Single-photon sources

INSTITUTE OF PHYSICS PUBLISHING REPORTS ON PROGRESS IN PHYSICS

Rep. Prog. Phys. 68 (2005) 1129–1179 doi:10.1088/0034-4885/68/5/R04

Single-photon sources

Brahim Lounis1 and Michel Orrit2

1 C P M O H, Universite Bordeaux I, 351 Cours de la Liberation F-33405, Talence Cedex,France2 MoNOS, Leiden Institute of Physics, Huygens Laboratory, Postbus 9504, NL-2300 RA,Leiden, The Netherlands

Received 3 November 2004Published 21 April 2005Online at stacks.iop.org/RoPP/68/1129

Abstract

The concept of the photon, central to Einstein’s explanation of the photoelectric effect, isexactly 100 years old. Yet, while photons have been detected individually for more than50 years, devices producing individual photons on demand have only appeared in the lastfew years. New concepts for single-photon sources, or ‘photon guns’, have originated fromrecent progress in the optical detection, characterization and manipulation of single quantumobjects. Single emitters usually deliver photons one at a time. This so-called antibunchingof emitted photons can arise from various mechanisms, but ensures that the probability ofobtaining two or more photons at the same time remains negligible. We briefly recall basicconcepts in quantum optics and discuss potential applications of single-photon states to opticalprocessing of quantum information: cryptography, computing and communication. A photongun’s properties are significantly improved by coupling it to a resonant cavity mode, either inthe Purcell or strong-coupling regimes. We briefly recall early production of single photonswith atomic beams, and the operation principles of macroscopic parametric sources, whichare used in an overwhelming majority of quantum-optical experiments. We then review thephotophysical and spectroscopic properties and compare the advantages and weaknesses ofvarious single nanometre-scale objects used as single-photon sources: atoms or ions in the gasphase and, in condensed matter, organic molecules, defect centres, semiconductor nanocrystalsand heterostructures. As new generations of sources are developed, coupling to cavities andnano-fabrication techniques lead to improved characteristics, delivery rates and spectral ranges.Judging from the brisk pace of recent progress, we expect single photons to soon proceed fromdemonstrations to applications and to bring with them the first practical uses of quantuminformation.

(Some figures in this article are in colour only in the electronic version)

0034-4885/05/051129+51$90.00 © 2005 IOP Publishing Ltd Printed in the UK 1129

http://dx.doi.org/10.1088/0034-4885/68/5/R04

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1130 B Lounis and M Orrit

Contents

Page1. Introduction 11312. Photon optics 1133

2.1. Quantized radiation field 11332.2. Different flavours of light 11342.3. Antibunching and second-order correlation function 11372.4. Indistinguishable photons 11392.5. Summary: characteristics of the emitted photons 1141

3. Possible uses of single photons 11423.1. Measurement of weak absorptions 11433.2. Random number generation 11443.3. Quantum information processing 1144

4. Coupling to cavities 11544.1. Weak-coupling regime 11544.2. Strong-coupling regime 1156

5. Early and macroscopic sources 11585.1. Faint laser pulses 11585.2. Entangled photon pairs 1159

6. Microscopic single-photon sources 11616.1. Excitation schemes 11616.2. Atoms, ions in gas phase 11626.3. Organic molecules 11636.4. Colour centres 11666.5. Semiconductor nanocrystals 11686.6. Self-assembled quantum dots and other heterostructures 11706.7. Summary 1172

7. Conclusion and outlook 1173Acknowledgments 1175References 1176

Single-photon sources 1131

1. Introduction

Nearly one century after the discovery of its basic principles, quantum mechanics remainsenshrouded in mystery and paradox (Bouwmeester et al 2000). Very early on, several ofits counter-intuitive predictions were pointed out and gave rise to sometimes conflictinginterpretations. Fortunately, quantum mechanics has also, meanwhile, become the mostaccurately tested of all physical theories. Strange though it may seem to us, macroscopicbeings, it has spawned a variety of clever schemes for storing, transporting and processinginformation. Wave–particle duality is a case in point. A beam of particles, say electrons,sometimes behaves as a wave. Reciprocally, a classical wave such as light sometimes showsa ‘grainy’ nature and can be described as consisting of particles, called photons. A ‘number-state’ is a quantum state in which the number of these particles, n, is a precisely fixed integer.We are familiar with the number-states of electrons in atoms and molecules, but number-statesof light are more exotic. Except for the trivial vacuum-state (n = 0), number-states of lightare unfamiliar and very hard to obtain in practice. A stabilized laser delivers light pulses thatseem extremely regular in amplitude, but this is only because they contain so many photonsthat their number fluctuations become negligible on a macroscopic scale. Indeed, such pulsesare just superpositions of many number-states with different ns. Restricting fluctuations in thenumber of photons seems easiest for states with n = 1, single-photon states. Recent progressin the fabrication, manipulation and characterization of individual nano-objects, moleculesand nanocrystals has opened new routes for the production of number-states containing onesingle photon. The purpose of this paper is to review current ideas and recent developmentsin the field of single-photon sources.

The idea of manipulating single photons did not arise immediately from quantumprinciples. The quantization of electromagnetic energy was first proposed by Planck (1900)to explain the thermodynamics of black-body radiation, and the concept of the photon wasintroduced by Einstein, just one century ago (Einstein 1905), to interpret the photoelectriceffect. However, it was later realized that the photoelectric effect results from the quantumnature of atoms and can be fully interpreted in a semi-classical frame, where matter is quantized,but light remains a classical wave. Only much later, in the 1960s, and mainly under Glauber’simpulsion (Glauber 1965), did researchers begin to look for specific quantum properties oflight that could not be understood in a classical frame. A crucial step in this process wasthe correlation experiment of Hanbury Brown and Twiss (1956), who proposed the use ofcoincidences between two detectors to study the coherence properties of astrophysical sources.Their experiments, and the subsequent discovery of the laser, led to a deep analysis of thesecond-order coherence of light (Born and Wolf 1980), which was later reformulated in termsof quantum-mechanical operators (Mandel and Wolf 1995). Field interference, a classicalconcept in wave optics, was generalized to intensity interference, i.e. to quantum effectsinvolving numbers of photons instead of amplitudes of electric fields. Number-states, andsingle-photon states in particular, then appear as special cases of squeezed light, i.e. lightfor which quantum fluctuations have been reduced on one component, or quadrature, at theexpense of enhanced fluctuations on the conjugate component (Walls and Milburn 1994). Wereview some basic notions of quantum optics in section 2, in order to define the concepts andnotation used in the remainder of this paper.

Assuming that the challenging problem of producing number-states of light has beensolved, what new kinds of experiments do they open? The motivation for the production ofsingle photons is examined in section 3. The detection of weak absorption signals is in generalhampered by photon noise, i.e. fluctuations of the number of photons. The low noise levelof a single-photon source would be very valuable for such measurements. Another appealing


Figure 1. Although a classical or Schrodinger wave is equally divided by a beam splitter, eachsingle photon has to follow either the transmitted or the reflected path. The quantum ‘which-path’decision is forced by measurement, here with photon-counting detectors.

application is quantum cryptography, more precisely quantum key distribution (QKD). Inthis case, the well-defined character of the single-photon state is exploited to detect and foileavesdropping on the transmission of a secret key between two parties, which they can use laterfor encryption. An eavesdropper can escape detection while attempting to extract informationfrom a statistical mixture of number-states, but not from a well-prepared number-state. Single-photon sources are crucial components in recent proposals for quantum computation by meansof linear optics (Knill et al 2001), the only nonlinear step being the detection.

The first successful generation of single photons (Clauser 1974) was based on a cascadetransition of calcium atoms. Each excited atom delivers a couple of two photons with differentcolours. A photon at one of the wavelengths is detected after spectral filtering, and is usedfor the conditional detection of its companion at the other wavelength. Each single photonis thus a ‘herald’ for the presence of its companion. One-photon states present a peculiaranticorrelation effect, which does not exist for a classical wave. If we send a one-photonstate on a beam splitter and place photon-counting detectors on the reflected and transmittedbeams, we never observe any coincidence between counts measured by the two detectors, asthis would violate energy conservation (see figure 1). As a consequence of the principles ofquantum mechanics, the wavefunction of the photon has to collapse onto either one or the otherof the two detectors. This absence of coincidences of detection events on the two detectors hasbeen dubbed photon antibunching and is utterly irreconcilable with a classical description oflight. It was first observed in the fluorescence from an attenuated sodium atomic beam, whereat most one atom was present in the excitation focus at any time (Kimble et al 1977, Walls1979, Cresser et al 1982). The cascade source was instrumental for tests of Bell’s inequalities(Clauser and Shimony 1978, Aspect et al 1981) and for the demonstration that classical effectssuch as interference fringes could be observed with individual photons (Grangier et al 1986).Although the cascading calcium atoms and the faint sodium beam were the first sources ofsingle photons, their brightness was very low, and a further drawback was that the operation ofthe source was limited by the density and transit times of the atoms and could not be controlled.From the mid-1980s, single ions in traps provided long observation times with one and thesame ion (Diedrich and Walther 1987). These long series of antibunched photons came closerto a ‘photon gun’, i.e. an ideal device delivering photons one by one. At about the same time,pairs of correlated photons (twin photons) were obtained at high rates by parametric down-conversion (Burnham and Weinberg 1970, Hong and Mandel 1986). When a short laser pulseis sent into a nonlinear crystal, it generates pairs of signal and idler photons, which are highlycorrelated in space and time. Provided the probability of generating two pairs at the same timeremains negligible, such correlated pairs can be used as sources of heralded single photons. Tothis day, the parametric sources are the workhorses of quantum-optics experiments. These andthe early atomic sources will be briefly described and discussed in section 5. A microscopicemitter of light, say an atom, can only realize its full potential as a single-photon source if it


is coupled to a resonant cavity, which can fulfil several functions. A cavity can enhance thespontaneous emission rate and thereby the rate of photon production, it can channel the emittedphotons into a well-defined spatial mode to improve collection efficiency and to match furtherutilization, or it can restrict the spectral range of the emission. The principles of coupling to acavity are reviewed in section 4.

In the early 1990s, sensitive detection of fluorescence led to the detection of singleorganic molecules (Orrit and Bernard 1990) and soon thereafter of single semiconductorheterostructures (Birotheau et al 1992, Brunner et al 1992). Some of the quantum-opticsexperiments conducted earlier on atoms and ions in the gas phase could then be reproduced incondensed matter, where single objects are much easier to manipulate. For example, photonantibunching was detected in the fluorescence of single molecules (Basche et al 1992) andin the emission of other individual nano-objects in condensed matter. Nano-objects weresoon proposed as possible sources of single photons (De Martini et al 1996, Kitson et al1998). Antibunching is an essential ingredient for the production of single photons, but aswe discuss in section 6, the photophysical mechanisms by which it arises are different for thedifferent kinds of emitters. Yet, antibunching alone is by no means sufficient for obtainingsingle photons on demand. Perfect antibunching means that the probability of emitting two ormore photons at the same time is nil, but we also require that the source emit one photon withcertainty. For this, two more conditions must be fulfilled:

(i) the emission quantum yield must be unity, or as close to it as possible, which is the casefor most of the emitters we discuss here.

(ii) The excited state of the emitter must be prepared with certainty. Various schemes forpreparing the emitting state have been proposed, for example, sweeping the sharp, low-temperature resonance of a single molecule through resonance with a laser by an externalelectric field (Brunel et al 1999) or exciting a single molecule with a short pump pulse(Lounis and Moerner 2000), or injecting single charge carriers with certainty in a quantumwell at sub-Kelvin temperatures (Kim et al 1999). Those early demonstrations havenow been extended to many different systems which could become useful emitters ofsingle photons. They include semiconductor nanocrystals (Lounis et al 2000, Michleret al 2000a, Messin et al 2001), colour centres in diamond (Brouri et al 2000, Kurtsieferet al 2000), single quantum dots in various geometries (Michler et al 2000b, Santori et al2001) and several more. Section 6 is devoted to a review of the photophysical propertiesof these various systems and of the spectroscopic characteristics of the photons they emit.As a summary, a table provides a comparison of their properties.

Finally, section 7 concludes the review, and tries to identify promising directions andbottlenecks for the future development of single-photon sources.

2. Photon optics

In this section, we briefly review basic notions of quantum optics relevant to the properties anduses of single-photon sources. Complete presentations can be found in textbooks on quantumoptics (Loudon 1983, Walls and Milburn 1994, Mandel and Wolf 1995).

2.1. Quantized radiation field

A mode of the electromagnetic field (wave vector �k, angular frequency ωk , polarization �εperpendicular to �k) in a cavity with volumeL3 can be directly quantized as a harmonic oscillator.Operators a+

�k,�ε and a�k,�ε, respectively, create and annihilate one quantum—or photon—in this


mode. The electric field at position �r now becomes an operator, written as the sum of twoHermitian-conjugate terms:

�E(�r) = �E(−)(�r) + �E(+)(�r). (1)

With inclusion of the time-dependence in the Heisenberg picture, the field operator takesthe form

�E(−)(�r, t) = i∑�k,�ε

√hωk

2ε0L3�εa+

�k,�εei(�k·�r−ωkt). (2)

The statistical properties of light in a given state can be characterized in various ways.Neglecting spatial coherence aspects, we only consider the three following functions of time,ordered by increasing complexity and interest:

(i) the instantaneous intensity,

I (t) = 2ε0c〈E(−)(t)E(+)(t)〉, (3)

(ii) the first-order correlation function (or correlation function of the field), where the averageis supposed to be taken over a stationary state of light,

g(1)(τ ) = 〈E(−)(t + τ)E(+)(t)〉〈E(−)(t)E(+)(t)〉 (4)

(iii) the second-order correlation function, classically written as

g(2)(τ ) = 〈I (t + τ)I (t)〉〈I (t)〉2

(5)

and whose proper quantum-mechanical expression is (Loudon 1983)

g(2)(τ ) = 〈E(−)(t)E(−)(t + τ)E(+)(t + τ)E(+)(t)〉〈E(−)(t)E(+)(t)〉2

. (6)

2.2. Different flavours of light

In order to better grasp the original features of a single-photon source with respect to standardlight sources, it is interesting to compare first the statistical properties of light emitted by ablack body or a lamp (thermal light) and by a stable laser (coherent light).

(a) Thermal light. In conventional macroscopic sources such as lamps, many independentemitters contribute to the signal. Therefore, the field is a superposition of many incoherentwaves. We assume all of them to have the same formal time-dependence (e.g. a dampedexponential in the case of a high-pressure sodium lamp), but with a random phase and delay:

�E =∑

i

eiϕi �E0(t − ti).

The first-order correlation function of such a field is identical to that of a single emitter. TheFourier-transform of the field correlation function is the spectrum of the macroscopic source,i.e. that of an individual emitter. For such a superposition of many independent sources, onemay average over phases in the numerator of equation (6), which leaves only two series ofterms. One then finds the following relation between g(2)(τ ) and g(1)(τ ):

g(2)(τ ) = 1 + [g(1)(τ )]2. (7)

The value g(2)(0) = 2 (following from g(1)(0) = 1) shows that thermal light has large intensityfluctuations. Indeed, the total field strongly fluctuates around zero, its most likely value.


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m

Single-Photon Source

Coherent Source

p(m

)

Figure 2. Probability distribution of the number of photons for three sources with an averagephoton number 〈n〉 = 1. The thermal source presents large number fluctuations due to the Bose–Einstein statistics of black-body radiation. The coherent light source presents a Poisson distribution,narrower than that of thermal light, but with still strong number fluctuations, called photon noise.An ideal squeezed-light source delivers a number-state with m = 1. A single-photon source canmatch this distribution by delivering single photons at regular time intervals.

The fluctuations may be seen as arising from the boson character of photons. The characteristictime of the fluctuations is related to the spectrum of the thermal source. The number of photonsin a given mode follows the Bose–Einstein distribution of black-body radiation (see figure 2):

pth(m) = 〈n〉m(1 + 〈n〉)m+1

, (8)

where 〈n〉 is the average number of photons in the mode. Note that, for thermal light, the statewith zero photons (m = 0) always has the largest probability of occupation. This distributionis therefore very far from that of a desired source of single photons, which should present asharp maximum for m = 1.

