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Photons and (artificial) atoms: an overview of optical spectroscopytechniques on quantum dotsA. N. Vamivakas a; M. Atatüre aa Cavendish Laboratory, University of Cambridge, Cambridge, UK

Online publication date: 15 December 2009

To cite this Article Vamivakas, A. N. and Atatüre, M.(2010) 'Photons and (artificial) atoms: an overview of opticalspectroscopy techniques on quantum dots', Contemporary Physics, 51: 1, 17 — 36To link to this Article: DOI: 10.1080/00107510903298198URL: http://dx.doi.org/10.1080/00107510903298198

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Photons and (artificial) atoms: an overview of optical spectroscopy techniques on quantum dots

A.N. Vamivakas and M. Atatüre*

Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, UK

(Received 27 March 2009; final version received 6 August 2009)

In most branches within experimental physics technical prowess lies at the heart of many seminal works. From theobservation of the photoelectric effect and the ultraviolet catastrophe that led to the development of quantummechanicsto the first transistor that shaped the modern age of electronics, significant physical insight has been achieved on theshoulders of technical advances and progress. Research on self-assembled quantum dots may be a drop in the sea ofphysics, but it still is no exception to this trend, and more physical insight continues to be revealed as the tools of thetrade get increasingly more complex and advanced. This article is written primarily for senior undergraduate studentsand first year graduate students of experimental physics involving optically active quantum dots. More often than not,we have seen students shuffling through journal articles trying to relate the reported physics to the used experimentaltechniques. What we want to cover here is not in any way the history or the recent progress in quantum dot research –there are an ample number of topical books and review articles for that – but rather to highlight a selection of optics-based measurement techniques that have led to significant progress in our understanding of quantum dot physics as wellas their applications in the last two decades. We hope a basic survey of the relevant optical spectroscopy techniques willhelp the newcomers in connecting the dots between measurements and physics.

Keywords: quantum dots; excitons; spectroscopy; quantum optics; quantum information science

1. Introduction

Quantum dots have become a system of study in a broadrange of disciplines in a relatively short time. Theincredible progress in synthesis, growth and fabricationquality fed further the advances in optical investigationsin physics, biology and chemistry. Particular to quantumphysics, quantum dots allow optical studies of confinedcharge and spin systems and in parallel studies onengineering light–matter interaction and even thesuppression of spontaneous emission. We start belowwith an introduction to the growth of quantum dots andthen follow with a discussion of the various opticallyactive charge complexes that a quantum dot cansupport. We then proceed with a handful of opticaltechniques lined up vaguely with increasing technicaldifficulty and chronological appearance. We finish witha selection of applications, mainly driven by quantuminformation science, in order to highlight how muchexperimental progress in quantum information science isindeed driven by a reinterpretation of results obtainedvia conventional optical spectroscopy.

2. Quantum dots: from growth to energy levels

Advances in material science have enabled the growthof heterostructures exhibiting inhomogeneity on a

length scale relevant for influencing the spectrum of amaterial’s excitations. Quantum dots (QDs) areheterostructures engineered to provide three-dimen-sional spatial confinement for electronic excitations.The confinement yields a discrete spectrum of QDenergy eigenstates and it is not uncommon to refer toQDs as artificial atoms. In practice, there exist anumber of quantum confined physical systems thatexhibit a discrete electronic spectrum – interfacefluctuation QDs that form at the gallium arsenide/aluminum gallium arsenide (GaAs/AlGaAs) barrierboundary in a quantum well, core–shell cadmiumselenide/zinc sulphide (CdSe/ZnS) nanocrystalsformed through both colloidal methods and quantumdots grown by Metallic Organic Vapour PhaseEpitaxy (MOVPE), electrically defined QDs via gateelectrodes patterned on two-dimensional electron gasyielding precise control over the local electrostaticpotential and self-assembled QDs grown by Molecu-lar Beam Epitaxy (MBE). Here, we will focus mainly,but not exclusively, on indium arsenide (InAs)/GaAsself-assembled QDs in portraying the optical techni-ques used to date. In this section, before looking intothe characteristic electronic structure of QDs, we willdiscuss the material science advances that haveresulted in the growth of self-assembled QDs byMBE.

*Corresponding author. Email: [email protected]

Contemporary Physics

Vol. 51, No. 1, January–February 2010, 17–36

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Epitaxial growth is a process where a new crystal isgrown over a host crystal surface via layer-by-layeratomic deposition [1]. Epitaxial techniques are capableof depositing high quality semiconductors with anabrupt change in material composition having mono-layer (*3 Å) accuracy. The formation of InAs/GaAsQDs is a natural process and is the manifestation of astrain-driven phase transition that occurs when com-bining two materials of different lattice constantsduring one material growth cycle. Every material hasits own lattice constant and this commonly leads toformation of strain on two layers constituting anabrupt interface. Figure 1(a) illustrates the two typicalcases of strain release: a monolayer-thick materialembraces a lattice constant dictated by the hostmaterial or a sufficiently thick material recovers itsown lattice constant resulting in strain release viadislocations and defects at the interface. The formationoccurs exactly during the transitional period linkingthe two regimes of strain release. If the latticeconstants are significantly different (e.g. 7% mismatchbetween la and lb, as is the case for GaAs and InAslattices), the epitaxial growth of InAs with the GaAslattice can not be sustained for more than twomonolayers of growth. At one point, the newly formedlayer goes through a phase transition forming minia-ture islands, very much like mercury droplets do on asmooth flat surface. Further growth with the samematerial as the handle wafer, in this case GaAs, caps

the QDs and protects them from the surroundingenvironment. After growth, the height of the QDs istypically 4–5 nm, as determined by cross-sectionalscanning tunnelling microscopy image of Figure 1(b)[2]. We emphasise that, although the self-assembledQDs exhibit pristine optical properties, the in-planeQD distribution is disordered and extensive efforts arestill made today in this field in order to achieve bettercontrol over the island size distribution and location ofnucleation.

The MBE growth process results in strong three-dimensional carrier confinement for electrons in QDsresulting in quantisation of energy states. However,QDs are composed of around 105 atoms, and thus forma mesoscopic system with arbitrary shape and composi-tion which differ from QD to QD. The distribution inshape and composition combined with the strain profileexperienced by the QD all influence the single particleQD energy levels in the form of inhomogeneousbroadening. In addition to material properties, ifmultiple charges are confined in the QD the Coulombinteraction between the quantum confined carriers hasto be taken into account when calculating the multi-particle energy levels. All the previous complicationsmake an analytical determination of QD propertiespractically impossible and modelling typically relies onperturbative or numerical methods. Even with all thesecomplications it is striking that the roughly 105 InAsatoms in the GaAs matrix conspire to exhibit a discreteatomic-like energy spectrum.

The InAs/GaAs QDs covered in this work aresemiconductors in bulk (three-dimensional) form.Therefore, to solve for the energy levels of QDs, it isnatural to start from the bulk material properties anddetermine the consequences of reducing the system’sdimensionality. For a phenomenological, but satisfac-tory, prediction of bulk semiconductor band struc-tures, we resort to a perturbative k � p model. In k � psingle-particle wavefunctions and energy eigenvaluesare assumed to be known at k ¼ 0 and the banddispersion is obtained in the small k approximationaround the � -point [3]. These perturbative methodscan also be applied to quantum dots since the k-vectordistribution of confined charges is concentratedaround k ¼ 0. Figure 2 shows a schematic of theband structure of bulk GaAs with relevant parametervalues at room temperature. The band structure ofInAs looks esentially identical, but, the values of theindicated parameters differ significantly from GaAs.

Excitation of an electron across the bandgap leavesan empty electronic state in the otherwise electron-filled valence band. These holes can equally be treatedas positively charged particles with modified mass andg-factor. The lowest conduction band has to a verygood approximation parabolic dispersion around the

Figure 1. (a) An illustration of lattice constant mismatchfor two materials grown by MBE for strained thin layers anddislocated thick layers. (b) Cross-sectional scanningtunnelling microscopy of a self-assembled InAs QD grownby MBE [2].

