic system, to control spatial and temporal properties of a signal
pulse. We use two Raman absorption resonances in rubidium vapor separated by a
few MHz to achieve a rapidly tunable slow-light system. We control the slow-light
characteristics all-optically by tuning the frequency and power of a coupling beam.
A dual absorption slow-light system is known to cause less pulse broadening than
a single transmission resonance system, and thus, a tunable double absorption
system is advantageous. We use a four-wave mixing process to demonstrate pulse
storage in rubidium vapor for times much greater than the pulse width. We
demonstrate storage of both the temporal and spatial profile of the pulse. We
overcome the diffusion of spatial information during the storage in warm atomic
vapor by storing the Fourier transform of the image instead of an image with a
flat phase. The Raman absorption resonance is also used to control the transverse
refractive index profile of the signal beam. The refractive index of the signal
interacting with a coupling beam in a Raman process is dependent on the coupling
beam intensity. We use a first order Laguerre-Gaussian (LG01) coupling beam
to create a waveguide like transverse refractive index profile. We demonstrate
propagation of a focused signal beam for lengths much greater than the Rayleigh
length. Finally, we demonstrate a dual absorption atomic prism, which is capable
ix
of spatially separating spectral lines that are 50 MHz apart and which can precisely
measure frequency fluctuations. This simple prism is a valuable spectral filtering
tool for a variety of atomic experiments.
x
Table of Contents
Curriculum Vitae iii
Acknowledgments vi
Abstract viii
List of Figures xii
Foreword 1
1 Background 3
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Interactions in a Lambda System . . . . . . . . . . . . . . . . . . 7
1.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2 Double Raman Absorption Slow Light 27
2.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 33
xi
2.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3 Stopped Light Using Four Wave Mixing 44
3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4 All Optical Waveguiding Using Raman Absorption 61
4.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5 Slow Light Prism Spectrometer 78
5.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6 Conclusions 89
Bibliography 94
A Derivation of Susceptibility for a Lambda System 109
xii
List of Figures
1.1 Various configurations for the interaction of three atomic energy
levels and two optical fields. . . . . . . . . . . . . . . . . . . . . . 5
1.2 Three level Λ system . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Lines shapes for various detunings . . . . . . . . . . . . . . . . . . 10
1.4 Four wave mixing in double-Λ system . . . . . . . . . . . . . . . . 14
2.1 Dependence of C1 and C2 on ∆ . . . . . . . . . . . . . . . . . . . 29
2.2 (a) & (b) The energy level diagram for vapor cells VC1 and VC2
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3 The outline of the setup used in the experiment. . . . . . . . . . . 34
2.4 Variation of absorption line shapes with coupling beam power. . . 36
2.5 Variation of fit parameters with coupling beam power. . . . . . . 36
2.6 Transmission of the signal through a vapor cell in the presence of
two coupling beams differing in frequency.Green curve is a refer-
ence absorption with single coupling beam frequency. The cou-
pling beams are separated in frequency by 1 MHz and 2 MHz for
the transmission curves in red and blue respectively. . . . . . . . . 37
xiii
2.7 The signal pulse out of a vapor cell with two coupling beams dif-
fering in frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.8 The transmission profile of the probe corresponding to the observed
delay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.9 Reference pulse and the delayed pulse. . . . . . . . . . . . . . . . 39
2.10 Measurement of turn-on and turn-off times of Raman absorption. 41
2.11 Beating in the cw signal beam resulting from coupling beam fre-
quency change. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.12 The experimental observation of slow light switching. . . . . . . . 42
3.1 (a) Experimental setup. (b) Four wave mixing energy levels. (c)
Representation of the synchronized timing of signal (dashed red),
delayed signal (blue) and coupling (black) beams. . . . . . . . . . 50
3.2 Measured (asterisks) and theoretical (solid) steady state signal (blue)
and idler (red) intensities as a function of signal detuning. . . . . 52
3.3 Demonstration of four-wave mixing slow light and stored light . . 54
3.4 Idler pulse peak power as a function of coupling input power. . . . 55
3.5 CCD camera capture of the signal intensity profile at the object
plane and at the vapor cell (Fourier plane). . . . . . . . . . . . . . 56
3.6 Input signal profile (a) and the time evolution of measured (b) and
calculated (c) transverse images. . . . . . . . . . . . . . . . . . . 57
3.7 Theoretical time evolution of stored ground state coherence of Rb
atoms. The inset shows a close up of the time evolution near zero
crossover points of electric field amplitude. . . . . . . . . . . . . . 58
xiv
4.1 Beam propagation as a function of κ . . . . . . . . . . . . . . . . 63
4.2 Beam propagation as a function of coupling rabi frequency . . . . 66
4.3 Beam propagation as a function of Raman detuning . . . . . . . . 66
4.4 The experimental schematic for all optical waveguiding using atomic
rubidium vapor. The focusing scheme for control beam (black) and
signal (gray) is shown in the inset . . . . . . . . . . . . . . . . . . 69
4.5 The experimental plot of the variation in the transmission of signal
versus Raman detuning. . . . . . . . . . . . . . . . . . . . . . . . 70
4.6 The plot of refractive index of the signal, tuned -1.5 MHz away
from Raman resonance, as a function of coupling beam power. . . 71
4.7 (a) shows the spatial variation of refractive index. (b) shows the
plot of refractive index versus position along one of the axes. . . . 71
4.8 The Snap shots of the signal beam profile at the back of the vapor
cell with the coupling beam off (a) and on (b). Beam profiles along
the longer axis of the beams at the front face of the cell (c) and at
the back face of the cell(d). . . . . . . . . . . . . . . . . . . . . . 72
4.9 Plots of the signal beam size at the back face of the vapor cell versus
the control beam power . . . . . . . . . . . . . . . . . . . . . . . . 74
4.10 Plots of the signal beam size at the back face of the vapor cell versus
the Raman detuning . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.11 The plot of output signal power versus Raman detuning. . . . . . 75
4.12 The plot of output signal power versus input signal power. The
plot is nearly linear, the slope of the linear fit to data is 0.43. . . . 75
xv
5.1 The rubidium vapor cell can be approximated as a dispersing prism.
For our setup θp ≈ 79o, θ1 ≈ 20o and thus the geometrical param-
eter A is about 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.2 A schematic of the experimental setup. . . . . . . . . . . . . . . . 81
5.3 The camera images for different modulation frequencies. . . . . . 83
5.4 One dimensional intensity scans for different modulation frequencies. 84
5.5 Plot of deflection as a function of frequency. Circles represent the
experimental data and solid line is the linear fit. The slope of the
line is 1.95 µm/MHz. . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.6 (a) Four wave mixing beams are split at the prism and imaged at
the camera. (b) The one dimensional scan of the intensity profile. 86
1
Foreword
All the research included in this thesis was done under supervision of my thesis
advisor Prof. John C. Howell. In addition, several graduate students collaborated
with me on projects which resulted in this work. Their individual contribution to
this thesis is elaborated in the following paragraphs.
The research in chapter 2 was performed in collaboration with Dr. Ryan M.
Camacho and Prof. John C. Howell. Ryan and I worked on experiments which
resulted in figures 2.8, 2.9 and 2.12. Ryan created all three of these figures. Rest of
the work, both experimental and analytical, was done by me with guidance from
both Ryan and Prof. Howell. Most of this work was published in Physical Review
A 82, 053807 (2010), which I wrote with the help of Ryan and Prof. Howell.
The research in chapter 3 was performed in collaboration with Dr. Ryan M.
Camacho and Prof. John C. Howell. Ryan and I worked on experiments which
resulted in all the experimental plots. Ryan created figures 3.1(b), 3.2, 3.3 and
3.4 and the rest of the figures were created by me. This chapter was adapted
from two published journal papers, Nature Photonics 3, 103 (2009) and Physical
Review Letters 100, 123903 (2008). The Nature Photonics paper was written by
Ryan with inputs from me and Prof. Howell and the PRL paper was written by
me with inputs from Ryan and Prof. Howell.
2
The research in chapter 4 was performed in collaboration with David J. Starling
and Prof. John C. Howell. David and I worked on experiments which resulted in
all the experimental plots. I created all the figures in this chapter. Most of this
work was published in Physical Review Letters 102, 123602 (2009), which I wrote
with the help of David and Prof. Howell.
The research in chapter 5 was performed in collaboration with Steven M.
Bloch, David J. Starling, Joseph S. Choi and Prof. Howell. Steve worked on the
experimental setup with the help of Joe. I worked on experiments with Steve,
David and Joe which resulted in all the experimental plots. I created all the
figures in this chapter. I wrote a paper based on this work with the help of Steve,
David, Joe and Prof. Howell, which we submitted to Physical Review Letters and
is under review.
3
1 Background
1.1 Introduction
The study of interactions between optical fields and matter has a long history
and has resulted in the discovery of many physical processes. The optical prop-
erties of matter such as absorption, reflection, refraction, scattering, dispersion
etc., can all be explained by the properties of the atoms and molecules with which
the light is interacting. The advent of quantum mechanics resulted in the under-
standing of atomic structure and electronic energy levels inside an atom [Schiff,
1968]. The quantum nature of light was used to explain physical processes like
black body radiation, photoelectric effect, etc. It was postulated by Bohr that an
electron jumps from one electronic state of an atom to another electronic state
by absorbing or emitting one quantum of energy hν. The quantum theory also
enabled us to understand the spectroscopic properties of various elements. A
thorough understanding of matter-light interactions requires us to quantize both
the optical field and matter. However, one can explain many interesting phenom-
ena in a semi-classical regime wherein we quantize just the matter and consider
4
the optical field as a classical electromagnetic wave satisfying Maxwell’s equa-
tions. Following this approach one can derive the refractive index of a material
as a function of light frequency, obtain the absorption coefficient and understand
many nonlinear optical processes [Boyd, 2003].
The invention of laser provided optical physicists a powerful, highly coherent,
nearly monochromatic source which opened up the fields of quantum optics and
nonlinear optics. A single mode laser interacting with a two-level atom leads to
several interesting phenomena like resonance florescence, Autler-Townes splitting,
Ramsey fringes, self-induced transparency etc [Allen and Eberly, 1987]. A power-
ful laser can also lead to nonlinear effects like parametric down conversion, higher
harmonic generation, Kerr effect, Raman effect etc [Boyd, 2003]. It also proved a
highly effective spectroscopic tool enabling several techniques like saturation ab-
sorption spectroscopy, time-resolved laser spectroscopy, laser-induced florescence
spectroscopy etc [Demtroder, 2002].
While the coherent interactions of an optical field with a two-level atom lead us
to many applications, the interactions of multiple optical frequencies with multiple
levels of atom result in an even richer variety of physical phenomenon. In such
a system, the optical properties of the medium can be altered by one or many
optical fields. The simplest of such a multi-level system is a three-level atom
interacting with two optical frequencies where the optical interaction between two
of the energy levels of the atom is forbidden. The three-level system can be in the
form of a cascade, a “V” or a “Λ” shaped energy diagram as shown in 1.1.
The three-level system has been used to demonstrate various phenomena.
Atomic cascade systems have been used to demonstrate selective reflection of
the probe beam in a pump-probe scheme [Schuller et al., 1993; Amy-Klein et al.,
5
Cascade/
LadderV Λ
E1
E2
E1 E2 E2E1
1
2
3
1
2
3
1
2
3
Figure 1.1: Various configurations for the interaction of three atomic energy levelsand two optical fields.
1995]. Effects like electromagnetically induced transparency (EIT) and lasing
without inversion have also been demonstrated in a three-level cascade atomic
system [Xiao et al., 1995; Sellin et al., 1996]. Based on the polarization interfer-
ence between one photon and two photon processes in such a system, ultrafast
modulation spectroscopy has been demonstrated in sodium vapor [Fu et al., 1995].
Lasing without inversion has also been predicted in a V type three-level system
[Zhu, 1992]. It was shown that spontaneous emission can be canceled in a V type
system, resulting in inversion without lasing [Scully et al., 1989]. Comparison of
electromagnetically induced transparency in cascade, V and Λ types systems has
been carried out by Fulton et al. [Fulton et al., 1995].
This thesis focuses on the applications of interactions in three-level lambda
system for slow light and stopped light. It was shown by Gray et al. [Gray et al.,
1978] that the population of the three-level lambda system can be trapped in
the lower ground states, an effect known in literature as “coherent population
trapping” (CPT). One can use magnetic sub-levels or hyperfine levels and an
excited state of alkali atoms to form a lambda system. The three-level lambda
6
system has been studied extensively in the context of electromagnetically induced
transparency (EIT) [Fleischhauer et al., 2005]. The narrow resonance feature of
EIT results in steep dispersion which has been used to obtain ultra-slow group
velocities [Harris et al., 1992; Hau et al., 1999]. The ground state atomic coherence
induced by two optical fields is long lived and this property has been exploited to
store optical information [Phillips et al., 2001]. Using appropriate pulse shapes of
the two optical fields, one can coherently transfer populations between two ground
states by stimulated Raman adiabatic process (STIRAP) [Bergmann et al., 1998].
Coherent preparation of the atomic lambda system also proved helpful in en-
hancing several nonlinear processes in the atomic systems. Harris et al. have
shown that a four-wave mixing process can be enhanced by electromagnetically
induced transparency [Harris et al., 1990]. Preparation of the lambda system in a
maximally coherent state can enhance the frequency conversion efficiencies [Jain
et al., 1996]. Hemmer et al. demonstrated an efficient phase conjugation scheme
using coherent population trapping [Hemmer et al., 1995]. Schmidt and Imamoglu
proposed a scheme for a giant cross phase modulation making use of the steep
dispersion resulting from electromagnetically induced transparency [Schmidt and
Imamoglu, 1996]. This giant Kerr effect means that one can achieve large phase
shift in one optical mode using only few photons of another optical mode. In fact
it is predicted by Lukin and Imamoglu that two slowly propagating photons can
induce large phase shift on each other [Lukin and Imamoglu, 2000].
This chapter is organized as follows. In section 1.2, we introduce a lambda
system and briefly discuss some of the interactions relevant to this thesis like
electromagnetically induced transparency, Raman absorption and four-wave mix-
ing. In section 1.3 we provide the motivation behind the thesis. We introduce
7
1
2
3
∆
δ
Γ/2 Γ/2
γ
Signal Coupling
Figure 1.2: Three level Λ system
the concepts of slow light stopped light and all-optical waveguide and discuss the
advantages of using off-resonant interactions in rubidium for these problems. In
section 1.4 we present an outline for the thesis.
1.2 Interactions in a Lambda System
In this section we discuss some of the consequences of coherent interactions of
two optical fields with three-level lambda system. We first introduce a lambda
atomic system. The energy level diagram of a lambda system is shown in Fig.
1.2. The two lower energy levels are optically coupled to an excited state by the
“signal” and the “coupling” beams. The dipole interaction between the two lower
energy levels is forbidden. The excited state life time is given by 1/Γ while the
ground state decoherence rate is given by γ. In a typical lambda system, Γ ≫ γ.
The signal of frequency ωs is detuned from 1 → 2 transition frequency ω12
by the single photon detuning ∆. The coupling frequency, ωc, is detuned from
3 → 2 transition frequency ω32 by ∆− δ where δ is the two photon detuning. The
two photon effects are usually observed when δ is of the order of γ. The Rabi
8
frequencies of signal and coupling beams are denoted by Ωs and Ωc respectively.
