Single-photon sources

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Single-photon sources

To cite this article: Brahim Lounis and Michel Orrit 2005 Rep Prog Phys 68 1129

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Single-photon sources

Brahim Lounis¹ and Michel Orrit²

¹ C P M O H, Universite Bordeaux I, 351 Cours de la Lib´ eration F-33405, Talence Cedex,´ France

² MoNOS, Leiden Institute of Physics, Huygens Laboratory, Postbus 9504, NL-2300 RA, Leiden, The Netherlands

Received 3 November 2004

Published 21 April 2005

Online at stacks.iop.org/RoPP/68/1129

Abstract

The concept of the photon, central to Einstein’s explanation of the photoelectric effect, is exactly 100 years old. Yet, while photons have been detected individually for more than 50 years, devices producing individual photons on demand have only appeared in the last few years. New concepts for single-photon sources, or ‘photon guns’, have originated from recent progress in the optical detection, characterization and manipulation of single quantum objects. Single emitters usually deliver photons one at a time. This so-called antibunching of emitted photons can arise from various mechanisms, but ensures that the probability of obtaining two or more photons at the same time remains negligible. We briefly recall basic concepts in quantum optics and discuss potential applications of single-photon states to optical processing of quantum information: cryptography, computing and communication. A photon gun’s properties are significantly improved by coupling it to a resonant cavity mode, either in the Purcell or strong-coupling regimes. We briefly recall early production of single photons with atomic beams, and the operation principles of macroscopic parametric sources, which are used in an overwhelming majority of quantum-optical experiments. We then review the photophysical and spectroscopic properties and compare the advantages and weaknesses of various single nanometre-scale objects used as single-photon sources: atoms or ions in the gas phase and, in condensed matter, organic molecules, defect centres, semiconductor nanocrystals and heterostructures. As new generations of sources are developed, coupling to cavities and nano-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 from demonstrations to applications and to bring with them the first practical uses of quantum information.

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

Contents

1. Introduction 1131
2. Photon optics 1133
   2.1. Quantized radiation field 1133
   2.2. Different flavours of light 1134
   2.3. Antibunching and second-order correlation function 1137
   2.4. Indistinguishable photons 1139
   2.5. Summary: characteristics of the emitted photons 1141
3. Possible uses of single photons 1142
   3.1. Measurement of weak absorptions 1143
   3.2. Random number generation 1144
   3.3. Quantum information processing 1144
4. Coupling to cavities 1154
   4.1. Weak-coupling regime 1154
   4.2. Strong-coupling regime 1156
5. Early and macroscopic sources 1158
   5.1. Faint laser pulses 1158
   5.2. Entangled photon pairs 1159
6. Microscopic single-photon sources 1161
   6.1. Excitation schemes 1161
   6.2. Atoms, ions in gas phase 1162
   6.3. Organic molecules 1163
   6.4. Colour centres 1166
   6.5. Semiconductor nanocrystals 1168
   6.6. Self-assembled quantum dots and other heterostructures 1170
   6.7. Summary 1172
7. Conclusion and outlook 1173
   Acknowledgments 1175
   References 1176

1. Introduction

Nearly one century after the discovery of its basic principles, quantum mechanics remains enshrouded in mystery and paradox (Bouwmeester et al 2000). Very early on, several of its counter-intuitive predictions were pointed out and gave rise to sometimes conflicting interpretations. Fortunately, quantum mechanics has also, meanwhile, become the most accurately tested of all physical theories. Strange though it may seem to us, macroscopic beings, it has spawned a variety of clever schemes for storing, transporting and processing information. 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 shows a ‘grainy’ nature and can be described as consisting of particles, called photons. A ‘numberstate’ 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-states of light are more exotic. Except for the trivial vacuum-state $(n=0)$ , number-states of light are unfamiliar and very hard to obtain in practice. A stabilized laser delivers light pulses that seem extremely regular in amplitude, but this is only because they contain so many photons that their number fluctuations become negligible on a macroscopic scale. Indeed, such pulses are just superpositions of many number-states with different ns. Restricting fluctuations in the number of photons seems easiest for states with $n=1$ , single-photon states. Recent progress in the fabrication, manipulation and characterization of individual nano-objects, molecules and nanocrystals has opened new routes for the production of number-states containing one single photon. The purpose of this paper is to review current ideas and recent developments in the field of single-photon sources.

The idea of manipulating single photons did not arise immediately from quantum principles. 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 was introduced by Einstein, just one century ago (Einstein 1905), to interpret the photoelectric effect. However, it was later realized that the photoelectric effect results from the quantum nature ofatoms 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’s impulsion (Glauber 1965), did researchers begin to look for specific quantum properties of light that could not be understood in a classical frame. A crucial step in this process was the correlation experiment of Hanbury Brown and Twiss (1956), who proposed the use of coincidences 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 the second-order coherence of light (Born and Wolf 1980), which was later reformulated in terms of quantum-mechanical operators (Mandel and Wolf 1995). Field interference, a classical concept in wave optics, was generalized to intensity interference, i.e. to quantum effects involving numbers of photons instead of amplitudes of electric fields. Number-states, and single-photon states in particular, then appear as special cases of squeezed light, i.e. light for which quantum fluctuations have been reduced on one component, or quadrature, at the expense of enhanced fluctuations on the conjugate component (Walls and Milburn 1994). We review some basic notions of quantum optics in section 2, in order to define the concepts and notation used in the remainder of this paper.

Assuming that the challenging problem of producing number-states of light has been solved, what new kinds of experiments do they open? The motivation for the production of single photons is examined in section 3. The detection of weak absorption signals is in general hampered by photon noise, i.e. fluctuations of the number of photons. The low noise level of a single-photon source would be very valuable for such measurements. Another appealing application is quantum cryptography, more precisely quantum key distribution (QKD). In this case, the well-defined character of the single-photon state is exploited to detect and foil eavesdropping on the transmission of a secret key between two parties, which they can use later for encryption. An eavesdropper can escape detection while attempting to extract information from a statistical mixture of number-states, but not from a well-prepared number-state. Singlephoton sources are crucial components in recent proposals for quantum computation by means of linear optics (Knill et al 2001), the only nonlinear step being the detection.

Figure 1. Although a classical or Schrodinger wave is equally divided by a beam splitter, each ¨ single 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.

The first successful generation of single photons (Clauser 1974) was based on a cascade transition of calcium atoms. Each excited atom delivers a couple of two photons with different colours. A photon at one of the wavelengths is detected after spectral filtering, and is used for the conditional detection of its companion at the other wavelength. Each single photon is thus a ‘herald’ for the presence of its companion. One-photon states present a peculiar anticorrelation effect, which does not exist for a classical wave. If we send a one-photon state on a beam splitter and place photon-counting detectors on the reflected and transmitted beams, we never observe any coincidence between counts measured by the two detectors, as this would violate energy conservation (see figure 1). As a consequence of the principles of quantum mechanics, the wavefunction of the photon has to collapse onto either one or the other of the two detectors. This absence of coincidences of detection events on the two detectors has been dubbed photon antibunching and is utterly irreconcilable with a classical description o light. It was first observed in the fluorescence from an attenuated sodium atomic beam, where at most one atom was present in the excitation focus at any time (Kimble et al 1977, Walls 1979, 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 effects such 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 of single photons, their brightness was very low, and a further drawback was that the operation of the 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 the same ion (Diedrich and Walther 1987). These long series of antibunched photons came closer to 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 downconversion (Burnham and Weinberg 1970, Hong and Mandel 1986). When a short laser pulse is sent into a nonlinear crystal, it generates pairs of signal and idler photons, which are highly correlated in space and time. Provided the probability of generating two pairs at the same time remains negligible, such correlated pairs can be used as sources of heralded single photons. To this day, the parametric sources are the workhorses of quantum-optics experiments. These and the early atomic sources will be briefly described and discussed in section 5. A microscopic emitter 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 the spontaneous emission rate and thereby the rate of photon production, it can channel the emitted photons into a well-defined spatial mode to improve collection efficiency and to match further utilization, or it can restrict the spectral range of the emission. The principles of coupling to a cavity are reviewed in section 4.

In the early 1990s, sensitive detection of fluorescence led to the detection of single organic molecules (Orrit and Bernard 1990) and soon thereafter of single semiconductor heterostructures (Birotheau et al 1992, Brunner et al 1992). Some of the quantum-optics experiments conducted earlier on atoms and ions in the gas phase could then be reproduced in condensed matter, where single objects are much easier to manipulate. For example, photon antibunching was detected in the fluorescence of single molecules (Basche´ et al 1992) and in the emission of other individual nano-objects in condensed matter. Nano-objects were soon proposed as possible sources of single photons (De Martini et al 1996, Kitson et al 1998). Antibunching is an essential ingredient for the production of single photons, but as we discuss in section 6, the photophysical mechanisms by which it arises are different for the different kinds of emitters. Yet, antibunching alone is by no means sufficient for obtaining single photons on demand. Perfect antibunching means that the probability of emitting two or more photons at the same time is nil, but we also require that the source emit one photon with certainty. 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 case for most of the emitters we discuss here.