(b) Coherent light. The classical picture of laser light is a wave with constant amplitude andphase. The first- and second-order correlation functions of such a wave are obviously bothequal to unity. In other words, there are no fluctuations in this classical description.

The quantum representation of such a wave is a so-called coherent state, |α〉, which canbe written in the basis of number-states (or Fock states) with n photons according to

|α〉 = e−|α|2/2∑

n

αn

√n!

|n〉. (9)


Note that |α〉 is an eigenstate of the annihilation operator a (and therefore of operator E(+)) witheigenvalue α. This property allows us to replace quantum-mechanical averages by classicalfunctions. Coherent states are thus sometimes called quasi-classical states. The number ofphotons in a coherent state is a variable that fluctuates according to a Poisson distribution. Theprobability of finding m photons in the mode is given by (see figure 2)

p(m) = 〈n〉mm!

e−〈n〉 (10)

with the average number of photons

〈n〉 = |α|2

and the variance

�n2 = 〈n2〉 − 〈n〉2 = 〈n〉. (11)

These statistics are very different from those of thermal light. The maximum probability is tofind 〈n〉 photons in the mode. In coherent states, the number of photons obeys the central limittheorem for large numbers. The associated physical picture is that of independent particles,corresponding to a classical limit. The resulting Gaussian noise is called shot noise (or photonnoise) and is an absolute minimum for the noise of a macroscopic laser. Again, the probabilitydistribution is very different from the desired one.

(c) Squeezed light. Being a harmonic oscillator, an electromagnetic mode can be describedby two non-commuting operators analogous to position and momentum for a one-dimensionalparticle. These complementary or conjugated operators can be chosen as the electric fieldand vector potential or, in a classical wave representation, as the amplitude and phase.Complementarity in quantum mechanics requires that fluctuations of conjugate variablesfollow Heisenberg relations. Therefore, quantum mechanics authorizes so-called squeezedstates of light, where fluctuations in one of these variables can be reduced (as comparedwith those of a coherent state), provided fluctuations of the conjugate variable are increased(Walls and Milburn 1994, Bachor 1998). Squeezed light with reduced amplitude fluctuations(or equivalently photon number fluctuations) must thus present enhanced phase noise.

An ideal amplitude-squeezed source would be one delivering a regular stream of photonsat regular time intervals, and a single-photon source is a good example of it. The fluctuationsof the number of photons emitted by a squeezed source are weaker than those of a coherentstate. They are sub-Poissonian, and their deviation from Poisson statistics can be characterizedby the time-dependent Mandel parameter:

Q(T ) = 〈n2〉T − 〈n〉2T

〈n〉T − 1, (12)

where the notation 〈· · ·〉T stands for ‘averaged over a time-interval T ’. The Mandel parameterof a Poisson source is nil at all times and that of an ideal single-photon source deliveringphotons at regular time intervals is −1. The Mandel parameter is related to the second-ordercorrelation function by (Short and Mandel 1983)

Q(T ) = 2〈I (t)〉T

∫ T

0dτ

∫ τ

0dτ ′(g(2)(τ ′) − 1), (13)

where 〈I (t)〉 is the average emission rate (in units of photons per second).


2.3. Antibunching and second-order correlation function

Sub-Poissonian statistics were first demonstrated in the fluorescence of a very faint atomicbeam excited by a continuous laser (Kimble et al 1977). Because the atomic beam is faint, atmost one atom is in the laser beam at a given time, and most of the time the laser focus is empty.Fluorescence therefore arises from single atoms. In contrast to the Poisson distribution in anattenuated laser beam, where two photons can be separated by any delay, very short delays arenot allowed for photons emitted by a single atom. The photons have a tendency to be emittedone by one, i.e. they are ‘antibunched’1.

Antibunching may arise from different mechanisms, as will be discussed in section 6. Fortwo-level systems such as atoms, the observation of one photon projects the system into theground state, whence no second photon can be immediately emitted. To emit a second photon,a new excitation–emission cycle is necessary, which requires an average delay. Molecules,quantum dots and other complex objects in condensed matter always possess higher excitedlevels, which can be reached resonantly via absorption of a second photon by the first excitedstate. In most cases, however, relaxation back to the first excited state is so fast that noemission from these higher excited states can be detected. As a consequence, since onlyemission between the lowest two levels can be detected, the complex system behaves as atwo-level emitter.

With ideal detectors, antibunching would be easy to demonstrate, by simply recording thestream of fluorescence photons with a time-resolution better than the fluorescence lifetime.Unfortunately, real photon-counting detectors have their limitations. The dead-time is thatperiod of blindness of the detector just after detection of one photon, which arises from there-loading time of the photomultiplier dynodes or semiconductor junction. Dead-times canbe as long as tens of nanoseconds. On even longer timescales, stray charges released uponthe detecting avalanche can lead to spurious counts called after-pulses. The practical way ofeliminating these factitious events and the blind periods is to split the detection beam and usetwo detectors, one on the transmitted beam, the other on the reflected beam (see figure 1).

Coincidences between the two detectors are then insensitive to dead-times and after-pulses. Such setups were first proposed by Hanbury Brown and Twiss (1956) and Twiss et al(1957) to measure light coherence (see figure 3). A histogram of coincidence events givesthe distribution of pairs of consecutive photons, whereas the second-order correlation functiongives the distribution of all pairs of photons. These two distributions, related by a simpleequation, are nearly equal for short delays (Reynaud 1983, Lounis et al 2000, Verberk andOrrit 2003). For perfectly antibunched light, the correlation function therefore cancels forzero time:

g(2)(τ = 0) = 0.

Note that this relation is totally incompatible with the classical equation (5), for any classicalstationary random function of time must satisfy the Schwartz inequality 〈I (t ′)I (t)〉 � 〈I (t)2〉.The important feature of antibunching for us is that the single emitter ‘picks’ a photon out ofthe coherent beam and redirects it towards the optical mode used for detection and it does this‘one photon at a time’. The single atom thus works as a source of single photons. In otherwords, it acts as a nonlinear filter, eliminating the shot noise of the laser, albeit for short timeintervals, and only for those photons redirected into the detection mode.

1 The sub-Poissonian character of this emission only applies during the time an isolated atom crosses the excitationbeam. On longer timescales, as the atoms themselves enter and leave the excitation beam according to Poisson statistics,coincidences of two photons emitted by two different atoms may be observed. Therefore, the antibunching effect onlymanifests itself by an increase in the correlation function with delay, starting from unity for short delays. A similareffect is observed in fluorescence correlation spectroscopy, for diffusing fluorescent dye molecules in solution.


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Figure 3. (a) Schematic diagram of a Hanbury Brown and Twiss experiment. Coincidencesbetween counts on the two detectors give access to the (second-order) correlation function. For along coherence lifetime and high resonant exciting intensity (b), Rabi oscillations appear aroundthe antibunching dip. For non-resonant excitation, or a very short coherence lifetime (particularlyat room temperature), only an exponential dip appears (c). A pulsed excitation gives rise to periodicemission of single photons (d). In that case, the zero-delay peak is suppressed by antibunching,i.e. by the which-detector decision each single photon has to make.

Figure 3 shows two examples of a second-order correlation function of the antibunchedlight from a single emitter excited continuously. If the emitter presents a long coherence time,e.g. for emitters at cryogenic conditions, damped Rabi oscillations appear at sufficiently highexcitation intensity (figure 3(b)). For most room-temperature emitters, however, the coherencetime is extremely short (often shorter than picoseconds). The correlation function presents adip for zero delay and tends exponentially to unity, its long-time value (figure 3(c)).

An ideal single-photon emitter can also be excited by a pulsed source. Each pump pulsewill give rise to the emission of a single photon. The correlation function of the pulsed laser, asmeasured in a Hanbury Brown and Twiss setup, is a Dirac comb of short peaks (each peak beingthe convolution of the pulse envelope with itself), appearing at all integer times of the period T

of the pulses. The single-photon emitter delivers an individual photon after each pulse, with adecaying exponential probability. The correlation function is again a Dirac comb of peaks, butwith the zero-time peak missing (see figure 3(d)). Since only one photon is produced, therecan be no coincidence between the two detectors for zero delay. The shape of the correlationpeaks is again the convolution of the decaying exponential, with itself (exp(−|t − nT |/τf),where τf is the decay time of the emitter and n �= 0 is the order of the correlation peak).

To conclude this discussion, let us remark on two important features of the spontaneousemission process (neglecting dephasing, which will be considered in the next section).

(i) Spontaneous emission by a system (say, an atom) prepared at t = 0 in its excited statee leads to creation of one single photon in a wavepacket (Cohen-Tannoudji et al 1992).


The final state is a coherent superposition of states with the photon in a continuum ofpossible modes k (with nearly the same energy), described by the wavefunction:

|ψ(t)〉 = ce(t)|e, 0〉 +∑

k

ck(t)|g, 1k〉

with ce(t) = exp(−t/2τf − iωet) and

ck(t) = Vk/h

ωk − ωe + i/2τf

(1 − exp

[− t

2τf+ i(ωk − ω0)t

]).

hωe and hωk are the energies of the initial and final states, Vk the dipole coupling matrixelement and τf is the spontaneous emission lifetime.

(ii) The decaying exponentials show that the emission process is continuous in the aboverepresentation. Only a subsequent measurement of the photon by a detector will projectstate |ψ(t)〉 into one of the modes k. This process appears as a quantum jump.Alternatively, it is possible to consider the emission process itself as a quantum jump.In so-called quantum Monte Carlo simulations, the evolution of the system is obtained byintroducing sudden random jumps, and by renormalizing the wavevector after each jump(Dalibard et al 1992, Gardiner et al 1992, Carmichael 1993, Hegerfeldt 1993).

2.4. Indistinguishable photons

A number of quantum-optics experiments, for example in quantum computing, rely oninterference between two single photons. A single-photon source producing photons at pre-determined times can be used to generate pairs of photons if a suitable pumping sequenceis applied and a propagation delay is introduced between successively emitted single photons(Santori et al 2002). Pure spontaneous emission by an ideal two-level system leads to perfectlyindistinguishable photons. However, dephasing and spectral diffusion, which arise from fastand slow fluctuations of the transition frequency, complicate this picture. Hereafter, we brieflydiscuss some possible experiments with two-photon states, and the specific properties of thesingle-photon source they require.

Fearn and Loudon (1987) and Hong et al (1987) pointed out that two photons incident atthe same time on the two input ports of a 50%/50% beam splitter interfere in such a way that theyboth exit from one of the output ports, as illustrated in figure 4. This effect is a consequence ofthe Bose–Einstein statistics followed by photons. The two photons can be seen as ‘coalescing’,i.e. they both reach together either one of the two detectors in the output arms. When the delaybetween photons is varied, the rate of coincidences on the two output detectors drops forzero delay. In order to give rise to a fully destructive interference, the two photons mustbe completely indistinguishable, i.e. they must be in exactly the same mode. If the photonsare described by space–time wavepackets, the two wavepackets must be identical for the twophotons, and they must arrive simultaneously on the beam splitter. Let us consider a sourceemitting all photons in the same spatial mode and discuss only the frequency wavepacket. Ifthe spectrum of a single-photon source is Fourier-transform-limited, i.e. if each photon canbe described by the same coherent wavepacket, two photons will be indistinguishable. Inmany cases, however, the spectrum of a source is broader than the Fourier-transform of thetime-profile of the emitted pulse. This broadening, arising from fluctuations of the opticalresonance frequency, can be described as dephasing or spectral diffusion, depending on therate and amplitude of the fluctuations (Anderson 1954, Kubo 1954, Skinner and Reilly 1994).

(i) Fast and weak fluctuations (dephasing): The loss of coherence of the emitter follows frommany ‘collision’ events with a bath, leading to a gradual loss of phase with the dephasing


ReflectedPhoton

TransmittedPhoton

BeamSplitter

+ =0

(a)

(c)

(b)

Figure 4. Photon coalescence. Two single-mode, but otherwise independent, photonssimultaneously enter the two ports of a 50–50 beam splitter. They give rise to transmitted orreflected amplitudes at the ouput ports. The two paths in (c) interfere destructively. Only paths (a)and (b) contribute to the observed amplitudes, so that if one photon is transmitted, the other onemust be reflected. Both photons emerge at the same output port, as if they had coalesced.

(or decoherence) time, T2, shorter than twice the fluorescence lifetime, T1:

1

T2= 1

2T1+

1

T ∗2

,

where T ∗2 characterizes pure dephasing processes arising from interactions with the bath

(e.g. via collisions in gas phase, or phonons in condensed matter). In that case, individualphotons can be seen as short incoherent wavepackets emitted at random times during thelong fluorescence lifetime. Since different wavepackets will be randomly shifted in time,the photon interference effect will be reduced. The depth of the two-photon interferenceeffect will no longer be 100% but only T2/2T1 (Bylander et al 2003).

(ii) Slow and strong fluctuations (spectral diffusion): In the other limit of slow and large-amplitude fluctuations, the optical frequency drifts or jumps comparatively slowly in thespectrum (hence the phrase ‘spectral diffusion’). Two emitted photons separated by a timeinterval longer than the spectral diffusion time will be distinguishable in principle, becausetheir frequencies will differ and they will not interfere (Santori et al 2002). However, ifthe delay between the emission times of the two photons is short enough, slow spectraldiffusion processes may be neglected.

(iii) An additional loss of indistinguishability arises from incoherent pumping. If the relaxationprocess between excitation and emission is not short compared with the coherence lifetime(Kiraz et al 2004), an additional delay is due to relaxation jitter. This effect is importantin the case of self-assembled quantum dots whose lifetimes are shortened by the Purcelleffect (see sections 4 and 6).

A different quantum-mechanical feature of photon pairs is the possibility of entanglement.The global quantum state of an entangled system, say of two particles, remains non-separable,even when the two particles are widely separated from each other and unable to interact or


communicate in any way. Experiments with entangled two-photon states will be discussed insection 3.

2.5. Summary: characteristics of the emitted photons

So far, we generally discussed the properties of light from the point of view of photons in agiven single mode. To be used in practice, photons have to leave the source and should ratherbe described by wavepackets spread over several modes. The wavepackets themselves arecharacterized by a number of parameters, which can be very different for different types ofsingle-photon sources and which we enumerate below.

(i) Spatial mode: To be used in interference, telecommunication or computationexperiments, photons have to be collected and collimated into a given spatial mode. Unlessthe emitter undergoes spatial motion or reorientations, a solid-state source based on amicroscopic emitter will in general emit all its photons in the same spatial mode. Mobileemitters such as atoms may deliver different spatial wavepackets at different times or maycouple with variable efficiency to the fixed spatial mode of a cavity. For an efficient utilization,the spatial mode of the photon at the output of the source must be matched to those of allsubsequent optical components, in particular that of the detector.

(ii) Spectrum: The desirable spectrum of a single-photon source depends on the applicationand on the further use of photons. For example, the propagation of photons in opticaltelecommunication systems would favour the three main transmission windows of silica fibres,around 840, 1320 and 1550 nm. To fully exploit the quantum properties of a single-photonsource, the detector must be a photon-counter with a high detection yield, which is not readilyavailable in the near-infrared domain. The cheapest and most convenient detectors today arebased on silicon, whose detection efficiency peaks around 750 nm and does not extend muchbeyond 1000 nm on the IR side. Photon-counting devices based on III–V semiconductors nowbecome available in the near-IR window (Hiskett et al 2000, Rarity et al 2000, Stucki et al2001).

(iii) Emission lifetime: Most of the single-photon sources we consider here are based onspontaneous emission. The excited state of an individual quantum system (an atom, a moleculeor a quantum dot) is created in a rapid process, and the subsequent emission follows on a longertimescale. Therefore, the spontaneous emission lifetime eventually limits the rate at which thesingle photons can be emitted and is an important feature of the source. The emission lifetimein vacuum τf(1) is a characteristic of the emitter. It is determined by the transition frequency,ωeg, and by the transition dipole moment, µeg, between ground and excited states:

τ−1f (1) = 4

3

µ2eg

4πε0h

(ωeg

c

)3. (14)

If the emitter is placed in a large bulk of dielectric with refractive index n, the emissionlifetime is significantly modified. A first correction factor arises from the different density ofphoton modes in the dielectric, as pointed out by Nienhuis and Alkemade (1976), yielding theoften-used relation

τf(n) = 1

nτf(1).