For the remainder of the thesis we assume that Ωs ≪ Ωc. We also assume that
the atom is initially in the state |1〉. Under these assumptions the susceptibility
at the signal frequency is given by
χ(∆, δ,Ωc) = βδ − iγ
(δ − iγ)(∆− iΓ/2)− |Ωc|2/4, (1.1)
where β = Nµ2/hǫ0 where N is the number density and µ is the transition dipole
moment. The above equation has been derived in the appendix A. The above
equation for susceptibility can be used to explain many physical phenomena in
this thesis and is a good starting point for the discussion that follows in this
section. We note that in the limit of Ωc = 0, we obtain the usual susceptibility
for a two level system.
The real and imaginary parts of the susceptibility, χ′ and χ′′, can be written
as,
χ′ = β∆
∆2 + Γ2/4
[
1 +
(
2δ0 −|Ωc|24∆
)
δ′
δ′2 + γ′2− 2γ0
γ′
δ′2 + γ′2
]
, (1.2)
χ′′ = βΓ/2
∆2 + Γ2/4
[
1 + 2δ0δ′
δ′2 + γ′2−
(
2γ0 −|Ωc|22Γ
)
γ′
δ′2 + γ′2
]
. (1.3)
In the above, δ0 = |Ωc|2∆/(4∆2 + Γ2), γ0 = |Ωc|2Γ/(8∆2 + 2Γ2), δ′ = δ − δ0
and γ′ = γ + γ0. The first terms in each of the above equations are the real
and the imaginary parts of the susceptibility χ0 when the coupling beam is off.
The second and the third terms have dispersive and absorptive functional forms.
The single photon detuning ∆ determines how each term contribute to the total
susceptibility.
We can observe many qualitative features of signal transmission from the above
9
equations. There is a broad absorptive feature of width Γ/2 resulting from the
signal interaction with 1 → 2 transition. Within that feature there is another
narrow feature of width γ′ centered at δ = δ0. The narrow feature resulting from
two photon interaction can be transmissive or absorptive depending on ∆.
Consider the case where ∆ = 0. The real and imaginary parts of the suscep-
tibility at signal frequency for this case are given as,
χ′(∆ = 0) = −β|Ωc|2Γ2
δ
δ2 + (γ + |Ωc|2/2Γ)2, (1.4)
χ′′(∆ = 0) =2β
Γ
[
1− |Ωc|22Γ
γ + |Ωc|2/2Γδ2 + (γ + |Ωc|2/2Γ)2
]
. (1.5)
The first term in the imaginary part of the susceptibility corresponds to absorption
of signal in the absence of the coupling beam. The coupling beam results in
decrease in absorption. The result is a Lorentzian transparency profile in the
background of absorption with width given by γ + |Ωc|2/2Γ. Here |Ωc|2/2Γ is
power broadening term and in the case where power broadening is not dominant,
the width of transparency is given by the ground state decoherence rate γ. This
coupling beam induced transparency is known in literature as electromagnetically
induced transparency (EIT). Choosing appropriate coupling beam intensity, it is
possible to achieve zero absorption for δ = 0.
Now consider the case where ∆ >> Γ. In this case the susceptibility can
be approximated as zero for the signal frequency with no coupling beam. With
the coupling beam, the real and imaginary parts of the susceptibility at signal
10
−100 0 100
0.5
1
0−100 0 100
−1
0
1
−100 0 1000
2
4
x 10−3
−100 0 100
−2
0
2
x 10−3
−5 0 50
3
x 10−5
−5 0 5
−7
−6
−5
x 10−5
Real(χ)Imag(χ)
Frequency (KHz) Frequency (KHz)
∆ = 0
∆ = 100xΓ/2
∆ = Γ/2
(a)
(f)
(e)
(d)
(c)
(b)
Figure 1.3: Lines shapes for various detunings
frequency are approximately given by
χ′(∆ >> Γ) = β|Ωc|24∆2
δ′
δ′2 + γ′2, (1.6)
χ′′(∆ >> Γ) = β|Ωc|24∆2
γ′
δ′2 + γ′2. (1.7)
The absorptive line profile is approximately Lorentzian, with a width of γ′ and the
strength proportional to |Ωc|2/∆2. The line center is determined by the frequency
of the coupling beam, and the width and the depth of absorption are determined
by the intensity of the coupling beam. Thus, by changing the properties of the
coupling beam one can alter the properties of the absorption. We call this coupling
beam induced absorption as Raman absorption.
11
The imaginary and real parts of susceptibility for various single photon de-
tunings are plotted in Fig. 1.3. The plot of imaginary part of susceptibility as a
function of δ show that for ∆ = 0, we have a Lorentzian transmission feature, for
∆ = Γ/2 the transmission profile line shape is dispersive and for ∆ ≫ Γ/2, we
have a Lorentzian absorption.
1.2.1 Electromagnetically Induced Transparency
We have seen in the previous discussion that for ∆ = 0, the signal experi-
ences electromagnetically induced transparency. EIT was investigated first by
Harris in 1990 [Harris et al., 1990]. It was shown that EIT can enhance nonlinear
susceptibility while reducing signal absorption. EIT was soon demonstrated in
strontium vapor [Boller et al., 1991] and in lead vapor [Field et al., 1991]. It was
pointed out that the narrow transmission feature of EIT is also accompanied by
steep linear dispersion which results in optical pulse delays [Harris et al., 1992;
Xiao et al., 1995; Kasapi et al., 1995]. It was also shown that signal and coupling
pulses of arbitrary shape evolve into a matched pulse shape after propagating cer-
tain characteristic distance through EIT medium [Harris, 1993; Harris, 1994]. Hau
et. al demonstrated ultra slow propagation (17 m/s) of signal pulses in an ultra
cold atomic gas [Hau et al., 1999]. Enhancement of various nonlinear processed by
EIT has been investigated by various groups (for review, see [Fleischhauer et al.,
2005]).
In order to have qualitative understanding of EIT, consider the three-level
lambda system shown in Fig. 1.2. The coherence ρ12 can be written in the form
12
of a geometric series as follows.
ρ12 =Ωs
2
1
∆− iΓ/2× (1.8)
[
1 +
(
Ωc
2
1
δ − iγ
Ω∗c
2
1
∆− iΓ/2
)
+
(
Ωc
2
1
δ − iγ
Ω∗c
2
1
∆− iΓ/2
)2
+ ...
]
.
The first term in the above represents the probability for the transition 2 → 1. The
subsequent terms represents the probability for the transitions 2 → 3 → 2 → 1,
2 → 3 → 2 → 3 → 2 → 1, and so on. The combination of all the terms for ∆ = 0
and δ = 0 results in the atomic coherence,
ρ12 =iΩs
Γ
(
1− |Ωc|2/2Γγ + |Ω|2/2Γ
)
. (1.9)
For γ << |Ω|2/2Γ, ρ12 = 0. Thus, for the resonant lambda system, the polariza-
tion at the signal frequency is zero when Raman resonance condition is satisfied.
This means that the signal will propagate as if the medium is absent.
This effect can also be understood in a dressed state picture. It has been
shown that, in an appropriate basis, the ground states can be written in terms of
a “bright-state” and a “dark-state” [Fleischhauer and Manka, 1996]. In this basis,
only bright state is coupled to the excited state by an effective Rabi frequency
while the dark state is sees no interaction. If the medium is prepared in such a
dark state, the signal is completely decoupled from atoms and hence propagates
without any losses.
One characteristic feature of EIT is the existence of dark state polaritons
[Fleischhauer and Lukin, 2000]. It was shown that the quantum signal field in the
EIT medium propagates as quasi-particles and that these particles, when trapped
in the medium, transfer their pulse shape and quantum state to the atomic ground
13
state coherence. Trapping of the dark state polariton occurs when the coupling
beam is adiabatically turned off. One can retrieve the signal by turning on the
coupling beam at a later time. Pulse storage was first demonstrated in a cold
atomic cloud by Liu et. al [Liu et al., 2001], in warm atomic vapor by Phillips et.
al [Phillips et al., 2001] and in a solid by Turukhin et. al [Turukhin et al., 2001].
The storage of quantum states of light was demonstrated by Van Der Wal et. al
[van der Wal et al., 2003].
1.2.2 Raman Absorption
For the far detuned case, the system can be treated like a two-level system
[Gerry and Eberly, 1990]. If Ωs and Ωc are Rabi frequencies of signal and coupling
fields respectively, the effective Rabi frequency coupling level 1 and level 3 for far
detuned case is ΩsΩc/∆. We observe Lorentzian absorption line shapes similar
to two-level atoms and it has been predicted that the three-level system produce
similar non-classical effects as a two-level atom.
The Raman absorption resonance in an off-resonant lambda system received
considerable attention recently in the context of slow light and fast light research.
Mikhailov et al. have reported pulse advancements of about 300 µs when the signal
pulse is tuned to the center of Raman absorption (However, the signal pulse is
attenuated because of absorption) [Mikhailov et al., 2004]. Knappe et al. have
studied signal line shapes for the off-resonant lambda system in terms of simple
parameters [Knappe et al., 2003]. More recently these absorption were used to
achieve gradient echo memory (GEM) [Hetet et al., 2008]. In this scheme the
Raman resonance is broadened using a magnetic field that varies linearly along
the light propagation direction and reversing that gradient results in photon echo.
14
1
2
3
δ
Γ/2 Γ/2
γ
Coupling
Signal Idler
∆1∆2
Figure 1.4: Four wave mixing in double-Λ system
Reim et al. demonstrate a memory capable of more than a GHz signal bandwidth
using Raman resonance [Reim et al., 2010].
1.2.3 Four-wave Mixing
Consider the double lambda system shown in Fig. 1.4. The coupling beam in
the lambda system discussed in the previous sections can also act on the ground
state |1〉 and scatter idler photons. The four-wave mixing process (FWM) grows
with propagation distance and at high optical densities it can dominate the other
processes discussed previously. The transmission line shape of the signal under
FWM conditions look like a single Lorentzian transmission, like EIT, and the
dispersion within the gain profile can give rise to slow light.
Four-wave mixing in the double lambda configuration has been studied by
various research groups. Generation of squeezed states of light using four-wave
mixing was proposed by Reid and Walls [Reid and Walls, 1985]. It was soon
demonstrated in sodium vapor by Slusher et. al [Slusher et al., 1985]. Four wave
mixing has also been studied in optical fibers and many applications like frequency
15
conversion and amplification are based on degenerate four-wave mixing [P., 2001].
More recently, four-wave mixing in rubidium vapor was used and strong relative
intensity squeezing was achieved [McCormick et al., 2007; McCormick et al., 2008].
It was also shown that the signal and idler pulsed propagate slowly in such a
medium [Boyer et al., 2007]. Quantum mechanical aspects of FWM were also
demonstrated [Marino et al., 2009; Pooser et al., 2009a; Pooser et al., 2009b].
1.3 Motivation
Controlling optical properties of a material is desirable for many applications.
For example, the control of refractive index of some crystals by the electric field
is an enabler for high speed optical modulators. This work deals with all-optical
control of temporal and spatial propagation of optical pulses in rubidium vapor.
By optically tuning the dispersion characteristics of rubidium vapor we can control
the group velocity of an optical signal and the transverse shape of the beam as it
propagates through the medium. This optical control has been applied to achieve
slow light, stopped light and guided light.
1.3.1 Slow Light
The field of slow light research deals with the control of the group velocity
of an optical pulse through various means. In a dispersive medium the group
velocity is given by
vg =c
n0 + ν dndν
, (1.10)
where n0 is the refractive index at the central frequency of the pulse, ν0. We call
the denominator of the right hand side of the above equation as group index ng.
16
For highly dispersive material we can approximate ng as ν dndν. Controlling ng by
various means, we can control the group velocity of the pulse. For positive ng,
the pulse is delayed and hence we call it slow light. Similarly, for negative values
of ng, we will have fast light.
One can obtain very large group indices in the vicinity of a sharp transmission
at the signal frequency. Such sharp transmissions can be achieved using variety of
nonlinear effects in atomic media [Boyd, 2003] or by fabricating specially designed
photonic structures [Baba, 2008]. In this work we deal with the former case, that
is, obtaining slow-light by tailoring the refractive index of the medium. From
the Kramers-Kronig relations we know that sharp change in the refractive index
within a frequency range is accompanied by sharp change in the transmission
properties at those frequencies. Thus narrow spectral features result in slow or
fast light.
The ability to control the group velocity of a light pulse is central to the
practical realizations of all optical communication systems [Ku et al., 2002]. After
an initial demonstration of ultra slow group velocities of light pulses by Hau et al.
[Hau et al., 1999], several groups have demonstrated slow-light in a wide variety of
media like atomic vapors [Kash et al., 1999; Budker et al., 1999; Camacho et al.,
2006; Harris et al., 1992; Kasapi et al., 1995; Budker et al., 1999; Hau et al., 1999;
Liu et al., 2001; Yang et al., 2007; Boyer et al., 2007], solid materials [Turukhin
et al., 2001; Bigelow et al., 2003a; Bigelow et al., 2003b; Khurgin, 2005; Gehring
et al., 2006; Zhu and Gauthier, 2006a; Shumakher et al., 2006], optical fibers
[Okawachi et al., 2005], liquid crystals [Residori et al., 2008], photonic crystals
[Baba, 2008] etc. Slow-light has applications in many diverse fields like optical
communications [Ku et al., 2002], interferometry [Shi et al., 2007; Shi and Boyd,
17
2008], sensing [Shahriar et al., 2007], etc.
An ideal slow light system must have large delay-bandwidth product with fast
tunability, large bandwidth, low distortion and low absorptive losses [Boyd, 2003;
Milonni, 2005; Khurgin, 2010]. The relative importance of each of the charac-
teristic varies with the application. For example, very large dispersion is more
important for a slow light based interferometer than very wide bandwidth. In
contrast, for an optical buffer, the large delay-bandwidth product is more impor-
tant than large absolute delay.
For the purposes of optical communication systems and other applications,
rapid tunability of the group velocity of a light pulse is advantageous. For example,
one could use such a system to synchronize a pulse train by tailoring the group
velocity of each pulse in the pulse train. Such an application demands the group
velocity switching times to be less than the signal pulse width. We can also use it
to get single photons on demand, by slowing down the single photons and turning
off the delay at appropriate time. For applications such as slow light enhanced
Fourier transform interferometry [Shi et al., 2007], the time required to determine
the spectrum of an unknown signal is directly proportional to the group velocity
switching time.
Camacho et al. [Camacho et al., 2006] have shown that one can achieve multi-
ple pulse delays by operating between two Lorentzian absorption resonances. Such
a double Lorentzian system is shown to dramatically reduce absorptive broaden-
ing and cancel the dispersive broadening, and thus is advantageous over a single
Lorentzian system like an electromagnetically induced transparency (EIT) based
system. Slow-light is primarily limited by absorptive broadening in addition to
absorptive losses in such a double Lorentzian system. The depth and line-width of
18
each of the resonance along with the separation between two resonances determine
the system bandwidth, the group delay and the absorptive losses.
Camacho et al. use naturally occurring Rubidium or Cesium resonances to
achieve double absorption system. In such a system, line-width and the depth of
each of the resonance is fixed and the slow light tuning is achieved primarily by
changing the number density of the system. By using Raman absorption lines,
one can have optical control over the properties of the absorption line and thus an
optical control over the delay and bandwidth. The double absorption slow light
is also demonstrated in fibers using anti-stokes absorption resonances by Zhu and
Gauthier [Zhu and Gauthier, 2006a].