(ii) The excited state of the emitter must be prepared with certainty. Various schemes for preparing the emitting state have been proposed, for example, sweeping the sharp, lowtemperature resonance of a single molecule through resonance with a laser by an external electric 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 quantum well at sub-Kelvin temperatures (Kim et al 1999). Those early demonstrations have now been extended to many different systems which could become useful emitters of single photons. They include semiconductor nanocrystals (Lounis et al 2000, Michler et al 2000a, Messin et al 2001), colour centres in diamond (Brouri et al 2000, Kurtsiefer et al 2000), single quantum dots in various geometries (Michler et al 2000b, Santori et al 2001) and several more. Section 6 is devoted to a review of the photophysical properties of 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 and bottlenecks 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 and uses of single-photon sources. Complete presentations can be found in textbooks on quantum optics (Loudon 1983, Walls and Milburn 1994, Mandel and Wolf 1995).

2.1. Quantized radiation field

A mode of the electromagnetic field (wave vector $\vec{k},$ angular frequency ${\omega }_k$ , polarization $\vec{\epsilon }$ perpendicular to $\vec{k})$ in a cavity with volume $L^3$ can be directly quantized as a harmonic oscillator. Operators $a_{\vec{k},\vec{\epsilon }}^{+}$ and $a_{\vec{k},\vec{\epsilon }},$ respectively, create and annihilate one quantum—or photon—in this mode. The electric field at position $\vec{r}$ now becomes an operator, written as the sum of two Hermitian-conjugate terms:

$$\vec{E}(\vec{r})={\vec{E}}^{(-)}(\vec{r})+{\vec{E}}^{(+)}(\vec{r}).\text{\hspace{1em}(1)}$$

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

$${\vec{E}}^{(-)}(\vec{r},t)=\mathrm{i}\sum _{\vec{k},\vec{\epsilon }}\sqrt{\frac{\hslash {\omega }_k}{2{\epsilon }_0{L}^3}}\vec{\epsilon }{a}_{\vec{k},\vec{\epsilon }}^{+}{\mathrm{e}}^{\mathrm{i}(\vec{k}\cdot \vec{r}-{\omega }_{k}t)}.\text{\hspace{1em}(2)}$$

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,

$$I(t)=2{\epsilon }_{0}c⟨E^{(-)}(t)E^{(+)}(t)⟩,\text{\hspace{1em}(3)}$$

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

$$g^{(1)}(\tau )=\frac{⟨E^{(-)}(t+\tau )E^{(+)}(t)⟩}{⟨E^{(-)}(t)E^{(+)}(t)⟩}\text{\hspace{1em}(4)}$$

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

$$g^{(2)}(\tau )=\frac{⟨I(t+\tau )I(t)⟩}{⟨I(t){⟩}^2}\text{\hspace{1em}(5)}$$

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

$$g^{(2)}(\tau )=\frac{⟨E^{(-)}(t)E^{(-)}(t+\tau )E^{(+)}(t+\tau )E^{(+)}(t)⟩}{⟨E^{(-)}(t)E^{(+)}(t){⟩}^2}.\text{\hspace{1em}(6)}$$

2.2. Different flavours of light

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

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

$$\vec{E}=\sum _i{\mathrm{e}}^{\mathrm{i}{\phi }_i}{\vec{E}}_0(t-t_i).$$

The first-order correlation function of such a field is identical to that of a single emitter. The Fourier-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, one may average over phases in the numerator of equation (6), which leaves only two series of terms. One then finds the following relation between $g^{(2)}(\tau )$ and $g^{(1)}(\tau )$

$$g^{(2)}(\tau )=1+[g^{(1)}(\tau ){]}^2.\text{\hspace{1em}(7)}$$

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

Figure 2. Probability distribution of the number of photons for three sources with an average photon number $⟨n⟩=1$ . The thermal source presents large number fluctuations due to the Bose– Einstein statistics ofblack-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 can match this distribution by delivering single photons at regular time intervals.

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

$$p_{\mathrm{th}}(m)=\frac{⟨n{⟩}^m}{(1+⟨n⟩{)}^{m+1}},\text{\hspace{1em}(8)}$$

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

(b) Coherent light. The classical picture of laser light is a wave with constant amplitude and phase. The first- and second-order correlation functions of such a wave are obviously both equal 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 can be written in the basis of number-states (or Fock states) with n photons according to

$$|\alpha ⟩={\mathrm{e}}^{-|\alpha {|}^2/2}\sum _n\frac{{\alpha }^n}{\sqrt{n!}}|n⟩.\text{\hspace{1em}(9)}$$

Note that α is an eigenstate of the annihilation operator a (and therefore of operator $E^{(+)})$ with eigenvalue α. This property allows us to replace quantum-mechanical averages by classical functions. Coherent states are thus sometimes called quasi-classical states. The number of photons in a coherent state is a variable that fluctuates according to a Poisson distribution. The probability of finding m photons in the mode is given by (see figure 2)

$$p(m)=\frac{⟨n{⟩}^m}{m!}{\mathrm{e}}^{-⟨n⟩}\text{\hspace{1em}(10)}$$

with the average number of photons

$$⟨n⟩=|\alpha {|}^2$$

and the variance

$$\Delta n^2=⟨n^2⟩-⟨n{⟩}^2=⟨n⟩.\text{\hspace{1em}(11)}$$

These statistics are very different from those of thermal light. The maximum probability is to find n photons in the mode. In coherent states, the number of photons obeys the central limi theorem 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 photon noise) and is an absolute minimum for the noise of a macroscopic laser. Again, the probability distribution is very different from the desired one.

(c) Squeezed light. Being a harmonic oscillator, an electromagnetic mode can be described by two non-commuting operators analogous to position and momentum for a one-dimensional particle. These complementary or conjugated operators can be chosen as the electric field and vector potential or, in a classical wave representation, as the amplitude and phase. Complementarity in quantum mechanics requires that fluctuations of conjugate variables follow Heisenberg relations. Therefore, quantum mechanics authorizes so-called squeezed states of light, where fluctuations in one of these variables can be reduced (as compared with 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 photons at regular time intervals, and a single-photon source is a good example of it. The fluctuations of the number of photons emitted by a squeezed source are weaker than those of a coherent state. They are sub-Poissonian, and their deviation from Poisson statistics can be characterized by the time-dependent Mandel parameter:

$$Q(T)=\frac{⟨n^2{⟩}_T-⟨n{⟩}_T^2}{⟨n{⟩}_T}-1,\text{\hspace{1em}(12)}$$

where the notation $⟨\cdot \cdot \cdot {⟩}_T$ stands for ‘averaged over a time-interval $T^{\prime }$ . The Mandel parameter of a Poisson source is nil at all times and that of an ideal single-photon source delivering photons at regular time intervals is 1. The Mandel parameter is related to the second-order correlation function by (Short and Mandel 1983)

$$Q(T)=\frac{2⟨I(t)⟩}{T}{\int }_0^T\mathrm{d}\tau {\int }_0^{\tau }\mathrm{d}{\tau }^{\prime }(g^{(2)}({\tau }^{\prime })-1),\text{\hspace{1em}(13)}$$

where $⟨I(t)⟩$ is the average emission rate (in units of photons per second).

2.3. Antibunching and second-order correlationfunction

Sub-Poissonian statistics were first demonstrated in the fluorescence of a very faint atomic beam excited by a continuous laser (Kimble et al 1977). Because the atomic beam is faint, at most 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 an attenuated laser beam, where two photons can be separated by any delay, very short delays are not allowed for photons emitted by a single atom. The photons have a tendency to be emitted one by one, i.e. they are ‘antibunched’¹.

Antibunching may arise from different mechanisms, as will be discussed in section 6. For two-level systems such as atoms, the observation of one photon projects the system into the ground 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 excited levels, which can be reached resonantly via absorption of a second photon by the first excited state. In most cases, however, relaxation back to the first excited state is so fast that no emission from these higher excited states can be detected. As a consequence, since only emission between the lowest two levels can be detected, the complex system behaves as a two-level emitter.