Moreover, the induced dipoles in the surrounding dielectric also participate in the couplingto radiation. In the approximation of the spherical Lorentz cavity, and if the polarizability ofthe object is not too different from that of the surrounding matrix (Schuurmans et al 2000,


Vallee et al 2005), the lifetime in a dielectric with index n becomes (Imbusch and Kopelman1981)

τf(n) = 1

n

(3

n2 + 2

)2

τf(1). (15)

At a higher level of approximation, the local-field correction depends both on the exact structureof the dielectric around the emitter and on the geometry. In complex geometries, e.g. when theemitter is placed close to an interface (Lukosz and Kunz 1977) or in a particle whose size iscomparable with the wavelength, the local field correction depends on details of the structureand must be calculated numerically.

In addition to these short-range effects, the lifetime can be considerably altered if theobject is placed in a high-finesse cavity, which may amplify or reduce vacuum fluctuations atthe emission frequency, thereby ‘forcing’ or ‘blocking’ spontaneous emission (see section 4).

(iv) Emission linewidth and indistinguishability: The frequency spectrum of a source isthe Fourier-transform of the correlation function of the electric dipole operator. Electronicoscillations cannot live longer than the emission lifetime, T1, determined by spontaneousemission and other population relaxation processes, but their coherence can be furthershortened by dephasing (or decoherence), arising from perturbations of the oscillationfrequency through interactions with a bath. In the absence of slow spectral diffusion, theresulting frequency linewidth (full-width at half-maximum),

�ν = 1

2πT1+

1

πT ∗2

, (16)

takes its minimum possible value only when dephasing is negligible, i.e. when T ∗2 = ∞. In

that case, one has a lifetime-limited linewidth. For most systems in condensed matter at roomtemperature, the dephasing time is by several orders of magnitude shorter than the excited statelifetime, i.e. the linewidth is very far from being lifetime-limited.

Many applications of single-photon sources require two-photon states. Two photonsconsecutively emitted by the same source, or two photons from independent emitters, candisplay number-interference or coalescence, provided they are sufficiently indistinguishable(see section 2.4). Indistinguishability requires lifetime-limited sources, i.e. emitters for whichdephasing and spectral diffusion are negligible.

(v) Polarization state: It is determined by the microscopic nature of the emitter and by theway it is coupled to the emission mode. For example, for a linear dipole emitter such as asingle molecule, the polarization pattern of the emitted photons depends on the orientation ofthe dipole moment (Fourkas 2001, Novotny et al 2001). If the dipole lies in the focal planeof the collection objective, the polarization is close (but not identical) to linear, parallel to thedipole, for all photons across the whole spatial mode. If the dipole is perpendicular to the focalplane, the polarization in the collected mode is radial (Fourkas 2001).

3. Possible uses of single photons

A reliable and bright source of single photons would open up a number of applications inspectroscopy and quantum optics. Hereafter, we review some proposed uses and recentexperiments using single photons. These demonstrations are a strong motivation to developcompact and efficient sources. We have little doubt, however, that the availability of suchsources will lead to completely new, and today unsuspected, applications.


3.1. Measurement of weak absorptions

With squeezed light, fluctuations of a given quadrature are reduced with respect to those ofcoherent light (Kimble and Walls 1987, Drummond 1999). A single-photon source deliversamplitude-squeezed light, which implies that its noise is reduced for intensity measurements(Xiao et al 1987, Polzik et al 1992). As an example, we consider the measurement of a weakabsorption signal with an ideal single-photon source, as compared with a coherent light source(a laser).

We first consider the coherent source, delivering a Poisson distribution (see section 2.2(b),equation (10)) with a large average number M of photons. These photons are transmitted by anabsorbing medium (intensity transmission coefficient T ) and measured by a photon-countingdetector (quantum yield x). The distribution of the detected counts is also Poissonian, withthe average number of detected counts (supposed to be large)

SC = xMT

and with the shot noise (standard deviation)

NC =√

xMT .

To measure the transmission coefficient, we repeat this measurement without the absorberand take the ratio of detected counts. Adding the variances (or squared standard deviations)of the two signals, we obtain the following variance for the transmission coefficient measuredwith the coherent source:

�T 2C = 1

xMT (T + 1).

If we now repeat this experiment with an ideal single-photon source, the average number ofdetected counts is the same, but the distribution of photon counts is now binomial. Each photonemitted by the source is either detected with probability xT or goes undetected with probability(1 − xT ). The probability of observing n counts for M emitted photons is therefore

px(n) =[

n

M

](xT )n(1 − xT )M−n. (17)

The difference with the coherent case is that we are now certain of the number of photonssent into the absorber and that we get additional information from the undetected counts. Thestandard deviation of the distribution (17) is less than that of a Poisson distribution and leadsto the following noise:

NSP =√

xMT (1 − xT ).

For the transmission measurement, the variance becomes

�T 2SP = 1

xMT (T + 1 − 2xT ).

With a similar result for the reference measurement, the ratio of the variances for the squeezedto coherent cases is

�T 2SP

�T 2C

= 1 − 2xT

1 + T, (18)

which shows that, as expected, the noise with the single-photon source is always less than withthe laser. The noise reduction is nearly total for high detection quantum yields (x ≈ 1) and forweakly absorbing samples (T ≈ 1). A perfect single-photon source associated with a perfectdetection thus would give access to arbitrary small absorptions, impossible to measure with alaser source because of photon noise.


3.2. Random number generation

Random numbers are key ingredients at many levels of information processing, fromcomputational methods (Monte Carlo simulations (Alspector et al 1989) for example) tothe broader field of cryptography (for the generation of encrypting keys, see section 3.3.5).There are generally two approaches to random number generation. Pseudorandom generatorsrely on numerical algorithms implemented on a computer. Physical random generators usethe randomness or noise of a physical observable (e.g. that of an electronic device). Inboth approaches, systematic errors and perturbations lead to deviations from a truly randomdistribution of numbers.

A clear physical source of randomness is the elementary collapse or decision process inquantum mechanics. Rarity and collaborators (Rarity et al 1994) proposed a physical quantum-mechanical random generator based on the 50 : 50 splitting of a beam of single photons. Eachindividual photon coming from the light source and going through the 50 : 50 beam splitterhas an equal probability of being reflected or transmitted. Quantum theory predicts that, foreach photon, the ‘which-way’ decision is truly random and independent of history or otherexperimental parameters. A continuous stream of random numbers at a rate of 1 Mbit s−1

could be produced with no correlations between the generated numbers (Jennewein et al 2000,Stefanov et al 2000) using attenuated LED light sources. Efficient single-photon sources willin principle improve the generation rate. However, care has to be taken of experimental details.The detector’s after-pulses and dead-times may introduce strong correlations or anticorrelationsbetween successive bits.

3.3. Quantum information processing

Quantum mechanics offers original and powerful schemes for processing information, whichcould never be implemented in classical devices. In the broad field of quantum computing,teleportation and networking, the storage of quantum information in the states of individualphotons would present specific advantages (Bouwmeester et al 2000, Di Vincenzo 2000):

(i) Photons are in principle identical and indistinguishable, and they are weakly coupled tothe environment.

(ii) A big problem in quantum computing is the transportation of quantum informationfrom processor to processor. Being propagating particles, photons can transport theinformation.

(iii) If, moreover, quantum computing can be performed on photons, they can serve as ‘flyingqubits’ (Chuang and Yamamoto 1995, Turchette et al 1995, Knill et al 2001), which couldconsiderably simplify the processing architecture.

The experimental development of large quantum information processors remains a dream.Bright and reliable sources of single photons would be of great help in identifying technicalbottlenecks and developing quantum components with simple hardware. Below, we reviewrecent theoretical and experimental advances in quantum information processing, in the hopethat individual photons will lead to improved schemes and developments.

3.3.1. Quantum bit. The fundamental unit of quantum information is the quantum bitor ‘qubit’. Qubits are quantum two-level systems which can be prepared in a coherentsuperposition of their eigenstates |0〉 and |1〉:

|ψ〉 = α|0〉 + β|1〉, (19)


whereα andβ are complex amplitudes. A measurement projects that state into either eigenstate,|0〉 or |1〉, with respective probabilities |α|2 and |β|2 = 1 − |α|2. Although a qubit may at firstsight seem a rather trivial generalization of a classical bit, the continuous complex amplitudesα and β also convey potential information and introduce qualitatively new possibilities. Thepower of quantum information rests on quantum-mechanical superpositions, which open aninfinity of qubit-states that do not correspond to the classical values 0 or 1.

Among the many proposed physical means of attaining quantum bits (Monroe 2002), aninteresting one is single photons. For example, a qubit can be a photon with two eigenstatesrepresented by horizontal |H 〉 or vertical |V 〉 polarizations in a given basis. Logical statescan also be coded as physical qubits using one photon in one of two output modes of a beamsplitter: |0〉L ≡ |1〉1|0〉2 and |1〉L ≡ |0〉1|1〉2.

3.3.2. Quantum entanglement, EPR paradox and Bell’s inequality. What happens if wetry to encode two bits of information onto two quantum particles, for example two photonslabelled 1 and 2? The classical approach would be to code one bit of information onto eachphoton separately (one bit per photon), leading to four possibilities ‘00’, ‘01’, ‘10’ and ‘11’to which are associated the four states |H1H2〉, |H1V2〉, |V1H2〉 and |V1V2〉. These are purestates, or product states, where, for example in the |H1V2〉 state, the first photon is horizontallypolarized and the second one is vertically polarized.

However, quantum mechanics offers different ways of encoding information into twoqubits. The superposition principle allows the construction of the following pure state:

|ψ〉 = α|H1H2〉 + β|H1V2〉 + γ |V1H2〉 + δ|V1V2〉. (20)

The important new feature of this superposition is ‘entanglement’, i.e. in this new quantumstate the two particles are no longer separable, even if their spatial wavepackets are physicallyseparated! A perturbation or a measurement at the position of one particle instantaneouslymodifies the whole wavefunction, as also at the remote position of the other particle. Aspointed out by Einstein et al (1935), the instantaneous character of the quantum measurementat first sight appears inconsistent with relativistic causality, seemingly involving some ‘spookyaction at-a-distance’. Indeed, the quantum correlations arising from entanglement are strongerthan classical correlations, for example, those which would arise from hidden variables (Bell1965, Clauser and Shimony 1978).

There are four possible states with maximum entanglement, called Bell states, for a pairof photons with two possible orthogonal polarizations:

|ψ+〉 = 1√2(|H1V2〉 + |V1H2〉),

|ψ−〉 = 1√2(|H1V2〉 − |V1H2〉),

|ϕ+〉 = 1√2(|H1H2〉 + |V1V2〉),

|ϕ−〉 = 1√2(|H1H2〉 − |V1V2〉).

(21)

(Note that the labels 1, 2 correspond to additional quantum numbers, e.g. the wavevector,which allow us to distinguish the photons. Therefore, both symmetric and antisymmetricsuperpositions are allowed.)

Each Bell state represents a coherent superposition of two possibilities. In the former pairof states, |ψ+〉 and |ψ−〉, the two qubits are different, and in the latter pair, |ϕ+〉 and |ϕ−〉, theyare the same. As the photons are ‘entangled’ in these superpositions, the two-photon Bell states


cannot be written as products of single-photon states. In state |ψ+〉, for example, photon 1can be in state |H 〉 and photon 2 in state |V 〉 or vice-versa, but the question of which photonhas which polarization is clearly meaningless. All of the information is therefore distributedamong two qubits, and none of the individual systems carries any information.

Consider a pair of entangled photons prepared in one of the Bell states, say |ψ+〉, flyingapart along two different directions 1 and 2 (modes 1 and 2). Their polarization is measuredby two widely separated observers, Alice and Bob, who receive photon 1 and 2, respectively.Whatever polarization basis Alice chooses to measure the polarization of her photon, she willobtain ‘0’ or ‘1’ with equal probability, the actual result being completely random. If Bobchooses the same basis, he will always obtain the opposite result. If Alice knows that Bob usedthe same basis, she can deduce his result from her own with certainty, even though she hasno means to communicate with him. However, a similar outcome would arise if the quantumstate had been pre-determined by a hidden variable. In Einstein’s view, such hidden variableswere required to explain the EPR paradoxon, which proved the incompleteness of quantummechanics. In 1965, Bell investigated possible correlations for a thought experiment in whichAlice and Bob choose bases that are at oblique angles. Bell proved that hidden-variable theoriespredict that the probabilities of the different measurements satisfy a certain inequality (Bell1965). Bell’s inequality was later proved experimentally to be violated, in agreement withstandard quantum-mechanical predictions.

A first series of experimental tests of Bell inequalities used photon pairs produced in atomicradiative cascades (Clauser and Shimony 1978, Aspect et al 1981) (section 5.2.1). Althoughthese early setups were far from ideal (they used single-channel polarizers), their results agreedwith quantum mechanics. In the early 1980s, a new experiment performed by Aspect et al(1982a, 1982b) used two-channel polarizers, as in the ideal EPR thought experiment. It gavean unambiguous violation of Bell’s inequalities by tens of standard deviations, and a perfectagreement with quantum mechanics.

The second generation of tests began in the late 1980s (Ou et al 1988, Shih and Alley1988) and used photon pairs generated in nonlinear crystals by spontaneous parametric down-conversion (SPDC) (section 5.2.2). The remarkable feature of parametric sources is theproduction of two narrow beams of correlated photons that can be fed into optical fibres,making tests possible with great separations between the source and detectors (4 km (Tapsteret al 1994), and later tens of kilometres (Tittel et al 1998)). All these experiments haveconfirmed the predictions of quantum mechanics. Yet, from a strictly logical point of view,they still leave two subtle loopholes.

The first one is the possibility of information exchange from one polarizer to the other, byunspecified processes. Bell proposed to randomly rotate the polarizers during the flight of thephotons to make communication between the polarizers causally impossible. This loopholehas now been closed, first in an experiment by Aspect et al (1982a) where the directions ofpolarization analysis were switched after the photons left the source, and more recently byZeilinger’s group (Weihs et al 1998), who used sufficiently remote measurement stations,ultrafast and random analyser settings and two completely independent data acquisitionsystems.

The second loophole arises from the weakness of the detection yield in all experiments sofar. One always assumes that the recorded pairs are an unbiased sample of the ensemble. Toclose that loophole, an ultimate experiment would need a much higher collection and detectionefficiency of photon pairs, which could be achieved with an efficient source of single photons.A first step in that direction is the recent generation of polarization-entangled photon pairswith a quantum-dot source (section 6.6). This experiment relies on two crucial features ofthe single-photon source, namely its ability to suppress multiphoton pulses and its ability


1 EPR Source

2 3

BOB

Bell-StateMeasurement

ALICE

InputState

1 EPR Source

2 3

UnitaryTransformation

Quantum Channel

TeleportedState

Classical Channel

Figure 5. Polarization teleportation. Alice performs a Bell-state measurement between the qubitshe wants to teleport (photon 1) and one (photon 2) of the photons of an EPR-entangled pairdelivered by an auxiliary source (EPR-source). The other photon of the EPR-pair (3) conveysthe desired qubit to Bob, either directly or after Bob performs a unitary transformation, whoseparameters depend on Alice’s measurement and which Alice communicates to him via a classicalchannel (Bouwmeester et al 1997).

to generate consecutively two indistinguishable photons. The next step is to ‘collide’ thesephotons with suitable polarizations at two conjugated input ports of a non-polarizing beamsplitter. In such a way, the violation of Bell’s inequality was demonstrated with pairs ofindependent photons (Fattal et al 2004b).

To show that Bell’s inequality is violated in a two-particle experiment, it is necessaryto perform a statistical experiment: the violation cannot be demonstrated with a singlemeasurement. However, this does not apply to three-photon entangled states such as(|H1H2H3〉 + |V1V2V3〉)/

√2, the Greenberger, Horne and Zeilinger (GHZ) states (Greenberger

et al 1989). There, the contradiction between the Einstein–Podolsky–Rosen assumptions andquantum mechanics arises even for an individual event. A single measurement (of either threevertically polarized photons or of a horizontal polarization for two photons and a verticalpolarization for the third one) will discriminate with certainty between quantum mechanicsand a local hidden-variable theory. The Innsbruck group has realized this experiment usingsources of photon pairs from parametric down-conversion (Pan et al 2000). The preparationof GHZ states with a single-photon source would provide clearer violation and open series ofmore accurate tests.