Double Raman absorption is also useful for applications requiring rapid change
of group index. A slow light based Fourier transform interferometer, for example,
requires a scan of group index of the slow light medium and the interferometer
output is recorded as function of group index which is then Fourier transformed
to obtain the signal spectrum. Shi et al [Shi et al., 2007] demonstrated such an
interferometer using naturally occurring double absorption resonance slow light
system. A rapidly tunable slow light system lessens the spectral determination
time and the steeper refractive index gradient in Raman absorption based system
improves the spectral resolution by at least two orders of magnitude.
1.3.2 Stopped Light
“Stopped light” usually refers to the interconversion of electromagnetic fields
and long-lived atomic coherences. It allows for recording coherent signals for later
retrieval even at very low light levels. The initial work by Liu et al.[Liu et al.,
2001] and Phillips et al. [Phillips et al., 2001] stimulated additional research with
19
a recent demonstration of storage times in excess of one second [Longdell et al.,
2005]. Stopped light may be useful for applications in remote sensing, image
processing, and quantum information.
Typically, the pulses used in stopped light experiments are several kilometers
long in free space. However, because the stopped light medium usually ranges
from a few tens of microns up to several centimeters, slow light is first used to
spatially compress the optical pulses inside the medium. Slow light is achieved
when a steep linear dispersion is obtained in a medium, leading to a large group
index ng = n + ω ∂n∂ω
and a small group velocity vg = c/ng, where n is the phase
index and ω is the angular frequency. Slow group velocities have been achieved in
a variety of media, including atomic vapors [Harris et al., 1992; Kasapi et al., 1995;
Budker et al., 1999; Hau et al., 1999; Liu et al., 2001; Yang et al., 2007; Boyer
et al., 2007] and solids [Bigelow et al., 2003a; Bigelow et al., 2003b; Khurgin, 2005;
Gehring et al., 2006; Zhu and Gauthier, 2006b; Okawachi et al., 2006; Shumakher
et al., 2006].
The ability of rubidium to preserve the ground state coherence for times much
longer than the pulse width enables us to store the signal and retrieve it at a
later time. The motivation here is to store the signal information like spatial and
temporal characteristics and not to store the energy of the pulse. We employ a
four-wave mixing process in rubidium (we elaborate more about the process later)
to store the signal as well as spontaneously generated idler pulses. The signal and
idler beams produced in such a process are known to be number squeezed. Storing
such fields is fundamentally interesting and may be useful in applications where
two correlated fields need to be stored. In addition, we demonstrate a scheme in
which we store the spatial mode information of the signal beam in warm vapor
20
overcoming diffusion. We believe that storing images may have applications in
image processing, quantum information and remote sensing.
1.3.3 Guided Light
An all optical waveguide refers to a waveguide whose transverse refractive
index profile is set by the interaction of an optical control beam with the medium.
The properties of the waveguide can be altered or modified to fit a particular
experimental requirement simply by changing the properties of the control beam.
Several schemes have been proposed to achieve all optical wave guiding and there
have been some experimental realizations. Moseley et al. have proposed [Moseley
et al., 1996] and realized [Moseley et al., 1995] focusing and defocusing of the
signal beam using electromagnetically induced focusing. There are several other
papers which address Raman focusing [Walker et al., 2002; Yavuz et al., 2003;
Shverdin et al., 2004; Proite et al., 2008]. Truscott et al. have achieved optical
wave guiding [Truscott et al., 1999] and their scheme is analyzed in detail by
Kapoor et al. [Kapoor and Agarwal, 2000] and Anderson et al. [Andersen et al.,
2001]. In their scheme the control beam modifies the refractive index of the
medium by pumping the ground state rubidium atoms to an excited state. There
are several other schemes proposed for waveguiding [Shpaisman et al., 2005], and
more recently image guiding [Firstenberg et al., 2009], using electromagnetically
induced transparency in lambda and double lambda systems.
Nonlinear optical properties of various gases within an optical waveguide struc-
ture have received considerable recent attention [Benabid et al., 2002; Ghosh et al.,
2006]. The use of an optical waveguide enables one to have high intensities at
lower optical powers over distances much greater than the diffraction length. The
21
combination of higher intensities, longer interaction lengths and higher atomic
densities within the mode volume results in efficient nonlinear processes [Ben-
abid et al., 2002]. Benabid et al. demonstrated an efficient stimulated Raman
scattering process in a hollow-core photonic crystal fiber filled with hydrogen gas
requiring two orders of magnitude less control beam power than any other pre-
viously reported experiments [Benabid et al., 2002]. Efficient nonlinear processes
have been demonstrated by injecting rubidium vapor in such a waveguide [Ghosh
et al., 2006] and also by using the small optical mode area of a tapered-nano-fiber
(TNF) placed in rubidium vapor [Spillane et al., 2008].
Previous proposals and demonstrations of an all optical waveguide use the
sharp refractive index variation near an EIT line. This means that the signal is at
the atomic resonance frequency. This limits the application of such a waveguide
because of the signal losses due to absorption. If the signal is guided in the core of
the donut shaped control beam, it is advantageous to have the signal in the trans-
parent region. Off-resonant Raman absorption discussed in the previous section
can be used to create an all optical wave guide. The dependence of signal refrac-
tive index on the control beam intensity enables us to control the spatial mode of
the signal. One can choose a spatial mode of the coupling beam which results in a
quadratic dependence of signal refractive index on the radial coordinate. Such a
quadratic index medium is known to confine the optical mode within small radial
bounds. By changing the frequency or power of the coupling beam we can tune
the size of the signal mode propagating in the medium.
Propagation of the focused signal beam for lengths greater than the diffrac-
tion length can lead to efficient nonlinear processes which depend on the signal
intensity. For example, a weak pulse of light can produce enhanced AC-Stark
22
shift in the atoms which may be useful for the single photon non-demolition mea-
surement schemes. One can also obtain enhanced nonlinear optical frequency
conversion using the compressed mode inside the waveguide.
1.3.4 Rubidium Prism
Prisms have played a fundamental role in our understanding [Newton, 1704]
and manipulation of light. The dispersing power of the prism is a function of the
material’s frequency-dependent index of refraction. The faster the index changes
as a function of frequency, the better the dispersing power. For example, in the
visible region of the spectrum, BK7, a popular prism material, has dn/dν ≈
5×10−17 Hz−1. We recently showed using precision deflection measurements that
the frequency sensitivity of a BK7 prism could reach approximately 100 kHz/√Hz
using about 1 mW of light [Starling et al., 2010]. The low dispersing power of
these standard prisms limits their usefulness to systems that exhibit large spectral
changes or for broadband applications. Gratings can improve the dispersing power
by a couple orders of magnitude, but even this is too small for many applications.
There has been an ever increasing demand for spectral resolution and disper-
sion. Common techniques for high resolution spectroscopy include Fourier trans-
form interferometry [Hariharan, 2003] and high finesse cavities [Fortier et al.,
2006]. However, these systems are single mode and must be scanned to determine
other frequencies in the system. Optical frequency combs are used as efficient
frequency counters and find applications in high precision optical spectroscopy
[Udem et al., 2002]. In addition, the teeth of the frequency comb can be used
in “Direct Frequency Comb Spectroscopy” (DFCS) [Stowe et al., 2008]. However
since the teeth are closely spaced in frequency, one needs a highly dispersive el-
23
ement to separate individual teeth spatially. Bartels et al. [Bartels et al., 2009]
have overcome this by making the comb less spectrally dense and making use
of regular dispersing elements. With high dispersion one can spatially separate
dense optical frequency combs. A spatially dispersing element can also be used as
an efficient frequency filter for applications like entanglement filtering in quantum
information [Okamoto et al., 2009].
The primary advantage of using prisms is their ability to unambiguously de-
termine spectral lines [Demtroder, 2002]. The dispersing power of a prism can be
improved by using light frequencies near a material resonance (see for example
[Wood, 1988; Marlow, 1967]). For example, Finkelstein et al. showed, using the
resonance enhancement of dispersion (of a single absorption resonance), that a
mercury vapor prism could resolve the Raman lines of CO2 [Finkelstein et al.,
1998]. Previous experiments have focused on a single resonance, typically a single
absorption [Lin et al., 1996; Finkelstein et al., 1998; Zheltikov et al., 2000] or sin-
gle transmission resonance [Sautenkov et al., 2010]. The difficulty in working with
single absorption resonances is the nonlinear dispersion, the strong absorption and
strong frequency-dependent absorption, and the inability to resolve many spectral
lines.
The steep dispersion in a slow light system can be used to achieve highly disper-
sive prism that can be used as an experimental tool for off-resonant experiments
in rubidium vapor. Experiments involving several optical frequencies in the same
optical mode require a spectral filter. We present a rubidium vapor prism which is
capable of spatially resolving closely spaced frequencies. In addition, it is capable
of measuring frequency fluctuations with an accuracy better than 1 part in 1013.
This rubidium prism is capable of separating signal, idler and coupling beams in a
24
four-wave mixing process or signal and coupling beams in an off-resonant Raman
process.
1.4 Thesis Outline
In this chapter, we have reviewed some of research work centered around pro-
cesses in a three-level lambda system. In the following chapters we discuss the
experimental results that are central to this thesis.
In chapter 2, we present a slow light system based on dual Raman absorption
resonances in warm rubidium vapor. Each Raman absorption resonance is pro-
duced by a coupling beam in an off-resonant Λ system. This system combines
all-optical control of the Raman absorption and the low dispersion broadening
properties of the double Lorentzian absorption slow light. The bandwidth, group
delay and central frequency of the slow-light system can all be tuned dynami-
cally by changing the properties of the coupling beam. We demonstrate multiple
pulse delays with low distortion and show that such a system has fast switching
dynamics and thus fast reconfiguration rates.
In chapter 3, we demonstrate a four-wave mixing based storage of signal pulses
in warm rubidium vapor for times much longer than the pulse width. We first
show the results which demonstrate storage of temporal information of a single
transverse mode pulse. The signal is stored in and retrieved from the long-lived
ground state atomic coherences. We show that the signal and spontaneously
generated idler can be stored simultaneously in the medium and can be retrieved
by just turning off and turning on the coupling beam. We also show that the
retrieved signal pulse power varies linearly with the retrieval beam demonstrating
25
that a coherently prepared vapor generate signal and idler more efficiently than an
unprepared medium. We also present an experimental realization of the storage
of multiple transverse modes of the signal in the rubidium vapor. We show that
an image impressed onto a 500 ns pulse can be stored and retrieved up to 30 µs
later. The primary limitation in storing multiple transverse modes in hot vapor
is atomic diffusion. The image storage is made robust to diffusion by storing the
Fourier transform of the image.
In chapter 4, we present experimental results demonstrating the signal beam
propagation with a small spot size over several diffraction lengths. We use the
off-resonant coherent Raman absorption in warm atomic rubidium vapor to create
a medium whose refractive index changes along the radial direction quadratically.
Such a medium is known to guide light and we demonstrate such a waveguide
whose properties are controlled by a low power Laguerre Gaussian coupling laser
beam. We first present the basic theoretical predictions of a lossless quadratic
index medium and then show that the spot size of the signal as it propagates in
the medium can be controlled using a coupling beam. We investigate the behavior
of the signal beam size as we vary coupling beam parameters. We also show that
the coupling efficiency of the signal beam into the waveguide varies linearly with
the signal power.
In chapter 5, we present a tool for off-resonant atomic experiments. We present
an atomic prism spectrometer with five orders of magnitude greater dispersing
power than a standard glass prism. The prism utilizes the steep linear dispersion
between two strongly absorbing hyperfine atomic resonances of rubidium. We
show that the number of resolvable spectral lines is proportional to the slow light
delay-bandwidth product. We resolve spectral lines 50 MHz apart and realize
26
a spectral sensitivity of 20 Hz/√Hz via precision deflection measurements. To
demonstrate the practicality of this setup, we spatially separate collinear pump,
signal and idler beams resulting from a four-wave mixing process. We believe this
prism will have applications in quantum information, in spatially separating sev-
eral teeth of an optical frequency comb and as a spectral filter in atomic resonance
experiments.
In chapter 6, we make some concluding remarks with suggestions for the pos-
sible future work.
In Appendix A, we present a simple theoretical model that predicts some of
the key features of the transmission of the signal beam through a lambda system
in the presence of a coupling beam. We derive the susceptibility of the signal beam
in the presence of coupling beam using density matrix equations and discuss the
role of single photon detuning in the shape of signal transmission line profile.
27
2 Double Raman Absorption
Slow Light
In this chapter we present a fast, optically tunable double absorption slow
light system where the absorption resonance for the signal is artificially created
by a coupling beam in such a three-level off-resonant Λ system. We show that
the properties of each of the absorption lines can be changed optically by tuning
the properties of the coupling beams which create them and consequently acheive
optical tunability of slow light.
The resonant Λ system has been studied extensively in the context of EIT
[Fleischhauer et al., 2005]. The Raman absorption line produced by an off-
resonant Λ system has been studied and used to produce fast-light [Mikhailov
et al., 2004; Knappe et al., 2003]. Such a process has also been used by Reim et
al. to store very weak signal pulses [Reim et al., 2010]. Proite et al. create a gain
line and an absorption line using two coupling lasers in order to obtain enhanced
refractive index [Proite et al., 2008]. Here we propose to use two absorption lines
in order to realize a double Lorentzian slow light system. We also study the tran-
sient response to the switching of the coupling beam properties and show that
one can dynamically and rapidly tune the group delays. This system combines
28
all optical control of the Raman absorption and the low dispersion broadening
properties of the double Lorentzian absorption slow light. The bandwidth, group
delay and central frequency of the slow-light system can all be tuned dynami-
cally by changing the properties of the control beam. We demonstrate multiple
pulse delays with low distortion and show that such a system has fast switching
dynamics and thus fast reconfiguration rates.
2.1 Theory
The susceptibility for a three-level lambda system has been derived in the
Appendix A. The real and imaginary parts of susceptibility at the signal frequency
can be written as
χ′(∆, δ,Ωc) = χ′0 + C1
δ′
δ′2 + γ′2− C2
γ′
δ′2 + γ′2, (2.1)
χ′′(∆, δ,Ωc) = χ′′0 + C2
δ′
δ′2 + γ′2+ C1
γ′
δ′2 + γ′2. (2.2)
The effective detuning and the effective line width are given by δ′ = δ−δ0 and
γ′ = γ+γ0 respectively where δ0 = |Ωc|2∆/(4∆2+Γ2) and γ0 = |Ωc|2Γ/(8∆2+2Γ2).
The constants C1 = β |Ωc|2
4∆2−Γ2/4
(∆2+Γ2/4)2and C2 = β |Ωc|2
4∆Γ
(∆2+Γ2/4)2.
Fig. 2.1 illustrates the dependence of C1 and C2 on ∆. Solid line in the plot
shows 2C1/Γ, dashed line shows 2C2/Γ and dotted line shows normalized χ′′0. C2
is zero for ∆ = 0, is maximum at ∆ = Γ/2√3 and falls back to zero for large
∆. C1 is negative for ∆ < Γ/2 and positive for ∆ > Γ/2. We also see that
for ∆ >> Γ/2, C2 approaches zero faster than C1 and this is the region we are
interested in to produced an Lorentzian-like absorption line.
29
0 1 2 3 4−6
−4
−2
0
2
∆/Γ
2C1/Γ
2C2/Γ
n’’/n’’max
Figure 2.1: Dependence of C1 and C2 on ∆
3
∆ ∆
Γ/2 Γ/2 Γ/2 Γ/2
γγ
δ1 δ2
Ωc1 Ωc2Ωs
VC1 VC2
(a) (b)
Ωs
22
1 1
3
Figure 2.2: (a) & (b) The energy level diagram for vapor cells VC1 and VC2respectively.