With ideal detectors, antibunching would be easy to demonstrate, by simply recording the stream of fluorescence photons with a time-resolution better than the fluorescence lifetime. Unfortunately, real photon-counting detectors have their limitations. The dead-time is that period of blindness of the detector just after detection of one photon, which arises from the re-loading time of the photomultiplier dynodes or semiconductor junction. Dead-times can be as long as tens of nanoseconds. On even longer timescales, stray charges released upon the detecting avalanche can lead to spurious counts called after-pulses. The practical way of eliminating these factitious events and the blind periods is to split the detection beam and use two 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 afterpulses. 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 gives the distribution of pairs of consecutive photons, whereas the second-order correlation function gives the distribution of all pairs of photons. These two distributions, related by a simple equation, are nearly equal for short delays (Reynaud 1983, Lounis et al 2000, Verberk and Orrit 2003). For perfectly antibunched light, the correlation function therefore cancels for zero time:

$$g^{(2)}(\tau =0)=0.$$

Note that this relation is totally incompatible with the classical equation (5), for any classical stationary random function of time must satisfy the Schwartz inequality $⟨I(t^{\prime })I(t)⟩⩽⟨I(t{)}^2⟩$ The important feature of antibunching for us is that the single emitter ‘picks’ a photon out of the 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 other words, it acts as a nonlinear filter, eliminating the shot noise of the laser, albeit for short time intervals, and only for those photons redirected into the detection mode.

(a)

(b)

(d)

Figure 3. (a) Schematic diagram of a Hanbury Brown and Twiss experiment. Coincidences between counts on the two detectors give access to the (second-order) correlation function. For a long coherence lifetime and high resonant exciting intensity (b), Rabi oscillations appear around the antibunching dip. For non-resonant excitation, or a very short coherence lifetime (particularly at room temperature), only an exponential dip appears (c). A pulsed excitation gives rise to periodic emission 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 antibunched light 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 high excitation intensity (figure 3(b)). For most room-temperature emitters, however, the coherence time is extremely short (often shorter than picoseconds). The correlation function presents a dip 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 pulse will give rise to the emission of a single photon. The correlation function of the pulsed laser, as measured in a Hanbury Brown and Twiss setup, is a Dirac comb of short peaks (each peak being the 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 a decaying exponential probability. The correlation function is again a Dirac comb of peaks, bu with the zero-time peak missing (see figure 3(d)). Since only one photon is produced, there can be no coincidence between the two detectors for zero delay. The shape of the correlation peaks is again the convolution of the decaying exponential, with itself $(\exp (-|t-nT|/{\tau }_{\mathrm{f}})$ where ${\tau }_{\mathrm{f}}$ is the decay time of the emitter and $n\ne 0$ is the order of the correlation peak).

To conclude this discussion, let us remark on two important features of the spontaneous emission 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 state e 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 of possible modes k (with nearly the same energy), described by the wavefunction:

$$|\psi (t)⟩=c_{\mathrm{e}}(t)|e,0⟩+\sum _k{c}_k(t)|g,1_k⟩$$

with $c_{\mathrm{e}}(t)=\exp (-t/2{\tau }_{\mathrm{f}}-\mathrm{i}{\omega }_{\mathrm{e}}t)$ and

$$c_k(t)=\frac{V_k/\hslash }{{\omega }_k-{\omega }_{\mathrm{e}}+\mathrm{i}/2{\tau }_{\mathrm{f}}}\left(1-\exp \left[-\frac{t}{2{\tau }_{\mathrm{f}}}+\mathrm{i}({\omega }_k-{\omega }_0)t\right]\right).$$

$\hslash {\omega }_{\mathrm{e}}$ and $\hslash {\omega }_k$ are the energies of the initial and final states, $V_k$ the dipole coupling matrix element and ${\tau }_{\mathrm{f}}$ is the spontaneous emission lifetime.

(ii) The decaying exponentials show that the emission process is continuous in the above representation. Only a subsequent measurement of the photon by a detector will project state $|\psi (t)⟩$ 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 by introducing 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 on interference between two single photons. A single-photon source producing photons at predetermined times can be used to generate pairs of photons if a suitable pumping sequence is 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 perfectly indistinguishable photons. However, dephasing and spectral diffusion, which arise from fast and slow fluctuations of the transition frequency, complicate this picture. Hereafter, we briefly discuss some possible experiments with two-photon states, and the specific properties of the single-photon source they require.

Fearn and Loudon (1987) and Hong et al (1987) pointed out that two photons incident at the same time on the two input ports ofa 50%/50% beam splitter interfere in such a way that they both exit from one of the output ports, as illustrated in figure 4. This effect is a consequence of the 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 delay between photons is varied, the rate of coincidences on the two output detectors drops for zero delay. In order to give rise to a fully destructive interference, the two photons must be completely indistinguishable, i.e. they must be in exactly the same mode. If the photons are described by space–time wavepackets, the two wavepackets must be identical for the two photons, and they must arrive simultaneously on the beam splitter. Let us consider a source emitting all photons in the same spatial mode and discuss only the frequency wavepacket. If the spectrum of a single-photon source is Fourier-transform-limited, i.e. if each photon can be described by the same coherent wavepacket, two photons will be indistinguishable. In many cases, however, the spectrum of a source is broader than the Fourier-transform of the time-profile of the emitted pulse. This broadening, arising from fluctuations of the optical resonance frequency, can be described as dephasing or spectral diffusion, depending on the rate 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 from many ‘collision’ events with a bath, leading to a gradual loss of phase with the dephasing (or decoherence) time, $T_2$ , shorter than twice the fluorescence lifetime, $T_1\mathrm{i}$ :

Figure 4. Photon coalescence. Two single-mode, but otherwise independent, photons simultaneously enter the two ports of a 50–50 beam splitter. They give rise to transmitted or reflected 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 one must be reflected. Both photons emerge at the same output port, as if they had coalesced.

$$\frac{1}{T_2}=\frac{1}{2T_1}+\frac{1}{T_2^{\ast }},$$

where $T_2^{\ast }$ 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, individual photons can be seen as short incoherent wavepackets emitted at random times during the long fluorescence lifetime. Since different wavepackets will be randomly shifted in time, the photon interference effect will be reduced. The depth of the two-photon interference effect will no longer be 100% but only $T_2/2T_1$ (Bylander et al 2003).

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

(iii) An additional loss of indistinguishability arises from incoherent pumping. If the relaxation process 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 important in the case of self-assembled quantum dots whose lifetimes are shortened by the Purcell effect (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 in section 3.

2.5. Summary: characteristics ofthe emitted photons

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

(i) Spatial mode: To be used in interference, telecommunication or computation experiments, photons have to be collected and collimated into a given spatial mode. Unless the emitter undergoes spatial motion or reorientations, a solid-state source based on a microscopic emitter will in general emit all its photons in the same spatial mode. Mobile emitters such as atoms may deliver different spatial wavepackets at different times or may couple 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 all subsequent optical components, in particular that of the detector.

(ii) Spectrum: The desirable spectrum of a single-photon source depends on the application and on the further use of photons. For example, the propagation of photons in optical telecommunication 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-photon source, the detector must be a photon-counter with a high detection yield, which is not readily available in the near-infrared domain. The cheapest and most convenient detectors today are based on silicon, whose detection efficiency peaks around 750 nm and does not extend much beyond 1000 nm on the IR side. Photon-counting devices based on III–V semiconductors now become available in the near-IR window (Hiskett et al 2000, Rarity et al 2000, Stucki et al 2001).

(iii) Emission lifetime: Most of the single-photon sources we consider here are based on spontaneous emission. The excited state of an individual quantum system (an atom, a molecule or a quantum dot) is created in a rapid process, and the subsequent emission follows on a longer timescale. Therefore, the spontaneous emission lifetime eventually limits the rate at which the single photons can be emitted and is an important feature of the source. The emission lifetime in vacuum ${\tau }_{\mathrm{f}}(1)$ is a characteristic of the emitter. It is determined by the transition frequency, ${\omega }_{\mathrm{eg}},$ , and by the transition dipole moment, ${\mu }_{\mathrm{eg}},$ between ground and excited states:

$${\tau }_{\mathrm{f}}^{-1}(1)=\frac{4}{3}\frac{{\mu }_{\mathrm{eg}}^2}{4\pi {\epsilon }_0\hslash }{\left(\frac{{\omega }_{\mathrm{eg}}}{c}\right)}^3.\text{\hspace{1em}(14)}$$

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

$${\tau }_{\mathrm{f}}(n)=\frac{1}{n}{\tau }_{\mathrm{f}}(1).$$

Moreover, the induced dipoles in the surrounding dielectric also participate in the coupling to radiation. In the approximation of the spherical Lorentz cavity, and if the polarizability of the 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 Kopelman 1981)

$${\tau }_{\mathrm{f}}(n)=\frac{1}{n}{\left(\frac{3}{n^2+2}\right)}^2{\tau }_{\mathrm{f}}(1).\text{\hspace{1em}(15)}$$

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

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

(iv) Emission linewidth and indistinguishability: The frequency spectrum of a source is the Fourier-transform of the correlation function of the electric dipole operator. Electronic oscillations cannot live longer than the emission lifetime, $T_1.$ , determined by spontaneous emission and other population relaxation processes, but their coherence can be further shortened by dephasing (or decoherence), arising from perturbations of the oscillation frequency through interactions with a bath. In the absence of slow spectral diffusion, the resulting frequency linewidth (full-width at half-maximum),

$$\Delta \nu =\frac{1}{2\pi T_1}+\frac{1}{\pi T_2^{\ast }},\text{\hspace{1em}(16)}$$

takes its minimum possible value only when dephasing is negligible, i.e. when $T_2^{\ast }=\infty$ . In that case, one has a lifetime-limited linewidth. For most systems in condensed matter at room temperature, the dephasing time is by several orders of magnitude shorter than the excited state lifetime, i.e. the linewidth is very far from being lifetime-limited.