18 A.N. Vamivakas and M. Atatüre


� -point, as indicated by the red curve in Figure 2. Thewavefunctions for this band have s-wave charactersustaining a twofold spin-degeneracy with [SjjSz] ¼[1/2jj + 1/2]. The valence band wavefunctions havep-wave character that would normally sustain a sixfoldspin-degeneracy forming a [3/2jj + 3/2, + 1/2] quad-ruplet and [1/2jj + 1/2] doublet. However, spin-orbit coupling in these semiconductors causes the[1/2jj + 1/2] doublet to be separated in energy formingwhat is referred to as the split-off band (Figure 2).Further, upon including the influence of other bands,even the fourfold degeneracy of the [3/2jj + 3/2,+1/2] states is lifted for k 6¼ 0 forming the heavy-hole and the light-hole bands with near-parabolicnegative curvature dispersion as seen in Figure 2.

When the dimensionality of the system is reducedsuch that the effective Bohr radius becomes compar-able to the physical extent of the confining material,quantum confinement strongly influences the densityof states, band dispersion and degeneracies. In the caseof QDs, the dimensionality is zero resulting inmotional confinement along all three directions.Therefore, a set of discrete energy levels arise withlevel spacings determined by the, not necessarily equal,confinement strength along each direction. In fact, dueto their particular lens-like topology (see Figure 1(b)),the QDs considered here display strongest motionalconfinement along the growth (z) axis. Therefore, themain features of the energy levels of these QDs can beseen by simply considering a strong confinement alongthe z direction with a two-dimensional quasi-parabolicconfinement in the two remaining directions. Agenerally accepted approach to quantifying the QDenergy levels and the corresponding wavefunctionsrelies on pseudopotential theory [4]. A nice tutorial

discussion of pseudopotential theory, with illustrationsof the QD electronic excitation wavefunctions, can befound in the review by Zunger [5].

From the optics perspective an important featureof the quantum confinement is that although theenergy spectrum of the QD is altered when comparedto the bulk semiconductor, the electrons and holes thatbecome trapped in the QD inherit the spin structure ofthe bulk semiconductor. This determines the optical(polarisation) selection rules for transitions betweenQD electron and hole states mediated by a photon.Explicitly, focusing on the conduction band and heavyhole valence band, we can specify the QD electron andhole spin. The QD levels derived from the conductionband levels sustain their twofold spin degeneracy,while the QD levels derived from the valence bandstates display a confinement-induced splitting intoheavy-hole and light-hole doublets.

We qualitatively established the energy levels ofelectrons and holes confined in all three dimensions insemiconductor QDs. We now identify the energy scalesof common InAs/GaAs QD charge configurations thatare probed optically. The simplest charge configura-tion linked to an optical emission is the neutralexciton X0 (see Figure 3), i.e. a single electron–holepair occupying the lowest discretised energy levels

Figure 2. A simplified band structure illustration for III–Vsemiconductors such as GaAs and InAs with the typicallyaccepted values for key energy scales.

Figure 3. Neutral exciton (X0), biexciton (XX) transitionsunder excitonic level splitting and negatively charged trion(X17) transitions under magnetic field along the growth axisfor a typical quantum dot. The wavy arrows indicate photonmediated transitions between the states. For the negativelycharged trion (right illustration) the up (down) arrowrepresents the electron spin projection of þ1/2 (71/2)along the growth direction and the solid up (down) triangleis the hole projection of þ3/2 (73/2). The ground state of thetrion transition is a single electron with its spin projection upor down. Each transition is decorated with a symbolindicating the emitted photons polarisation – pX (pY) forhorizontal (vertical) and sþ (s7) for right (left) circularlypolarised photons. The direction of linear polarisation(horizontal and vertical) is defined with respect to themajor and minor axis of the elliptical QD base (as opposedto circular) due to strain-induced anisotropy of the dotgeometry.

Contemporary Physics 19


within the original conduction and valence bands. Theelectron in the conduction band can have spinquantum number [SjjSz] ¼ [1/2jj + 1/2]. The heavyhole in the valence band has spin [JjjJz] ¼ [3/2jj + 3/2].By addition of angular momentum, a single electron–hole pair in the QD can end up in any one of fourspin-state combinations. The total angular momentumof these combinations being DJ ¼ +1 or DJ ¼ +2,each doublet is degenerate. In an optical transitionangular momentum must be conserved, and this isreflected in the polarisation of the emitted photons.Recombination via a single-photon emission processcan only occur for the DJ ¼ +1 exciton doublet, sincesingle photons carry angular momentum +1. Theangular momentum conservation is reflected in theemitted photon’s polarisation. Specifically, photonscarrying þ1 (71) angular momentum are left (right)hand circularly polarised and are denoted with thesymbol sþ (s7). The exciton doublet DJ ¼ +1 that islinked to photon emission is called bright, while theremaining optically inactive doublet is called dark (theDJ ¼ +2 excitons). The polarisation selection rulesalso constrain the set of excitons that may be createdoptically to the DJ ¼ +1 doublet.

Of course, the real world is not simple! Thepreviously mentioned shape nonuniformity and strainact to coherently mix the bright DJ ¼ +1 excitondoublet via the electron–hole exchange interaction.This interaction couples the spins of the electron andhole confined in the QD and depends sensitively on thestructural symmetry of the QD. The electron–holeexchange serves to both break the DJ ¼ +1 excitondoublet’s degeneracy and alter the polarisation of theemitted photons from circular to linear, indicated bypX/pY in Figure 3. This new polarisation basis, which isdefined along the major and minor axis of the ellipticalQD base, led to the phrase X – Y splitting to denotethis effect. Due to exchange interaction, an electron–hole pair once created in DJ ¼ þ1 state will precesscoherently between DJ ¼ +1 spin configurations. Are-diagonalised Hamiltonian after including this inter-action leads to new eigenstates with the degeneracy oftheir energies lifted in proportion to the interactionstrength. Typical energy scale for the DJ ¼ +1 excitondoublet fine structure splitting is *10 meV for self-assembled InAs/GaAs QDs. We will see this fine-structure splitting has consequences for applicationsinvolving photon emission in later sections of thisarticle. We direct the reader to [6] for a completediscussion of electron–hole exchange interaction.

The next QD charge complex we discuss is twoelectrons and one hole. We call this singly chargedexcitonic QD excited state a trion, see Figure 3 (rightdiagram), and label it as X17. In forming the trioncomplex, Pauli’s principle forces the electron pair to

form a spin singlet state where the closest triplet state hasenergy much higher than typical ambient temperature(4 K). Since the resident hole can have either spin up orspin down, each QD has two trionic transitions that areenergetically degenerate. Due to Coulomb interactionsin this three-body problem, the recombination energy ismodified with respect to the original neutral X0

excitonic transition energy (ignoring fine structure) byDE ¼ Eee7Eeh – the direct energy due to electron–electron and electron–hole Coulomb interaction [7] asdictated by the wavefunctions via the form

Emn /ð�1Þð1�dm;nÞe2

4pe0er

ZZdrdr0

jcmðrÞj2jcnðr0Þj

2

jr� r0j : ð1Þ

In the InAs/GaAs QDs considered here the result is atotal shift of DE ¼ 6 meV to lower energy for the trionictransitions. In contrast to the neutral exciton where theelectron–hole spin exchange breaks the twofold degen-eracy, the electronic spin singlet is immune to electron–hole exchange and the two trion states remain degen-erate. In this case, the polarisation of the emitted photonis in the circular basis and the handedness is determinedthe direction of the resident hole spin.

The situation is conceptually similar when there aretwo electron–hole pairs present in the QD referred toas the biexciton (XX) shown in Figure 3 (middlediagram). The shift in the transition energy for abiexcitonic transition can once more be determined bythe energy difference between the initial and finalstates, DE ¼ 2Eee þ2Ehh7 Eeh, and is on the order of2 meV for InAs QDs. Ultimately, every chargecombination results in a distinct spectral signaturedue to the Coulomb interaction, and we refer thereader to [7] for a detailed explanation of this approachfor direct and exchange type interactions.

It is good to note here that the relevant energyscales for each mechanism considered are well defined.The optical transitions occur at eV range while directCoulomb interactions within a QD are at tens of meV.The fine structure such as X–Y splitting as we will seelater is on the order of tens of meV, which is still muchlarger than the characteristic transition linewidth of1 meV. While each quantum dot can have vastlydifferent emission energies due to inhomogeneity inthe quantum dot ensemble, the relative energy shiftsare conveniently rather robust. With an understandingof common QD charge complexes, we can begin toaddress how the tools of optical spectroscopy revealphysical properties of the QD states.