30
Now consider the case where the signal has to propagate through two identi-
cally prepared vapor cells VC1 and VC2 as shown in Fig. 2.2. The effective suscep-
tibility at the signal frequency is given by the sum of susceptibilities in each of the
cells. The control beams in the two cells only differ by frequency 2δc. The effec-
tive susceptibility is given by χeff (∆, δ,Ωc, δc) = χ(∆, δ−δc,Ωc)+χ(∆, δ+ δc,Ωc)
where χ(∆, δ,Ωc) is given by Eqn. (1). Expanding the refractive index around
δ = |Ωc|2∆/(4∆2+Γ2), we obtain the following expressions for real and imaginary
parts of the refractive index.
n′ = n′0 −
C1
δc
[
δ′
δc+
(
δ′
δc
)3
+ ...
]
− C2γ′
δ2c
[
1 + 3
(
δ′
δc
)2
+ ...
]
, (2.3)
n′′ = n′′0 −
C2
δc
[
δ′
δc+
(
δ′
δc
)3
+ ...
]
+C1γ
′
δ2c
[
1 + 3
(
δ′
δc
)2
+ ...
]
. (2.4)
For the off-resonant case, the absorption line shape can be approximated to a
Lorentzian and thus we can obtain a double Lorentzian system. From the above
equations, we can derive the group index, defined as ng = n′0 + ωsdn/dω, to be
ng = n′0+ωsC1/δ
2c . The relevant slow light parameters for a propagation distance
L, the group delay τd and absorption αL are given as
τd =βωsL
c
|Ωc|24∆2
1
δ2c, (2.5)
αL =βωsL
c
( |Ωc|24∆2
)2Γ
2δ2c. (2.6)
Following the approach of Camacho et al. [Camacho et al., 2006], we can
write the dispersive and absorptive broadening for a pulse with Gaussian envelop
31
exp(−t2/(2T 20 )) after propagation through the medium of length L as
T 2a = T 2
0 +12ωsLC1γ
′
cδ4c= T 2
0 +12τdγ
′
δ2c, (2.7)
T 2d = T 2
0 +
(
3ωsC1L
δ4c cT20
)2
= T 20 +
(
3τdδ2cT
20
)2
. (2.8)
We define the bandwidth, B, of the system as the bandwidth of the pulse
for which the absorptive broadening is equal to two. Thus the bandwidth of the
system is given as B = δc/(2√αL). Also, from above equations we note that
τd = αL/γ0 and hence the time delay for one absorption length is simply the
inverse of the power broadened line width of the Raman absorption resonance.
The delay-bandwidth product for one absorption length is thus given by
τd ×B =δc2γ0
. (2.9)
The bandwidth of the system can be tuned simply by changing the frequency
difference between two control beams and the delay-bandwidth product can be
changed by changing the line width, γ0, of the Raman resonance. γ0 can be
tailored by changing the single photon detuning, ∆, or by changing the power of
the control beams. Thus we obtain all optical control over the bandwidth and the
delay bandwidth product.
In order to study the time response of the system, we need to solve the dynamic
equations for atomic coherence. In the weak perturbation regime for the probe,
we write the following equation for the ground state coherence ρ21.
32
ρ21 + i(∆ + δ − iΓ/2− iγ) ˙ρ21
−[
(∆− iΓ/2)(δ − iγ)− |Ωc|2/4]
ρ21 +Ωs
2(δ − iγ) = 0 (2.10)
We first solve the above equation for the case where the control beam is turned
on at time t = 0. The solution is of the form
ρ21(t) =
[
λ−
λ+ − λ−eλ+t − λ+
λ+ − λ−eλ−
t
]
(ρ(ss)21 − ρ
(0)21 ) + ρ
(ss)21 , (2.11)
where ρ(ss)21 is the steady state coherence and ρ
(0)21 is the steady state coherence
without the control beam on. The constants λ+ and λ− for large ∆ are given by
λ+ = i|Ωc|24∆
− |Ωc|2Γ8∆2
, (2.12)
λ− = −i
(
∆+|Ωc|24∆
)
− Γ
2
(
1− |Ωc|24∆2
)
. (2.13)
For |Ωc|2
4∆2 ≪ 1 we can see that exp(λ−t) is rapidly decaying compared to exp(λ+t)
Hence the characteristic transient time for turning on the slow-light medium is
given by reciprocal of real part of λ+.
τ (on) =8∆2
|Ωc|2Γ= 1/γ0. (2.14)
In order to calculate the turn off time, we consider the case where the coherence
is ρ(ss)21 for t < 0 and pump is turned off at t = 0. The time evolution of the
33
coherence for t > 0 is given by
ρ21(t) =(
ρ(ss)21 − ρ
(0)21
)
e−(i∆+Γ/2)t + ρ(0)21 . (2.15)
The characteristic turn off time in this case is just the excited state life time.
τ (off) = 2/Γ. (2.16)
Lastly, for the case where the frequency of the control beam is changed without
changing its intensity, we state without explicitly writing the equations that the
characteristic time for the transients to decay is same as τ (on). The coherence also
has a beat note during this transient time oscillating at the difference frequency
of the initial and final control beams.
We see that the turn on time τ on is the inverse of the Raman absorption line
width and the turn off time is the inverse of the homogeneous line width of the
excited state. Hence τ off ≪ τ on. τ on is also equal to the time delay obtained for
one absorption length. Thus for a system with a delay-bandwidth product of one
we can change the slow light parameters at the bit rate of the pulse stream and we
can turn off the slow light within the time much less than the pulse width. Such
fast dynamics makes this scheme an attractive candidate for pulse synchronization
techniques.
2.2 Experimental Setup
We demonstrate the dual Raman absorption slow light using warm atomic ru-
bidium vapor. The outline of the experimental setup is depicted in Fig. 2.3. The
34
DLaser
795nm
λ/2
PB1 PB2 PB3 PB4 PB5
AO
1
AO2 AO3
VC1 VC2 PD
C1 C2
S
BS
Figure 2.3: The outline of the setup used in the experiment.
795 nm laser comprises of a narrow line-width tunable diode laser followed by a
tapered amplifier to obtain high power, narrow line-width laser beam. The fre-
quency of the laser is tuned to the required frequency of the coupling beam. Signal
“S” is obtained by frequency shifting part of the the laser beam by 3.035GHz by
double passing it through a tunable 1.5GHz acousto-optic modulator (AOM1).
The other part of the laser beam is further split and sent through two 80 MHz
AOMs resulting in two coupling beams “C1” and “C2”. The frequency difference
between C1 and C2 is controlled by changing the RF frequency fed to these AOMs.
The coupling beams C1 and C2 are combined with the signal at the polarizing
beam splitters PB2 and PB4 in front of the vapor cells VC1 and VC2 respectively.
C1 is filtered from signal at PB3 before VC2 and C2 is filtered at PB5 placed after
the vapor cell VC2. The signal is measured at the photo detector PD.
VC1 and VC2 are identical 5 cm long vapor cells placed inside hollow mu
metal tubes to block stray magnetic fields. The vapor cells are heated by strip
heaters and the current passing through the strip heaters is identical for each
cell to ensure similar temperatures for both vapor cells. Each cell along with the
mu metal tubing and the heaters is placed inside a teflon tube enclosed by anti-
reflection coated windows at each end. The vapor cells contain both rubidium
35
isotopes in their natural abundance. In addition we also have 20 torr neon in each
cell which acts as a buffer gas. The temperature of each vapor cell is about 80 C
resulting in a number density of about 1012 cm−3 in each cell.
2.3 Experimental Results
2.3.1 Single Absorption Characteristics
Fig. 2.4 shows the experimental observations of coupling beam induced ab-
sorption line shapes. We measure the signal transmission through one of the vapor
cells shown in Fig. 2.3. The two photon Raman detuning is changed by changing
the modulation frequency at the AOM1. The observed transmission as a func-
tion of Raman detuning is then divided by transmission with the coupling beam
off to obtain normalized transmission. The natural logarithm of the normalized
transmission gives the absorption coefficient times the length of propagation, αL.
Fig. 2.4 shows the absorption lines for three different coupling powers 3 mW
(squares), 6 mW (circles), 10 mW (diamonds). The solid lines show the fitted line
for the experimental data shown by markers. We fit the experimental data to the
function given by Eqn. 2.4 with C1,C2,δ0,γ0 and n′′0 as fit parameters.
Fig. 2.5(a) shows the variation of fit parameters C1 (squares), C2 (circles)
and Fig. 2.5(b) show the variation of δ0 (circles) and γ0 (squares) with coupling
beam power. From the definitions, we expect a linear dependence of C1 and C2
with coupling beam intensity. There is a clear linear dependence upto 10 mW.
Beyond 10 mW, four-wave mixing begins to affect the line shape and so we observe
nonlinear behavior beyond 10 mW. We can also see that the absorption and the
line width increase linearly with the increasing coupling power. At low coupling
36
−0.2 −0.1 0 0.1 0.2
0
1
2
3
4
Raman Detuning (MHz)
αL
Figure 2.4: Variation of absorption line shapes with coupling beam power.
0 5 10 15 200
2
4
0 5 10 15 200
20
40
60
Coupling Power (mW)
Fre
qu
en
cy (
kH
z)
(a)
(b)
Figure 2.5: Variation of fit parameters with coupling beam power.
37
−2 −1 0 1 20
0.2
0.4
0.6
0.8
1
1.2
Raman Detuning (MHz)
Tra
nsm
issio
n (
arb
. u
nits)
Figure 2.6: Transmission of the signal through a vapor cell in the presence of twocoupling beams differing in frequency.Green curve is a reference absorption withsingle coupling beam frequency. The coupling beams are separated in frequencyby 1 MHz and 2 MHz for the transmission curves in red and blue respectively.
beam powers where the we can ignore power broadening, the line width is relatively
constant. This gives the ground state decoherence rate γ for our system. Thus
we can tune the absorption properties just by changing the coupling beam power.
We can also change the center of absorption just by changing the frequency of the
coupling beam by tuning the AOM.
2.3.2 Demonstration of Slow Light
In order to obtain double absorption slow light we first introduce two coupling
beams at two frequencies and the signal into just one vapor cell. Fig. 2.6 shows
the observation of line shapes. The green curve shows the line shape with just one
coupling beam frequency. The red curve shows the transmission with two coupling
38
−40 −20 0 20 40
0
0.05
0.1
0.15
0.2
Time(ms)
Am
plit
ud
e (
Arb
. U
nits)
Figure 2.7: The signal pulse out of a vapor cell with two coupling beams differingin frequency.
beam frequencies separated by 1 MHz and the blue curve shows the transmission
with frequency separation of 2 MHz. The powers in each of the coupling beam
are equal. We see that having two coupling beam frequencies in one vapor cell
reduces the depth of absorption in each of the absorption compared to having just
one frequency. This is due to competing nonlinear effects producing frequencies
at the multiples of the coupling frequency difference. We can see evidence of this
effect in the red curve where we have small absorptions on either sides of the
double absorption feature we expect. A signal pulse tuned to a frequency between
two absorption resonances exhibit beating at the difference frequency of the two
coupling beams at the output. Fig. 2.7 demonstrate this effect. Gray curve is the
reference pulse and black curve is the pulse shape at the output of the cell.
We need to use two vapor cells with a single coupling beam in each to overcome
the above mentioned problems. The transmission profile for this case is shown in
39
Tra
nsm
issio
n (
arb
. u
nits)
−15 −10 −5 0 5 10 150
10
20
30
40
50
Frequency (MHz)
Figure 2.8: The transmission profile of the probe corresponding to the observeddelay.
−0.4 −0.2 0 0.2 0.4 0.6 0.80
0.2
0.4
0.6
0.8
1
µs)
Inte
nsity
Time (
Figure 2.9: Reference pulse and the delayed pulse.
40
Fig. 2.8 with 3 MHz of frequency difference between two coupling beams. Fig
2.9 shows the experimental observation of the delay. The frequency difference
between two coupling beams in VC1 and VC2 is 3 MHz. The alignment of the
coupling and signal beams is adjusted for each vapor cell to obtain the Raman
absorption dip. The signal pulse with full width at half maximum (FWHM) of
about 179µs is tuned to the center of transparency in between the absorption dips.
We observe a delay of about 374ns with the coupling beams on. This corresponds
to a delay bandwidth product of about 2 (bandwidth here is measured as FWHM
width of the pulse). The FWHM of the output pulse is about 262ns and thus a
broadening of factor 1.4. Thus we demonstrate slow group velocity with relatively
less pulse broadening using the two coupling induced absorption dips.
2.3.3 Slow Light Switching
First we will measure transient times for the change in absorption of the signal
because of the changes in coupling beam properties. Fig. 2.10 shows the exper-
imental observations of switching times. In order to measure turn off time, the
signal pulse, tuned to the center of Raman absorption, is sent through the medium
with the coupling beam on and as the pulse is propagating inside the medium,
coupling beam is turned off. In the Fig 2.10, dotted curve on the top shows the
reference pulse and on the bottom shows the output pulse in the presence of cou-
pling induced absorption. The dashed line shows the coupling level. The solid
line on the top shows the turn-off characteristics of the absorption and on the
bottom the turn-on characteristics. Upper solid curve shows that the pulse has
lower transmission before turning off the coupling beam because of Raman ab-
sorption. After the coupling beam is turned off, the signal transmission increases
41
−20 −10 0 10 20
Time(µs)
Figure 2.10: Measurement of turn-on and turn-off times of Raman absorption.
to match with the dotted line within 50ns and the rise time is limited in this case
by the rise time of the AOM used to switch coupling beam. The turn on time is
measured by having the coupling beam initially off and turning it on as the signal
is inside the medium as shown in the lower curve of Fig. 2.10. The turn on time
is measured to be about 0.75 µs.
The transients for the change in frequency of the coupling beam has similar
time constant and we see modulation in the signal during transition time. Fig.
2.11 shows the transient characteristics of the signal transmission for the change
in coupling beam frequency. Coupling beam frequency is initially tuned to Raman
resonance so that signal experiences absorption. Then we change the frequency
of the coupling beam so that Raman detuning is far off resonance where the sig-
nal experiences no absorption. During the transition time from being absorptive
to being transmissive, the signal experiences beating in its amplitude with a fre-
quency equal to the difference in the coupling beam frequency.
Fig. 2.12 shows the experimental observation of slow light switching. The
42
−30 −20 −10 0 10 20
0
0.1
0.2
0.3
0.4
0.5
0.6
Time (µs)
Tra
nsm
issio
n (
arb
. u
nits)
Figure 2.11: Beating in the cw signal beam resulting from coupling beam fre-quency change.
2 4 6 80
0.02
0.04
0.06
0.08
0.1
0.12
Tra
nsm
issio
n(a
rb. u
nits)
Time(µs)
Figure 2.12: The experimental observation of slow light switching.
43
dotted line is the input pulse without any delay and the broken line is the output
pulse with delay. The black line demonstrates the slow light switching. As the
pulse is compressed inside the vapor cell, we turnoff the control fields as shown by
the black dashed line. We turn off the slow propagation of the pulse all optically
within 50ns demonstrating high speed switching of slow light.