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

(v) Polarization state: It is determined by the microscopic nature of the emitter and by the way it is coupled to the emission mode. For example, for a linear dipole emitter such as a single molecule, the polarization pattern of the emitted photons depends on the orientation of the dipole moment (Fourkas 2001, Novotny et al 2001). If the dipole lies in the focal plane of the collection objective, the polarization is close (but not identical) to linear, parallel to the dipole, for all photons across the whole spatial mode. If the dipole is perpendicular to the focal plane, 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 in spectroscopy and quantum optics. Hereafter, we review some proposed uses and recent experiments using single photons. These demonstrations are a strong motivation to develop compact and efficient sources. We have little doubt, however, that the availability of such sources 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 of coherent light (Kimble and Walls 1987, Drummond 1999). A single-photon source delivers amplitude-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 weak absorption 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 an absorbing medium (intensity transmission coefficient T ) and measured by a photon-counting detector (quantum yield x). The distribution of the detected counts is also Poissonian, with the average number of detected counts (supposed to be large)

$$S_{\mathrm{C}}=xMT$$

and with the shot noise (standard deviation)

$$N_{\mathrm{C}}=\sqrt{xMT}.$$

To measure the transmission coefficient, we repeat this measurement without the absorber and 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 measured with the coherent source:

$$\Delta T_{\mathrm{C}}^2=\frac{1}{xM}T(T+1).$$

If we now repeat this experiment with an ideal single-photon source, the average number of detected counts is the same, but the distribution of photon counts is now binomial. Each photon emitted 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

$$p_x(n)=\left[\begin{array}{c}n\\ M\end{array}\right](xT{)}^n(1-xT{)}^{M-n}.\text{\hspace{1em}(17)}$$

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

$$N_{\mathrm{SP}}=\sqrt{xMT(1-xT)}.$$

For the transmission measurement, the variance becomes

$$\Delta T_{\mathrm{SP}}^2=\frac{1}{xM}T(T+1-2xT).$$

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

$$\frac{\Delta T_{\mathrm{SP}}^2}{\Delta T_{\mathrm{C}}^2}=1-2x\frac{T}{1+T},\text{\hspace{1em}(18)}$$

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

3.2. Random number generation

Random numbers are key ingredients at many levels of information processing, from computational methods (Monte Carlo simulations (Alspector et al 1989) for example) to the 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 generators rely on numerical algorithms implemented on a computer. Physical random generators use the randomness or noise of a physical observable (e.g. that of an electronic device). In both approaches, systematic errors and perturbations lead to deviations from a truly random distribution of numbers.

A clear physical source of randomness is the elementary collapse or decision process in quantum mechanics. Rarity and collaborators (Rarity et al 1994) proposed a physical quantummechanical random generator based on the 50 : 50 splitting of a beam of single photons. Each individual photon coming from the light source and going through the 50 : 50 beam splitter has an equal probability of being reflected or transmitted. Quantum theory predicts that, for each photon, the ‘which-way’ decision is truly random and independent of history or other experimental 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 wil in 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 anticorrelations between successive bits.

3.3. Quantum information processing

Quantum mechanics offers original and powerful schemes for processing information, which could 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 individua photons would present specific advantages (Bouwmeester et al 2000, Di Vincenzo 2000):

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

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

(iii) If, moreover, quantum computing can be performed on photons, they can serve as ‘flying qubits’ (Chuang and Yamamoto 1995, Turchette et al 1995, Knill et al 2001), which could considerably 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 technical bottlenecks and developing quantum components with simple hardware. Below, we review recent theoretical and experimental advances in quantum information processing, in the hope that individual photons will lead to improved schemes and developments.

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

$$|\psi ⟩=\alpha |0⟩+\beta |1⟩,\text{\hspace{1em}(19)}$$

where α and $\beta$ are complex amplitudes. A measurement projects that state into either eigenstate, 0 or 1 , with respective probabilities $|\alpha {|}^2$ and $|\beta {|}^2=\bar{1}-|\alpha {|}^2$ . Although a qubit may at first sight seem a rather trivial generalization of a classical bit, the continuous complex amplitudes α and $\beta$ also convey potential information and introduce qualitatively new possibilities. The power of quantum information rests on quantum-mechanical superpositions, which open an infinity 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), an interesting one is single photons. For example, a qubit can be a photon with two eigenstates represented by horizontal $|H⟩$ or vertical $|V⟩$ polarizations in a given basis. Logical states can also be coded as physical qubits using one photon in one of two output modes of a beam splitter: $|0{⟩}_L\equiv |1{⟩}_1|0{⟩}_2$ and $|1{⟩}_L\equiv |0{⟩}_1|1{⟩}_2$ ₂.

3.3.2. Quantum entanglement, EPR paradox and Bell’s inequality. What happens if we try to encode two bits of information onto two quantum particles, for example two photons labelled 1 and $2?$ The classical approach would be to code one bit of information onto each photon separately (one bit per photon), leading to four possibilities ${}^{\cdot }00^{\cdot },{}^{\cdot }01^{\cdot },{}^{\cdot }10^{\cdot }$ and ${\cdot }_{11},$ to which are associated the four states $|H_1{H}_2⟩,|H_1{V}_2⟩,|V_1{H}_2⟩$ and $|V_1{V}_2⟩$ . These are pure states, or product states, where, for example in the $|H_1{V}_2⟩$ state, the first photon is horizontally polarized and the second one is vertically polarized.

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

$$|\psi ⟩=\alpha |H_1{H}_2⟩+\beta |H_1{V}_2⟩+\gamma |V_1{H}_2⟩+\delta |V_1{V}_2⟩.\text{\hspace{1em}(20)}$$

The important new feature of this superposition is ‘entanglement’, i.e. in this new quantum state the two particles are no longer separable, even if their spatial wavepackets are physically separated! A perturbation or a measurement at the position of one particle instantaneously modifies the whole wavefunction, as also at the remote position of the other particle. As pointed out by Einstein et al (1935), the instantaneous character of the quantum measurement at first sight appears inconsistent with relativistic causality, seemingly involving some ‘spooky action at-a-distance’. Indeed, the quantum correlations arising from entanglement are stronger than classical correlations, for example, those which would arise from hidden variables (Bell 1965, Clauser and Shimony 1978).

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

$$\begin{array}{l}|{\psi }_{+}⟩=\frac{1}{\sqrt{2}}(|H_1{V}_2⟩+|V_1{H}_2⟩),\\ |{\psi }_{-}⟩=\frac{1}{\sqrt{2}}(|H_1{V}_2⟩-|V_1{H}_2⟩),\\ |{\phi }_{+}⟩=\frac{1}{\sqrt{2}}(|H_1{H}_2⟩+|V_1{V}_2⟩),\\ |{\phi }_{-}⟩=\frac{1}{\sqrt{2}}(|H_1{H}_2⟩-|V_1{V}_2⟩).\end{array}\text{\hspace{1em}(21)}$$

(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 antisymmetric superpositions are allowed.)