3. Optical spectroscopy techniques

In the previous section we highlighted the mostrelevant excitonic complexes in QD optics and their



relevant energy scales. How excitons and photonscouple to each other can be presented in many ways.We will make a distinction between two methods ofoptical excitation. The first approach, which we referto as nonresonant excitation, is to use a light source,typically a laser, with an energy that is larger than theenergy of the relevant excitonic transition. The lasercreates exciton population in either higher energy QDexcitons or in the bulk of the host semiconductormatrix. The higher energy excitons then nonradiativelyrelax, giving off energy through carrier scattering andphonons, and populate the lowest energy availableexciton state. The second approach is to use a laserwith energy equal to the excitonic state of interestmuch like in atomic physics. In the latter case we willsay the laser is resonant to conform to the languageused in the field. Of course either method of excitationlinks real states of the system and is resonant; but werather use the term resonant to indicate a laser that hasenergy commensurate with the relevant excitontransition.

3.1. Optics of quantum dots: nonresonant excitation

3.1.1. Exciton spectrum – photoluminescence

A quantum dot may host discretised electronic levels,but the surrounding semiconductor matrix introducesa continuum of filled valence and available conductionband states. Therefore, an optical field can generate anensemble of electron–hole pairs in the vicinity of aquantum dot. Typically, most of these pairs recombinequickly to yield photon generation at the bandgapenergy of the semiconductor, while occasionally, acombination of electrons and holes may be capturedinto the discrete exciton levels of the quantum dot. Forthis to happen, the excess energy of the excitons has tobe taken away by carrier scattering or phonons.Exactly how this relaxation occurs and how it dependson external parameters such as temperature andmagnetic field has formed a whole branch of researchon quantum dots, especially in the 1990s. Thisrelaxation process happens at relatively fast timescales,i.e. tens of picoseconds and removes any coherencewith the excitation laser. Therefore, generation ofexcitons in a quantum dot via bandgap excitation canbe treated typically by an incoherent driving fieldbetween quantum dot states. In addition, the spinorientations of the optically excited electrons and holesare affected during the relaxation processes yieldingonly a residual correlation to the original excitationlaser polarisation. Radiative recombination of anelectron–hole pair in the quantum dot consequentlyreveals information on the quantised energy levelsand the optical selection rules. Therefore, micro-

photoluminescence (mPL), i.e. measuring the spectrumof QD light emission under continuum excitation, hasbeen an essential workhorse of quantum dot researchover the years.

In a typical mPL setup, a schematic can be found inFigure 6(a), Section 3.1.3, laser light is directed by abeamsplitter to an objective which focuses the lightonto the QD sample (the triangle in the illustration).The numerical aperture of the objective and the laserwavelength determine the focal volume the laserexcites. The luminescence emitted from the sample iscollected by the same objective and is directed througha pinhole (RP in the illustration). The function of thepinhole is to limit the sample volume from whichluminescence is collected. Selection of the objectivenumerical aperture and pinhole diameter result indiffraction limited focal volumes of less than 1 mm3

which, provided the sample density is low enough(1–10 dots per mm2), can enable single QD spectro-scopy. After the pinhole, a flip mirror (FM in theillustration) directs the luminescence to an imagingspectrometer that is able to resolve the spectral contentof the luminescence.

We present in Figures 4(a) and (b) two of the firstreported ensemble and d-function-like spectra ofphotoluminescence from InAs/GaAs quantum dots[8,9]. Figure 4(a) displays the broad emission of a QDensemble at the low energy tail of the spectrum alongwith the sharper bulk luminescence at 1.38 eV. In themeasurements presented in Figure 4(b), to probe

Figure 4. (a) Photoluminescence from ensemble of InGaAsQDs [8]. (b) Top three traces are mPL spectra recorded fromthree different sample locations at 10 K. The bottom trace isthe sum of 20 spectra [9]. (c) mPL from a single well-isolatedInAs QD showing the individual transitions. The solid circlesrepresent electrons and the open circles are holes.



individual QDs the sample surface was etched tocontain small mesas separated by 15 m m. The mesaslimited the spatial region of excitation and collectionrevealing signatures of single QD emission althoughthe sample density resulted in *250 QDs within thediffraction limited collection spot size. The etched mesais similar in effect to an appropriately selected pinholeplaced in a conjugate image plane of the microscope.Figure 4(c) is a similar mPL measurement on a well-isolated single InAs QD displaying the individualoptical transitions with the corresponding chargeconfigurations.

3.1.2. Exciton lifetime – time correlated photoncounting

Although photoluminescence is able to determine thetransition energies between optically active quantumdot electronic states, it is not able to associate a timescale with these transitions. Assuming a stable groundstate, it is natural to invert the linewidth measured withthe spectrometer to obtain the transition timescale.Unfortunately, for most QDs and a single spectro-meter, the transition linewidth is narrower than theinstrument’s spectral resolution (a typical resolution is30 meV), hence numerous publications historicallyhave used the term resolution-limited emission overthe years. It is clear from these considerations that inorder to assess the dynamics of the light emissionprocess a new technique is needed – one with temporalresolution.

One approach with temporal resolution is togenerate an ensemble of electron–hole pairs in thesurrounding matrix by a short nonresonant laser pulse,typically on the order of a few hundred femtosecond toa few picosecond duration. In bulk semiconductorswith direct bandgap, excitonic lifetime is on the orderof a few picoseconds. When confined in all three spatialdimensions, as in a quantum dot, the lifetime ofexcitonic complexes is enhanced by about three ordersof magnitude. Due to the unequal recombination ratesof excitons in extended versus confined systems, onlythe excitons captured by the quantum dot remain evenafter a mere tens of picoseconds following the initialexcitation pulse. This allows the quantum dot excitonto decay in the absence of any other excitations, andthus free from excited carrier induced effects. QDemission, spectrally filtered from all other excitonicrecombinations, can then be detected by photoncounting photodetectors. A registry of photon arrivaltime delays with respect to each excitation laser pulsethus builds up a temporal histogram of detectionevents. This histogram can be interpreted as theprobability that a QD exciton remains alive (or QDremains in the excited state) over a timescale.

Figure 5 presents data from a single InAs QDmeasured as a function of average pulse power for a 3ps wide excitation pulse [10]. At the lowest pumppower (bottom curve), the exciton (or excited state)lifetime is revealed, and the fit decay time is 0.8 ns. Foran ideal two-level system, such a measurement willproduce a single exponential decay of width directlyrelated to the excited state lifetime, and thus thenatural linewidth of the transition through a Fouriertransform. As the average pulse power is increased,there is a pronounced shift in the start time for theexciton decay. This is in fact a very nice signature ofmulti-excitonic effects taking place at high pumppowers. The probability of capturing one exciton isalready high, so an additional exciton capture withinthe recombination time becomes considerable. Conse-quently, exciton decay is necessarily delayed by thecharacteristic biexciton decay time. It is important tonote here that not only the radiative decay, but alsoany other process that results in exciton populationloss will affect this histogram. For example, if there is anonradiative channel for exciton decay that is largerthan the radiative lifetime, the measured histogrammay exhibit a double exponential where the decay ofeach exponential is dictated by the radiative andnonradiative decay rates. Quantum dot systems, suchas CdSe/ZnS core/shell colloidal QDs, do possess finitenonradiative decay rates [11], where one mechanismleading to nonradiative decay in CdSe/ZnS is Augerrecombination – a process whereby an exciton pairrecombines to form a more energetic electron. In thecase of InAs QDs with effectively no appreciable

Figure 5. The photoluminescence intensity from a singleInAs QD, at 20 K, as a function of time after excitation witha 3 ps laser pulse. The inset indcates the average power forthe excitation pulse. The lowest power decay curve fits to anexcited state lifetime of 800 ps [10].



nonradiative decay channels, another mechanism doeslead to biexponential decay in lifetime measurements –bright and dark exciton mixing. A clear study of thiseffect can be found in [12,13].