2.4 Summary
We have presented a double absorption slow light system based on the Raman
absorption dip. The strength, the width and the center of the absorption can be
modified all optically by changing the properties of the coupling beam. In addition
we can even tune the number density and the single photon detuning to further
modify the properties of the slow light system. The wide range of independent
controls makes this slow light system versatile. However the limitation to this
versatility is imposed by the competing nonlinear four-wave mixing process which
dominates at high atomic number densities. This limitation is dealt to some
extent in this paper by careful alignment of the probe and coupling beams in
order to have a phase matching favoring absorption and suppressing four-wave
mixing. Also we avoid the problem of multi-wave mixing resulting from having
two coupling beams and the probe beam in one vapor cell by using two vapor
cells.
44
3 Stopped Light Using Four
Wave Mixing
In this chapter we present experiments that demonstrate the ability to store
light using two Λ resonances simultaneously, and hence record information from
two distinct optical frequencies for later retrieval. The four-wave mixing process
discussed in chapter 1 is used to obtain the double lambda system. The signal
beam and spontaneously generated idler beam are stored simultaneously in warm
rubidium vapor. As in EIT based slow light, light pulses are mapped onto a
coherent polarization (spin) wave in an atomic ensemble, and then retrieved by
converting the spin wave back into an optical pulse. The four-wave mixing gain
on Raman resonance, and absorption away from Raman resonance results in large
pulse compression allowing for storage of a complete waveform. In addition, the
parametric process generates a frequency-shifted idler pulse that may also be
stored and retrieved simultaneously with the signal pulse.
Many experiments have been performed where optical information is stored
in one lambda transition and retrieved with another [Zibrov et al., 2002; Wang
et al., 2005; Chen et al., 2006]. In these experiments, a collective atomic spin
wave is coherently prepared by injecting both write beams and stokes beams
45
simultaneously, and then anti-Stokes beam is read out using a read beam whose
frequency is chosen to complete a four-wave mixing process. In the present scheme,
the signal and idler generated in the four-wave mixing are known to be correlated.
The storage of such fields may have important implications in image processing,
quantum information, and remote sensing, where two correlated fields may need
to be preserved for later use. If macroscopic fields are stored, one may imagine
retrieving only small fractions of the coherence multiple times, allowing for the
production of weak correlated fields at regular intervals.
Recently there has been some interest in extending the ideas of single trans-
verse mode slow and stopped light to multiple transverse modes where the atomic
medium delays and stores the spatial profiles of the stored pulses [Jain et al., 1995;
Camacho et al., 2007; Pugatch et al., 2007; Zhao et al., 2008; Shuker et al., 2008].
One of the difficulties of using hot vapors as the storage medium for stopped light
experiments is the diffusion of atoms during storage period. Recently, Pugatch et
al. [Pugatch et al., 2007] showed that an optical vortex with a phase singularity
in the transverse spatial profile can be stored in an atomic medium despite strong
diffusion. Subsequently there has been some theoretical work done to make image
storage in hot vapors robust to diffusion [Zhao et al., 2008].
We demonstrate the storage and retrieval of a transverse image in a hot
atomic vapor using a combination of electromagnetically induced transparency
(EIT)[Harris et al., 1990] and four-wave mixing (FWM) techniques. We overcome
the adverse effects of diffusion by storing the Fourier transform of the image in
the stopped light medium rather than the image itself. While the optical phase
in the object plane is constant, the phase in the Fourier plane varies in such a
way that atoms with opposite phase destructively interfere during diffusion [Zhao
46
et al., 2008]. This is similar to the storage of the Laguerre-Gauss beams [Pugatch
et al., 2007] under diffusion where atoms on opposite sides of the phase singularity
have relative phases of π and destructively interfere.
3.1 Theory
From Appendix A, we know the linear susceptibility at the signal frequency
for a lambda system. The linear polarization at signal frequency can be written
as,
P (1)s = Nµ12
Ωs
2
∆R
∆s∆R − Ω2c
4
(3.1)
The quantities ∆s = ∆s − iΓ/2 and ∆R = ∆s − ∆c − iγ are the complex single
photon and two photon (Raman) detunings where Γ and γ represent the transverse
excited and longitudinal ground-state decay rates respectively, N is the atomic
number density, and Ωj = Ej · µj/h represents the Rabi frequency induced by
electric field amplitdue Ej via the dipole matrix element µj.
The nonlinear polarization at the signal frequency is proportional to the idler
field and is responsible for four-wave mixing:
P (NL)s =
Nµ12
8
ΩiΩ2c
∆1∆2∆3 − Ω2c∆1/4
, (3.2)
where ∆1 = ∆c+ω12− iΓ/2 ,∆2 = ∆c+ω12−∆i− iγ, and ∆3 = 2∆c+ω12−∆i−
iΓ/2 are the complex one, two and three photon detunings along this excitation
pathway.
The total steady state polarization oscillating at the signal frequency is then
Ps = P(1)s +P
(NL)s . A similar result may be derived for the polarization oscillating
47
at the idler frequency:
P(NL)i =
Nµ32
8
ΩsΩ2c
∆s∆R∆1 − Ω2c∆1/4
, (3.3)
which, except for being proportional to Ωs instead of Ωi, is identical to the non-
linear contribution (P(NL)s ) to the polarization at the signal frequency when the
idler beam is generated in a parametric process, which guarantees the equivalence
of ∆s and ∆3 as well as ∆R and ∆2.
The coupled amplitude equations for signal and idler for perfect phase match-
ing can thus be written as:
∂Ωs
∂z= gsΩi − αsΩs, (3.4)
∂Ωi
∂z= giΩs. (3.5)
The gain coefficients gs and gi can be written as:
gs =ωs
2cnsℑ(χFWM(ωs))|Ec|2, (3.6)
gi =ωi
2cniℑ(χFWM(ωi))|Ec|2. (3.7)
Solving the coupled amplitude equations we obtain the following solution for Ωs(z)
and Ωi(z):
Ωs(z) = Ωs(0)
[
cosh(ξz)− αs
2ξsinh(ξz)
]
e−αsz/2, (3.8)
Ωi(z) = Ωs(0)gigsξ
sinh(ξz)e−αsz/2, (3.9)
where ξ =√
α2s/4− gigs
48
The steady state coherence between ground states |1〉 and |3〉 can be written
as:
ρ13 =ΩsΩc
∆s∆R − Ω2c
+ΩiΩc
∆1∆2 − Ω2c∆1
∆3
. (3.10)
We note that the ground state coherence set up in the medium has two distinct
terms, one proportional to signal field amplitude and one proportional to the idler
field amplitude, and that each contribution identifies a unique excitation pathway.
As we adiabatically turn off the coupling beam, all coherences associated with the
excited state decay within excited state lifetime which is very fast. The ground
state coherence decays much slower. When we turn on the coupling beam at a
later time before ρ13 decays we resume the gain process and obtain signal and
idler.
We now model storing a transverse spatial profile of the signal in the atomic
vapor. Consider the system shown in Fig. 3.1. Suppose we have a 4f imaging
system such that the vapor cell is placed in between two lenses as shown in the
figure. In this way the transverse intensity profile at the vapor cell is the Fourier
transform of the object. The second lens images the object onto a camera.
We adopt the diffusion model of Pugatch et al. to simulate image diffusion
in the Fourier plane. We first determine the Fourier transform of the field in the
back focal plane of L1. Let Eo be the field at the object plane which is also the
front focal plane of a spherical lens of focal length f . At the back focal plane of
the lens, the field Ef is given by the Fourier transform of the field in the object
plane:
Ef = F (Eo) =1√λf
∫ ∞
−∞
Eo exp(−i2π
λfξu)dξ, (3.11)
where ξ and u are the coordinates of object plane and Fourier transform plane
49
respectively. The ground state atomic coherence at the time of storage is given
by ρ13 = gΩs, where g is the nonlinear coupling coefficient and Ωs is the Rabi
frequency of the signal field. The time evolution of the atomic coherence is given
by
∂ρ13(u, t)
∂t= D
∂2ρ13∂u2
− ρ13Tc
(3.12)
where Tc is ground state decoherence time of the coherence ρ13, and D is the
diffusion coefficient of the atoms. Assuming a constant pump intensity along the
transverse dimension of the cell, the only spatial dependence of the ground state
coherence comes from the signal field amplitude.
The field at the image plane, Ei, is recovered by inserting Eq. (1) into the
diffusion equation [ Eq. (2)] and taking another Fourier transform, which upon
integration gives
∂
∂tEi(x, t) = −(
1
Td
+1
Tc
)Ei(x, t), (3.13)
with a solution given by
Ei(x, t) = Ei(x, 0) exp
[
−t(1
Td+
1
Tc)
]
, (3.14)
where Td = λ2f2
(2π)2Dx2 is the diffusion time constant and x is the coordinate in the
image plane.
There are two features worth noting in Eq. 3.14. First, each spatial point in
the image decays exponentially in time, with a time constant given by 1/Td +
1/Tc. This means that dark areas of the image remain dark for appreciable times
compared to the temporal pulse length. Second, since Td falls off as 1/x2, the
central portion of the image has maximum storage time. We can increase the
diffusion time Td by making the image smaller or making the focal length of the
50
T T
Input Signal
Pump Pulse
Retrieved Signal
1 2
Delayed Signal
ObjectRb
Vapor CellPBS PBS Image
Camera
Signal
3.035 GHz Acousto-Optic
Modulator
Laser 795 nm
Pump
L1 λ
2
L2
(a)
(c)
Ωs
Ωi
ΩC
ΩC
|1
|3
|2∆s ∆c
∆ i
(b)
Figure 3.1: (a) Experimental setup. (b) Four wave mixing energy levels. (c)Representation of the synchronized timing of signal (dashed red), delayed signal(blue) and coupling (black) beams.
imaging lens larger. In either case, the Fourier transformed spatial profile at
the vapor cell would be larger, requiring a correspondingly larger coupling beam
diameter and vapor cell.
3.2 Experimental Setup
A schematic of the experimental setup is shown in Fig. 3.1(a). An external
cavity diode laser followed by an amplifier is the source for coupling beam. The
signal is generated by frequency shifting part of the coupling by 3.035 Ghz using
51
an acousto-optic modulator. The full-width at half maximum signal and coupling
field intensity spatial profiles for pulse storage experiments were approximately
500 µm and 300 µm respectively. For image storage experiments, the 1/e2 beam
diameter of the coupling is approximately 4 mm and that of signal approximately
1 mm at the object plane. The powers of coupling and signal beams are approx-
imately 12 mW and 300 µW respectively. The coupling and signal beams are
orthogonally polarized and are combined using a polarizing beam splitter (PBS)
before the cell. The coupling is filtered from the signal using another PBS after
the cell. We use the D1 transitions of 85Rb to create a Λ configuration. The cou-
pling is detuned by 700 MHz to the blue of the optical transition connecting the
F = 2 ground state to the F ′ = (2, 3) excited states, and the signal is set 3.035
GHz (the ground state hyperfine splitting) to the red of the coupling.
A 12.5 cm long Rb vapor cell with natural isotopic abundance containing 20
torr neon buffer gas is placed in a magnetically shielded oven and heated to ap-
proximately 180 C, yielding a number density of approximately 1013 atoms/cm3.
A 4f imaging system is used to image the object on to the camera as well as place
the Fourier plane at the cell. It consists of two lenses, L1 and L2, each of focal
length of f = 500 mm separated by a distance 2f . The object is placed at the
front focal plane of L1 and the image is obtained at the back focal plane of L2.
The vapor cell is placed at the back focal plane of L1 (the Fourier plane). The
diameter of the cell is 1 cm and the transverse diameter of the signal beam is
chosen such that the profile of Fourier image fits in the cell. The coupling beam
is orthogonally polarized to the signal to filter out the coupling. In addition to
polarization filtering, we also performed a temporal filtering correlation measure-
ment using a 100 MHz detector (3 dB roll off). A 25 µm slit is placed in the focal
52
−0.1 −0.05 0 0.05 0.1 0.150
0.2
0.4
0.6
0.8
1
Ω (
arb
. u
nits)
s 2
δ (MHz)r
Figure 3.2: Measured (asterisks) and theoretical (solid) steady state signal (blue)and idler (red) intensities as a function of signal detuning.
plane and a bucket detector is placed behind the slit. The position of the slit is
scanned in the image plane and the temporal intensity profile of the retrieved light
pulse hitting the bucket detector is recorded is measured on a 1.5 GHz scope.
Our slow light scheme is based on a combination of EIT and FWM in a Λ
system which consists of two lower energy levels coupled to a common higher
energy level of the atom by two electromagnetic fields. The relevant energy levels
of 85Rb are shown in Fig. 1(b). To obtain a highly transparent region which
also exhibits steep dispersion, the pump and signal lasers are detuned several
hundred MHz from the zero velocity class in a Doppler broadened vapor. The
signal experiences both FWM gain and EIT when its frequency is tuned to the
two-photon Raman resonance. As a note, optical alignment, buffer gas pressures,
laser detunings etc. all affect the transmission and dispersion. The rapid change
in the transmission profile near Raman resonance leads to a steep dispersion and
slow group velocity. Once the signal has been compressed inside the cell, we turn
53
off the pump, storing the image. The transverse spatial profile of the signal is
mapped onto the long-lived ground state coherence of the 85Rb atoms. The signal
is retrieved by turning on the pump at a later time. The timing of the pulses is
illustrated in Fig. 1(c). The retrieved signal intensity falls off exponentially with
storage time due to diffusion and decoherence.
3.3 Experimental Results
To demonstrate pulse storage and retrieval we take the object and lenses out
of the setup. The signal beam is pulsed using a separate AOM (not shown) and
the coupling beam is turned off while the pulse is compressed within the cell and
then turned on at a later time. A plot of the measured signal and idler intensities
as a function of signal detuning around Raman resonance is shown in Fig. 3.2
for the case of 3 mW coupling field intensity and 50 µW of signal field intensity
before entering the cell. Accompanying the experimental data are theory plots,
generated by inserting Eqs. 3.8 and 3.9. We used the parameters Ωs(0) = 1,
z = 10 cm, N = 2 × 1011 atoms/cm3, Γ = 36 Mhz, γ = 10 Khz, Ωc = 7 Mhz,
∆c = 700 Mhz, ∆i = 2∆c + ω12 − ∆s and have included 600 Mhz of doppler
broadening for theory plots. The theory curves agree well with the data except
for a frequency offset of approximately 0.25 Mhz to the blue (not shown), which
may be due to buffer gas suppression of linear stark shifts [Nagel et al., 1999].
Figure 3.3 shows the delay of a 1.4 µs signal pulse and the simultaneous gener-
ation of a idler pulse (a), as well the storage and retrieval of both signal and idler
pulses 20 (b) and 120 (c) µs later. Part (a) of the figure shows an example of an
incident signal pulse (black), delayed signal pulse (blue) and generated idler pulse
(red) for the case when the coupling field (3 mW) remains on for the duration of
54
0
1
2
3
4
5P
ea
k In
ten
sity (
% o
f in
pu
t p
uls
e)
0 10 20 30
0 10 120 130
x 1/20 x 3
0 10 20 30
(a)
(b)
(c)
Time (µs)
Figure 3.3: Demonstration of four-wave mixing slow light and stored light
the measurement. In parts (b) and (c) of the figure, the coupling field (dashed
line) is turned off approximately 500 ns after the peak of the signal pulse enters
the medium, and then turned back on 20 µs and 120 µs later, respectively. The
retrieved pulse waveforms correspond to that fraction and shape of each pulse
that was in the medium when the coupling field was turned off, demonstrating
that the stored coherence contains information about each waveform that may be
separately retrieved
The maximum storage time observed was approximately 500 µs. We note that
the entire signal pulse waveform is retrieved and is broadened temporally by a
factor of two. Also of note is that only a fraction of the idler pulse is recovered,
indicating that each pulse retains its individual waveform during storage and
retrieval. This means that the retrieval process is not simply incoherent scattering,
55
101
102
103
100
101
102
Pump Retrieval Power (µW)
Sig
na
l P
ow
er
(µW
)
Figure 3.4: Idler pulse peak power as a function of coupling input power.
as might be expected by storing the signal pulse in the absence of four-wave mixing
gain and then retrieving near a four-wave mixing resonance. If that were the case,
then one would expect similar pulse shapes for both the signal and idler beams.