3.3.3. Quantum teleportation. In 1993, Bennett and colleagues described a remarkableprotocol for transporting a quantum state from one location to another (Bennett et al 1993), aprotocol that succeeds even though the quantum state is and remains completely unknown, andeven undetermined, on both sides. Such quantum teleportation makes use of entanglement,the distinctive feature of quantum mechanics. In the proposed scheme, Alice wants toteleport an unknown quantum state, for example, the polarization state of a single photon|ψ〉1 = α|H1〉 + β|V1〉, to Bob (see figure 5). They both agree to share a maximally entangledpair of qubits, known as the ancillary pair, of photons 2 and 3.

The entangled ancillary pair can for example be prepared in the state |ϕ+〉23 =(1/

√2)(|H2H3〉 + |V2V3〉), one of the four possible Bell states of (21). The overall state

| 〉123 of the three photons is then

| 〉123 = (α|H1〉 + β|V1〉)(|H2H3〉 + |V2V3〉)/√

2

∝ |ϕ+〉12(α|H3〉 + β|V3〉) + |ϕ−〉12(α|H3〉 − β|V3〉)+|ψ+〉12(α|V3〉 + β|H3〉) + |ψ−〉12(α|V3〉 − β|H3〉). (22)


π

tψ

aψ

‘1’

‘3’

‘2’

‘4’

L0 L

0

L0

L1

L1

L1

tψC

D

BS1

BS2

Figure 6. Single-mode teleportation. Alice’s target qubit | t〉 is now defined by a superpositionbetween modes 1 and 2. Similarly, an auxiliary qubit | a〉 is encoded in modes 3 and 4. Alicemixes modes (2) and (3) in a beam splitter (BS2) and measures the outputs with detectors C andD. Whenever she records a single count in detector C, her qubit is now encoded in modes (1) and(4), ready for transmission (after Fattal et al (2004)).

Alice then performs a joint Bell-state measurement on the photon 1 she wants to teleport andone of the ancillary photons (photon 2): She mixes them in a 50/50 beam splitter and measuresthe output ports with polarization-sensitive detectors. She randomly obtains one of the fourpossible Bell results given in (21). This measurement projects the other ancillary photon intoa quantum state uniquely related to the original one according to equation (22). Bob’s lonequbit, being correlated with the state measured by Alice, now carries the quantum informationα and β. After Alice tells Bob by classical means which Bell state she measured, Bob performsa prescribed local qubit manipulation on his one qubit to replicate the initial state α|H 〉+β|V 〉.For example, if Alice reports a measurement of |ϕ+〉, Bob does nothing to his qubit. If Aliceinstead reports a measurement of |ψ−〉, Bob flips his qubit and adds a π -phase shift to one ofthe polarizations with a half-wave plate. Similar flips and phase-shifts can be performed forthe other two measurements of Alice. In this way, the qubit α|H 〉 + β|V 〉 is ‘teleported’ fromAlice to Bob without disturbing or measuring it.

The first experimental demonstrations of this quantum teleportation protocol used singlephotons and entangled photon pairs generated by parametric down-conversion (Bouwmeesteret al 1997).

Another teleportation protocol is the probabilistic single-mode teleportation (Lombardiet al 2002). In its simplest form, it involves two qubits, a target and an ancilla, each defined bya single photon occupying two optical modes (see section 3.3.1 and figure 6). The targetqubit can a priori be in an arbitrary state |ψt〉 = α|0〉L + β|1〉L where the logical |0〉Land |1〉L states correspond to the physical states of the output modes of a beam splitter:|1〉1|0〉2 and |0〉1|1〉2, respectively, in a dual mode representation. The ancilla qubit is preparedwith a beam splitter (BS1) in the coherent superposition |ψa〉 = (1/

√2)(|0〉L + |1〉L) =

(1/√

2)(|1〉3|0〉4 + |0〉3|1〉4).One mode of the target (mode 2) is mixed with one mode of the ancilla (mode 3) by means

of a beam splitter (BS2), for subsequent detection in photon counters C and D. For a givenrealization of the procedure, if only one photon is detected at detector C, and none at detectorD, then we can infer the resulting state for the output qubit composed of modes (1) and (4)α|0〉L + β|1〉L = α|1〉1|0〉4 + β|0〉1|1〉4, which is the initial target qubit state.

Similarly if D clicks and C does not, the output state is α|0〉L − β|1〉L = α|1〉1|0〉4 −β|0〉1|1〉4, which again is related to the target state by an additional phase shift of π , which canbe introduced to retrieve the initial state |ψt〉 (Giacomini et al 2002). Half of the time, eitherzero or two photons are present at counters C or D, and the teleportation procedure fails. Thesingle-mode protocol thus leads to teleportation in half of the attempts. Measurements and


( )( )VVHH +2

1

controlout

controlin

targetout

targetin

ψ ψ

ψ ψ

Figure 7. An all-optical CNOT gate combining two qubits. In the present case, the qubits arepolarization-encoded, in the HV basis, and an auxiliary |ϕ+〉 Bell state is required. The gate hasoperated successfully only when certain combinations of events are observed on the four detectors.The gate is therefore probabilistic (after Pittman et al (2001)).

postselection of the successful attempts lead to a faithful teleportation of the target qubit onmodes (1) and (4).

This scheme has been demonstrated experimentally very recently with a semiconductorsingle-photon source (Fattal et al 2004a). These authors obtained a 80% rate of success in thetransfer of the coherence to the transmitted modes. They attributed the 20% of failures to aresidual distinguishability of the single photons.

An improved version of this teleportation scheme using more ancilla photons is thecornerstone of a recent proposal for efficient linear-optics quantum computation (see nextsection).

3.3.4. Linear-optics quantum computing. We mentioned earlier the strong potentialadvantages of optics in the race towards quantum computing (Milburn 1988). Photons areeasily manipulated and, as the electro-magnetic environment at optical frequencies can beregarded as vacuum, their decoherence is negligible. Unfortunately, logical gates combiningtwo optical qubits require strong interactions between single photons, i.e. huge nonlinearcoefficients, well beyond current technology. An elegant solution to this problem was recentlyproposed in theoretical papers (Knill et al 2001, Ralph et al 2001). It is based on passive linearoptics, photodetectors and single-photon sources. The only nonlinearity is introduced by thephotodetection. Hereafter, we briefly describe its main ingredients.

An important example of a quantum logic gate is the so-called controlled-NOT (CNOT)gate, which has been shown to be a universal gate for quantum computers in the same way thatthe classical NAND gate is a universal gate for conventional computers (Sleator and Weinfurter1995, Nielsen and Chuang 2000). A CNOT gate has two inputs (a control qubit and a targetqubit) and operates in such a way that the NOT operation (bit flip) is applied to the targetqubit, provided the control qubit has a logical value of 1. Such a logic operation is inherentlynonlinear because the state of one quantum particle must be able to control the state of theother.

An example of proposed schemes which shows the basic idea of a linear optics quantumcomputing (LOQC)-type CNOT gate is illustrated in figure 7. In this scheme (Pittman et al2001), the quantum logic operation is performed using polarization-encoded qubits. In additionto the control and target photons carrying qubits, an ancillary pair of entangled photons isinjected into a device which contains only linear optical elements. This linear device givesthree possible types of detection outcomes, each of which is signalled by a unique combinationof detection events. For one set of events, we know that the control and target photons arein the desired logical output state. In a second set of events, the control and target are not inthe desired output state, but they can be brought into it by real-time corrections of the type


(a)

α U α

(b) α

U α

B

ZX{Φ

Probabilistic

(c)

α

U α

B

{ΦX’ Z’

A different entangled state

Deterministic

Figure 8. Quantum computing requires the application of a unitary transformation U on a qubit (a).In a quantum teleportation process (b) and (c), it is equivalent to apply U on the output bit (b) oron one of the entangled auxiliary qubits (c). This makes it possible to prepare the gate with theproper entangled pair and only then to teleport the computation qubit with the proper pair, in adeterministic process (c) (after Ralph et al (2001)).

discussed for quantum teleportation and known as feedforward control (Pittman et al 2002).For the third set of events, the control and target photons have been lost or are in a logicalstate that cannot be corrected. The first experimental demonstration of quantum CNOT gatefor different photons is based on this scheme (Gasparoni et al 2004).

These LOQC logic gates are probabilistic devices which fail with a relatively highprobability (e.g. 75% for the example in the figure). They work upon repeated attempts,after the measurement of two photons has indicated with certainty when the logic operationhas succeeded. Yet, the low rate of success of such nondeterministic gates makes themimpractical for large computations because the probability of success of a large computationdecays exponentially with the number of logic operations.

Gottesman and Chuang (1999) have proposed a gate operation based on teleportationwhich can solve this difficulty. They showed that performing the logic gate unitary operationU on one of the ancillary photons in a teleportation experiment is equivalent to applying Udirectly to the teleported qubit (see figure 8). The probabilistic step can thus be transferred tothe preparation of the proper ancillary state. Deterministic teleportation (section 3.3.3) is thenapplied to the precious target qubit once the proper ancillary pair has been obtained. The ‘right’ancillary state is thus prepared ‘off-line’ and stored as a resource for the next computation step.

The last remaining issue in a linear optical scheme is the probability of success of theBell measurements required in the teleportation protocol, which can never reach unity withlinear optics. Knill et al (2001) have showed that, with the appropriate entangled ancillary n

photons state, the teleportation step succeeds with near certainty. Their nearly deterministicteleportation protocol requires only linear optics, photon counting and fast feedforward.

3.3.5. Quantum cryptography. Quantum cryptography has been one of the earliest examplesof quantum processing, and is the closest to practical realizations. As it is one of the maincurrent motivations for building bright and reliable single-photon sources, we examine it insome detail here.


Table 1. Example of Alice’s emissions and Bob’s measurements in the two different polarizationbases. Only correlated bits, obtained when emission and measurement were done in the same basis,are kept to distil the sift-key (bottom line).

Alice’s basis ⊕ ⊗ ⊗ ⊕ ⊗ ⊗ ⊕ ⊗ ⊕ ⊕ ⊗ ⊕A-bit value 0 0 0 0 1 0 1 1 0 1 0 1Polarization V F F V S F H S V H F H

Bob’s basis ⊕ ⊕ ⊕ ⊗ ⊗ ⊗ ⊕ ⊕ ⊗ ⊕ ⊗ ⊗B-bit value 0 1 0 1 1 0 1 0 0 1 0 1Same basis Y N N N Y Y Y N N Y Y NSift-key 0 1 0 1 1 0

Quantum cryptography, or more precisely quantum key distribution (QKD), is a securemethod for the distribution of a secret key between two distant partners, Alice and Bob (Bennettand Brassard 1984, Gisin et al 2002). In its ideal version, where experimental limitations aredisregarded, its security is absolute. This safety arises from the impossibility of measuring anunknown quantum state without modifying it. Thus, eavesdropping by spy Eve necessarilyintroduces detectable errors in the communication between Alice and Bob.

(a) The BB84 protocol. The first protocol for QKD, proposed by Bennett and Brassard (1984,1985), is a four-quantum-state scheme. It uses two conjugate bases for the polarization ofsingle photons, a straight basis ⊕, with horizontal |H 〉 and vertical |V 〉 basis states, and adiagonal basis ⊗, with |F 〉 and |S〉 as basis states. Each state can encode a qubit in the properbasis, and one conventionally attributes the binary value 0 to states |V 〉 and |F 〉 and the value 1to states |H 〉 and |S〉. The two bases are chosen so that any two state vectors from differentbases have the same overlap, e.g. |〈H | S〉|2 = 1

2 .Alice sends individual photons with random polarization states to Bob. Bob measures

the incoming polarization states using one of the two bases, also at random. Whenever they(by chance) use the same basis, they get perfectly correlated results. When the bases chosenare different, the results are uncorrelated. By exchanging the information of which bases theyhave used, without revealing their results, Bob and Alice know exactly which bits are perfectlycorrelated and which ones are not. This information can be exchanged over a public channelsince the outcome of Alice’s or Bob’s measurements is unknown to Eve. Uncorrelated bits(approximately 50% of the total) are eliminated. The obtained sift-key is random, i.e. neitherAlice nor Bob can influence the result, but both of them know it with certainty. A schematicview of the principle can be seen in table 1.

The sift-key shared by Alice and Bob may still contain errors due to technicalimperfections, as well as possible eavesdropping. The last step of the protocol is to useclassical algorithms first to correct the errors and then to reduce Eve’s information on the finalkey. This process is called privacy amplification (Bennett et al 1988, 1995, Maurer and Wolf1999).

(b) Security of quantum cryptography. A practical and secure QKD protocol must beguaranteed against ‘smart’ eavesdropper Eve, whose technology is limited only by the laws ofphysics. In particular, Eve is supposed to possess perfect detectors and to be able to forwardsignals to Bob without any loss. Alice and Bob, on the other hand, may only use currenttechnology.

If Eve would just intercept a qubit sent by Alice to Bob, Bob would easily see that hemisses it, and he could simply tell Alice to disregard hers. For each intercepted qubit, Eve must


therefore send an ersatz-qubit to Bob. Ideally, she would like to send this qubit in its originalstate, keeping a copy for herself. However, according to the no-cloning theorem, even ‘evil’Eve cannot produce a perfect copy of an unknown quantum system (Dieks 1982, Wootters et al1982). She thus cannot at the same time keep a copy of the qubit and forward it intact to Bob.

The natural eavesdropping method for Eve is the intercept-resend strategy (Bennett et al1992). She measures each qubit in one of the two bases (choosing randomly ⊕ or ⊗). Then,she sends as an ersatz-qubit the eigenstate corresponding to her result. In about half of thecases, Eve will have been lucky and chosen the basis compatible with the state prepared byAlice. Then, neither Alice nor Bob can notice her action. In the other cases, Eve has useda different basis from Alice’s. Alice and Bob discover discrepancies in their bit sequences,indicating perturbation of the transmitted photons. Altogether, when Eve uses the intercept-resend strategy, she gets 50% of the information, while Alice and Bob find a 25% error rate intheir sift-key. They detect Eve’s action by simply comparing a random sample of transmittedbits to earlier tests of their line and detectors.

In practice, however, Alice’s and Bob’s source, transmission line and detectors areimperfect. Alice must therefore send more information than in the above example to make surethat Bob can extract the sift-key. This opens new possibilities for Eve to intercept informationwhile camouflaging the ensuing perturbation in the noise and defects of the communicationand detection hardware. As we briefly discuss hereafter, a single-photon source makes it moredifficult for Eve to hide her actions in noise.

(i) Faint laser pulses: The vast majority of QKD experiments use faint laser pulses, witha Poisson distribution of the number of photons. There is a non-zero probability of havingmore than one photon in a pulse, even for very weak pulses with average photon number〈n〉 = µ � 1.

If we endow Eve with unlimited technological power within the laws of physics, thefollowing photon number splitting attack (PNS attack) is in principle possible (Brassard et al2000, Lutkenhaus 2000):

(1˚) Eve counts the number of photons per pulse, using a photon number quantumnondemolition (QND) measurement; (2˚) she blocks the single-photon pulses, (3˚) selects thetwo-photon pulses and splits them, storing one photon in a quantum memory and forwardingthe other one to Bob using a perfectly transparent quantum channel; (4˚) she waits until Aliceand Bob publicly reveal the bases used and correspondingly measures the photons stored inher quantum memory (Felix et al 2001). In this way, Eve has obtained full information onBob’s bits without introducing any error.

The unique constraint on the PNS attack is that Eve’s presence should not be noticed; inparticular, Eve must ensure that the rate of photons received by Bob is not modified.

Thus, the PNS attack can be performed when the losses that Bob expects because of thefibre are equal to those introduced by Eve’s blocking photons, that is when TABµ < µ2/2, orwhen the transmission of the Alice–Bob line is less than µ/2.