Each Bell state represents a coherent superposition of two possibilities. In the former pair of states, $|{\psi }_{+}⟩$ and $|{\psi }_{-}⟩$ , the two qubits are different, and in the latter pair, $|{\phi }_{+}⟩$ and $|{\phi }_{-}⟩$ , they are 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 1 can be in state H and photon 2 in state V or vice-versa, but the question of which photon has which polarization is clearly meaningless. All of the information is therefore distributed among 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₊ , flying apart along two different directions 1 and 2 (modes 1 and 2). Their polarization is measured by 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 will obtain ‘0’ or ‘1’ with equal probability, the actual result being completely random. If Bob chooses the same basis, he will always obtain the opposite result. If Alice knows that Bob used the same basis, she can deduce his result from her own with certainty, even though she has no means to communicate with him. However, a similar outcome would arise if the quantum state had been pre-determined by a hidden variable. In Einstein’s view, such hidden variables were required to explain the EPR paradoxon, which proved the incompleteness of quantum mechanics. In 1965, Bell investigated possible correlations for a thought experiment in which Alice and Bob choose bases that are at oblique angles. Bell proved that hidden-variable theories predict that the probabilities of the different measurements satisfy a certain inequality (Bell 1965). Bell’s inequality was later proved experimentally to be violated, in agreement with standard quantum-mechanical predictions.

A first series of experimental tests of Bell inequalities used photon pairs produced in atomic radiative cascades (Clauser and Shimony 1978, Aspect et al 1981) (section 5.2.1). Although these early setups were far from ideal (they used single-channel polarizers), their results agreed with 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 gave an unambiguous violation of Bell’s inequalities by tens of standard deviations, and a perfect agreement with quantum mechanics.

The second generation of tests began in the late 1980s (Ou et al 1988, Shih and Alley 1988) and used photon pairs generated in nonlinear crystals by spontaneous parametric downconversion (SPDC) (section 5.2.2). The remarkable feature of parametric sources is the production 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 (Tapster et al 1994), and later tens of kilometres (Tittel et al 1998)). All these experiments have confirmed 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, by unspecified processes. Bell proposed to randomly rotate the polarizers during the flight of the photons to make communication between the polarizers causally impossible. This loophole has now been closed, first in an experiment by Aspect et al (1982a) where the directions of polarization analysis were switched after the photons left the source, and more recently by Zeilinger’s group (Weihs et al 1998), who used sufficiently remote measurement stations, ultrafast and random analyser settings and two completely independent data acquisition systems.

The second loophole arises from the weakness of the detection yield in all experiments so far. One always assumes that the recorded pairs are an unbiased sample of the ensemble. To close that loophole, an ultimate experiment would need a much higher collection and detection efficiency 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 pairs with a quantum-dot source (section 6.6). This experiment relies on two crucial features of the single-photon source, namely its ability to suppress multiphoton pulses and its ability to generate consecutively two indistinguishable photons. The next step is to ‘collide’ these photons with suitable polarizations at two conjugated input ports of a non-polarizing beam splitter. In such a way, the violation of Bell’s inequality was demonstrated with pairs of independent photons (Fattal et al 2004b).

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

To show that Bell’s inequality is violated in a two-particle experiment, it is necessary to perform a statistical experiment: the violation cannot be demonstrated with a single measurement. However, this does not apply to three-photon entangled states such as $(|H_1{H}_2{H}_3⟩+|V_1{V}_2{V}_3⟩)/\sqrt{2}$ , the Greenberger, Horne and Zeilinger (GHZ) states (Greenberger et al 1989). There, the contradiction between the Einstein–Podolsky–Rosen assumptions and quantum mechanics arises even for an individual event. A single measurement (of either three vertically polarized photons or of a horizontal polarization for two photons and a vertical polarization for the third one) will discriminate with certainty between quantum mechanics and a local hidden-variable theory. The Innsbruck group has realized this experiment using sources of photon pairs from parametric down-conversion (Pan et al 2000). The preparation of GHZ states with a single-photon source would provide clearer violation and open series of more accurate tests.

3.3.3. Quantum teleportation. In 1993, Bennett and colleagues described a remarkable protocol for transporting a quantum state from one location to another (Bennett et al 1993), a protocol that succeeds even though the quantum state is and remains completely unknown, and even undetermined, on both sides. Such quantum teleportation makes use of entanglement, the distinctive feature of quantum mechanics. In the proposed scheme, Alice wants to teleport an unknown quantum state, for example, the polarization state of a single photon $|\psi {⟩}_1=\alpha |H_1⟩+\beta |V_1⟩$ , to Bob (see figure 5). They both agree to share a maximally entangled pair of qubits, known as the ancillary pair, of photons 2 and 3.

The entangled ancillary pair can for example be prepared in the state $|{\phi }_{+}{⟩}_{23}=$ $(1/\sqrt{2})(|H_2{H}_3⟩+|V_2{V}_3⟩)$ , one of the four possible Bell states of (21). The overall state $|\Psi {⟩}_{123}$ of the three photons is then

$$\begin{array}{rl}& |\Psi {⟩}_{123}=(\alpha |H_1⟩+\beta |V_1⟩)(|H_2{H}_3⟩+|V_2{V}_3⟩)/\sqrt{2}\\ & \propto |{\phi }_{+}{⟩}_{12}(\alpha |H_3⟩+\beta |V_3⟩)+|{\phi }_{-}{⟩}_{12}(\alpha |H_3⟩-\beta |V_3⟩)\\ & +|{\psi }_{+}{⟩}_{12}(\alpha |V_3⟩+\beta |H_3⟩)+|{\psi }_{-}{⟩}_{12}(\alpha |V_3⟩-\beta |H_3⟩).\end{array}\text{\hspace{1em}(22)}$$

Figure 6. Single-mode teleportation. Alice’s target qubit $|{\Psi }_{\mathrm{t}}⟩$ is now defined by a superposition between modes 1 and 2. Similarly, an auxiliary qubit $|{\Psi }_{\mathrm{a}}⟩$ is encoded in modes 3 and 4. Alice mixes modes (2) and (3) in a beam splitter (BS2) and measures the outputs with detectors C and D. 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 and one of the ancillary photons (photon 2): She mixes them in a 50/50 beam splitter and measures the output ports with polarization-sensitive detectors. She randomly obtains one of the four possible Bell results given in (21). This measurement projects the other ancillary photon into a quantum state uniquely related to the original one according to equation (22). Bob’s lone qubit, being correlated with the state measured by Alice, now carries the quantum information α and $\beta .$ . After Alice tells Bob by classical means which Bell state she measured, Bob performs a prescribed local qubit manipulation on his one qubit to replicate the initial state $\alpha |H⟩+\beta |V⟩$ For example, if Alice reports a measurement of $|{\phi }_{+}⟩$ , Bob does nothing to his qubit. If Alice instead reports a measurement of $|{\psi }_{-}⟩$ , Bob flips his qubit and adds a π-phase shift to one of the polarizations with a half-wave plate. Similar flips and phase-shifts can be performed for the other two measurements of Alice. In this way, the qubit $\alpha |H⟩+\beta |V⟩$ is ‘teleported’ from Alice to Bob without disturbing or measuring it.

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

Another teleportation protocol is the probabilistic single-mode teleportation (Lombardi et al 2002). In its simplest form, it involves two qubits, a target and an ancilla, each defined by a single photon occupying two optical modes (see section 3.3.1 and figure 6). The target qubit can a priori be in an arbitrary state $|{\psi }_{\mathrm{t}}⟩=\alpha |0{⟩}_L+\beta |1{⟩}_L$ where the logical $|0{⟩}_L$ and $|1{⟩}_i$ _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 prepared with a beam splitter (BS1) in the coherent superposition $|{\psi }_{\mathrm{a}}⟩=(1/\sqrt{2})(|0{⟩}_L+|1{⟩}_L)=$ $(1/\sqrt{2})(|1{⟩}_3|0{⟩}_4+|0{⟩}_3|1{⟩}_4)$

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 given realization of the procedure, if only one photon is detected at detector C, and none at detector D, then we can infer the resulting state for the output qubit composed of modes (1) and (4) $\alpha |0{⟩}_L+\beta |1{⟩}_L=\alpha |1{⟩}_1|0{⟩}_4+\beta |0{⟩}_1|1{⟩}_4$ , which is the initial target qubit state.

Similarly if D clicks and C does not, the output state is α $|0{⟩}_L-\beta |1{⟩}_L=\alpha |1{⟩}_1|0{⟩}_4-$ $\beta |0{⟩}_1|1{⟩}_4$ , which again is related to the target state by an additional phase shift of $\pi ,$ which can be introduced to retrieve the initial state $|{\psi }_{\mathrm{t}}$ (Giacomini et al 2002). Half of the time, either zero or two photons are present at counters C or D, and the teleportation procedure fails. The single-mode protocol thus leads to teleportation in half of the attempts. Measurements and postselection of the successful attempts lead to a faithful teleportation of the target qubit on modes (1) and (4).