It is also of note that the intensity of the retrieved pulses varies linearly with
coupling retrieval power, as shown for the idler beam (the signature of four-wave
mixing) in Fig. 3.4. The pulses were stored using approximately 3.5 mW of
coupling power, and then retrieved using progressively smaller coupling powers.
Such a scheme may prove useful in cases where low background noise is needed for
precision low light level measurements using optical nonlinearities, since the strong
beams used to prepare the media are turned off long before the weak scattering
beams enter the media. It has recently be shown [McCormick et al., 2007], for
example, that -7.1 dB of relative intensity squeezing may be achieved in a similar
system before storage, and the present scheme may be beneficial for exploring
the entanglement properties between the stored signal and idler fields and in the
generation of narrowband entangled pairs of photons.
56
Image plane Fourier plane
−2
−1
0
1
2
Po
sitio
n (
mm
)
Figure 3.5: CCD camera capture of the signal intensity profile at the object planeand at the vapor cell (Fourier plane).
Having demonstrated temporal pulse storage, we now present the storage of
transverse spatial profile of the signal. We now use the 4f imaging system to
achieve this task. The intensity profile of the signal at the object plane and its
Fourier transform at the vapor cell are shown in 3.5. Figure 3.6 shows the input
image (a), as well as the retrieved (b) and calculated (c) image profiles for several
storage times. The theory plots are generated by using Eq. 3.14 to propagate the
measured input image with a diffusion coefficient of 10 cm2/s. The object used
is an amplitude mask containing a 5 bar test pattern. We note that the image
contrast remains high even for the longer storage times, even though the wings of
the image decay faster than the central part, as predicted earlier.
The physical mechanism responsible for preserving the image can be under-
stood in terms of the phase distribution of the stored optical wavefront and the
diffusion of the atoms (shown graphically in Fig. 3.7). Each atom in the field
acquires the local coherence set by the signal and pump fields. As the atoms dif-
57
Input Image
1
Position (mm)−1 0 1
0
1
(a)
−1 0 1 −1 0 1
Position (mm) Position (mm)
1 µs
Retrieved Images
TheoryExperiment Time
4 µs
16 µs
32 µs
(b) (c)
Figure 3.6: Input signal profile (a) and the time evolution of measured (b) andcalculated (c) transverse images.
58
−1 0 1
−1
0
1
0
Reference10 µs30 µs
Position (mm)
Am
pli
tud
e
Figure 3.7: Theoretical time evolution of stored ground state coherence of Rbatoms. The inset shows a close up of the time evolution near zero crossover pointsof electric field amplitude.
fuse in the Fourier plane, atoms of opposite phase tend to destructively interfere
preserving the high contrast. This is similar to the topological stability of stored
Laguerre-Gauss beams as demonstrated by Pugatch et al. and in good agreement
with the theoretical predictions by Zhao et al.
We note that our theoretical model does not account for the finite numerical
aperture of the 4f imaging system. The size of the numerical aperture in our
system is set by the size of the pump beam at the cell, which has a 1/e2 intensity
diameter of approximately 4 mm. Since this is larger than the spatial extent of all
relevant features in the Fourier transformed image (see Fig. 2), this approximation
is valid. In addition, we have assumed that all modes of spatial diffusion for the
59
prepared atoms remain within the pump beam diameter, so that higher order
modes which exit from and then return to the pump beam during storage do not
cause interference [Xiao et al., 2006]. Since the diffusion length for the longest
storage time is given by√Dt ≈ 170 µm, and the closest relevant feature in the
Fourier transform of the image is farther than 500 µm from the edge of the pump
beam, this is also a reasonable assumption. As a note, we also performed numerical
integration over the relevant finite dimensions of our experiment which produced
negligible errors.
Figure 3.7 shows the evolution of the ground state coherence under the con-
ditions of decoherence and diffusion. The inset of the figure shows a section of
plot containing zero crossover points. We see that the zeros of ρ13 are unchanged,
though the amplitude on either side of zero crossovers decreases with time. As the
atoms with positive and negative phases have equal probability of reaching the
zero crossover point, the retrieved fields from such atoms at those points tend to
destructively interfere, maintaining a zero in the field amplitude. At other points,
the same interference process results in a decrease in amplitude while maintaining
the field profile.
3.4 Summary
In summary, we demonstrated pulse storage and retrieval in rubidium vapor
using four-wave mixing process. We have shown that the four-wave mixing gain
enables us to achieve complete pulse localization within the warm rubidium vapor
cell and allows us to store full wave form. We also demonstrated that the four-wave
mixing efficiency can be improved by coherently preparing the medium ahead of
time using storage.
60
We also demonstrated slowing and storage of an arbitrary transverse image
in a hot atomic vapor, and shown that the retrieved image is robust to atomic
diffusion. We achieved this by storing the Fourier transform of the image instead
of an image with flat phase front. This remarkable feature allows the coherent
storage of spatial information even in doppler broadened media with large diffusion
constants.
61
4 All Optical Waveguiding
Using Raman Absorption
Off-resonant Raman absorption discussed in chapter 2 in the context of slow
light can be used to create an all optical wave guide. The refractive index depen-
dence on the coupling beam power has been exploited to reduce diffraction of the
focused signal beam. We show that the signal beam propagates with a small spot
size over several diffraction lengths. This all optical waveguide is imprinted by a
low power Laguerre Gaussian coupling laser beam. The refractive index at the
annulus of the donut control beam is lower than that at the core for signal fre-
quencies tuned to the blue of Raman resonance. We also show that the coupling
efficiency of the signal beam into the waveguide varies linearly with the signal
power.
4.1 Theory
We first review some of the properties of a Gaussian beam propagating in a
medium whose refractive index decreases quadratically with the radial coordinate
as we move away from the optical axis. The electric field distribution of a Gaussian
62
mode propagating in the z direction is given by
E(r, z) = E0ω0
ωexp
[
−i(kz + φ)− ikr2/2q]
, (4.1)
where φ(z) is the phase shift due to the geometry of the beam and q is the complex
parameter describing the Gaussian beam. q is defined as
1
q=
1
R− i
λ
πω2. (4.2)
R in the above equation is the radius of curvature of the phase front, ω(z) is
the radius of the beam at a location z and ω0 is the beam radius at the beam
waist. As the beam propagates in free space from the beam waist, the spot size
and the radius of curvature vary according to the following equations:
ω2(z) = ω20
[
1 +
(
λz
πω20
)2]
, (4.3)
R = z
[
1 +
(
πω20
λz
)2]
. (4.4)
We have a much simpler transformation relation for the beam parameter q and
it is given by,
q2 = q1 + z, (4.5)
where q1 and q2 are the beam parameters at two spatial locations separated by
the distance z along the propagation direction.
The transformation relation when the beam passes a converging thin lens of
focal length f is given by,
1
q2=
1
q1− 1
f. (4.6)
63
0 2 4 6 8 100
50
100
150
200
250
300
κ = 0
κ = 10
κ = 20
κ = 30
κ = 50
z (cm)
Sig
na
l S
ize
(m
icro
ns)
Figure 4.1: Beam propagation as a function of κ
It is known that in the media where the refractive index varies quadratically
with the transverse position, the transformation for the beam parameter can be
written in terms of an ABCD matrix as,
q2 =Aq1 +B
Cq1 +D. (4.7)
It has been shown by Yariv and Yeh [Yariv and Yeh, 1978] that for the lossless
quadratic index medium, whose refractive index is given by n = n0[1− (κ2/2)r2],
64
the transformation relation is given by,
q(z) =cos(κz)q0 + sin(κz)/κ
−sin(κz)κq0 + cos(κz). (4.8)
Fig. 4.1 shows the change in signal beam size for various values of κ. We
show the change for a propagation distance of 10 cm. The initial width of the
beam is assumed to be 100 µm. κ = 0 corresponds to the case of the free space
propagation. We can see that the beam expands to more than 2.5 times its initial
size in free space. As the value of κ increases, we see that the divergence of
the beam is reduced. For higher values of κ the signal size exhibits oscillatory
behavior. For each κ, the mode is confined to a finite transverse size or, in other
words, the beam is guided within that transverse cross-section. If we can control
the properties of κ, we have a control over the properties of the waveguide.
Next we show that in the Raman system, we can control κ all optically by the
power, size and frequency of the control beam. Consider the coupling beam in the
Laguerre-Gaussian LG-01 mode (donut mode) whose intensity can be written as,
Ic(r) =2P
πw2c
r2
w2c
exp(−2r2
w2c
), (4.9)
where P is the coupling beam power and wc is its width.
When r << wc, we can approximate the above expression as,
Ic(r) =2P
π
r2
w4c
. (4.10)
From above, the square of the coupling beam Rabi frequency can be written
65
as,
|Ωc|2 =4µ2
ch2ǫ0
P
πw4c
r2. (4.11)
Using the above coupling beam rabi frequency, the off-resonant refractive index
discussed in chapter 2 can be written as,
n = 1 +β
2
|Ωc|24∆2
δ
δ2 + γ2= 1 +
β
2
µ2
ch2ǫ0
P
πw4c
δ
∆2(δ2 + γ2)r2. (4.12)
Note that we ignored power broadening and the AC stark effect in the above
equation because the region of interest is the center of the donut beam where the
coupling beam intensity is very small. We can see from the above equation that
we have a medium with a refractive index that changes quadratically with the
radial coordinate.
From Eqn. 4.12, we can write κ2 as,
κ2 = −βµ2
ch2ǫ0
P
πw4c
δ
∆2(δ2 + γ2). (4.13)
We can see from the above equation that for negative values of δ, κ is real
and hence we have an oscillatory solution discussed in the previous section. For
positive values of δ, the refractive index at the core of the LG-01 beam is lower
than at the annulus and hence we have a diverging solution. We also note that the
value of κ2 is proportional to P and hence higher values of coupling beam power
results in better mode confinement. Also note that the value of κ2 is inversely
proportional to w4c and hence the waveguiding is strongly dependent on the size
of the coupling beam.
Fig. 4.2 shows the dependence of output signal size on coupling beam power.
66
0 10 20 30 40 500
100
200
300
400
500
600
700
800
900
1000
100 microns
75 microns
50 microns
25 microns
Coupling Power (mW)
Ou
tpu
t S
ign
al S
ize
(m
icro
ns)
Input Signal Size
Figure 4.2: Beam propagation as a function of coupling rabi frequency
−5 0 5
x 105
0
500
1000
1500
2000
2500
100 microns
75 microns
50 microns
25 microns
Raman Detuning (MHz)
Ou
tpu
t S
ign
al S
ize
(m
icro
ns)
Input Signal Size
Figure 4.3: Beam propagation as a function of Raman detuning
67
Each of the curve is plotted for different input signal beam widths. Eqns. 4.8 and
4.13 are used to simulate the beam size after propagating a distance of 10 cm. We
use the coupling beam width of 800 µm, β of 105 and a single photon detuning of
1.5 GHz for the simulation. We also assumed that δ = −γ = 100 KHz. We can
observe that the behavior of the signal output size is also dependent on its initial
size. Smaller initial size results in faster oscillations in its size.
We can also see from Eqn. 4.13 that κ2 has a dispersive relation with the
Raman detuning. Fig. 4.3 depicts this behavior. The output signal size is plotted
against the Raman detuning for various input signal sizes. The coupling beam
power is assumed to be 10 mW. We see that signal size is smaller for negative δ
and larger for positive δ. The size approaches the free space value for large δ on
either side of zero Raman detuning. The change is faster for smaller input signal
beam size.
4.2 Experimental Setup
The experimental setup shown in Fig. 4.4 consists of a 795 nm external cavity
tunable diode laser followed by a tapered amplifier. The beam is split in two at
a 50/50 beam splitter. One beam acts as the signal after frequency shifting it by
about 3.035 GHz to the red by double passing it through a 1.5 GHz acousto-optic
modulator. The other beam acts as the coupling beam which is sent through a
spatial filter in order to clean up its mode and is followed by a charge one spiral
phase plate, resulting in a first order Laguerre Gaussian beam. The orthogonally
polarized coupling and signal are then combined at a polarizing beam splitter.
We use a configuration where the coupling focuses at the back face and the
68
signal focuses into the core of the donut coupling beam at the front face of the
vapor cell. We use the configuration of focusing coupling beam rather than a
collimated beam because we want to show that the signal beam follows the size
of the coupling beam. The transmission properties of the beams and two photon
characteristics are observed by taking off the spiral phase plate and the lenses L1
and L2. The coupling is filtered at another polarizing beam splitter after the cell.
We image the back face of the cell with a 4f imaging system to determine the
size of the signal. The anti-reflection coated vapor cell is 5 cm long and contains
a natural abundance of rubidium isotopes with a 20 torr neon buffer gas. The
vapor cell is placed inside a magnetically shielded oven and is maintained at a
temperature of about 80 C which results in number densities of approximately
1012 cm−3. We have a positive single photon detuning, ∆, of about 500 MHz.
4.3 Experimental Results
The plot showing the transmission of the signal beam as a function of Raman
detuning is shown in Fig. 4.5. The coupling and the signal are tuned to be
about 500 MHz to the blue of Rb85 F = 2 to F′ = (2,3) and F = 3 to F′ =
(2,3) D1 transitions respectively. Both coupling and signal are collimated and are
co-propagating. The line shape is similar to what we obtained in chapter 2.
The dispersion of the medium is obtained by applying Kramers Kronig rela-
tions on the observed transmission profile. We obtain dispersion profile in this
manner for various coupling beam powers. Fig. 4.6 shows the plot of the varia-
tion of the refractive index versus coupling beam power at a fixed signal frequency
with δ = -1.5 MHz. We can see that the refractive index decreases with increasing
coupling beam intensity. We choose the signal frequency to be close to Raman
69
0 0
Laser
795 nm
AOM
λ/2
PBS PBS
Rb
Signal
Coupling
0
0
DCamera
2ff f
Spiral
Phase
Plate
L1L2
L3 L4
50/50
BS
Figure 4.4: The experimental schematic for all optical waveguiding using atomicrubidium vapor. The focusing scheme for control beam (black) and signal (gray)is shown in the inset
70
Raman Detuning (MHz)-3 -2 -1 0 1 2 3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Tra
nsm
issio
n (
AU
)
Figure 4.5: The experimental plot of the variation in the transmission of signalversus Raman detuning.
resonance such that there is good contrast in refractive index for higher and lower
coupling beam powers and away from the absorption dip shown in Fig 4.5. We
found that we have optimum guiding at a δ = -2 MHz.