Alice may want to decrease the number of two-photon pulses by attenuating her pulsesfurther, but she will be limited by the dark counts of Bob’s detectors:

µTABηdet > pdark,

where pdark is the dark count during the time slot and ηdet the detection yield. As shown byBrassard et al (2000), in order to eschew a PNS-attack in the presence of dark counts, thetransmission efficiency, TAB, of the line from Alice to Bob should obey the following criterion:

TAB >pdark

ηdetµ+

µ

2ηdet.


Figure 9. Comparison of theoretical (curves) and experimental (symbols) quantum communicationrates as functions of transmission losses for a faint laser source (- - - -, +×) and for a single-photonsource (——, ×). The device based on the single-photon source tolerates larger losses. Figurereproduced with the authors’ permission, from Waks et al (2002).

The optimal choice for Alice’s rate of emission is therefore µ = √2pdark, which leads to the

combined transmission-and-detection limit,

TAB >

√2pdark

ηdet. (23)

(ii) Single-photon source: Alice now uses an ideal single-photon source. As Eve cannot applythe PNS attack, she must use the simple intercept-resend attack. She thus necessarily inducesan additional error rate. The only way for her to remain unnoticed is to restrict her measurementrate to less than the dark count rate, pdark, of Bob’s detectors. The condition for security ofthe communication will be that the rate of bits received by Bob, TABηdet, be larger than hisdetectors’ dark counts probability, pdark, which may hide Eve’s measurements (Brassard et al2000):

TAB >pdark

ηdet. (24)

For low dark count probabilities, it is clear from equations (23) and (24) that the single-photonsource permits more attenuation, i.e. a longer fibre, than an attenuated laser source.

The reliability and secrecy of the key transmission can be improved by applying error-correcting and privacy-amplifying algorithms to the transmitted bits (Bennett et al 1995).Lutkenhaus (2000) gives analytical expressions of the minimum amount of transmission toensure secrecy, taking these algorithms into account, and compares the cases of attenuatedlaser pulses and of a single-photon source under these conditions.

(c) Experimental realizations. The first experimental QKD was demonstrated in 1989, butthe results were published only in 1992 (Bennett et al 1992). In this first experiment withfaint laser pulses, 30 cm of air separated Alice and Bob. Since then, considerable progresshas been booked. The first demonstration of a QKD over a 1 km fibre has been reported byMuller et al (1993). Since then, several prototypes have been proposed and demonstratedover distances longer than 20 km. The free-space transmission of quantum-encrypted signalsreached first a distance of 150 m as reported in Jacobs and Franson (1996) and then exceeded1 km under favourable conditions such as night-time (Buttler et al 1998, 2000, Gorman et al


2001). Rapid progress in open-air cryptography is fuelled by the perspective of transmittingquantum-encrypted signals via satellite.

To date, two groups have used a single-photon source in a QKD link which includes sifting,error-correction and privacy amplification. Both systems were free-space demonstrations overshort distances, 1 m (Waks et al 2002) and 50 m (Beveratos et al 2002). Attenuators were usedin the 1 m experiment to demonstrate the effect of further channel losses (potentially longerdistances). Both experiments show that single-photon devices can provide completely securecommunication in conditions under which this would be impossible with attenuated laserpulses. This is illustrated in figure 9, where the number of secure bits per pulse is reported asa function of the channel loss.

4. Coupling to cavities

4.1. Weak-coupling regime

Spontaneous emission is not an intrinsic property of an isolated emitter but rather a propertyof the coupled system of the emitter and the electromagnetic modes in its environment. Theradiative transition rate, γ , of an atom from an excited, initial state |e〉 to a lower energy, finalstate |g〉 depends on the density of available photon states, ρ(ω), at the transition frequency, ω,and on the atom–vacuum field dipole interaction Hamiltonian. Thus, by altering ρ(ω) (placingthe atom in a cavity or near an interface), the spontaneous emission rate can be enhanced orsuppressed (Purcell 1946, Drexhage 1974).

First experimental demonstrations of the inhibition and enhancement of a spontaneousemission rate were carried out in the 1970s with molecules such as europium complexes inLangmuir–Blodgett films deposited on metallic mirrors (Drexhage et al 1968, Kuhn 1970,Chance et al 1974). Later, in the mid-1980s, atoms were coupled to single mirrors, planarcavities or spherical Fabry–Perot resonators (Kleppner 1981, Goy et al 1983, Gabrielse andDehmelt 1985, De Martini et al 1987, Heinzen et al 1987). Advances in microfabricationtechniques enabled the construction of high-quality semiconductor micropost and microdiscmicrocavities in the late 1980s and early 1990s and triggered series of experiments on solid-statecavity-quantum-electrodynamics (CQED) (Yamamoto et al 1989, Jewell et al 1991, McCallet al 1992, Gerard et al 1998). Photonic-crystal structures (John 1987, Yablonovitch 1987)are promising candidates for strong modification of the spontaneous emission.

In order to efficiently generate single photons, the quantum emitter is coupled to a resonantmode of a high-quality factor cavity. The quality factor is defined as the ratio of the resonanceangular frequency to the cavity damping rate: Q = ω/γcav. Under these conditions, the atom–cavity interaction is described by the Jaynes–Cummings Hamiltonian (Jaynes and Cummings1963, Walls and Milburn 1994). The coupling parameter, g, between an emitter placed in aregion of maximum electric-field intensity in the cavity is given by

g = µeg

h

√hω

2ε0V, (25)

where ε0 is the vacuum dielectric constant, µeg the dipole moment of the transition and V thecavity mode volume. Depending on the ratio of the coupling parameter, g, to the cavity-fielddecay rate, γcav, and to the free-space decay rate of equation (14), γ0 = ω3µ2

eg/3πε0hc3,we can distinguish two regimes of coupling between the emitter and the cavity field: strongcoupling for g > γcav, γ0 and weak coupling for g < γcav, γ0 (Haroche 1992).

In the strong coupling case, the emitter is coherently coupled to the cavity field, andspontaneous emission is reversible. The presence of one excitation gives rise to ‘vacuum’


Figure 10. Scanning electron microscopy images of two types of cavities used to enhance thespontaneous emission of self-assembled quantum dots. Left: micropost cavity included betweentwo Bragg reflectors (top diameter 0.6 µm, height 4.2 µm (Pelton et al 2002, copyright 2002 by theAmerican Physical Society; reproduced with permission)). The emitters are placed in one of thecentral layers. Right: microdisc structure (diameter of the disc 5 µm, post height 0.5 µm (Michleret al 2000b, copyright 2000 AAAS; reproduced with permission)). The light is emitted in WGMs.The emitters are distributed throughout the material, but only those close to the mode’s antinodesare significantly coupled.

Rabi oscillation between the two coupled states, ‘excited emitter with empty cavity’ and‘ground-state emitter with one photon in the cavity’ (Brune et al 1996).

In the weak-coupling regime, the atomic excitation is irreversibly lost to the continuumof all available photon states, including the vacuum modes still accessible (leak modes) andthose of the cavity (cavity modes). When the cavity decay time is shorter than the spontaneousdecay time in vacuum (γcav > γ0, bad cavity limit), the spontaneous emission decay rate isenhanced and is given by Fermi’s golden rule (Yamamoto and Imamoglu 1999):

γ = 2π

h2 (hg)2ρcav(ω) + f γ0. (26)

We have separated the contribution of the cavity modes from that, f γ0, of all other leak modes(f , 0 < f < 1, is often close to unity). Assuming a normalized Lorentzian mode density witha width of ω0/Q for the cavity, ρcav(ω) ≈ 2Q/πω0, we deduce the following spontaneousemission enhancement:

γ

γ0= 3

4π2

λ3

VQ + f.

The first contribution to this ratio is called the Purcell factor (Purcell 1946):

FP = 3

4π2

λ3

VQ. (27)

If the Purcell factor is much greater than 1, the emitter will radiate much faster in the cavity thanin free space. The Purcell factor increases with Q/V only as long as the coupling parameter, g,is less than the decay rates of the system (γ and γcav). At that point, the coupled electron–cavitysystem enters the strong-coupling regime.

The fraction of the light emitted into the cavity is called the spontaneous emission couplingfactor β (Vuckovic et al 2002), and is related to the Purcell factor by

β = FPγ0

γ= FP

FP + f. (28)

Efficient cavity-coupled single-photon sources were demonstrated with single quantum dotsembedded in microcavities (Ohnesorge et al 1997, Gerard et al 1998, Solomon et al 2001). The


cavities consist of a high-refractive-index spacer containing the dot, sandwiched between twodielectric Bragg mirrors, as shown in figure 10. The confinement of light in these structures isachieved by the combined action of the distributed Bragg reflector (DBR) in the longitudinaldirection, along the post axis, and total internal reflection (TIR) in the transverse direction.Yamamoto’s group (Pelton et al 2002, Vuckovic et al 2003) reported enhancement of theemission rate from single quantum dots with a Purcell factor of FP ≈ 5.8, quality factors ofthe order of 1000 and mode volumes close to the fundamental limit, λ3. In this case β ≈ 80%of the emitted light was coupled into a single cavity mode. The majority of this light escapedinto a single-mode, nearly Gaussian travelling wave.

Michler et al (2000) embedded self-assembled InAs and InGaAs quantum dots inmicrodisc cavities to generate single photons. The whispering-gallery modes (WGMs) of thesecavities present higher quality factors (in the 104 range) but relatively larger mode volumesso that the Purcell factors were comparable (FP ≈ 6) with those obtained in the micropostcavities.

4.2. Strong-coupling regime

The generation of single photons in the strong-coupling regime of CQED (Law and Kimble1997) presents interesting new routes for quantum information processing. The reversibletransfer of quantum information between the emitter and the photons opens the way forquantum networks. In the all-optical network of Knill et al (2001), the nodes are linearoptical components. The quantum information is encoded in the number of photons flyingfrom node to node. The nodes perform gate operations based on quantum interferencebetween the indistinguishable photons. In a more general network, the nodes also serveas quantum memories storing information, e.g. in long-lived states of atoms located in anoptical cavity (Cirac et al 1997). The key requirement for such networks is their abilityto interconvert stationary and flying qubits and to transmit flying qubits between specifiedlocations (Di Vincenzo 2000).

The atom–cavity system, in particular, must be able to transfer quantum informationbetween atoms and photons in a coherent manner (Maıtre et al 1997, Brattke et al 2001). Itmust also act as an emitter and a receiver of single-photon states. These states must therefore begenerated by a reversible process. Sources based on incoherent excitation of the emitting stateof a quantum emitter do not meet this essential requirement. The reason is that the generationprocess, an electronic excitation followed by spontaneous emission, is irreversible and cannotbe described by a Hamiltonian evolution.

Recently, single-photon sources using single atoms strongly interacting with cavities havebeen demonstrated by the groups of Rempe and Kimble (Kuhn et al 2002, McKeever et al 2004).The photon production step is based on a stimulated Raman process driven by an adiabaticpassage (STIRAP) (Parkins et al 1993, Vitanov et al 2001) between two ground states of asingle atom. The stimulated emission process requires a strong coupling to a single mode of ahigh-finesse optical cavity (Law and Kimble 1997). A laser beam illuminates the atom slightlyoff-resonance, while the vacuum fluctuations of the cavity stimulate emission of the photontowards the other ground state of the atom. The adiabatic passage of STIRAP is obtained byvarying the pump laser intensity and is slow compared with the photon lifetime in the cavity.The field generated inside the cavity is therefore quickly transferred to the outside world.

Figure 11 illustrates the basic principle of this photon generation process with a lambdaatomic level-scheme. Single alkali atoms (85Rb for Rempe’s or 137Cs for Kimble’s groups) areprepared in |a〉, one of the two hyperfine sates of their S1/2 electronic ground state. The atomis located in a high-finesse optical cavity, which is near-resonant with the electronic transition


0,e

0,a

1,b 0,b

Pumplaser

Atom-cavitycoupling

STIR

AP

Rec

yclin

g la

ser

Photon emission

0,e

0,a

1,b 0,b

Γ

Figure 11. The stimulated Raman generation of single photons (STIRAP) is based on a �-schemeof atomic levels labelled |a〉, |b〉 and |e〉. The process starts by pumping close to resonance the(|a〉, |e〉) transition and stimulating the emission on (|e〉, |b〉) transition with a resonant cavity. If asingle atom is present in the cavity, a single photon is emitted in the cavity mode.

between states |b〉 and |e〉. Here, |b〉 is the second hyperfine state of the electronic groundstate and |e〉 is a hyperfine state of the electronically excited level P3/2. The state of the cavityis denoted by |n〉, where n is the number of photons. Inside the empty cavity, the productstates |e, 0〉 and |b, 1〉 are coupled by the electric dipole interaction with coupling parameterg. When a pump laser pulse close to |a〉 → |e〉 resonance is applied with a Rabi frequency�p(t), the product states |a, 0〉, |e, 0〉 and |b, 1〉 of the atom–cavity system are coupled. AtRaman-resonant excitation, when the detunings of the pump-laser and the cavity from therespective atomic transitions are equal, one of the three eigenstates of the coupled atom–cavitysystem has no contribution of the excited state |e〉 and is therefore not affected by spontaneousemission. The expression of this dark state, |ψdark〉, is given by

|ψdark〉 = 2g|a, 0〉 + �p(t)|b, 1〉√4g2 + �2

p(t). (29)

The dark state |ψdark〉 is now used to generate a single photon inside the cavity. This is achievedby establishing a large atom–cavity coupling constant, g, before turning on the pump pulse. Inthis case, the system’s initial state, |a, 0〉, coincides with |ψdark〉. Provided the pump pulse risesslowly, the system’s state vector adiabatically follows any change in |ψdark〉 and for a losslesscavity a smooth transition from |a, 0〉 to |b, 1〉 is attained as soon as �p � g. The singlephoton generated in the cavity mode leaves it through the output mirror, and the final stateof the coupled system, |b, 0〉 is then reached. As this state is not coupled to the one-photonmanifold, the atom cannot be excited again, which limits the number of photons per pumppulse and atom to one.

To emit a sequence of photons from one and the same atom, the system must be transferredback to |a, 0〉 once an emission has taken place. Recycling laser pulses are applied to the atombetween consecutive pump pulses. The recycling pulses are resonant with the |b〉 → |e〉transition and pump the atom to state |e〉. From there, it decays spontaneously to the initialstate, |a〉. If an atom that resides in the cavity is now exposed to a sequence of laser pulses,alternately triggering single-photon emission and repumping the atom in its initial state, asequence of single-photon pulses is produced.

In the experiment with 85Rb, atoms are released from a magneto-optical trap and passthrough the TEM00 mode of a high-finesse cavity (finesse ∼60 000, distance between mirrors∼1 mm, Q ≈ 108). In contrast to solid-state single-photon sources, this source operates undernon-stationary conditions because atoms enter and leave the cavity randomly. Photons are


emitted only in the presence of an atom in the cavity, and each atom generates only a fewsingle photons in the cavity during its short residence time. If more than one atom is present,the number of simultaneously emitted photons can exceed one, which requires that the averagenumber of atoms be very low (Kimble 2003). To circumvent this problem, Kimble’s groupemployed a single Cs atom trapped in the optical cavity. After being released from a magneto-optical trap, atoms are cooled in the cavity region and then loaded into a far-off-resonancedipolar trap (McKeever et al 2004) which localizes them within the mode of the high-finessecavity (finesse ∼4 × 105, distance between mirrors ∼42 µm, Q ≈ 4 × 107). The lifetime of atrapped atom in the presence of the driving fields (pumping �p(t) and repumping fields) limitsthe number of generated photons to about 104, a few hundreds of which are detected.

5. Early and macroscopic sources

In our discussion of possible applications of single photons, we have seen that even a smallprobability of generating two- or multi-photon states could open serious security breaches inquantum cryptography or completely disrupt the operation of an all-optical quantum computer.Faint laser pulses and parametric sources of entangled pairs deliver Poisson distributions ofphotons (or of photon pairs), from which multi-photon events can never be entirely suppressed.Nevertheless, such sources are much easier to build and operate than single-photon sources.In this section, we briefly recall the principle of operation and properties of these early andmacroscopic sources of few-photon states in order to compare them with single-photon sources.

5.1. Faint laser pulses

Single-photon number-states may be approximated by coherent states with a very low averagephoton number 〈n〉. They can easily be developed using only standard pulsed lasers (suchas inexpensive semiconductor laser diodes) and calibrated attenuators. As we have seen insection 2.2(b), the probability of finding n photons in such a coherent state follows Poissonstatistics (equation (10)).