3.1.3. Two-time field and intensity correlations

Time-correlated photon counting allows us to measurethe emission timescale of the photons, but reveals verylittle about their coherence properties. In order tounderstand the extent of excitonic and carrier induceddecoherence mechanisms taking place prior to orduring the exciton recombination, we need to use atechnique where the signal is directly linked with thedegree of coherence of the QD emission. A first step toquantify the degree of coherence in the emitted photonstream is to interferometrically measure two-time fieldcorrelations, where temporal correlations in the lightbeam are revealed by interfering the field with itstime delayed replica. One approach, illustrated inFigure 6(a), is to use a Michelson interferometer whichmaps path length difference between the two inter-ferometer arms to time delay. Mathematically, this canbe represented as

ID Dxð Þ ¼ 2� Iavg 1þ< g Dxð Þ½ �ð Þ; ð2Þ

where

g Dxð Þ ¼ exp �i2kDx� 2g Dxj jð Þ: ð3Þ

Dx is path length difference between the two arms ofthe Michelson interferometer and g is the totaldamping rate of the transition (we have assumed thetransition lineshape is Lorentzian). In Equation (2),Iavg is the average intensity that reaches the detector.While the first term in the parentheses of Equation (2)reveals the average intensity of light and does notexhibit relative optical path length dependence, thesecond term, in Equation (2), modifies the detectedsignal to the extent that the parts of the optical fieldat two different times remain correlated. Figure 6presents this measurement on photons emitted from asingle self-assembled indium phosphide QD [14]. InFigure 6(b) widefield images of QD emission arepresented for the open circles decorating the interfer-ogram in Figure 6(c). The decay of the interferogramenvelope for large path length differences, Figure 6(d),reveals the damping rate g defined in Equation (2) willresult in an exciton with an emission linewidth of186 meV. The inverse of this quantity yields thecoherence time of the emitted photons. Zwiller et al.indeed extract a coherence time of 140 ps, muchshorter than the measured excited state lifetime of1.2 ns, allowing for an accurate quantification of the

dephasing mechanisms. One notable source of dephas-ing in this measurement is the optical charging of theQD environment that results from above bandgapnonresonant excitation. Specifically, photoexcited elec-tron hole pairs, a portion of which relax to occupy theQD exciton states, may also become trapped at defect

Figure 6. (a) Experimental setup. Flip mirrors (FM) areused to direct the emitted photons to either to aspectrometer, to a Michelson interferometer, or to aHanbury-Brown and Twiss correlator. The latter two arebuilt around nonpolarising beam splitters (BS). A removablepinhole (RP) can be used to select a single dot. A narrowbandpass filter (F) is tilted to transmit single spectral lines.(b) Images taken through the Michelson interferometershowing several dots with varying intensity as the mirror isscanned. (c) Single-dot photoluminescence intensity for thedot marked by an arrow in (b) as a function of mirrorposition. The circles indicate the three positions where theimages were taken. (d) The envelope of the exciton emissioninterferogram measured at 6 K with a 5 mm relative pathdifference between the two arms. The inset is the intensitycorrelation, g(2)(t), measurement on the same exciton [14].



sites in the vicinity of the QD. These local chargescreate a varying Stark field (uncontrolled electric fieldat the location of the QD) which acts to shift the QDtransition energy. If the time-scale of the measurementsamples a number of local defect charge configurationsthen the QD transition energy will shift throughout thecourse of the measurement. The shift in the energy willact to smear out the single-photon interference fringesand will lead to a reduction in the measured contrast,i.e. extracted photon coherence time.

Two-time measurements can also be carried oneorder higher, where the correlations in optical intensityrather than the field are considered at two differenttimes (in Figure 6(a) the HBT box). In this case, thesignal arises due to intensity fluctuations and does notdepend on interference or phase relations within thefields. Therefore, historically second-order correlationsof strong light beams were obtained from correlatingphotodetector current outputs i(t) after detection, inthe form of

gð2ÞðtÞ ¼ iðtÞiðtþ tÞh i= iðtÞh i iðtþ tÞh i : ð4Þ

A truly uncorrelated pair of photocurrent measure-ments gives unity value for this function, and anydeparture from unity value indicates correlation (oranti-correlation) within a characteristic memory time-scale of the source. Consequently, Equation (4) can bewritten in the form g(2)(t) ¼ 1 þ ZC(t). For intenseoptical beams detected by photodiodes, the registeredcurrent is proportional to the intensity, therefore thesecond-order optical correlations are mapped conve-niently to the current. The scenario is quite similar forsingle photon light levels where a temporal registry ofindividual detection events carries similar correlationsto the incident photon stream. For example, coherentlight (such as laser emission) exhibits no correlation(Z ¼ 0) and g(2)(t) ¼ 1 for all delay times whereas thememory present in single photon emission from a two-level system results in anti-correlation, g(2)(t) 5 1 withZ ¼ (71) and C(t) ¼ exp(7gjtj). In order to performsecond-order correlation measurements on single QDemission lines, the Hanbury-Brown Twiss (HBT)experimental arrangement, shown in Figure 6(a), isused. The collected QD emission is split to be detectedby two photon counting detectors. Photon detectiontimes are then registered per detector. A count isrecorded conditioned on the previous detection of aphoton – a coincidence count. A histrogram of countsis plotted as a function of the delay time betweensuccessive detection events using time–amplitude con-version. Figure 7 shows intensity (photon) correlationmeasurements performed on the emission undernonresonant excitation for an ensemble of InAs QDs(panel 1) and two individual InAs QDs (panels 2/3)

[15]. The pump power in panel 2 (3) is at the onset(below) the power necessary to saturate the QDemission. In panel 1, there is no correlation in thephoton emission from the ensemble of QDs, and thereis no deviation from unity as a function of delay time.Panels 2/3 show that the detector outputs areuncorrelated in all time scales except around zerotime difference, where detection events are anti-correlated. The QD emission therefore is antibunched,or has a degree of temporal order. This is aconsequence of the anharmonicity of the energy levelsinvolved in the QD emission. In these systems only onephoton can be generated within a radiative lifetime.Therefore, the two-time dependence of coincidencecounts also reveals the temporal profile of photonemission. For a two-level system with Lorentzianspectrum, with a linewidth of g, we expect to see anexponential decay in the time domain, and photoncorrelation measurements indeed reveal a symmetricdouble exponential profile. It is important to note herethat this technique measures the probability of two-photons being generated sequentially, therefore it isimmune to deviations from ideal configurations. Forexample, if the experimental setup includes imperfectelements or has mechanical instability, parameters

Figure 7. The data presents the measured distribution ofcoincidence counts and a fit of the correlation function. Panel1 is for a high density QD sample with more than one QDemitting within the focal volume of the collection optic. Themeasured g(2)(t) does not exhibit antibunching. Panels 2 and3 are for a single InAs QD at two different pump powers.Panel 2 (3) is taken at 125 W cm72 (66 W cm72). Panel 3exhibits a pronounced antibunching dip [15].



extracted from an interference-based measurementwould be significantly affected, but not the observedphoton correlation function. Alternatively, we mightbe dealing with less than ideal atomic-like states wherepure (elastic) dephasing and nonradiative decaymechanisms of the excited state may be non-negligible.While previously discussed lifetime measurementswould only provide the total excited state decaycharacteristics, photon correlation measurements, pro-vided the excitation method can be classified asincoherent, will reveal only the radiative part andremain immune to all other decay and dephasingchannels [16]. In experimental physics such powercomes rarely from such a simple arrangement. Anotherpoint we wish to emphasise here is that while two-timecorrelation measurements to date have been performedusing two independent detectors, identical measure-ments could also be performed using a single detectorwith sufficiently fast response and timing capability.Therefore, there still is a lot of attention on thedevelopment of single-photon sensitive, photon-num-ber resolving and high temporal resolution detectortechnology [17], where superconducting detectors seemto offer promising capabilities for this direction [18].

The power of photon correlation measurementsdoes not stop here at single photon generation, and canreveal a lot more about the internal dynamics of multi-exciton recombination. We have seen previously thatthe individual recombination processes of neutral andcharged excitons as well as biexcitons have welldistinguished energy shifts (see Figure 4(c)) and havealready seen such consequences on lifetime measure-ments (see Figure 5, top panel). Therefore, the intensitycorrelations between any two QD emission lines can bemeasured as well. Figure 8 shows such measurementsfor biexciton–exciton cascade decay [19]. The break-down of symmetry around the zero time delay betweencoincident detection events reveals the one-sidedordering of the cascade process. The probability todetect exciton decay is increased on the condition thata photon from biexciton decay has been detected, whilethe probability to detect a photon from biexcitonicdecay is essentially nil upon detection of a photon fromexciton decay. This technique therefore gives directinformation on the emission time ordering, plays acrucial part in identification of the observed spectrallines and allows us to identify metastable intermediatestates during the cascade.