Fig. 4.7(a) shows the refractive index profile along the transverse plane. White
indicates higher refractive index and black, along the ring of the donut, is lower
refractive index. The intensity profile at the front of the vapor cell is captured
with a camera and Eqn. 4.12 is used to obtain the refractive index. Fig. 4.7(b)
shows the index profile along one of the axes of the beam. We see that we have a
refractive index contrast of about 10−5 between maximum and minimum refractive
indices. We use a single photon detuning of 500 MHz, a Raman detuning of -1
71
0 5 10 15 20 25 30−16
−14
−12
−10
−8
−6
−4
−2
0
2x 10 −7
Coupling Beam Power (mW)
Refractive Index - 1
Figure 4.6: The plot of refractive index of the signal, tuned -1.5 MHz away fromRaman resonance, as a function of coupling beam power.
-792 -342 108 558
9.6
9.65
9.7
9.75
x 10−4
Position (micro meters)
Re
fra
ctive
In
de
x -
1
(c) (b)(a)
Figure 4.7: (a) shows the spatial variation of refractive index. (b) shows the plotof refractive index versus position along one of the axes.
72
−500 0 500
−50
0
50
100
150
−500 0 500
−50
0
50
100
150
Inte
nsity (
AU
)
Position(microns) Position(microns)
(a) (b)
(c) (d)
(a) (b)
Figure 4.8: The Snap shots of the signal beam profile at the back of the vapor cellwith the coupling beam off (a) and on (b). Beam profiles along the longer axis ofthe beams at the front face of the cell (c) and at the back face of the cell(d).
MHz and 30 mW coupling beam power in Eqn. (4.12) to obtain the refractive
index profiles from a camera snap shot of intensity profile.
Note that the integrated intensity of the black curve is approximately 43%
of the gray curve, indicating a good coupling of power into the waveguide. The
coupling beam power is 18 mW. The coupling beam intensity in (c) and (d) is
normalized to fit in the figure while the signal intensities in (d) are relative.
Fig. 4.8 shows the main result of the experiment. The coupling beam is
converging along the cell and has a focus at the back face of the cell. The snap
73
shots of the signal beam with and without the coupling beam at the back face
of the vapor cell are shown in Fig. 4.8(a) and Fig. 4.8(b). The signal is focused
into the “core” of the coupling beam at the front face as shown in Fig. 4.8(c).
The black dashed line is the measured coupling beam intensity profile, the black
dotted line is the measured signal intensity and the solid black line is the gaussian
fit to the measured signal intensity. The gaussian width of the signal is 56 µm. In
the absence of the coupling beam, the signal beam diverges along the length of the
cell as shown by the gray curve in Fig. 4.8(d). In Fig. 4.8(d), the black dashed
line is the measured coupling beam profile, gray and black dotted lines are the
measured signal beam profiles when the coupling is off and on respectively and
the solid gray and black lines are gaussian fits to the dotted lines. The gaussian
widths of gray and black curves are 102 µm and 35 µm respectively. When the
coupling is on, the signal is guided along its core and so the signal beam diameter
is smaller at the back face of the cell. For example, the Gaussian width of the
signal at the back face is 35 µm with an 18 mW input coupling (black curve of
Fig. 4.8(d)). The integrated intensity of the black curve is approximately 43%
of the gray curve implying that there is a good coupling of the signal power into
the waveguide. Note that the peak intensity of the black curve is more than that
of the gray curve. The coupling beam intensity in Fig. 4.8(c) and Fig. 4.8(d)
is normalized to fit in the figure while the signal intensities in Fig. 4.8(d) are
relative. The laser beams are slightly elliptical and so the axis mentioned above
is along the longer axis.
We saw in Figs. 4.2 and 4.3 the variation of the output signal beam size with
the coupling beam power and the two photon Raman detuning. Fig. 4.9 and Fig.
4.10 shows the experimental observation of the dependence. We see from Fig. 4.9
74
0 5 10 15 2030
40
50
60
70
80
90
100
Coupling Power (mW)
Sig
na
l B
ea
m W
idth
(m
icro
ns)
Figure 4.9: Plots of the signal beam size at the back face of the vapor cell versusthe control beam power
−5 0 530
40
50
60
70
80
90
100
110
120
Sig
na
l B
ea
m S
ize
(m
icro
ns)
Raman Detuning (MHz)
Figure 4.10: Plots of the signal beam size at the back face of the vapor cell versusthe Raman detuning
75
−5 0 540
60
80
100
120
140
160
180
200
220
Sig
na
l P
ow
er
(AU
)
Frequency (MHz)
Figure 4.11: The plot of output signal power versus Raman detuning.
50 100 150 200 250 300 350 400 450 500
40
60
80
100
120
140
160
180
200
220
Input Signal Power (micro Watts)
Ou
tpu
t S
ign
al P
ow
er
(mic
ro W
atts)
Figure 4.12: The plot of output signal power versus input signal power. The plotis nearly linear, the slope of the linear fit to data is 0.43.
76
the oscillatory behavior predicted in the theory. The gray curve beam size along
the horizontal and the black curve shows the beam size along the vertical. Fig.
4.10 shows the expected dispersive looking plot of the output beam size versus
Raman detuning. We have waveguiding when the Raman detuning is negative
and divergence for positive detuning. It also shows that we can have waveguiding
for a range of frequencies over a bandwidth of few MHz. This means that we can
potentially guide optical pulses with bandwidths of a few megahertz. Fig 4.11
shows the output powers for different frequencies. We see that in the frequency
range where we have good waveguiding, we also have relatively good transmission
of the signal power.
In order to verify that the waveguide is the result of the Raman absorption and
not due to other nonlinear effects like four-wave mixing, we measure the output
signal power for various input signal powers as shown in Fig. 4.12. We see that the
plot is linear and hence the waveguiding effect is linear in signal power. This also
means that the waveguiding effect is not due to self focusing effect. Instead it is
due to coupling beam dependent focusing. The slope of the linear fit is about 0.43,
which means that we couple about 43% of input signal power into the waveguide.
This implies that we can expect to guide light of very low power signal without
significant loss.
4.4 Summary
In summary, we use the intensity dependent refractive index resulting from a
Raman transition in a Λ system to create an all optical waveguide. We are able to
transmit about 43% of the power along the waveguide, for lengths much greater
than the diffraction length, using a low power control beam.
77
This all optical waveguide can be used to achieve efficient nonlinear processes
at very low light levels. For example, in the case of naturally abundant rubidium,
we can use one isotope to guide the signal and the other isotope as a medium for
the nonlinear processes such as two photon absorption, Stark shift, Kerr effect, etc.
Lukin and Imamoglu [Lukin and Imamoglu, 2000] suggested the use of rubidium
isotopes for two simultaneous, independent nonlinear processes to achieve large
Kerr nonlinearities. One can also optimize the waveguide to increase the band-
width and allow for multiple frequency waveguiding. Finally, one can use this
waveguide as a building block for an all optical beam-coupler and beam-splitter,
similar to solid-state waveguide devices.
78
5 Slow Light Prism
Spectrometer
In this chapter, we present a rubidium vapor prism spectrometer that operates
in the transparent region between two strongly absorbing resonances with six
orders of magnitude greater dispersing power than a standard glass prism. Unlike
the earlier proposals which utilize enhanced refraction in the absorption region,
this scheme utilizes the steep linear dispersion in the transparent region between
two resonances. Such a transparent region also gives rise to slow light and has
been studied in various systems recently [Khurgin, 2010]. We show that the
number of resolvable spectral lines in such a system is proportional to slow light
delay-bandwidth product. The delay bandwidth product for double absorption
slow light in rubidium has been shown to be nearly 100 [Camacho et al., 2006;
Camacho et al., 2007] and thus is advantageous over electromagnetically induced
transparency (EIT) [Fleischhauer et al., 2005] based slow light prisms [Sautenkov
et al., 2010] where the delay bandwidth products are typically less than 1. Our
slow light prism is capable of measuring optical frequency fluctuations with a
precision of 20 Hz/√Hz and can spatially resolve frequencies of about 50 MHz.
79
θ1
θp
θ2 θ3
θ4
x0
n(ω)Rb Prism
Figure 5.1: The rubidium vapor cell can be approximated as a dispersing prism.For our setup θp ≈ 79o, θ1 ≈ 20o and thus the geometrical parameter A is about2.
5.1 Theory
First we compute the dispersing power of the double Lorentzian prism then
show that the number of resolvable lines between the resonances is proportional
to the delay bandwidth product. Consider a double absorption slow light medium
[Camacho et al., 2006] with an angled interface with air as shown in Fig. 5.1.
Assuming that the index of refraction of air is unity, the change in the direction of
the beam at the interface is small and sin(θ1) ≈ θ1, we can obtain the exit angle of
the ray propagating through the prism as, sin(θ4) = n(ν) sin(θp− θ1/n(ν)), where
θp is the apex angle of the prism and θ1 is the angle made by the ray with the
normal of the first surface of the prism as shown in Fig 5.1. For n(ν) ≈ 1, the
angular dispersion of frequencies can be written as
dθ4dν
= Aλτ
L, (5.1)
where the geometrical parameter A = tan(θp−θ1)+θ1. In the above equation, we
assumed a steep linear dispersion and replaced dn/dν with ng/ν where ng is the
group index. For a medium of length L and a group delay of τ , ng/ν = λτ/L where
λ is the wavelength. The group delay in such a system is approximately given by
80
τ = αL/Γ where α is the absorption coefficient at the center of the transparency
and Γ is the full width at half maximum of each absorption [Camacho et al., 2006].
We now compute the number of spatially resolvable frequencies. We place a
lens of focal length f near the exit face of the prism, the displacement of the beam
in the focal plane is ∆y = ∆θ(ν)f . For a beam with the Gaussian diameter of D
before the lens, the Fourier transform limited diameter of the beam at the detector
is given by, d = 4λπ
fD. In order to estimate the spatial resolution, we calculate the
amount of frequency shift needed for one beam waist displacement of the beam
at the detector. Setting ∆y = d, we find
δν =d
fA
L
λτ. (5.2)
Suppose ∆ν is the bandwidth of the prism, the maximum deflection for the system
is given by,
∆ymax = fλAτ∆ν
L. (5.3)
We see that the maximum deflection depends on the delay-bandwidth product
over unit length. We are interested to have ∆ymax/d >> 1 in order to spatially
separate large number of frequency components.
In order to maximize ∆ymax/d we need a slow light system with large delay
bandwidth product like a double Lorentzian absorption system. The bandwidth
of the system is governed by the separation between the two absorptions and
the delay is dependent on the optical depth. The hyperfine absorption lines in
alkali metals can provide the required double absorption resonances. Our demon-
stration in this paper uses the dispersion between two such rubidium absorption
resonances.
81
Laser 780 nm λ/2
Fiber EOM
LensHeated
Rb Vapor cell Camera
Figure 5.2: A schematic of the experimental setup.
The simplified model discussed above can predict our experimental results.
However for more accuracy, we need to consider the effect of centroid shift due
to differential absorption across the transverse cross-section of the beam in the
prism. The centroid of the Gaussian beam is shifted by αL0w2/4x0, where α is
the absorption coefficient, L0 is the propagation distance for the centroid when
there is no absorption, w is the Gaussian beam spot size (D/2) and x0 is the
distance of the beam from vertex of the prism. We note that L and τ in all the
equations correspond to the length of propagation and delay for the centroid of
the exit beam.
5.2 Experimental Setup
The schematic of the experimental setup is shown in Fig. 5.2. A narrow
line width external cavity diode laser at 780 nm is tuned between the two Rb85
hyperfine resonances of the D2 line and coupled into a fiber electro-optic modulator
(EOM) which is driven by an arbitrary waveform generator. A λ/2 wave plate is
used to control the efficiency of the sideband creation in the EOM. The light is
82
then focused through an angled hot vapor cell onto a camera a distance 38 cm
away. The vapor cell, 7.5 cm long with angled windows, has rubidium with natural
isotopic abundance. The light enters the vapor cell near the edge of one of the
windows and exits through the curved surface as shown in Fig. 5.2. This creates
an effective prism of length 6 mm along the center of the exit beam. The vapor cell
is placed approximately 3 cm away from the focusing lens. The Gaussian spot size
of the beam at the vapor cell is about 1.6 mm and the beam is at approximately
3.2 mm from the apex of the prism. The deflection is observed on the camera
through a coarse change of the laser frequency or by modulation of the EOM. A
removable 50/50 beam splitter is put in the path of the beam after the vapor cell
to monitor the optical depth of the cell.
5.3 Experimental Results
As we change the frequency of the laser within the transparent region between
the two Rb85 hyperfine resonance frequencies, we see a shift in the position of the
beam at the camera. The effective focal length of the focusing lens is chosen such
that the displacement of the beam for maximum possible frequency shift is within
the active area of the camera. We note that the displacement of the beam as well
as its focal spot size increase for longer focal distance. The temperature of the
vapor cell is tuned such that we have about 25% transmission at the center of the
transparent region. Even though the deflection of the beam increases for higher
temperatures of the vapor cell, the effective bandwidth of the system decreases
due to increased absorption. At our working temperature the bandwidth of our
system is about 1 GHz. The Gaussian diameter of the focal spot at the camera is
about 90 µm.
83
ν = 0 MHz
ν = 50 MHz
ν = 100 MHz
ν = 300 MHz
ν = 500 MHz
Figure 5.3: The camera images for different modulation frequencies.
The frequency dependent deflection is quantified by first turning off the EOM
and tuning the frequency of the laser to the center of the transparency. Turning on
the EOM results in frequency side bands. Different frequency bands in the signal
are spatially separated after the rubidium vapor cell and the resultant spatial
distribution of intensities is recorded at the camera. Fig. 5.3 shows the camera
images for different RF modulation frequencies. The central spot in each of the
images is the zeroth order (unmodulated) frequency followed by the first order
and second order side bands. Frequency sidebands up to second order are clearly
visible for the modulation frequency of 200 MHz. The first order side bands are
visible up to the modulation frequency of 550 MHz. The intensity dependence
on the position for different modulation frequencies is shown in the Fig. 5.4.
Even though the phase modulation efficiency of the EOM is constant for the RF
frequencies we used, the frequency dependent absorption causes the change in
84
0 0.5-0.5 1-1 1.5-1.5
Position (mm)
Figure 5.4: One dimensional intensity scans for different modulation frequencies.
85
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0 200 400 600-200-400-600
Def
lect
ion (
mm
)
Frequency (MHz)
DataLinear Fit
Figure 5.5: Plot of deflection as a function of frequency. Circles represent theexperimental data and solid line is the linear fit. The slope of the line is 1.95µm/MHz.
relative intensities. One can obtain the exact spectral information of the input
signal by correcting for the frequency dependent losses at the vapor cell. We also
note that for the modulation frequencies less than 50 MHz, the sidebands are not
well separated in space.
Fig. 5.5 shows that the displacement changes linearly with the modulation
frequency for the frequency range of about 1100 MHz. The circles show the
experimental data and the solid line its linear fit. From the slope of the linear
fit, we deduce that the displacement at the camera per unit frequency change is
about 1.95 µm/MHz. This is a relatively large displacement for a small change in
frequency. The maximum displacement of the beam ymax is measured to be 2.145
mm for a frequency range of 1100 MHz. This implies that dn/dν for the prism
is about 2.7 × 10−12 Hz−1. Using Eq. (5.3), for a delay of 26 ns and propagation
length of 6± 1 mm, we expect a maximum deviation of 2.6± 0.4 mm. From Eq.
86
0-0.2-0.4-0.6 0.2 0.4 0.6
Position (mm)
SignalIdler
Pump
Figure 5.6: (a) Four wave mixing beams are split at the prism and imaged at thecamera. (b) The one dimensional scan of the intensity profile.