For quantum cryptography, the reliability of the source is related to the probability, pmulti,that a non-empty weak coherent pulse contains more than one photon. These pulses can leadto fractional information leakage which can be exploited by a smart eavesdropper in a beamsplitter attack.

pmulti = pn�2

pn�1= 1 − p(0) − p(1)

1 − p(0)≈ 〈n〉

2, (30)

where pn�1 and pn�2 are the probabilities that a pulse contains at least one and at least twophotons. The value of pmulti can be made arbitrarily small. Weak pulses are thus extremelyconvenient and have indeed been used in the vast majority of quantum cryptographyexperiments. However, they have one major drawback. When 〈n〉 is small, most pulses areempty: p(n = 0) ≈ 1−〈n〉. In principle, the resulting decrease in bit rate could be compensatedby increasing the pulsed laser source rate (gigahertz modulation rates in telecommunicationslasers are achievable), but the problem of the detectors’ dark counts remains. Indeed, thedetectors must be active for all pulses, including the empty ones. Therefore, the dark countrate increases with the laser’s modulation rate, and the ratio of detected photons to dark counts(i.e. the signal-to-noise ratio) decreases with 〈n〉 (Gisin et al 2002). The optimal 〈n〉 valuedepends on the transmission losses and on assumptions on the eavesdropper’s technology(〈n〉 ≈ 0.1 is a typical value).


5.2. Entangled photon pairs

Two-photon states are the most popular entangled states in fundamental and applied quantumoptics. In a two-photon system, a pair of photons are generated in certain physical processes(atomic cascade or parametric down-conversion). The two photons may propagate away fromeach other in different directions, and the quantum states of each single photon may differconsiderably. Despite the distance between the subsystems, the pair will keep its correlationof space–time variables, spin variables or both. The correlations, usually defined in the form ofenergy conservation, momentum conservation and/or angular momentum conservation, arisein the process of generation of the pair.

An elegant way of creating pseudo-single-photon states is to use one photon of a correlatedpair as a trigger for its companion (Grangier et al 1986, Hong and Mandel 1986). In contrastto the faint laser sources discussed earlier, the second detector needs only be activated whenthe first one has detected a photon and not whenever a pump pulse is applied.

5.2.1. Atomic cascade. Entangled photon pairs can be obtained in certain atomic cascades(j = 0 → j = 1 → j = 0, j being the total angular momentum of the atomic states)because of the polarization correlation between the successive emitted photons. These sourceshave been used in the early experiments to test Bell’s inequalities (Clauser and Shimony 1978,Aspect et al 1981). For example, Aspect’s experiment used the 4p2 1S0 → 4s 4p 1P1 → 4s2 1S0

cascade in a calcium atom beam, which yields two polarization-correlated photons (λ1 =551 nm and λ2 = 423 nm) upon two-photon excitation from 4s2 1S0 to 4p2 1S0. A cascade rateof ∼5 × 107 s−1 could be achieved (Aspect et al 1982b).

In 1986, Aspect and co-workers (Grangier et al 1986) used this atomic cascade sourceto demonstrate the wave–particle duality of a photon by sending single photons on a beamsplitter. They observed a strong anticorrelation between the triggered detections on the twosides of the beam splitter, which was a clear signature of a particle-like behaviour of the singlephoton. When the output beams of the beam splitter were recombined on a second beamsplitter to form a Mach–Zehnder interferometer, interferences with a visibility of over 98%were observed when the optical path between the two arms of the interferometer was varied.This is consistent with a classical wave description of the photon. Only the quantum theory oflight is able to give a consistent description of both experiments.

Besides the heaviness and difficulty of operation of the experimental set-up, the atomiccascade sources suffer from a rather low brightness and from a loss of the polarizationcorrelation when the photons are emitted back to back (due to the recoil of the atom) (Santos1992).

5.2.2. Spontaneous parametric down-conversion (SPDC) sources. Photon pairs can begenerated by SPDC in non-centrosymmetric crystals such as KDP, BBO, LBO or LiNbO3.Nonlinear interaction of a pump laser pulse with the crystal occasionally splits one high-frequency pump photon into two lower-frequency, signal and idler photons (Shen 1984). Thefrequencies and wavevectors of the three photons satisfy the energy conservation and phase-matching conditions:

ωp = ωs + ωi, �kp = �ks + �ki. (31)

The process is said to be degenerate if the down-converted photons have the same frequency(e.g. ωs = ωi = ωp/2), and nondegenerate otherwise. In general, the photons leaving thecrystal propagate in non-collinear directions. However, under certain conditions the pair mayalso exit collinearly, in the same direction as the pump.


(a) (b) (c)

Figure 12. Three different types of parametric down-conversion schemes: (a) type-I phase-matching; (b) collinear degenerate type-II phase-matching, with two cones overlapping along thepump direction; (c) non-collinear degenerate type-II phase-matching—the cones intersect alongtwo symmetric directions. Correlated pairs are obtained at certain positions, indicated by thecrossing segments. With proper selection by pinholes, polarization-entangled states or Bell statescan be prepared.

The down-conversion process is called type-I if the signal and idler photons have identicalpolarizations. In the case of degenerate emission, a pair of photons with equal wavelengthsemerge in a cone centred on the pump beam and whose opening angle depends on the angle,θpc, between the optical axis of the crystal and the pump direction.

With type-II phase matching, the signal and idler photons have orthogonal polarizations.They are emitted into two cones, one ordinary, the other extraordinary polarized. In thecollinear situation (for a given incidence angle θpc), the two cones are tangent to one anotherexactly at the pump beam direction. If θpc is increased, the two cones tilt towards the pumpand intersect along two rays. Along the two directions (‘1’ and ‘2’), where the cones overlap,the light is described by a Bell entangled state:

|ψ〉 = (|H1, V2〉 + eiϕ|V1, H2〉)√2

, (32)

where H and V indicate horizontal and vertical polarization, respectively. The relative phase,ϕ, arises from the crystal birefringence. Its value can be varied using an additional birefringentphase shifter (or slightly rotating the nonlinear crystal), giving the possibility of preparing anyof the four EPR-Bell states (Shih 2003).

Figure 12 shows three examples of parametric sources based on type-I and type-II phasematching, in historical order. The first sources, used in the early two-photon experiments,were of the type shown in figure 12(a) (Hong et al 1987, Shih and Alley 1988). Collineartype-II SPDC was developed later by Shih and Sergienko (1994) and is shown in figure 12(b).In this set-up, a pinhole is usually used to select the overlap region between the two rings.The scheme shown in figure 12(c) was first developed by Kwiat et al (1995) to prepare Bellstates. This scheme has been used successfully to demonstrate many quantum entanglementphenomena (see section 3).

Phase-matching allows one to choose the wavelength and determine the bandwidth of thedown-converted photons. The latter is in general rather broad and varies from a few nanometresup to some tens of nanometres. For the nondegenerate case one typically gets a bandwith of5–10 nm, whereas in the degenerate case (where the central frequencies of both photons areequal), the bandwidth can be as large as 70 nm.

This photon-pair creation process is very inefficient (efficiency ≈ 10−7–10−11). Thenumber of photon pairs per mode is thermally distributed within the coherence time of thephotons and follows a Poissonian distribution for larger time windows (Walls and Milburn1994).


A typical rate of pair generation into single-mode optical fibres, corresponding to 10 mWcw pump power, is 107 pairs per second. The probability of having two pairs instead of one ina time slot of 1 ns is then about 1%. With a 10% detection yield on one of the fibres, therate of single-photon production in the other fibre is 106 per second. In order to obtain thesame brightness with attenuated laser pulses (with a comparable fraction of two-photon events,〈n〉 = 0.02), the repetition rate should be 50 MHz, leading to a 50 times higher rate of darkcounts than with the parametric source.

6. Microscopic single-photon sources

6.1. Excitation schemes

The macroscopic sources we just discussed give a Poisson distribution of photons and mustbe strongly attenuated to reduce the probability of obtaining two or more photons. The rateof emission of single photons is therefore limited. The working principles of the microscopicsources discussed in this section are profoundly different. These sources are built around asingle emitting nanometric object (a nano-object), producing photon distributions which arevery far from Poissonian. In most cases, for example, the probability density of emitting twophotons at the same time can be completely neglected, whereas it is still high for an attenuatedPoisson source with the same brightness. In most cases, the emission process is spontaneousand takes place after a rapid excitation of the emitter. In order to achieve emission at a highrepetition rate, and in order to avoid excitation events in which no photons are emitted, twoconditions have to be fulfilled:

(i) The emitting state must be prepared at the end of the excitation cycle with as high aprobability as possible (ideally, with certainty). A number of excitation schemes have beenproposed and can be gathered into two classes.

Incoherent pumping. These processes rely on a fast relaxation step to prepare the excited state.A generic example is the optical pumping of a molecule in a dye laser. A high-intensity opticalpulse pumps an organic dye molecule from its electronic ground state to a vibrationally excitedlevel of its excited electronic state. This short-lived state in turn quickly relaxes to the emittingstate. Because it lives about 1000 times longer than the vibronic state, the latter state canaccumulate a population from several pumping interactions. Repeated attempts at pumpingby the many photons of the exciting pulse thus efficiently transfer the ground state populationinto a 100% population of the emitting state. A variant of this scheme is electrical pumping ofa Coulomb blockade junction. Although many charge carriers can interact with the junction,only one will saturate it and prevent access to further charges. When an electron and/or holeare left to recombine, they will quickly relax to a single-exciton state, which will then emit asingle photon.

Coherent preparation. It is also possible to prepare the emitting state with resonant light, butone then has to separate the emitted photon from the excited ones, either temporally, by a time-delayed detection, or spectrally, by detecting a red-shifted photon emitted on a vibrational levelof the ground state. One possibility for resonant excitation is to apply a ‘pi-pulse’, i.e. a pulsewhose duration and intensity are so chosen that the vector state exactly evolves from the groundstate to the excited state during the excitation pulse (see, e.g. Shen (1984)). Another optionis rapid adiabatic passage: the laser frequency is swept through resonance with the two-levelsystem, quickly enough to avoid relaxation by spontaneous emission or dephasing but slowlyenough so that the state of the system adiabatically follows the frequency sweep. When both


conditions are fulfilled, the two-level system is brought very efficiently from the ground to theexcited state after one frequency sweep. Alternatively, the system’s resonance can be sweptthrough resonance with a fixed laser, as was demonstrated for an organic molecule (Brunelet al 1999). In a more elaborate version of the adiabatic passage requiring three levels in a�-scheme, the STIRAP, the photon is coherently emitted by a single atom into the mode of ahigh-Q cavity by a stimulated Raman process (see section 4.2).

(ii) The emission quantum yield of the nano-object must be as high as possible. Theemission yield primarily depends on the photophysics of the nano-object at hand, but it can beimproved if spontaneous emission is enhanced, for example, by the Purcell effect of a resonantcavity (see section 4). In the present section, we shall discuss separately the emission quantumyield for the main classes of nano-emitters that have been considered so far.

For each of the main classes of systems, we give a brief discussion of their optical andphotophysical properties (quantum yield, spectrum, lifetime, polarization, radiation pattern,antibunching and/or multi-excitation shifts, etc), which determine the characteristics of theemitted photons and the properties of the source. Where applicable, we shall indicate currentrealizations, with the schemes adopted for excitation, emission and collection.

6.2. Atoms, ions in gas phase

Atoms and ions are characterized by purely electronic eigenstates with hyperfine structure.In the absence of a Doppler effect and of collisions with residual gas, which is the case inpractice for cold atoms and trapped ions, the transitions are narrow and lifetime-limited. A bigadvantage of atoms is that their states are perfectly reproducible and well-known, includingthe hyperfine structure of the levels. With the right atom or ion, transitions can be foundthroughout the visible and near-IR parts of the spectrum (Cs at 852 nm, Rb at 780 nm).

Excitation schemes for atoms and ions often rely on multi-step processes between knownlevels. We already mentioned atomic cascades in section 5. In the STIRAP process discussedearlier (see section 4.2, Parkins et al (1993), Kuhn et al (2002), McKeever et al (2004)), anadiabatic passage between two hyperfine levels of the ground state is driven by a laser pulseclose to resonance with an excited state. The emitted photon is stimulated into the modeof a resonant high-finesse cavity. Since there is only one atom in the cavity and since onlyone passage is performed at each laser pulse, only one photon is produced in the cavity (seefigure 13). The incoherent pumping scheme does not apply to atoms. In contrast to emitters incondensed matter, they offer no possibility for fast relaxation between levels in the picosecondrange.

Typical radiative lifetimes of allowed atomic transitions are about 30 ns, correspondingto a linewidth of a few megahertz. The whole oscillator strength is concentrated in one singletransition, and because of the absence of non-radiative channels, the fluorescence quantumyield is unity.

One of the essential features of a single-photon emitter is that it should be stronglyanharmonic, i.e. two- or multi-excitation states should have very different emission propertiesfrom the singly excited state. This condition is easily satisfied for few-electron systemssuch as atoms and ions. They never emit two or more photons (with the same wavelength)simultaneously and consequently present strong antibunching. Simultaneous emission of twophotons in general indicates that two atoms are present in the detection volume (McKeeveret al 2004). The polarization and radiation pattern of the emitted photon are determined bythe angular momentum of the states involved in the transition.

After emission of a photon by an isolated atom or ion, no information is left in the mattersystem (except for a possible recoil, which is usually negligible). Therefore, the emitted


Figure 13. Example of a correlation function for a pulsed single-photon source based on a STIRAPprocess (data from McKeever et al (2004), copyright 2004 AAAS; reproduced with permission).

wavepackets will be transform-limited, and the photons will be perfectly indistinguishable.In the case of a trapped ion, indistinguishability requires that emission on the vibrationalsidebands should be negligible or eliminated.

Atoms are usually very stable under laser illumination. In contrast to large moleculesand semiconductor systems, they do not present long-lived dark states (or if they do, they canbe recycled out of them). The main challenge for the attainment of a single-photon sourceis in the isolation, manipulation and trapping of single atoms, which requires sophisticatedand expensive setups, including high-resolution stabilized lasers at several frequencies, andultra-high vacuum. The operation time of an atom-based single-photon source is limited bythe dwell time of the atom in the cavity mode, up to seconds for a cold atom in a trap. Afteran atom has been lost, a new one must be loaded.

In summary, atom- or ion-based sources of single photons present many advantagesas standard sources of pure and indistinguishable photons. They are nearly ideal for thedemonstration of quantum-optics experiments. Their main disadvantage is their complexityof operation.

6.3. Organic molecules

Some organic dyes and aromatic molecules emit fluorescence with a high quantum yieldand were the first condensed-matter systems for which antibunching was demonstrated. Thephotophysical properties of molecules make them interesting models for understanding thebehaviour of strongly confined electron systems such as colour centres and semiconductornanoparticles, to be discussed later. In the present section, we discuss separately thephotophysical properties of organic fluorophores at cryogenic and ambient temperatures andsingle-photon emission in more complex systems consisting of several coupled molecules(multichromophores, aggregates, polymers, etc).

6.3.1. Low temperatures. Organic molecules less than a nanometre in size can presentoptical transitions in the visible or near-IR. As compared with those of atoms, the eigenstatesof molecules in condensed matter involve vibrations and phonons, in addition to electronic


states. As a consequence, a transition to an excited electronic state, even if it starts from theground state of the solid at zero temperature, is distributed over a broad range of frequencies,corresponding to the creation of additional vibrations and phonons. At very low temperatures,however, the lowest-frequency transition connecting the ground vibrational states of the groundand excited electronic states is a very narrow line, called the zero-phonon line (ZPL) (Sild andHaller 1988). Zero-phonon linewidths are often lifetime-limited at low temperatures. Theintensity of the ZPL compared with the total intensity of the absorption spectrum is calledthe Debye–Waller factor. A typical value is 0.1, and it decreases quickly for temperatureshigher than that of liquid helium, while the width of the ZPL increases steeply because ofphonon-induced dephasing processes. A molecule also absorbs at a higher frequency than theZPL, towards vibrational excited states of the electronic excited state.