3.2. Optics of quantum dots: resonant excitation

Techniques presented in the previous section can givevaluable information on exciton capture and decaydynamics as well as properties of the generated opticalfields. A common feature in all of these studies is the

incoherent pumping of the quantum-dot transitionsthrough carrier generation in either the host matrixsuch as GaAs or the quasi-continuum states above thehigher-lying confined states of the quantum dot – whatwe have termed nonresonant excitation. This excita-tion method leads to photon-emission-time jitter, sinceit relies on an uncontrolled relaxation step to populatethe excited exciton state, and spectral wandering of thequantum-dot transition larger than the transition’slinewidth due to optical charging of the host semi-conductor matrix. Both effects reduce the usefulness ofnon-resonantly generated single photons in linear-optics quantum computing algorithms, even if thequantum dot is coupled to a cavity [20]. In an attemptto both address this previous shortcoming and providespectrally selective access to the quantum-dot electro-nic transitions, increasing attention has turned toresonant optical excitation. Noting all successfulquantum-information science (QIS) implementationson well-developed qubit candidates, such as trappedions, have relied on resonance scattering, it is clear thatresonant optical control of QD transitions is desirable.In the realm of resonant spectroscopy, we make afurther distinction between temporal and spectralmeasurements.

3.2.1. Temporal measurements

A single ultrafast pulse propagating through a materialmay be affected by both absorption and dispersion.Transient nonlinear optical spectroscopy, also knownas pump–probe spectroscopy, involves a sequence ofultrafast optical pulses, which are separated in time

Figure 8. Cross-correlation of the biexciton–excitoncascade emission. The experimental apparatus is identicalto Figure 6(a) except the detection of the exciton emission isconditioned on the measurement of biexciton emission. Torealise this experimentally, narrow band spectral filters(1–2 nm FWHM centred on either the exciton or biexcitonemission line) are placed in front of each APD [19].



and tuned to the spectral vicinity of an absorptionband in a material. Conditional on the opticalexcitations or an induced polarisation field due to thefirst pulse (usually called the pump) the propagation ofthe second pulse (i.e. the probe) may show a deviationfrom what one would expect from the single pulse case.Therefore, a dynamical response from the material canbe mapped out based on the time delay between thepump and probe pulses. The selection of the laser pulsespectral/temporal width depends on the desired experi-ment, but typical considerations for QD excitons arethe timescale of the dynamics limiting the pump–probedelay and the spectral separation of other excitonicresonances limiting the pulse bandwidth. The pulsespectral width (via Fourier transformation) dictates theshortest temporal separation between two pulses andconsequently sets the experiment’s temporal resolu-tion. Before continuing we highlight that nonlinearspectroscopy techniques result in weak signals (typi-cally 15 orders of magnitude with respect to theexcitation pulses) and the measurement of these smallsignals presents an experimental challenge [21]. A firststep toward signal recovery is to interfere the signalfield on the detector with a local oscillator field that iseither derived from a reference pulse that does notinteract with the QD or with one of the excitationpulses. Mixing the signal field with a second field on aphotodiode is referred to as heterodyne detection, aterm borrowed from radio wave engineering. Theadvantage of heterodyne detection is that the inter-ference term depends on the signal amplitude multi-plied by the conjugated strong local oscillator fieldamplitude and this acts to amplify the measured signal.In addition to interference it is common to modulatethe excitation pulses so that the signal is carried bydistinct spectral components in the measured photo-current which can then be accessed with a phase-sensitive lock-in detection system.

Within the family of pump–probe techniques usedfor ensemble and single QD spectroscopy, there is aparticularly elegant and powerful modality, where thepulse sequence itself generates a third-order nonlinearpolarisation in the QD which acts as a source of anadditional field that carries information related to thedynamics of the QD excitons. This polarisation fieldoscillates at a set of frequencies determined by centralfrequencies of the two laser pulses including a beatfrequency, and, if the pulses are non-collinear, thegenerated field may even propagate in a distinctdirection due to the phase matching requirement ofthe involved k -vectors. This counter-intuitive responseto a two-pulse sequence can be seen in the relevantdensity matrix equations of motion coupling popula-tions to coherences for spectrally shifted excitationpulses [22].

In the context of QD spectroscopy the initialmotivation for two-pulse degenerate transient FWMwas to circumvent the inhomogeneous broadeninginherent in ensemble QD measurements, and directlyaccess the dephasing time of a single QD exciton. Inthis spectroscopy modality the central frequency o0 ofa laser pulse is tuned to the mean frequency of theensemble X0 exciton resonance. As can be seen inFigure 9(a), the exciting laser pulse is split into threeparts – a pump, a probe and a reference pulse. Acontrollable time delay, td, is established between thepump and probe pulses. The pump (indexed with a 1)and probe (indexed with a 2) pulse each receive a small,but distinct, frequency up-shift typically with acousticoptic modulators to o1 ¼ o0 þ oRF1 and o2 ¼ o0 þoRF2. The pulse pair is subsequently directed to theQD sample exciting a third-order polarisation. It is atthis point, when the pulses excite the sample, wherethere are two variations in implementing FWM. In [23]the time-delayed pump pulse and probe pulseilluminate the sample with well-defined non-collineark-vectors, k1 and k2, where the general scheme is nicelyillustrated in Figure 9 [24]. The resultant polarisationradiates in a direction determined by the wavevectorcombination 2k27k1 and oscillates with a frequency2o27o1. The subsequent source field (resulting fromthe QD nonlinear polarisation) is mixed with areference laser pulse still at o0, but delayed by timetr, on a photodiode. A phase-sensitive lock-in amplifierdemodulates the FWM signal in the measured photo-current at 2oRF27oRF1, which is typically in the tensof MHz frequency range.

Another approach to FWM, which does notexploit the additional wavevector selectivity affordedby the previous approach, is presented in [25]. Againthe pump and probe pulse are time delayed, butinstead of illuminating along fixed, non-collinear,directions the collinear pump and probe pulses arefocused by a high numerical aperture objective ontothe sample as illustrated in Figure 9(b). The inducednonlinear polarisation radiation is collected by asimilar objective and mixed with the reference pulsedirectly on a photodiode. Unlike the previous examplewhere the signal was free from pump and probe pulsebackground, this second approach relies entirely on thelock-in amplifier to distinguish the signal of interestfrom the background pulses, which still modulatesthe photocurrent at a frequency 2oRF27oRF1. Thestringent k -vector considerations are not an issue forthis modified version, at the expense of reducedsensitivity in detection. Although these two modalitiesare slightly different with respect to signal acquisition,both benefit from the essential advantage of thetransient FWM signal; its immunity to inhomogeneousbroadening. Inhomogeneous broadening is an inherent



feature of QD emission spectra in high density QDsamples (when there are a number of optically activeQDs within the microscope focal volume) as a result ofthe QD size distribution. Specifically, when theinhomogeneous spectral broadening of the ensembleis considerably larger than the homogeneous dephas-ing rate of a single QD exciton, the polarisation willradiate a light pulse at exactly 2 times the pump–probedelay; the photon echo. Figure 10(a) presents themeasured echo pulse as a function of the probe delay tdfrom an ensemble of InAs QDs at a temperature of50 K. The important information is obtained fromplotting the integrated area of this light echo pulse as afunction of pump–probe delay. The strength of thephoton echo generated depends on the coherence ofthe ensemble, so the decay of the integrated area isdetermined by the total dephasing rate of a single QDexciton. Figure 10(b) is an example of the integratedecho pulse area FWM signal. The extracted dephasingrate of 630 ps when this measurement was repeated at7 K was the first of its kind to suggest the excited statedecay was predominantly the result of radiationbroadening.

The second transient nonlinear spectroscopy tech-nique we describe is degenerate pump–probe

Figure 9. Schematics of the setups for the two four-wave mixing modalities. (a) A pump and a probe pulse, at two distinctfrequencies, illuminate the sample along two directions. The phase matched signal, in a direction distinct from the pump andprobe, is mixed with a reference pulse on a photodiode. A lock-in amplifier demodulates the signal, at a frequency 2oRF27oRF1,from the photocurrent. (b) Same as (a) except the pump and the probe pulses are now focused and collected by high numericalaperture objectives.