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

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

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

3.3.4. Linear-optics quantum computing. We mentioned earlier the strong potential advantages of optics in the race towards quantum computing (Milburn 1988). Photons are easily manipulated and, as the electro-magnetic environment at optical frequencies can be regarded as vacuum, their decoherence is negligible. Unfortunately, logical gates combining two optical qubits require strong interactions between single photons, i.e. huge nonlinear coefficients, well beyond current technology. An elegant solution to this problem was recently proposed in theoretical papers (Knill et al 2001, Ralph et al 2001). It is based on passive linear optics, photodetectors and single-photon sources. The only nonlinearity is introduced by the photodetection. 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 that the classical NAND gate is a universal gate for conventional computers (Sleator and Weinfurter 1995, Nielsen and Chuang 2000). A CNOT gate has two inputs (a control qubit and a target qubit) and operates in such a way that the NOT operation (bit flip) is applied to the target qubit, provided the control qubit has a logical value of 1. Such a logic operation is inherently nonlinear because the state of one quantum particle must be able to control the state of the other.

An example of proposed schemes which shows the basic idea of a linear optics quantum computing (LOQC)-type CNOT gate is illustrated in figure 7. In this scheme (Pittman et al 2001), the quantum logic operation is performed using polarization-encoded qubits. In addition to the control and target photons carrying qubits, an ancillary pair of entangled photons is injected into a device which contains only linear optical elements. This linear device gives three possible types of detection outcomes, each of which is signalled by a unique combination of detection events. For one set of events, we know that the control and target photons are in the desired logical output state. In a second set of events, the control and target are not in the desired output state, but they can be brought into it by real-time corrections of the type 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 logical state that cannot be corrected. The first experimental demonstration of quantum CNOT gate for different photons is based on this scheme (Gasparoni et al 2004).

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) or on one of the entangled auxiliary qubits (c). This makes it possible to prepare the gate with the proper entangled pair and only then to teleport the computation qubit with the proper pair, in a deterministic process (c) (after Ralph et al (2001)).

These LOQC logic gates are probabilistic devices which fail with a relatively high probability (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 operation has succeeded. Yet, the low rate of success of such nondeterministic gates makes them impractical for large computations because the probability of success of a large computation decays exponentially with the number of logic operations.

Gottesman and Chuang (1999) have proposed a gate operation based on teleportation which can solve this difficulty. They showed that performing the logic gate unitary operation U on one of the ancillary photons in a teleportation experiment is equivalent to applying U directly to the teleported qubit (see figure 8). The probabilistic step can thus be transferred to the preparation of the proper ancillary state. Deterministic teleportation (section 3.3.3) is then applied 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 the Bell measurements required in the teleportation protocol, which can never reach unity with linear 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 deterministic teleportation protocol requires only linear optics, photon counting and fast feedforward.

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

Table 1. Example of Alice’s emissions and Bob’s measurements in the two different polarization bases. 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

$\oplus$

$\otimes$

$\otimes$

$\oplus$

$\otimes$

$\otimes$

$\oplus$

$\otimes$

$\oplus$

$\oplus$

$\otimes$

$\oplus$

A-bit value

0

0

0

0

1

0

1

1

0

1

0

1

Polarization

V

F

F

V

S

F

H

S

V

H

F

H

Bob's basis

$\oplus$

$\oplus$

$\oplus$

$\otimes$

$\otimes$

$\otimes$

$\oplus$

$\oplus$

$\otimes$

$\oplus$

$\otimes$

$\otimes$

B-bit value

0

1

0

1

1

0

1

0

0

1

0

1

Same basis

Y

N

N

N

Y

Y

Y

N

N

Y

Y

N

Sift-key

0

1

0

1

1

0

Quantum cryptography, or more precisely quantum key distribution (QKD), is a secure method for the distribution of a secret key between two distant partners, Alice and Bob (Bennett and Brassard 1984, Gisin et al 2002). In its ideal version, where experimental limitations are disregarded, its security is absolute. This safety arises from the impossibility of measuring an unknown quantum state without modifying it. Thus, eavesdropping by spy Eve necessarily introduces 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 of single photons, a straight basis , with horizontal H and vertical V basis states, and a diagonal basis , with F and S as basis states. Each state can encode a qubit in the proper basis, and one conventionally attributes the binary value 0 to states V and F and the value 1 to states H and S . The two bases are chosen so that any two state vectors from different bases have the same overlap, e.g. $\begin{array}{r}|⟨H\mid S⟩{|}^2=\frac{1}{2}\end{array}$

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 chosen are different, the results are uncorrelated. By exchanging the information of which bases they have used, without revealing their results, Bob and Alice know exactly which bits are perfectly correlated and which ones are not. This information can be exchanged over a public channel since 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. neither Alice nor Bob can influence the result, but both of them know it with certainty. A schematic view of the principle can be seen in table 1.

The sift-key shared by Alice and Bob may still contain errors due to technical imperfections, as well as possible eavesdropping. The last step of the protocol is to use classical algorithms first to correct the errors and then to reduce Eve’s information on the final key. This process is called privacy amplification (Bennett et al 1988, 1995, Maurer and Wolf 1999).

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

If Eve would just intercept a qubit sent by Alice to Bob, Bob would easily see that he misses 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 original state, 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 al 1982). 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 al 1992). 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 the cases, Eve will have been lucky and chosen the basis compatible with the state prepared by Alice. Then, neither Alice nor Bob can notice her action. In the other cases, Eve has used a 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 interceptresend strategy, she gets 50% of the information, while Alice and Bob find a 25% error rate in their sift-key. They detect Eve’s action by simply comparing a random sample of transmitted bits to earlier tests of their line and detectors.

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

(i) Faint laser pulses: The vast majority of QKD experiments use faint laser pulses, with a Poisson distribution of the number of photons. There is a non-zero probability of having more than one photon in a pulse, even for very weak pulses with average photon number $⟨n⟩=\mu \ll 1$

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

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

The unique constraint on the PNS attack is that Eve’s presence should not be noticed; in particular, 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 the fibre are equal to those introduced by Eve’s blocking photons, that is when $T_{\mathrm{AB}}\mu <{\mu }^2/2$ , or when the transmission of the Alice–Bob line is less than $\mu /2$

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

$$\mu T_{\mathrm{AB}}{\eta }_{\det }>p_{\mathrm{dark}},$$

where $p_{\mathrm{dark}}$ is the dark count during the time slot and ${\eta }_{\det }$ the detection yield. As shown by Brassard et al (2000), in order to eschew a PNS-attack in the presence of dark counts, the transmission efficiency, $T_{\mathrm{AB}}$ , of the line from Alice to Bob should obey the following criterion:

$$T_{\mathrm{AB}}>\frac{p_{\mathrm{dark}}}{{\eta }_{\det }\mu }+\frac{\mu }{2{\eta }_{\det }}.$$

Figure 9. Comparison of theoretical (curves) and experimental (symbols) quantum communication rates as functions of transmission losses for a faint laser source $(---,\ast )$ and for a single-photon source $(--,\times )$ . The device based on the single-photon source tolerates larger losses. Figure reproduced with the authors’ permission, from Waks et al (2002).

The optimal choice for Alice’s rate of emission is therefore $\mu =\sqrt{2p_{\mathrm{dark}}}$ , which leads to the combined transmission-and-detection limit,

$$T_{\mathrm{AB}}>\frac{\sqrt{2p_{\mathrm{dark}}}}{{\eta }_{\det }}.\text{\hspace{1em}(23)}$$

(ii) Single-photon source: Alice now uses an ideal single-photon source. As Eve cannot apply the PNS attack, she must use the simple intercept-resend attack. She thus necessarily induces an additional error rate. The only way for her to remain unnoticed is to restrict her measurement rate to less than the dark count rate, $p_{\mathrm{dark}}$ , of Bob’s detectors. The condition for security of the communication will be that the rate of bits received by Bob, $T_{\mathrm{AB}}{\eta }_{\det }.$ , be larger than his detectors’ dark counts probability, $p_{\mathrm{dark}}$ , which may hide Eve’s measurements (Brassard et al 2000):

$$T_{\mathrm{AB}}>\frac{p_{\mathrm{dark}}}{{\eta }_{\det }}.\text{\hspace{1em}(24)}$$

For low dark count probabilities, it is clear from equations (23) and (24) that the single-photon source 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 errorcorrecting and privacy-amplifying algorithms to the transmitted bits (Bennett et al 1995). Lutkenhaus (2000) gives analytical expressions of the minimum amount of transmission to¨ ensure secrecy, taking these algorithms into account, and compares the cases of attenuated laser pulses and of a single-photon source under these conditions.