Fluorescence is emitted from the lowest vibration state of the excited electronic statetowards vibrational levels of the ground state, giving rise to vibrational fluorescence lines(shifted to the red with respect to the ZPL by vibration frequencies, 100–3000 cm−1) andto phonon sidebands (width about 100 cm−1). Only the ZPL, the bluest emission at lowtemperatures, is narrow because its width is determined by the long lifetime of the excitedstate, nanoseconds. The width of the other vibronic lines is limited by the lifetime of thevibrations, about a few picoseconds.

The molecular transition is a linear dipole with an oscillator strength of about unity, whichcorresponds to a spontaneous emission lifetime of a few nanoseconds and to a ZPL linewidthof about 30 MHz. The fluorescence quantum yield is very close to unity (typically better than95% for ‘good’ fluorophores, for which non-radiative relaxation is small).

In contrast to atoms, there are many upper vibronic levels in molecules that can be reachedby absorption of a second or third photon. However, highly-excited singlet states relax veryrapidly towards the lowest excited singlet state in about a picosecond or less. This observationalfact is summarized by Kasha’s rule: only the lowest excited state of a given spin multiplicitycan efficiently emit light (Birks 1970). Multiple excitations therefore cannot lead to emissionof two or more photons. However, many photochemical channels open for highly excitedmolecular states, which promote the irreversible photodestruction of the molecule, speciallyunder ambient conditions.

The natural molecular fluorescence spectrum presents a number of lines and bands, spreadover several hundreds of wavenumbers. To obtain indistinguishable, lifetime-limited photons,one has either to select the emission of the ZPL, thus losing 80–90% of the fluorescence, or tostimulate the ZPL emission by coupling to a resonant cavity (section 4), which is not easy formolecular systems in condensed matter, and has never been done so far.

The stability of organic molecules is excellent (at least hours) in crystalline hosts at liquidhelium temperatures. Sometimes, even in crystals, defects can lead to frequency variations byspectral diffusion. Depending on the molecular compound and host crystal, excursions to atriplet lifetime can in some cases limit the molecular brightness (Bernard et al 1993).

Under low-temperature conditions, the fluorescence of single molecules shows deepantibunching and damped Rabi oscillations (Basche et al 1992). Aromatic molecules wereused for an early demonstration of a single-photon source in condensed matter (Brunel et al1999), as shown in figure 14.

6.3.2. Room temperature. Many of the above features apply to molecules at room temperatureas well, but thermal motions bring about several very important differences, which are listedhereafter.

Only very broad absorption and emission bands appear at room temperature. Their shapeand width depend on the matrix. A typical value for the linewidth is 300 cm−1 and is mainly


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Figure 14. Left: experimental measurements of the second-order correlation of photons deliveredby a single DBATT molecule in a molecular crystal at cryogenic temperatures (Brunel et al 1999).The molecule was excited by a rapid adiabatic passage through resonance with the exciting laser.Right: comparison with quantum Monte Carlo simulations. The residual events around zero delaystem from two successive emissions by the molecule, due to a double excitation (the passage wasnot rapid enough). This is confirmed by the antibunching dip at zero delay, more visible on thesimulations.

due to thermal fluctuations. Therefore, even in crystalline environments, lines of single organicmolecules at room temperature are never much narrower. Thermal fluctuations, i.e. vibrationsand phonons in solids, cause fast dephasing of the electronic oscillations. In soft molecularmatrices, the sharp transitions (ZPL) disappear altogether above 50 K. As a consequence offast dephasing, the emission is very far from lifetime-limited, by a factor of at least 10 000!

The spontaneous emission lifetime is usually not strongly temperature-dependent. Thefluorescence quantum yield usually decreases with temperature because of thermally inducedconformational changes of the molecule but remains better than 90% for many fluorescentdyes and aromatic molecules in rigid or viscous environments.

Molecular photostability is a much more serious issue. It is often severely limited atroom temperature because of the many photochemical processes that can be activated from theexcited state by thermal fluctuations. A chemically stable dye in a polymer matrix does notsurvive more than seconds the typical excitation intensities of single-molecule observations.Furthermore, the complexity and flexibility of the environment at room temperature allowsthe existence of metastable dark states, leading to extended dark periods, during whichemission stops (Zondervan et al 2003, 2004). However, if the molecules can be protected fromatmospheric oxygen and are rigidly held in a crystalline matrix, the stability can sometimes beextended to minutes or hours (Fleury et al 1998, Kulzer et al 1999), even under intense pulsedillumination, as has been shown for terrylene in p-terphenyl crystals (Lounis and Moerner2000).

The fluorescence of single molecules at room temperature also shows very strongantibunching (Ambrose et al 1997, Fleury et al 2000), due to the very short lifetime of


Figure 15. Fluorescence microscopy image of single terrylene molecules in a p-terphenyl crystalat room temperature. Each individual molecule can be excited by a short laser pulse and work asa stable source of single photons (Lounis and Moerner 2000).

higher excited levels. For systems presenting a high photo-stability, for example terrylenein a p-terphenyl crystal (see figure 15), single molecules can operate as single-photon sources(Lounis and Moerner 2000, Treussart et al 2002).

6.3.3. Multichromophoric and conjugated systems. Here, we only discuss the room-temperature properties of coupled systems of chromophores. In such complex assemblies,every individual chromophore may absorb the exciting light, but only one (or a very few)of them emits light at a time. At a low exciting intensity, the excitation energy of eachchromophore is transferred by hopping and relaxation towards sites with lower excitationenergy. At higher intensity, when two excitations meet, one of them is efficiently annihilatedand its energy released as vibrations and heat. This process, called singlet–singlet annihilation,is a consequence of Kasha’s rule and of the fast relaxation of high excited states. It veryefficiently prevents emission of two photons at the same time by a coupled system andexplains the strong antibunching observed in multiple chromophores (Hofkens et al 2003)and in conjugated polymers (Lee et al 2004).

Multichromophoric systems, molecular dye aggregates and conjugated polymers are allsubject to the same processes of photochemical destruction that limit the operation time ofsingle fluorophores at room temperature. For such large assemblies, however, the operationtime is lengthened because of the large number of available fluorophores in the assembly.When one fluorophore is bleached, and provided the product of the reaction is no quencherfor neighbouring sites, a new fluorophore becomes the lowest-energy trap and can resumeemission. Single polymer molecules have been in operation for as long as hours on a glasssurface before photobleaching eventually stopped the emission (Lee et al 2004).

6.4. Colour centres

Colour centres are defects of insulating inorganic crystals, for example alkali halides,presenting intense absorption and fluorescence bands or lines. They arise from localized


Figure 16. Atomic structure of an NV-centre in a diamond crystal. One of the carbon atomsis replaced by a nitrogen (N), and one of its neighbours is missing (vacancy V). The electronicstructure of the centre has not been completely elucidated yet, but presents a triplet ground state(copied from http://www.physik.uni-stuttgart.de/institute/pi/3/ags/jelezko/forschung/).

electronic states around interstitial or substitutional impurities, vacancies, inserted chargecarriers or combinations thereof. Although much is known from conventional ensemblespectroscopy about a huge number of colour centres in a variety of hosts (Crawford et al1972, Gellermann 1991), only a handful of systems have led to observations of individualdefects, and even fewer have been shown to possess the photophysical properties suitable forsingle-photon generation. Defects in low-band-gap semiconductors have been proposed assources of single photons (Strauf et al 2002) and will be briefly discussed in section 6.6. Here,we concentrate on the only two systems, both of them defects of diamond, which have beenproposed as single-photon sources. A major advantage of inorganic materials, and in particularof diamond, is their mechanical stiffness and stability. Although the electronic wavefunctionsand photophysical properties of colour centres are very similar to those of organic molecules,their photostability is much improved, both because of the rigidity of the diamond lattice andbecause of the protection it affords against small aggressive molecules such as oxygen.

6.4.1. Nitrogen-vacancy centre in diamond. The first single colour centre ever detected wasa nitrogen-vacancy (NV) centre in diamond (Gruber et al 1997), at room temperature. Thisdefect is a carbon vacancy situated next to a nitrogen atom, with an electron trapped (see ascheme of the structure in figure 16). The ground and optically excited states are triplets. Theluminescence spectrum of this defect is a broad band (more than 1000 cm−1, extending from640 to 720 nm), with a weak and rather narrow ZPL (at 640 nm, width about 50 cm−1), stillvisible at room temperature thanks to the stiffness of the diamond lattice (Gruber et al 1997).The breadth of the emission band indicates that the photons produced are very far from beinglifetime-limited and therefore are not indistinguishable.

The spontaneous emission lifetime in bulk is about 12 ns, corresponding to a reasonablystrong transition, and the quantum yield is close to 1. However, there is a dark state, presumablya singlet, responsible for bunching in the correlation function (Kurtsiefer et al 2000, Beveratoset al 2001, Jelezko et al 2002). Just as in organic fluorophores, multiple excitations areexcluded by the small size of the system and Kasha’s rule.

Diamond has a high refractive index, which seriously limits the fraction of luminescenceextracted in microscopy on a flat interface. An elegant solution to this collection problem is


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the use of nanocrystals (size 50–100 nm) containing one or a few centres for which no total-reflection effect takes place. This improves the collection efficiency by more than a factor of 2and additionally reduces the background. Because of the reduction in the surrounding indexof refraction, the luminescence lifetime is lengthened to 20–30 ns, depending on the individualnanocrystal. Nanocrystals offer the additional advantage of the ease of manipulation andsample preparation.

The stability of the NV-centres is excellent, even at room temperature, because of therigidity of the diamond lattice. However, the high peak intensities of a pulsed excitation canlead to complex dark states, possibly through charge transfer processes and photochemicalchanges (Dumeige et al 2004).

6.4.2. Nickel–nitrogen complex. Another colour centre of diamond was recently proposedas an alternative to the NV-centre (Gaebel et al 2004), with a similarly perfect photostability.The centre (NE8) consists of one nickel atom surrounded by four nitrogen atoms. It is foundat low concentrations in natural diamonds or can be created by implantation. The fluorescencespectrum displays an intense and narrow ZPL at 802 nm, a convenient wavelength forfibre communications. The linewidth, 20 cm−1 at room temperature, is much lower than thatof the NV-centre, and could be further narrowed by coupling to a cavity (section 4). The ZPLbears 70% of the total emitted intensity (the Debye–Waller factor is 0.7). The fluorescencelifetime is 11.5 ns, comparable with that of the NV-centre. However, a dark metastable statelimits the fluorescence intensity. Its lifetime is 170 ns, and it has a rather high production yieldof 20% from the excited state.

6.5. Semiconductor nanocrystals

Nanocrystals of II–VI semiconductors such as ZnS have been studied as individual emitters forseveral years (Empedocles et al 1996), but antibunching in their luminescence was discoveredonly recently (Lounis et al 2000, Michler et al 2000, Messin et al 2001). An example of anantibunching curve is shown in figure 17. Among the many possible materials and structureswith emission wavelengths spanning the visible and near-IR (Alivisatos 1996), the most


studied ones are core–shell CdSe–ZnS nanocrystals, a few nanometres in diameter. Theirsize-dependent tunable emission is situated in the yellow part of the spectrum (460–660 nm),and, due to their broad absorption continuum above the exciton transition, they can be excitedwith a variety of sources. The role of the ZnS shell, or capping layer, is to protect the coreexciton from surface defects and dangling bonds, which dramatically reduce the luminescenceefficiency (Hines and Guyot-Sionnest 1996, Dabbousi et al 1997).

The photophysics of semiconductor nanocrystals are akin to those of molecules and of bulksolids. Quantum confinement produces discrete exciton levels similar to atomic or molecularlevels, but the large size and large number of optically active electrons give rise to such complexfeatures as multi-excitation states, which have no equivalent in small molecules. Specificproperties, such as the fine excitonic structure, derive from the electronic band structure of thebulk material.

The luminescence of nanocrystals arises from radiative recombination of the lowest-lyingexciton state, with a lifetime of 20–30 ns at room temperature. This emission time lengthensconsiderably at low temperatures (Nirmal et al 1995, Labeau et al 2003). The spectrumis a single line (ZPL), with a weak phonon sideband. At room temperature, the ZPL isstrongly broadened to several hundreds of cm−1 (50 meV) by dephasing and spectral diffusionand is thus very far from lifetime-limited. The emission line narrows significantly at lowtemperatures, down to about 1 cm−1 (0.1 meV) in the best cases (Empedocles et al 1996),but never reaches the lifetime limit, probably because of spectral diffusion. Photoinducedcharge rearrangements in the passivation layer and the environment have been shown to leadto spectral shifts of the exciton line (Empedocles and Bawendi 1997). Spectral diffusion andthe very long luminescence lifetime are two weak points of nanocrystals for low-temperatureapplications as single-photon sources. The emitting transition dipoles are polarized in a plane,leading to a radiation pattern distinctly different from that of a molecule (Empedocles et al1999, Chung et al 2003). The emission quantum yield was recently shown to be very close tounity, but only during the bright periods of their blinking (Brokmann et al 2004b).

Being formed of thousands of atoms, semiconductor nanocrystals in principle supportmultiple excitations. However, all multi-excited states are nearly dark because of very efficientexciton–exciton annihilation (also called Auger processes (Klimov et al 2000)) similar to thesinglet–singlet and singlet–triplet annihilation in multi-chromophoric molecules and polymers.In a similar way, charged states of the nanocrystal are dark, since a ‘free’ electron or hole canvery efficiently dissipate the exciton energy by non-radiative relaxation. In a neutral, singly-excited nanocrystal, there are no levels in the gap to which electron or hole can relax. Therefore,a non-radiative transition requires coupling to a large number of phonons at once. The matrixelement for such a multi-phonon transition is exceedingly small. Radiative recombination is themain relaxation channel for mono-exciton states and therefore occurs with near-unity quantumyield. As soon as an additional charge carrier (or exciton) is present, however, its interactionwith the emitter exciton leads to highly excited electron or hole states, for which many one-phonon relaxation channels exist. The electronic energy is thus very efficiently dissipatednon-radiatively. The efficient annihilation of multi-excitations is the key mechanism of thestrong antibunching observed in nanocrystals and enables their use as single-photon sources(Lounis et al 2000, Brokmann et al 2004a).

Nanocrystals are much more photostable than organic molecules under similar conditions.This advantage, however, is partly offset by a major disadvantage, their blinking, in otherwords the strong fluctuations of their luminescence with time. The occasional transfer ofa charge-carrier out of the dot leaves a free charge which relaxes all further excitations,completely suppressing luminescence. These off-times follow power-law distributions (Kunoet al 2001) and can be extremely long. So far, blinking has been observed in all semiconductor


nanocrystals, and no clear strategy yet has emerged to control it. Semiconductor nanocrystalsare efficient emitters at ambient conditions. They are easy to manipulate and to couple toefficient collecting optics in a room-temperature microscope and have better stability thansingle organic chromophores. However, blinking limits their practical applications as sourcesof single photons.

6.6. Self-assembled quantum dots and other heterostructures

The core–shell nanocrystals we just discussed are grown in solution by the methods of chemicalsynthesis. Self-assembled quantum dots, on the other hand, are also islands of a low-band-gapsemiconductor embedded in a high-band-gap semiconductor, for example GaAs in AlGaAs,but they are grown epitaxially on single-crystalline substrates from molecular beams and/orreactive gases, by chemical vapour deposition. If one tries to grow a given material epitaxiallyon a (slightly) mismatched substrate, the elastic energy arising from the mismatch is atfirst compensated by surface forces for the first few monolayers but is soon relaxed by thegrowth of three-dimensional domains when more material is added. This transition, calledStranski–Krastanov instability, often appears after the first few ‘wetting’ monolayers havebeen deposited. With pure materials, high-quality substrates, ultra-high vacuum and well-controlled temperature conditions, the Stranski–Krastanov growth leads to uniformly sized,regularly spaced, three-dimensional islands of the deposited material (Moriarty 2001). Theresulting quantum dots are usually protected by a thick layer of the high-band-gap substratematerial, and they can support localized excitons. The well-controlled growth and environmentensure that the surface of the dots is very clean and that the radiative recombination ratesand the photostability are excellent. Quantum dots are often excited in the electron–holecontinuum. The excitons and/or charge carriers are trapped in the dots, where they recombineradiatively. For InAs in GaAs, the spectral range is typically 850–1000 nm, for InAs in InPit is 1300–1700 nm and for the II–VI semiconductors CdSe in ZnSe, it is around 550 nm.No significant phonon sideband can be seen in emission. At cryogenic temperatures, a singlequantum dot gives a narrow, close to Fourier-limited line, in a spectral domain depending on thematerial (Gammon et al 1996, Bayer and Forchel 2002, Zwiller et al 2004). The spontaneousemission rate is of the order of few nanoseconds but can be significantly reduced by the Purcelleffect if the dots are placed inside a resonant cavity (see section 4). The emission quantumyield is thought to be very close to unity.