Figure 10. Time resolved four-wave mixing data from anensemble of InAs QDs at 50 K. (a) The echo pulse as afunction of the reference pulse time delay tr. The time delaybetween the exciting pulse pair is varied in 400 fs steps from 0to 3.2 ps. (b) Time-integrated four-wave mixing obtained byintegrating the echo pulse area in (a) at different excitationintensities. (a) Corresponds to the 2Io trace in (b).Exponential fits to the traces in (b) yield the dephasing timeof a single QD exciton transition [25].



spectroscopy. Just like FWM, pump–probe alsointerrogates the induced third-order polarisation, but,in contrast to FWM, the pump–probe signal is derivedfrom the polarisation’s response at the probe field’sfrequency. Consequently, this alternative techniqueprobes essentially the occupation probabilities of theexcitonic states rather than being limited by theircoherences. In other words, the previous FWM signal(at the beat frequency of 2oRF27oRF1) will vanish ifthe coherence between the two states of the transitionis lost, while the probe response (at the probemodulation frequency oRF2) will still be visible aslong as the excitations are present, much like thelifetime measurements discussed before. The twopulses are delayed in time by td, but now the signalfield is mixed with the probe field on the photodiode. Alock-in amplifier filters the measured photocurrentisolating the signal at a frequency oRF2. In a pump–probe experiment the probe pulse transmission as afunction of delay time td is a direct measure of theexcited state lifetime: both radiative and nonradiativecontributions. A further twist on the pump–probetechnique results from plotting the probe transmissionat a fixed delay time td as the pump and probe centrefrequency are varied in unison. In this case the time-domain technique is able to unmask frequency domaininformation and probe the absorptive resonances ofthe QD. An example of the measured data in these twoapproaches is in Figure 11 for a single GaAs interfacial(fluctuation) QD [22]. In the top panel in Figure 11(a)two co-polarised 6 ps wide pulses, with a fixed 6 psdelay between the pulses, illuminate the sample. Theabsorption of the probe pulse, the transient differentialtransmission, is plotted as a function of the two pulses’centre frequency. The peaks in the absorption datareveal electronic transition energies in the QD.Figure 11(b) is the measured probe pulse area as thedelay between the pump and probe pulse is varied. Themeasured excited state decay time is 41 + 2 ps.Finally, the lower panel in Figure 11(b) presentsdifferential transmission with a continuous wave(CW) laser. The narrowband CW laser greatlyimproves the spectral resolution of the absorptionmeasurement, and as we will see in the coming section,has become an indispensable tool for selective excita-tion of QD transitions.

3.2.2. Spectral measurements

We introduced fundamental transitions per chargeconfiguration in the quantum dot in the first section(see Figure 3). The purpose of photoluminescencemeasurements was to map out the spectrum of allallowed transitions due to various charge combina-tions. Given a typical resolution of 30 meV, this

appears insufficient for resolving spin and anisotropyinduced fine structures. Likewise, even when theground state of the quantum dot is controlled to bein a particular charging configuration, we introducednothing to prevent the capture of uncontrolled chargecombinations under above-band-gap excitation.Therefore, while this particular type of excitationallows for mapping out the optical transitions, it stillrenders any systematic access to individual spin statesimpossible. Pump–probe techniques allow for precisemeasurement of the excitonic dynamics, but due to theextended bandwidth of the optical pulses, frequencyselectivity of individual transitions within the finestructure is limited. In order to address this short-coming, an alternative technique is utilised wheretemporal resolution is sacrificed for such spectralselectivity. Ultrafast pulse pairs are replaced by oneor more highly monochromatic single transverse andlongitudinal mode lasers with tuneable optical fre-quencies. We can now address a transition of interestdirectly and selectively among many allowing us to

Figure 11. Degenerate pump–probe spectroscopy data. (a)Copolarised pump and probe pulses with widths of 6 ps aredelayed by 6 ps and the transmission is measured as the pulsecentre frequency is varied. The dashed line is the zero signallevel. In the lower panel, a cw laser is tuned through the QDresonance. We will focus on the merits of this technique inthe next section. (b) The integrated probe pulse signal as afunction of its delay from the pump pulse. The fit exponentialdecay reveals an excited state lifetime of 41 + 2 ps [22].



study features such as optical selection rules andoscillator strengths by observing the scattered laserlight.

The signal in this case is still due to the interferenceof the background laser and the scattered dipoleradiation, where the transmitted laser field is measuredby photodetectors rather than the emitted photonsbeing detected by a spectrometer. If the laser field is inresonance with a quantum dot transition, the opticalfield scattered by the quantum dot interferes with thebackground optical field. The total field observed upontransmission through the quantum dot includes thesignature of the quantum dot response to the lightfield. All measurements are performed in the far-fieldso the phase difference is a result equivalent to theoverall Guoy’s phase acquired by the backgroundlaser. Consequently, we typically measure the absorp-tive response directly in intensity change. In short, themeasured optical field intensity will trace out thefollowing response:

IðDÞ=Ið1Þ ¼ 1� ao ð�2=4Þ=ðd2 þ �2=4Þ� �

; ð5Þ

where ao is the effective absorption strength deter-mined by the laser focus area and the quantumdot oscillator strength and is typically in the range of107371072. In order to see this response one has tohave control over the detuning d and this can beachieved in two ways: either by sweeping the opticalfrequency of the excitation laser with respect to the QDtransition or by shifting the quantum dot emissionenergy via DC Stark effect through an external electricfield.

Unlike photoluminescence measurements, the sig-nal strength here with respect to the background laserintensity is usually on the order of 1073 as dictated byao and lock-in based detection is required in order toeliminate this background. The modulation requiredfor the lock-in scheme is obtained by electronicallymodulating the external electric field with a squarewave signal. Normalising by the total power at thephotodetector one can calculate the deviation of thetransmissivity from the off-resonant value DT/T. Thisis called Stark-shift modulation spectroscopy andfurther experimental details of this technique can befound in [26]. Given that we are no longer limited bythe spectrometer spectral resolution of 30 meV, butrather the uncertainty of the applied gate voltage andthe spectral bandwidth of the applied laser, resonantscattering has significantly higher resolution, on theorder of 0.04 meV. The previously predicted fine struc-ture of each optical transition can now be observedclearly, as displayed in Figure 12(a). The black (grey)data set is obtained with a pX (pY) linearly polarised

laser. Figure 12(b) is the same measurement onX17 displaying a single peak due to degeneracy [27].Taking advantage of this high spectral resolution, QDtransition linewidths on the order of 1.3 meV have beenobserved [28]. In addition, this response can inaddition be controlled to yield both absorptive anddispersive lineshapes [29] depending on the relativephase between the laser and the QD scattering field.Nevertheless, the relatively modest signal to back-ground level limits how fast this measurement can becompleted. However, recent advances incorporatingsolid immersion lens technology [30] showed that themeasured signal strength can be increased furtherleading to higher detection bandwidth [31,32].

4. Recent achievements utilising these techniques

What we have presented so far is a survey of opticaltechniques utilised in the quantum dot research in thebroad sense. While each technique presents a com-plementing side to complete spectroscopy of quantumdots, some have in parallel been utilised to performsome key achievements. In this context, quantuminformation processing has benefited significantly fromthe re-interpretation of spectroscopic concepts andmeasurements. We give below a handful of examplesalong these lines, but emphasise that we are leaving outa vast area of quantum dot research on conventionaloptoelectronics technology, such as lasers and opticalswitches.

4.1. Deterministic source of single photons andquantum cryptography

One of the earliest applications of quantum informa-tion science was quantum cryptography, which drewits power of security from the indivisibility of a singlephoton. The information (the bit) is encoded on single

Figure 12. Differential transmission data for the neutralexciton (left panel) exhibiting the fine structure x – y splitting,in this case 27 meV, due to the electron–hole exchange.Differential transmission data for the trion exciton (rightpanel). The exchange splitting observed in panel (a) is zerosince the electrons are in a spin singlet [27].



photons thus ‘tapping’ the quantum channel wasquantum mechanically forbidden. The lack of a truesingle photon source prevented the use of the no-cloning principle, and had to revert once again to the‘difficulty’ of eavesdropping by using either heavilyattenuated laser beams with average photon numbermuch less than one, or down-conversion processes thatgenerate photon pairs with a small probability ofgenerating multiple photon pairs in well determineddirections. Sacrificing one of the photon pairs in down-conversion for timing gives a sense of heralding to thephotons in the other direction, but the probability tohave more than one photon in a pulse still remainsfinite. A compact, stable, reliable source of singlephotons was the missing link for relying on quantummechanical principles for ultimate cryptographic se-curity rather than assumed difficulty in practice ofbreaking the cryptographic code.