(c) Experimental realizations. The first experimental QKD was demonstrated in 1989, but the results were published only in 1992 (Bennett et al 1992). In this first experiment with faint laser pulses, 30 cm of air separated Alice and Bob. Since then, considerable progress has been booked. The first demonstration of a QKD over a 1 km fibre has been reported by Muller et al (1993). Since then, several prototypes have been proposed and demonstrated over distances longer than 20 km. The free-space transmission of quantum-encrypted signals reached first a distance of 150 m as reported in Jacobs and Franson (1996) and then exceeded 1 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 transmitting quantum-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 over short distances, 1 m (Waks et al 2002) and 50 m (Beveratos et al 2002). Attenuators were used in the 1 m experiment to demonstrate the effect of further channel losses (potentially longer distances). Both experiments show that single-photon devices can provide completely secure communication in conditions under which this would be impossible with attenuated laser pulses. This is illustrated in figure 9, where the number of secure bits per pulse is reported as a 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 property of the coupled system of the emitter and the electromagnetic modes in its environment. The radiative transition rate, $\gamma$ , of an atom from an excited, initial state $|e⟩$ to a lower energy, final state g depends on the density of available photon states, $\rho (\omega )$ , at the transition frequency, ω, and on the atom–vacuum field dipole interaction Hamiltonian. Thus, by altering $\rho (\omega )$ (placing the atom in a cavity or near an interface), the spontaneous emission rate can be enhanced or suppressed (Purcell 1946, Drexhage 1974).

First experimental demonstrations of the inhibition and enhancement of a spontaneous emission rate were carried out in the 1970s with molecules such as europium complexes in Langmuir–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, planar cavities or spherical Fabry–Perot resonators (Kleppner 1981, Goy et al 1983, Gabrielse and Dehmelt 1985, De Martini et al 1987, Heinzen et al 1987). Advances in microfabrication techniques enabled the construction of high-quality semiconductor micropost and microdisc microcavities in the late 1980s and early 1990s and triggered series ofexperiments on solid-state cavity-quantum-electrodynamics (CQED) (Yamamoto et al 1989, Jewell et al 1991, McCal et 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 resonant mode of a high-quality factor cavity. The quality factor is defined as the ratio of the resonance angular frequency to the cavity damping rate: $Q=\omega /{\gamma }_{\mathrm{cav}}$ . Under these conditions, the atom– cavity interaction is described by the Jaynes–Cummings Hamiltonian (Jaynes and Cummings 1963, Walls and Milburn 1994). The coupling parameter, g, between an emitter placed in a region of maximum electric-field intensity in the cavity is given by

$$g=\frac{{\mu }_{\mathrm{eg}}}{\hslash }\sqrt{\frac{\hslash \omega }{2{\epsilon }_{0}V}},\text{\hspace{1em}(25)}$$

where ${\epsilon }_0$ is the vacuum dielectric constant, ${\mu }_{\mathrm{eg}}$ the dipole moment of the transition and V the cavity mode volume. Depending on the ratio of the coupling parameter, g, to the cavity-field decay rate, ${\gamma }_{\mathrm{cav}:}$ , and to the free-space decay rate of equation (14), ${\gamma }_0={\omega }^3{\mu }_{\mathrm{eg}}^2/3\pi {\epsilon }_0\hslash c^3$ we can distinguish two regimes of coupling between the emitter and the cavity field: strong coupling for $g>{\gamma }_{\mathrm{cav}},{\gamma }_0$ and weak coupling for $g<{\gamma }_{\mathrm{cav}},{\gamma }_0$ (Haroche 1992).

In the strong coupling case, the emitter is coherently coupled to the cavity field, and spontaneous 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 the spontaneous emission of self-assembled quantum dots. Left: micropost cavity included between two Bragg reflectors (top diameter 0.6 µm, height 4.2 µm (Pelton et al 2002, copyright 2002 by the American Physical Society; reproduced with permission)). The emitters are placed in one of the central layers. Right: microdisc structure (diameter of the disc 5 µm, post height 0.5 µm (Michler et 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 antinodes are 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 continuum of all available photon states, including the vacuum modes still accessible (leak modes) and those of the cavity (cavity modes). When the cavity decay time is shorter than the spontaneous decay time in vacuum $({\gamma }_{\mathrm{cav}}>{\gamma }_0$ , bad cavity limit), the spontaneous emission decay rate is enhanced and is given by Fermi’s golden rule (Yamamoto and Imamoglu 1999):

$$\gamma =\frac{2\pi }{{\hslash }^2}(\hslash g{)}^2{\rho }_{\mathrm{cav}}(\omega )+f{\gamma }_0.\text{\hspace{1em}(26)}$$

We have separated the contribution of the cavity modes from that, $f{\gamma }_0.$ , of all other leak modes $(f,0<f<1$ , is often close to unity). Assuming a normalized Lorentzian mode density with a width of ${\omega }_0/Q$ for the cavity, ${\rho }_{\mathrm{cav}}(\omega )$ ≈ $2Q/\pi {\omega }_0$ , we deduce the following spontaneous emission enhancement:

$$\frac{\gamma }{{\gamma }_0}=\frac{3}{4{\pi }^2}\frac{{\lambda }^3}{V}Q+f.$$

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

$$F_{\mathrm{P}}=\frac{3}{4{\pi }^2}\frac{{\lambda }^3}{V}Q.\text{\hspace{1em}(27)}$$

If the Purcell factor is much greater than 1, the emitter will radiate much faster in the cavity than in 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 $(\gamma$ and ${\gamma }_{\mathrm{cav}})$ . At that point, the coupled electron–cavity system enters the strong-coupling regime.

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

$$\beta =\frac{F_{\mathrm{P}}{\gamma }_0}{\gamma }=\frac{F_{\mathrm{P}}}{F_{\mathrm{P}}+f}.\text{\hspace{1em}(28)}$$

Efficient cavity-coupled single-photon sources were demonstrated with single quantum dots embedded 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 two dielectric Bragg mirrors, as shown in figure 10. The confinement of light in these structures is achieved by the combined action of the distributed Bragg reflector (DBR) in the longitudinal direction, 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 the emission rate from single quantum dots with a Purcell factor of $F_{\mathrm{P}}\approx 5.8$ , quality factors of the order of 1000 and mode volumes close to the fundamental limit, ${\lambda }^3$ . In this case $\beta \approx 80\%$ of the emitted light was coupled into a single cavity mode. The majority of this light escaped into a single-mode, nearly Gaussian travelling wave.

Michler et al (2000) embedded self-assembled InAs and InGaAs quantum dots in microdisc cavities to generate single photons. The whispering-gallery modes (WGMs) of these cavities present higher quality factors (in the $10^4$ range) but relatively larger mode volumes so that the Purcell factors were comparable $(F_{\mathrm{P}}\approx 6)$ with those obtained in the micropost cavities.

4.2. Strong-coupling regime

The generation of single photons in the strong-coupling regime of CQED (Law and Kimble 1997) presents interesting new routes for quantum information processing. The reversible transfer of quantum information between the emitter and the photons opens the way for quantum networks. In the all-optical network of Knill et al (2001), the nodes are linear optical components. The quantum information is encoded in the number of photons flying from node to node. The nodes perform gate operations based on quantum interference between the indistinguishable photons. In a more general network, the nodes also serve as quantum memories storing information, e.g. in long-lived states of atoms located in an optical cavity (Cirac et al 1997). The key requirement for such networks is their ability to interconvert stationary and flying qubits and to transmit flying qubits between specified locations (Di Vincenzo 2000).