In contrast to nanocrystals, multiple excitations can coexist in self-assembled quantumdots and can also coexist with charge carriers, leading to characteristic bi- or multi-exciton linesand to trion or even multi-charged lines when an exciton is combined with an electron or a hole(Michler 2003). This profound difference between nanocrystals and self-assembled quantumdots can be traced back to two effects: (1˚) the higher purity and crystalline quality of the self-assembled dots, as less defects sites are available to promote non-radiative recombinationsand (2˚) the smaller size and the larger quantum confinement of nanocrystals, since Coulombinteractions between charge carriers and excitons are enhanced by confinement. Anotherimportant difference is that self-assembled quantum dots can also be excited via relaxation ofexcitons and charge carriers from the wetting layer and the substrate, which can replenish thedot with excitations at a high rate.

Multi-excitations give rise to photoluminescence lines which, at low temperatures, arered-shifted with respect to the single-exciton line. For a single-photon source, it is importantto make sure that multi-excitation lines are eliminated and that only photons from the single-exciton transition are selected. The biexciton line could also be selected as a source of singlephotons as it too shows antibunching (Thompson et al 2001, Zwiller et al 2002), but it requires


Figure 18. Second-order correlation function of a single-photon source based on a single self-assembled quantum dot, under pulsed excitation. The central peak has been completely suppressed(after Vuckovic et al (2003), copyright 2003, American Institute of Physics; reproduced withpermission).

a higher exciting intensity than the single exciton. An interesting alternative would be to use thebi-exciton and exciton lines to generate correlated pairs of single photons (Benson et al 2000).

Because of the multi-exciton lines, a single-photon source based on self-assembledquantum dots must work at a low temperature (about 5–10 K), and the mono-excitonicemission must be filtered spectrally, usually by means of a narrow-band spectral filter or amonochromator. Furthermore, at higher temperatures, the excitons escape towards the wettinglayers or the substrate, and the luminescence efficiency of the dot decreases. Figure 18 showsthe remarkable quality of the second-order correlation function of the emission of a singleself-assembled quantum dot. The radiation pattern and polarization of self-assembled quantumdots can be complex. They may involve different transitions and depend on the shape of theindividual dot and on its coupling to a cavity.

One of the most attractive features of self-assembled quantum dots is their narrow emissionline at very low temperatures (lower than 5 K), close to the Fourier-limit. Yamamoto’s groupin Stanford has demonstrated the large degree of indistinguishability (more than 80% for aparticular dot) of the emitted photons in a Hong–Ou–Mandel experiment (Santori et al 2002).Although they can occasionally blink (Santori et al 2001), quantum dots are very photostableand show no long-lived dark state.

Because of the high index of refraction of the embedding semiconductor, it is difficultto collect emission in the standard planar geometry. High extraction factors of the emittedphotons are achieved by coupling the dot to a cavity (see section 4), which has the additionaladvantage of enhancing the spontaneous emission rate and of collecting the photon in a well-defined mode. Several geometries have been proposed. In the micropillar geometry (Peltonet al 2002, Santori et al 2004), the dots have been grown between two Bragg mirrors, andthin pillars of this layered structure are etched from the original multi-layer. The microdiscgeometry (Michler et al 2000b) exploits WGMs of lithographically etched ‘mushrooms’. Thedots must be coupled to the field of the WGM to achieve efficient resonant enhancement ofthe spontaneous emission.


In conclusion, self-assembled quantum dots present many advantages as potential sourcesof single photons. They are bright, stable and nearly Fourier-limited at sufficiently lowtemperatures. The epitaxial growth and lithography methods facilitate the incorporation of thedots into resonant cavities, which improves the extraction factor and emission rate. However,both the rejection of multi-exciton lines and brightness require on the one hand liquid heliumtemperatures and on the other hand spectral filtering by a monochromator or filter, whichentails inevitable losses.

Other heterostructures such as quantum wells have also been used to deliver single photons.In the first experiments on semiconductor devices, a single electron and a single hole wereintroduced to recombine in a quantum well (Kim et al 1999). In that case, multi-excitationswere avoided because Coulomb blockade prevented more than one electron and more than onehole at the same time to enter the recombination region. However, Coulomb blockade in thecomparatively large structures necessitated temperatures of some tens of millikelvins, makinglight collection inefficient and the whole experiment difficult. The more recent demonstrationof an electrically driven single-photon source (Yuan et al 2002) based on a single quantum dotin a p–i–n junction (similar to a conventional light-emitting diode) only requires temperaturesaround 5 K. Again, a spectral selection makes it possible to isolate the single-exciton lineof a single quantum dot. Antibunching was demonstrated under pumping by short voltagepulses. A special kind of quantum dot can be created around a single impurity in a quantumwell (Strauf et al 2002), for example a nitrogen impurity in a ZnSe well. Such states areintermediate between colour centres and semiconductor quantum dots because the excitonwavefunction largely extends in the surrounding semiconductor host. Here also, cryogenictemperatures were required for the isolation of single defects, but no spectral filtering wasneeded to eliminate multi-excitation lines. Just as in the case of molecules or nanocrystals,the emitted light was naturally antibunched with an emission lifetime of 0.59 ns. Much of thespectroscopy of these systems is related to that of quantum dots.

6.7. Summary

Table 2 summarizes the main features of single quantum systems in view of applications assingle-photon sources. The numbers given in table 2 are only estimates providing a roughcomparison of the different systems. The choice of a given system will depend on the intendeduses of the single-photon source. For example, the indistinguishability of the emitted photonsrequires a Fourier-transform limited emission, which can only be achieved under conditionswhere the system is thoroughly protected from fluctuations of the environment. Only twopossibilities exist so far, either atoms or ions in ultra-high vacuum, and localized electronicstates in condensed matter at liquid-helium temperatures, in molecules or self-assembledquantum dots. Even at very low temperatures, semiconductor nanocrystals appear to be toosensitive to charge rearrangements and spectral diffusion to reach the Fourier-limit.

A single-photon source operated at room-temperature has the advantage of compactnessand convenience, together with a high efficiency of collection, thanks to efficient microscopeobjectives. Up to 20% of the total emission can be detected, while this efficiency usually doesnot exceed 1% in cryogenic experiments where no cavity is employed.

Some of the data given are obtained only when the source is coupled to a resonantcavity. Although any small object can in principle be coupled to a high-finesse cavity, solid-state semiconductor devices have the advantage that the cavity can be grown along with theemitting structures, which simplifies the whole production process and does not require latermanipulations.


Table 2. Comparison of characteristic features of different single quantum emitters as potentialsources of single photons. T = temperature; Col. cent. RT = colour centre room temperature;Q-dot = quantum dot; nanocryst. = nanocrystal; Max. rate of emiss. = maximum rate ofemission; Multi-excit. = multi-excitation; reproduc. = reproducible; Operat. temper. = operationtemperature; liq. = liquid.

Cold Ion in Molecule Molecule Col. III–V II–VIatom trap low T RT cent. RT Q-dot nanocryst.

Emission Sharp Sharp ZPL + broad Broad Broad line, Sharp Broadspectrum line line lines band band line line

Typical 10 MHz 10 MHz 30 MHz, 10 THz 1–30 THz 1 GHz 30 GHzlinewidth 30 GHz

Emission 15 ns 15 ns 4 ns 4 ns 10 ns 300 ps 30 nslifetime

Fourier-limited Yes Yes Yes, no No No Yes NoMax. rate 100 kHz 100 kHz 100 MHz 100 MHz 10 MHz 1 GHz 10 MHz

of emiss.Dark states No No Yes Yes Yes No YesMulti-excit. No No No No No Yes NoLong-term No, but Yes Yes No, in general Yes Yes No, blink

stability reproduc.Operat. temper. µK mK 1–10 K 300 K 300 K 1–30 K 300 KRequirements UHV, UHV, Liq. He Easy Easy MBE, Easy

lasers trap liq. He

7. Conclusion and outlook

This review has dealt with the preparation of single-photon states, in which the quantum stateof light is completely controlled. In terms of fluctuations and noise, lasers have broughttremendous progress over thermal light sources such as spectral lamps. However, light froman ideally stabilized laser, which can be described as a coherent state, is still subject to minimalfluctuations related to the Poisson distribution of quanta, appearing as photon noise in photon-counting detectors. Quantum mechanics allows states with even lower amplitude fluctuations toexist, squeezed light states. Single-photon states are a special case of number-states, i.e. statesin which the number of photons is precisely defined. We reported proposed uses, structuresand demonstrations of devices generating single photons on command, called single-photonsources.

The current proposals for applications of single-photon sources belong to different classes.In the first kind of experiments, the reduced amplitude noise is exploited to improve the accuracyof measurements done so far with stabilized lasers. A simple example is the measurementof extremely weak absorptions. The second series of applications takes advantage of theknowledge of the quantum state delivered by the source to detect potential tampering witha data stream. In quantum cryptography, two partners can exchange an encryption key as astream of quantum bits (photons) and detect eavesdropping on their communication line via theperturbation caused by the spy’s measurements. Single-photon sources improve the degreeof security of the quantum transmission by making it more difficult for the spy to hide hermeasurement actions. Quantum cryptography has now passed the demonstration stage andseems poised to become the first practical application of quantum information processing.

The third class of potential uses for single photons exploits entanglement, a key featureof quantum mechanics. In the last 15 years, a more and more precise roadmap for all-opticalquantum computing has been traced by a series of pioneering theoretical papers. The schemes


they propose are based on linear optics, the only nonlinearity required for computing beingcontained in the measurement process itself. Single-photon states play a central part in theselinear optical quantum computers. Bright and reliable sources of single photons would greatlyhelp to explore and demonstrate their working principles and, more generally, those of quantumteleportation, quantum networks and other fascinating quantum phenomena.

At present, however, such sources have only begun to appear. The first devices qualifyingas single-photon sources were faint atomic beams, producing single fluorescence photons,or atomic vapours with cascading two-photon transitions, yielding pairs of correlated singlephotons. The workhorses of today’s quantum-optics experiments remain the correlated pairs ofsingle photons obtained by degenerate parametric down-conversion of short pump pulses. Thisprocess is fairly bright and reliable and has the additional advantage of producing entangledpairs.

However, for all its advantages, parametric down-conversion only produces a Poissondistribution of photons, just as attenuated laser pulses would. To obtain single photons, onehas to either strongly attenuate a Poisson source, which strongly limits its brightness, or torely on a completely different architecture. By collecting the fluorescence photons from asingle quantum emitter, one ensures that no two photons are ever produced at the same time.This property called antibunching arises from various possible mechanisms, depending onthe emitters considered. Coupling the emitter to a resonant optical cavity makes it possibleto tailor the spontaneous emission towards the desired ends. In the weak coupling regime,the emission rate can be enhanced into the desired mode and often can be reduced for theleak modes. The strong-coupling regime makes other schemes possible, for example, theSTIRAP process, in which a single photon is produced in the cavity mode upon an adiabaticpassage.

We have described the various microscopic single emitters proposed so far: in gas phase,atoms and ions, and in condensed matter, organic molecules and polymers, colour centres,semiconductor nanocrystals and heterostructures. For each large class, we have compared theirphotophysical properties and their advantages and weaknesses and summarized them in table 2.Our main conclusion is that the emitter has to be chosen according to the application in view.If only a well-defined number of photons is important, as in the squeezed-light measurements,or for quantum cryptographic applications, simple devices operated at room temperature (withsingle molecules, colour centres or semiconductor nanocrystals) are sufficient. If, on theother hand, the application requires coherence between the single photons, as most quantumcomputation protocols do, the emitter must be extremely well shielded from fluctuations in itsenvironment. This condition leaves only two possibilities, either isolated atoms or ions in trapsin ultra-vacuum, or such emitters as molecules or self-assembled quantum dots in rigid matricesat cryogenic temperatures. In the latter class of solid-state sources, III–V semiconductorheterostructures benefit from all the knowhow developed for electronic microlithography,by which whole emitting structures, including the resonant cavity, can be produced in onemicrofabrication process.

Many technical problems must still be solved before single-photon sources become asstandard and user-friendly as lasers are. The simplest version of a photon gun, based on asingle diamond nanocrystal with a single colour centre at the focus of a microscope objective,already exists today. Such devices were already employed in early demonstrations of QKD.More sophisticated sources, in which successive photons could be brought to interfere, can inprinciple be built around single atoms or single quantum dots. However, the current setupsmust be downsized and simplified before routine use can be envisaged.

Even more importantly, all quantum processes consist of an intimate interaction(sometimes called transaction (Cramer 1986)) between emission and detection. As long as


its coherence has not been destroyed by interaction with a macroscopic bath, any quantumprocess must be considered as open to superposition and interference. Progress in the controlof quantum processes has been very fast lately on the sources’ side, but it cannot be separatedfrom progress on the detectors’ side. Higher collection and transmission efficiencies andhigher detection yields are indispensable for fully benefiting from the quantum cornucopia.While silicon detectors almost reach unity yield in the red part of the spectrum, efficientand dedicated detectors must be built for near-infrared communication wavelengths. Themanipulation of single photons has been made possible by the existence of single-photondetectors. The manipulation of higher number-states would benefit from detectors capable ofdistinguishing between multi-photon states, for example superconducting bolometers (Cabreraet al 1998). Quantum cryptography over large fibre distances will also require progress in fibretransmission.

Many quantum processing schemes rely on entangled pairs of particles. It is a challengingaim to produce entangled pairs of single photons from a compact, solid-state device. A possiblescheme would be the emission of polarization-entangled photons from a bi-exciton state ina single quantum dot (Benson et al 2000). Coupling to a resonant cavity can improve theefficiency and the characteristics of the single photons. The reduced mode volumes of photonic-crystal microcavities would be particularly interesting for reaching the strong-coupling regimein the solid state. With high-efficiency cavities, localized electronic states of molecules orimpurity-bound excitons in semiconductors could present a spectral luminescence purity closeto that of atoms.

Even with resonant-cavity enhancement, the spontaneous emission lifetimes of theemitters discussed in this paper are longer than 100 ps. Higher emission rates require shorterlifetimes. One could think of employing more efficient emitters. Large aggregates ofmolecules, for example J-aggregates, present enhanced fluorescence rates due to cooperativeinterference of the molecules in a coherence domain. However, the single-photon emissionrate would be limited by the relaxation of higher excited states, with lifetimes of the order of1 ps. Higher spontaneous emission rates would mean that two excitations could produce twophotons simultaneously. Inefficient emitters with high non-radiative relaxation and thereforevery short emission times, such as metal particles, would present the same problem. The easiestsolution is therefore to rely on cavity enhancement of the current emitters to decrease theiremission lifetimes down to some tens of picoseconds.

The potential uses of single-photon sources are many, as suggested by the current ideaswe have discussed. If the realization closest to us seems to be quantum cryptography, the fieldof quantum information processing is a quickly moving one, and new ideas keep croppingup. Last century’s biggest revolution in optics was the discovery of the laser. Today’s mostmundane uses of the laser are CD players and bar-code readers, two applications which couldhardly have been imagined 50 years ago, when the laser was invented. It is a safe bet thatroutine manipulation of quantum states will beget totally unsuspected ideas and devices. Thedevelopment of bright and reliable single-photon sources appears a most accessible step in thisdirection.

Acknowledgments

The authors thank Professor G Nienhuis for critical reading of the manuscript andsuggestions, as well as many other colleagues for helpful discussions, in particular: I Abram,G Abstreiter, J-M Gerard, E Giacobino, M Dahan, K Karrai, F Kulzer, W E Moerner, J-F Roch,P Tamarat, C Weisbuch and U Woggon.


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