We have discussed how intensity (photon) correla-tion measurements reveal the decay dynamics of theexcited states in quantum dots. In 2000, two groupsreported one of the first applications of this techniquein the realm of quantum information for the realisationof deterministic heralded single photon generation[33,34]. In these works, the above-bandgap nonreso-nant continuous-wave excitation laser was replaced byan ultrafast laser pulse train which generated excitons,in the vicinity of a quantum dot, at well known times.After each excitation pulse, a high density of photo-excited excitons was to ensure the probability tocapture at least one exciton inside the quantum dotapproached unity. While the number of excitonscaptured inside a quantum dot is statistically varying,the distinct exciton emission wavelengths due toCoulombic interactions allows one to spectrally filterout only the neutral exciton recombination. The resultis a single photon generated with unity probability at awell-defined wavelength, at discrete times manifested

by the excitation laser pulses. Figure 13(a) displays thephoton correlation measurement under these condi-tions. Unlike the continuous-wave excitation profile,we see uniform coincidences only at discrete timedelays indicating that the emission comes as a pulsetrain following the excitation laser pulse train. Theabsence of the central coincidence peak is due tothe lack of another photon emitted simultaneously at

Figure 13. (a) The top panel is the autocorrelation of a250 fs Ti:sapphire laser and the bottom panel is theautocorrelation of single photons emitted from the groundstate exciton of a QD embedded in a microdisc cavity underpulsed excitation. In the bottom panel (in contrast to the toppanel), the zero delay time is nearly zero. This is a result ofthe QD emitting one photon at a time. More importantly,upon pulsed excitation, the QD emits a single photon in well-defined time bins – the so-called photon turnstile [33]. (b) Thepolarisation of single photons emitted by a QD based photonturnstile is used exchange a secure key between Alice (thetransmitter) and Bob (the receiver). The key is used to encodeand decode a 140 6 141, 256 pixel colour bitmap of theStanford University’s memorial church (the top image). Theencoded message appears as white noise to all parties thatdetect the encoded photons without the key except for Bobwho has the correct key [35].

"



the same wavelength, verifying the single photonnature of this heralded emission process.

In 2002, Santori et al. implemented the firstquantum key distribution protocol using a singlequantum dot as a true single photon source [35].Figure 13(b) shows the original and decoded imagewhen the polarisation state of individual photonsemitted from a quantum dot are used to exchangethe key necessary to decode the image.

4.2. Going beyond single photons: polarisationentangled photon pairs

A cascade two-photon emission process with inter-mediate level degeneracy was the source of entangledphotons for the seminal work by Aspect et al., whereJ ¼ 0! J ¼ 1! J ¼ 0 type cascade decay of Calciumatoms was used for the first experimental demonstra-tion of bipartite entanglement and violation of localrealism models for quantum theory [36]. Once theanalogous cascade nature of the biexciton-excitondecay in quantum dots was revealed, the route togeneration of frequency and polarisation correlated(or entangled) two-photon states was established. Thetwo-photon field emitted via the biexciton–excitoncascade decay process, illustrated in the inset ofFigure 14(a), can be written as

ð o1HXX;o2HXj i þ o3VXX;o4VXj iÞ=21=2: ð6Þ

This is a maximally entangled state in the strict senseof the word. However, revealing the degree of, forexample, polarisation entanglement would be bur-dened by the simultaneous frequency entanglement.The frequency tag on the decay channel comes fromthe electron–hole exchange (discussed previously inrelation to Figure 12), labelled S in the inset ofFigure 14(a), which lifts the double degeneracy ofthe intermediate X0 excitonic energy levels. There-fore, removal of the which-path information for thetwo decay probabilities is necessary to revealpolarisation entanglement. We can list three ap-proaches to this problem two of which have shownsuccess.

The first method is to be oblivious! That is, let usfocus only on the subset of emission events that cannotreveal the path information. The finite spectral widthof excitons still allows a region of spectral overlap (andthus indistinguishability) where spectral pre-selectionof excitons in this overlap region, roughly in the middleof split X0 energy levels, will remove the informationon decay paths albeit the heavy cost of detection rates.Akopian and co-workers have done just that to showcorrelations that reach beyond the classical bounds for

the two photons generated by such a biexciton–excitondecay from a single quantum dot [37].

The second method approached the problem fromthe materials side: the excitonic exchange splitting to acertain extent originates from the shape anisotropy ofthe quantum dots. Therefore, physically altering theshape of the quantum dots after growth process has beencompleted will also alter this undesired level splitting.Stevenson et al. used post-growth annealing of the QDsample which led to a systematic reduction of theexchange splitting as a function of annealing time, as canbe seen in Figure 14(a) for two different QDs [38,39].Since every QD starts from a different splitting valuewithin a statistical distribution, a certain annealing rangeprovides a subset of QDs where exchange splitting iseffectively removed, and thus results in polarisationcorrelations that reach beyond the classical boundswithout sub-selecting the emission spectrum. To verifyentanglement in the emitted photon pairs, Figure 14(b)presents full state tomography of the emitted photons inthe linear polarisation basis. The inset of the figure is theresults of various entanglement tests performed on themeasured density matrix [40].

The final method we mention was in fact the firstone proposed, but is still yet to be realised: coupling anoptical transition to a cavity mode coherently broadensthe spectrum of the emitted photons due to a reductionof the radiative lifetime, known as the Purcell effect.We will not discuss emitter–cavity coupling in thisarticle; however, it suffices to note that the pathdistinguishability can be removed by increasing thespectral overlap of the two excitonic transitions viacavity-induced broadening [41]. This method not onlyrecovers polarisation-only entanglement, but furtherincreases the photon-pair generation rate – bothfeatures desirable for QIS applications.

4.3. A quantum gate with excitonic qubits

Another application we would like to highlight is thepossibility to use the QD exciton as the physicalrepresentation of a qubit. In 2003, the two-colourversion of the pump–probe transient nonlinear spec-troscopy technique was utilised along with the linearpolarisation selection rules of a quantum dot excitonand biexciton transitions in order to operate aquantum gate. We saw in the previous section thatthe cascaded decay process of a biexciton follows eitherof the two decay paths into the ground state. Here,Gammon et al. approached from the other directionwhere a large value for S is favoured and demonstratedthe conditionality of the biexciton generation onthe initial exciton polarisation [42]. The excitonicstates of Figure 15(a) can be interpreted via two-qubit logic states. A laser pulse resonant with the



exciton–biexciton transition was labelled as the ‘opera-tion’ and had an effect on the quantum dot excitationthat is conditional on the initial condition. Forexample, if the quantum dot was in the j00i or j01istate, the operation left the system unaltered due toeither frequency or polarisation selectivity. If, how-ever, the initial state was j10i the ‘operation’ pulseresulted in a p -excitation to the biexciton state eipj11i,while the initial state of j11i dropped to j10i throughstimulated emission. Consequently, one could con-struct a truth table for this effective two-qubitoperation that is analogous to a controlled rotation(CROT) gate. Verification of this operation andquantifying the gate’s fidelity still relied directly on

the transient pump–probe technique discussed earlier.In Figure 15(b) the reconstructed physical truth tablefor this excitonic gate is presented. The system isprepared in the various input states, the operationalpulse is applied, and the final state ofthe QD isdetermined. Ideal operation corresponds to the fourwhite bars having a value of 1 and the rest 0.