(5.2), using the measured focal spot size, we expect a spatial frequency resolution
δν of 37±6 MHz. These predictions are close to what we observed experimentally.
The sensitivity of the spectrometer is quantified using a similar configuration
to Fig. 5.2. Instead of a CCD camera we use a position sensitive quadrant detec-
tor. We use the EOM at a modulation frequency of 1 GHz and then adjust the
frequency of the laser such that one of the first order sidebands is in the transpar-
ent region between the two Rb85 hyperfine resonances. The central peak and the
other sidebands are in a highly absorbent region. The sideband frequency is then
modulated using an external arbitrary waveform generator. For this measurement
we modulated the frequency of the sideband by 10 kHz at an external rate of 60
kHz. The signal from the quadrant detector is low pass filtered at 100 kHz and
high pass filtered at 30 kHz. The power of the beam after passing through the
87
cell was roughly 300 µW. The sensitivity of the spectrometer is found to be ap-
proximately 20 Hz/√Hz, which corresponds to a signal to noise ratio of 1. This
is almost four orders of magnitude improvement over the standard glass prism
technique [Starling et al., 2010]. The theoretical limit for a shot noise limited
system is less than 1 Hz/√Hz, but we were limited by turbulent noise and other
technical noise.
To demonstrate the practical utility of this rubidium prism, we show that we
can spatially separate different frequencies resulting from a collinear four-wave
mixing process in Rb85. The signal and idler beams resulting from the four-wave
mixing process in rubidium have been shown to be number squeezed and are ex-
pected to be entangled. Thus, four-wave mixing in rubidium is a potential source
for narrow-band entangled photons. However, the residual pump noise in the sig-
nal and idler beams and the spontaneous emission noise in hot atomic vapor make
such experiments challenging. We can use the rubidium prism to separate differ-
ent frequency components spatially and reduce the amount of crosstalk between
the beams.
The signal beam is to the blue of the F = 3 → F ′ = 2, 3 Rb85 transition
and the pump beam, separated by 3.035 GHz in frequency from the signal beam,
is to the blue of the F = 2 → F ′ = 2, 3 Rb85 transition. Signal, idler and pump
beams at the output of a rubidium vapor cell are coupled into a fiber and at the
output of the fiber we place our rubidium prism. Signal and pump beams fall
in the highly dispersive frequency regions of the rubidium prism and hence are
deflected. The magnitude of deflection is different for signal and pump because
of differing refractive index of rubidium at those frequencies. The idler beam
which is about 6 GHz to the blue of signal beam is far away from any resonance
88
frequencies and hence it does not experience much deflection. Fig. 5.6(a) shows
the image at the detector and Fig. 5.6(b) its horizontal cross-sectional plot. The
central spot is the signal beam and the left and right spots are of pump and idler,
respectively. We thus see that each of the frequencies are well separated from one
another. We believe this prism will be an invaluable tool for spectral filtering in
atomic experiments.
5.4 Summary
To summarize, we demonstrated a highly dispersive atomic prism with a spatial
frequency resolution of 50 MHz. We show that the number of spatially resolv-
able spots is related to the slow light delay-bandwidth product, which makes a
double absorption system ideal for the prism. In addition, we made a precision
measurement of frequency fluctuations with a sensitivity of 20 Hz/√Hz which is
approximately a measurement of 1 part in 1013 using a very simple setup and thus
this prism may have applications in frequency metrology. We also demonstrated
a spatial separation of the pump, signal and idler beams from a four-wave mixing
process in rubidium. Number squeezed signal and idler beams have applications
in quantum metrology and hence this prism can be used to reduce the noise in
the signal and idler beams resulting from the crosstalk between the beams.
89
6 Conclusions
We have presented many experimental results in the preceding chapters re-
garding the optical control of temporal and spatial propagation of light. We used
Raman absorptions to obtain slow group velocities for the signal pulse. We used a
four-wave mixing process to store signal pulses in the atomic medium and we have
demonstrated a technique to store transverse information of pulse in vapor which
is robust to diffusion. We used coupling beam intensity dependent refractive index
to control the signal beam size as it propagates in rubidium vapor. Finally, we
demonstrated an atomic prism which can be used as a spectral filter in atomic
experiments. In this chapter, we summarize the results and provide an outlook
for further research.
In chapter 2, we demonstrated a slow light system based on dual Raman
resonances. Slow light based on dual absorption resonances are known to cause
low dispersive and absorptive broadening on signal pulses compared to a single
transmission resonance based slow light. We have shown that the bandwidth
of such a system is proportional to the frequency separation of the two Raman
absorptions and the group index is proportional to the strength of absorptions.
90
The turn-on time of the slow light is inversely proportional to the width of the
absorption and the turn-off time is proportional to the excited state life time. All
of these properties, except the excited state life time, can be controlled by the
coupling beam, and thus we have an ultra-tunable slow light system. Competing
nonlinear processes like four-wave mixing set a limit on the maximum achievable
strength of Raman absorption and thus set a limit for tunable range. We presented
experimental results which show the dependence of Raman absorption line shapes
on coupling beam powers. We have demonstrated experimentally a delay of more
than two pulse widths and slow light tuning times faster than a pulse width.
Tunable dual resonance slow light systems can have many applications. The
actual design parameters may vary according to application. For example, an
optical buffer may require large bandwidth and fast tuning times. In such a design,
we choose broad absorption resonances separated in frequency by the required
bandwidth. In interferometric and other sensing applications, we need large group
indices and in such designs we need two strong and narrow resonances which are
very close in frequency. Slow light based Fourier transform interferometer has
been demonstrated by Shi et al. [Shi et al., 2007]. The interferometer output as a
function of group index is first obtained and the signal spectrum is then retrieved
by Fourier transforming the output. The speed with which we determine the
spectrum can be dramatically improved by using optical tuning of the slow light.
The spectral resolution can also be improved by two orders of magnitude.
In chapter 3, we demonstrated four-wave mixing based stopped light. The
signal pulse and spontaneously emitted idler pulse are coherently mapped onto
the rubidium ground state coherence. Signal and idler pulses can be retrieved
before the ground state decay time, which in our case is shown to be about 120 µs.
91
We show that the retrieval efficiency is linear in retrieval beam power. The four-
wave mixing storage is interesting because of the quantum mechanical properties
of the signal and idler photons. More work needs to be done to explore the
possibility of enhanced correlated photon generation. This system is capable of
both amplifying and slowing the signal pulses and thus has applications for optical
communications.
We also demonstrate storage of multiple transverse modes of signal by storing
the Fourier transformed image instead of a flat phase image. This way the storage
is more robust to diffusion. We demonstrate the storage of a test image and show
that the dark areas of the image remain dark and the retrieved image is attenuated
more near the edges of the image than at the center. The central idea of this
scheme is that the atomic coherences carry phase information of the signal and
that oppositely phased coherences destructively interfere during diffusion.
The image storage scheme differs from other holographic storage devices. Even
though the image storage times for the atomic scheme is much less, the holographic
storage require much greater signal powers. The storage and retrieval of image
storage is also much faster. The scheme discussed here can potentially store
the transverse profile of a single photon and thus may be useful for quantum
information. More work needs to be done to demonstrate this possibility. This
is also useful for remote sensing and Lidar applications where fast storage and
retrieval of a reference image is required.
In chapter 4, we demonstrated waveguiding of the signal beam using Raman
absorption. The refractive index of the signal beam is shown to be dependent on
the coupling beam intensity. The first order Laguerre-Gauss coupling beam is used
to obtain quadratic dependence of the refractive index of the signal on the radial
92
coordinate. The signal beam size propagating inside the medium can be changed
by changing the power or frequency of the coupling beam. We experimentally
demonstrate waveguiding and also show the dependence of the output signal size
on coupling beam parameters.
The next step in this work is to demonstrate an improvement of the efficiencies
of the nonlinear processes. For example, we may use a sample with two isotopic
species, like in the case of rubidium, to achieve waveguiding using one species at
frequencies useful for nonlinear processes in another species. For instance, Raman
absorption resonance can be created for Rb87 at a frequency which is resonant to
Rb85 transition. We can then demonstrate degenerate EIT at a very low coupling
and signal beam powers.
In chapter 5, a sensitive atomic prism has been presented. We have shown
that the prism is capable of spatially separating spectral lines 50 MHz apart. This
property can be used to filter different optical frequencies in atomic experiments
involving many optical frequencies. For example, signal and coupling beams in
EIT experiments can be spatially separated after the vapor cell using this prism
instead of a polarizing beam splitter. This allows for exploring an EIT regime in
which signal and coupling beams have parallel polarizations. We demonstrated
that the signal, idler and coupling beams resulting from the four-wave mixing
process can be filtered using this prism. We believe that we can obtain more
spectral purity using this filter when compared to a Fabry-Perot etalon which we
used in chapter 3.
We also measured frequency fluctuations with an accuracy of 20 Hz/√Hz. Ac-
curate measurement of frequencies opens the door for many precision experiments.
For example, low powered optical signals can be measured in a nondestructive way
93
by measuring very small induced AC-stark shifts. One can also use this prism to
measure magnetic fields by converting magnetic field dependent resonances to
observable optical deflections. There is a possibility of tuning the properties of
the prism using an additional pump beam which can deplete the ground state
population to cause a decrease in refractive index and an increase in available
bandwidth.
To summarize, off-resonant interactions in rubidium discussed in this thesis
are very interesting and have potential use in fields like optical communications,
quantum information, precision measurements and sensing. We demonstrated
slow light, stopped light and guided light in this thesis and these demonstrations
lead us to more interesting experiments. Finally, the atomic prism presented in
this thesis can serve as a valuable tool for many atomic experiments.
94
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109
A Derivation of Susceptibility
for a Lambda System
In this appendix we derive susceptibility at the signal frequency in a three-
level atom in a Λ configuration. In a Λ system, two lower energy atomic states are
coupled optically with an exicted state. Optical coupling between two lower energy
states is forbidden. The resulting simultaneous interaction of the atom with two
optical modes has some interesting properties. For example, the transmission and
the refractive index of one optical mode is altered by the properties of the other
mode. We call the optical mode of our interest as signal beam and the optical
mode that controls the properties of the signal beam as control beam.
Consider the three-level Λ system shown in Fig. 1.2. The levels 1 and 2 are
coupled by the probe field Es
2e−i(ωst−kz) + c.c. The control field coupling states 2
and 3 can be written as Ec
2e−i(ωct−kz)+ c.c. The Rabi frequencies of signal and the
coupling beams are defined as. Ωs,c(t) = (µEs,c/h)e−i(ωs,ct−kz) = Ωs,ce
−i(ωs,ct−kz).
The detunings are defined as ∆ = ∆s = (ω2 − ω1)− ωs, ∆c = (ω2 − ω3)− ωc and
δ = ∆s −∆c. The Hamiltonian for this system is given by
110
H = hω1 |1〉 〈1|+ hω2 |2〉 〈2| + hω1 |3〉 〈3| −hΩs(t)
2|2〉 〈1| − hΩ∗
s(t)
2|1〉 〈2|
− hΩc(t)
2|2〉 〈3| − hΩ∗
c(t)
2|3〉 〈2| . (A.1)
Let the state of the system is given by |Ψ〉 = a |1〉+ b |2〉+ c |3〉. The evolution of
the state follows the Schrodinger’s equation, ih ∂∂t|Ψ〉 = H |Ψ〉. We can write this
in the matrix notation as follows:
ih∂
∂t
a
b
c
= h
ω1 −Ω∗s(t)/2 0
−Ωs(t)/2 ω2 −Ωc(t)/2
0 −Ω∗c(t)/2 ω3
a
b
c
. (A.2)
Using the transformation,
a
b
c
=
e−iω1t 0 0
0 e−iω2t 0
0 0 e−iω3t
a′
b′
c′
, (A.3)
we get,
∂
∂t
a′
b′
c′
= −i
0 −Ω∗se
−i∆st/2 0
−Ωsei∆st/2 0 −Ωce
i∆ct/2
0 −Ω∗ce
−i∆ct/2 0
a′
b′
c′
(A.4)
111
Using another transformation,
a′
b′
c′
=
1 0 0
0 ei∆t 0
0 0 eiδt
a′′
b′′
c′′
, (A.5)
we get,
ih∂
∂t
a′′
b′′
c′′
= −h
0 Ω∗s/2 0
Ωs/2 −∆ Ωc/2
0 Ωc/2 −δ
a′′
b′′
c′′
(A.6)
By comparing the equations A.2 and A.6, the effective Hamiltonian for this
system in this rotating frame is given by
Heff = −h
0 Ω∗s/2 0
Ωs/2 −∆ Ωc/2
0 Ω∗c/2 −δ
. (A.7)
Assuming that we have a closed system with the total spontaneous decay rate
from the excited state given by Γ and the ground state decoherence rate given by
γ, the decoherence matrix D is given by
D =
−Γρ22/2 Γρ12/2 γρ13
Γρ21/2 Γρ22 Γρ23/2
γρ31 Γρ32/2 −Γρ22/2
. (A.8)
Using the evolution equation for the density matrix, ρ = − ih[Heff , ρ]−D, the
equations of motion for the density matrix elements are given by,
112
ρ11 =Γ
2ρ22 − ℑΩ∗
sρ21 (A.9)
ρ22 = −Γρ22 + ℑΩ∗cρ23 + Ω∗
sρ21 (A.10)
ρ33 =Γ
2ρ22 − ℑΩ∗
cρ23 (A.11)
ρ23 = −[
i∆c +Γ
2
]
ρ23 + iΩc
2(ρ33 − ρ22) + i
Ωs
2ρ13 (A.12)
ρ21 = −[
i∆s +Γ
2
]
ρ21 + iΩs
2(ρ11 − ρ22) + i
Ωc
2ρ31 (A.13)
ρ31 = −(iδ + γ)ρ31 + iΩ∗
c
2ρ21 − i
Ωs
2ρ32 (A.14)
In the above equations ℑ(ρ) denotes the imaginary part of ρ. Assuming that
ρ11 ≈ 1 and ρ22 = ρ33 ≈ 0, the steady-state perturbative solution for ρ21 up to
first order in Ωs is,
ρ(ss)21 =
Ωs
2
δ − iγ
(δ − iγ)(∆− iΓ/2)− |Ωc|2/4. (A.15)
The susceptibility at the signal frequency is thus given by,
χ(∆, δ,Ωc) = βδ − iγ
(δ − iγ)(∆− iΓ/2)− |Ωc|2/4. (A.16)
where β = Nµ2/hǫ0 where N is the number density and µ is the transition
dipole moment.
113
The real and imaginary parts of the susceptibility can be written as,
χ′(∆, δ,Ωc) = β∆
∆2 + Γ2/4
[
1 +
(
2δ0 −|Ωc|24∆
)
δ′
δ′2 + γ′2− 2γ0
γ′
δ′2 + γ′2
]
,
χ′′(∆, δ,Ωc) = βΓ/2
∆2 + Γ2/4
[
1 + 2δ0δ′
δ′2 + γ′2−(
2γ0 −|Ωc|22Γ
)
γ′
δ′2 + γ′2
]
.
(A.17)
In the above, δ0 = |Ωc|2∆/(4∆2 + Γ2), γ0 = |Ωc|2Γ/(8∆2 + 2Γ2), δ′ = δ − δ0
and γ′ = γ + γ0. The first terms in each of the above equations are the real and
imaginary parts of the susceptibility χ0 when the coupling beam is off. The second
and the third terms have dispersive and absorptive functional forms. The single
photon detuning ∆ determines how each term contribute to the total susceptibility.