4.4. Initialisation and coherent rotation of quantumdot spin qubits

All nonresonant excitation mechanisms introducemultiple carriers into the quantum dot levels, whereasresonant excitations only create single excitons within

Figure 14. (a) In the inset, S depicts the fine structure splitting of the neutral exciton X0 (see Figure 3(a)) as a result of theelectron–hole exchange interaction. The data points represent how the fine structure splitting can be tuned through annealing twodifferent InAs QD samples for 5 min intervals at 6758C. For generating polarisation entangled photons from QD emission thethird data point for DOT B is of interest [39]. (b) The real and imaginary components of the measured density matrix for the two-photon state emitted from a QD with an exchange splitting less than 1.3 meV. Inset: The results of various entanglement testsperformed on the measured density matrix [40].



the quantum dot. The unprecedented spectral resolu-tion provided by the differential transmission techni-ques highlighted above allow direct access to thetransitions between the individual fine structure andZeeman levels. We showed in Figure 12 how this

accessibility can be used for spectroscopic measure-ments to reveal the strength of electron–hole ex-change interaction and quantum dot anisotropy. Inthe case of a single excess electron trapped in aquantum dot, we do not have a priori control, over or

Figure 16. (a) Experimental setup for pulsed optical rotation of QD spin. During each experimental cycle, one or two rotationpulses may be sent to the sample to observe Rabi oscillations or Ramsey interference, respectively. The time delay, t, betweenpairs of pulses is controlled by a retroreflector mounted on a computer-controlled translation stage. CW, continuous wave;QWP, quarter-wave plate; PBS, polarising beam splitter; SPCM, single-photon counting module; CCD, charge-coupled device; c,speed of light. (b) Reconstructed evolution of the Bloch vector. The curves trace out the tip of the Bloch vector in the one-pulse(Rabi oscillation) experiment. The colour scale indicates the length of the Bloch vector, which shrinks exponentially as the systemis evolved for longer times. Views are from the perspective of the x axis [44].

Figure 15. Controlled Rotation (CROT) gate based on QD biexciton and exciton transitions. (a) Panel A is a schematic of theQD states relevant to the CROT gate. Panel B is the excitation energy level diagram and Panel C is the CROT gatetransformation matrix (note j10i is transformed to 7 j11i). (b) The reconstructed physical truth table for the CROT gate.The operational pulse is a p -pulse tuned to the j10i 7 j11i. The system is prepared in the various input states, the operationalpulsed is applied, and the final state of the QD is determined. Ideal operation corresponds to the four white bars having a value of1 and the rest 0 [42].



knowledge of, its spin. In atomic physics, the conceptof optical pumping is used to create an imbalance inthe spin projection of the ground states of an atomicensemble. This process relies on the fact that uponexcitation to a higher electronic state, the atoms willdecay randomly into a set of ground states inaccordance with the strength of the correspondingtransition matrix elements. In charged quantum dots,such as the trion in Figure 3, the process can notwork in ideal conditions, but in the presence of statemixing mechanisms such as nuclear spin coupling andhole mixing, there is small but finite rate of decay intothe other electron ground state. Specifically, if aresonant laser drives the sþ transition, there is asmall, but finite probability for a spontaneousRaman spin flip process to take place andshelve the electron in the spin down ground state.The system then goes dark and no more light isscattered due to both frequency and polarisationselection rules. This exact idea, an outgrowth of thepreviously discussed DT technique, has been demon-strated in [43], where a QD electron was opticallypumped and thus prepared a given spin state with99.8% fidelity. This is the initial step of any QIPprotocol utilising single QD spins.

Finally, we briefly touch on a recent work thatdemonstrates coherent control of individual electronspin states using ultrafast optical pulses and opticallyinduced spin rotations. The spin of a singly chargedQD serves as the qubit. The sample is placed in astatic magnetic field perpendicular to the growthdirection (the Voigt geometry). If a single electron isloaded into the QD, it will coherently precess aroundthe applied perpendicular field at the Larmorfrequency. Visualised on the Bloch sphere the stateof the electron spin is traversing a great circle of thesphere ignoring any spin dephasing effects. Applica-tion of a circularly polarised ultrafast optical pulsered-detuned from the electron transition Stark shiftsthe two ground state electronic levels and results inspin precession around the direction parallel to theQD growth direction (perpendicular to the externallyapplied magnetic field). The key point is that theoptical pulse, via the dynamic Stark effect, acts as anultrafast effective magnetic field and induces coherentspin rotation. Figure 16(a) is the experimental setupemployed to optically evolve the electron spin andFigure 16(b) is the reconstructed evolution of theBloch vector. We direct the reader to [44,45] for amore detailed discussion of the two approaches, butthe main message here is that coherent opticalrotation of a single QD spin has now been observed10 years after the first photon correlation measure-ments indicating nonclassical light emission fromQDs.

5. Conclusion

Given that the word spectroscopy comprises two roots(one Latin and one Greek) meaning appearance-watching, one may be led to believe that spectroscopyindeed goes all the way back to the Greeks andRomans. Surprisingly, this powerful word itself wasnot coined until 1882 by Arthur Schuster in order toclassify studies focused on the spectral properties oflight–matter interaction. With this particular defini-tion, the first systematic spectroscopy experimentswere performed two centuries earlier. In 1666 Sir IsaacNewton demonstrated the multi-colour nature ofsunlight by using a glass prism which was readilyavailable from the local market as a ‘Fool’s Paradise’.His simple but ground-breaking findings initiated 350years of investigations performed on light absorbed,scattered, and emitted by matter. Today, spectroscopyhas generated a sea of knowledge on the nature of lightand its interaction with matter, where QD researchconstitutes a modest component. We tried to give abrief survey of the optical measurement techniquesutilised to investigate the physics of quantum dots.Today, thanks to these studies we now have a high-level of control over the physics of QDs and we canexpect to see a continuation of productivity inquantum dot research in the coming years with lessemphasis on observation and more on control. Theoptical techniques used for this progress, however, willlikely be a combination of what we discussed here,as the general principles of spectroscopy are here tostay.

Notes on contributors

Nick Vamivakas is a postdoctoralfellow in the Cavendish Laboratory atthe University of Cambridge since 2007supported by QIP IRC (GR/S82176/01)and EPSRC Grant no. EP/G000883/1.He received his Bachelor of Sciencedegree in 2001 from Boston UniversityECE Department and received hisPh.D. from the OCN Laboratory atBoston University in 2007 under the

supervision of Professors Swan, Unlu and Goldberg.

Mete Atatüre is a Lecturer in theCavendish Laboratory at the Univer-sity of Cambridge. He received hisBachelor of Science degree in 1996from Bilkent University Physics De-partment in Turkey. Then, he joinedthe Quantum Imaging Laboratory atBoston University for his Ph.D. studiesuntil 2002. From 2002 to 2007, heworked as a Postdoctoral Fellow inthe Quantum Photonics Group at ETH

Zurich. Current research efforts of his research group include



optical control of single and multiple quantum-dot spins,high-resolution spectroscopy of diamond-based emitters, andnanoplasmonics.

Suggested reading

(i) H. Haug and S.W. Koch, Quantum Theory of theOptical and Electronic Properties of Semiconductors, 3rded., World Scientific, Singapore, 2001, pp. 44–48.

(ii) R. Loudon, The Quantum Theory of Light, 3rd ed.,Oxford University Press, Oxford, 2004.

(iii) V.V. Mitrin, V.A. Kochelap, and M.A. Stroscio,Quantum Heterostructures: Microelectronics and Optoe-lectronics, Cambridge University Press, Cambridge,1999.

(iv) S. Mukamel, Principles of Nonlinear Optical Spectro-scopy, Oxford University Press, New York, 1995.

(v) U. Woggon, Optical Properties of SemiconductorQuantum Dots, Springer, Berlin, 1996.

(vi) L. Jacak, P. Hawrylak, and A. Wojs, Quantum Dots,Springer, Berlin, 1998.

(vii) D. Bimberg, M. Grundmann, N.N. Ledentsov,Quantum Dot Heterostructures, Wiley, Chichester,UK, 1999.

(viii) D.D. Awschalom, D. Loss, and N. Samarth, Semi-conductor Spintronics and Quantum Computation,Springer, Berlin, 2002.

(ix) P.M. Petroff, A. Lorke, and A. Imamoglu, Epitaxiallyself-assembled quantum dots, Phys. Today 54 (2001), pp.46–52.

(x) D. Gammon and D.G. Steel, Optical studies of singlequantum dots, Phys. Today 55 (2002), pp. 36–41.

(xi) M. Bayer, G. Ortner, O. Stern, A. Kuther, A.A.Gorbunov, A. Forchel, P. Hawrylak, S. Fafard, K.Hinzer, T.L. Reinecke, S.N. Walck, J.P. Reithmaier, F.Klopf, and F. Schäfer, Fine structure of neutral andcharged excitons in self-assembled In(Ga)As/(Al)GaAsquantum dots, Phys. Rev. B 65 (2002), 195315:1–23.

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