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

Recently, single-photon sources using single atoms strongly interacting with cavities have been demonstrated by the groups ofRempe and Kimble (Kuhn etal 2002, McKeever etal 2004). The photon production step is based on a stimulated Raman process driven by an adiabatic passage (STIRAP) (Parkins et al 1993, Vitanov $et$ al 2001) between two ground states of a single atom. The stimulated emission process requires a strong coupling to a single mode of a high-finesse optical cavity (Law and Kimble 1997). A laser beam illuminates the atom slightly off-resonance, while the vacuum fluctuations of the cavity stimulate emission of the photon towards the other ground state of the atom. The adiabatic passage of STIRAP is obtained by varying 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 lambda atomic level-scheme. Single alkali atoms ${(}^{85}$ Rb for Rempe’s or ${}^{137}\mathrm{Cs}$ for Kimble’s groups) are prepared in $|a⟩$ , one of the two hyperfine sates of their ${\mathrm{S}}_{1/2}$ electronic ground state. The atom is located in a high-finesse optical cavity, which is near-resonant with the electronic transition between states b and e . Here, b is the second hyperfine state of the electronic ground state and e is a hyperfine state of the electronically excited level $P_{3/2}$ . The state of the cavity is denoted by n , where n is the number of photons. Inside the empty cavity, the product states e, 0 and $|b,1⟩$ are coupled by the electric dipole interaction with coupling parameter $g.$ When a pump laser pulse close to $|a⟩\to |e⟩$ resonance is applied with a Rabi frequency ${\Omega }_{\mathrm{p}}(t)$ , the product states a, 0 , e, 0 and $|b,1⟩$ of the atom–cavity system are coupled. At Raman-resonant excitation, when the detunings of the pump-laser and the cavity from the respective atomic transitions are equal, one of the three eigenstates of the coupled atom–cavity system has no contribution of the excited state e and is therefore not affected by spontaneous emission. The expression of this dark state, ψ_dark , is given by

Figure 11. The stimulated Raman generation of single photons (STIRAP) is based on a -scheme of 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 a single atom is present in the cavity, a single photon is emitted in the cavity mode.

$$|{\psi }_{\mathrm{dark}}⟩=\frac{2g|a,0⟩+{\Omega }_{\mathrm{p}}(t)|b,1⟩}{\sqrt{4g^2+{\Omega }_{\mathrm{p}}^2(t)}}.\text{\hspace{1em}(29)}$$

The dark state $|{\psi }_{\mathrm{dark}}⟩$ is now used to generate a single photon inside the cavity. This is achieved by establishing a large atom–cavity coupling constant, g, before turning on the pump pulse. In this case, the system’s initial state, a, 0 , coincides with $|{\psi }_{\mathrm{dark}}⟩$ . Provided the pump pulse rises slowly, the system’s state vector adiabatically follows any change in $|{\psi }_{\mathrm{dark}}⟩$ and for a lossless cavity a smooth transition from a, 0 to $|b,1⟩$ is attained as soon as ${\Omega }_{\mathrm{p}}\gg g.$ . The single photon generated in the cavity mode leaves it through the output mirror, and the final state of the coupled system, b, 0 is then reached. As this state is not coupled to the one-photon manifold, the atom cannot be excited again, which limits the number of photons per pump pulse and atom to one.

To emit a sequence of photons from one and the same atom, the system must be transferred back to a, 0 once an emission has taken place. Recycling laser pulses are applied to the atom between 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 initial state, 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, a sequence of single-photon pulses is produced.

In the experiment with ${}^{85}\mathrm{Rb}$ , atoms are released from a magneto-optical trap and pass through the $\mathrm{TE}{\mathrm{M}}_{00}$ mode of a high-finesse cavity (finesse 60 000, distance between mirrors 1 mm, $Q\approx 10^8)$ ). In contrast to solid-state single-photon sources, this source operates under non-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 few single 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 average number of atoms be very low (Kimble 2003). To circumvent this problem, Kimble’s group employed a single Cs atom trapped in the optical cavity. After being released from a magnetooptical trap, atoms are cooled in the cavity region and then loaded into a far-off-resonance dipolar trap (McKeever et al 2004) which localizes them within the mode of the high-finesse cavity (finesse $\sim 4\times 10^5$ , distance between mirrors $\sim 42\mu \mathrm{m},Q\approx 4\times 10^7)$ . The lifetime of a trapped atom in the presence of the driving fields (pumping ${\Omega }_{\mathrm{p}}(t)$ and repumping fields) limits the number of generated photons to about $10^4$ , 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 small probability of generating two- or multi-photon states could open serious security breaches in quantum cryptography or completely disrupt the operation of an all-optical quantum computer. Faint laser pulses and parametric sources of entangled pairs deliver Poisson distributions of photons (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 and macroscopic 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 average photon number n . They can easily be developed using only standard pulsed lasers (such as inexpensive semiconductor laser diodes) and calibrated attenuators. As we have seen in section 2.2(b), the probability of finding n photons in such a coherent state follows Poisson statistics (equation (10)).

For quantum cryptography, the reliability of the source is related to the probability, $p_{\mathrm{multi}}.$ that a non-empty weak coherent pulse contains more than one photon. These pulses can lead to fractional information leakage which can be exploited by a smart eavesdropper in a beam splitter attack.

$$p_{\text{ multi }}=\frac{p_{n⩾2}}{p_{n⩾1}}=\frac{1-p(0)-p(1)}{1-p(0)}\approx \frac{⟨n⟩}{2},\text{\hspace{1em}(30)}$$

where $p_{n⩾1}$ and $p_{n⩾2}$ are the probabilities that a pulse contains at least one and at least two photons. The value of $p_{\mathrm{multi}}$ can be made arbitrarily small. Weak pulses are thus extremely convenient and have indeed been used in the vast majority of quantum cryptography experiments. However, they have one major drawback. When n is small, most pulses are empty: $p(n=0)\approx 1-⟨n⟩$ . In principle, the resulting decrease in bit rate could be compensated by increasing the pulsed laser source rate (gigahertz modulation rates in telecommunications lasers are achievable), but the problem of the detectors’ dark counts remains. Indeed, the detectors must be active for all pulses, including the empty ones. Therefore, the dark count rate 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 value depends 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 quantum optics. 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 from each other in different directions, and the quantum states of each single photon may differ considerably. Despite the distance between the subsystems, the pair will keep its correlation of space–time variables, spin variables or both. The correlations, usually defined in the form of energy conservation, momentum conservation and/or angular momentum conservation, arise in the process of generation of the pair.

An elegant way of creating pseudo-single-photon states is to use one photon of a correlated pair as a trigger for its companion (Grangier et al 1986, Hong and Mandel 1986). In contrast to the faint laser sources discussed earlier, the second detector needs only be activated when the 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 $(j=0\to j=1\to j=0,j$ being the total angular momentum of the atomic states) because of the polarization correlation between the successive emitted photons. These sources have 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 4 \mathrm { p } ^ { 2 } ^ { 1 } \mathrm { S } _ { 0 } 4 \mathrm { s } 4 \mathrm { p } ^ { 1 } \mathrm { P } _ { 1 } 4 \mathrm { s } ^ { 2 } ^ { 1 } \mathrm { S } _ { 0 } cascade in a calcium atom beam, which yields two polarization-correlated photons $({\lambda }_1=$ 551 nm and ${\lambda }_2=423$ nm) upon two-photon excitation from 4 \mathrm { s } ^ { 2 } ^ { 1 } \mathrm { S } _ { 0 } \mathrm { t o } 4 \mathrm { p } ^ { 2 } ^ { 1 } \mathrm { S } _ { 0 } . A cascade rate of $\sim 5\times 10^7{\mathbf{s}}^{-1}$ could be achieved (Aspect et al 1982b).

In 1986, Aspect and co-workers (Grangier et al 1986) used this atomic cascade source to demonstrate the wave–particle duality of a photon by sending single photons on a beam splitter. They observed a strong anticorrelation between the triggered detections on the two sides of the beam splitter, which was a clear signature of a particle-like behaviour of the single photon. When the output beams of the beam splitter were recombined on a second beam splitter 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 of light is able to give a consistent description of both experiments.

Besides the heaviness and difficulty of operation of the experimental set-up, the atomic cascade sources suffer from a rather low brightness and from a loss of the polarization correlation when the photons are emitted back to back (due to the recoil of the atom) (Santos 1992).

5.2.2. Spontaneous parametric down-conversion (SPDC) sources. Photon pairs can be generated by SPDC in non-centrosymmetric crystals such as KDP, BBO, LBO or $\mathrm{LiNb}{\mathrm{O}}_3$ Nonlinear interaction of a pump laser pulse with the crystal occasionally splits one highfrequency pump photon into two lower-frequency, signal and idler photons (Shen 1984). The frequencies and wavevectors of the three photons satisfy the energy conservation and phasematching conditions:

$${\omega }_{\mathrm{p}}={\omega }_{\mathrm{s}}+{\omega }_{\mathrm{i}},{\vec{k}}_{\mathrm{p}}={\vec{k}}_{\mathrm{s}}+{\vec{k}}_{\mathrm{i}}.\text{\hspace{1em}(31)}$$

The process is said to be degenerate if the down-converted photons have the same frequency $(e.g.{\omega }_{\mathrm{s}}={\omega }_{\mathrm{i}}={\omega }_{\mathrm{p}}/2)$ , and nondegenerate otherwise. In general, the photons leaving the crystal propagate in non-collinear directions. However, under certain conditions the pair may also exit collinearly, in the same direction as the pump.

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

The down-conversion process is called type-I if the signal and idler photons have identical polarizations. In the case of degenerate emission, a pair of photons with equal wavelengths emerge in a cone centred on the pump beam and whose opening angle depends on the angle, ${\theta }_{\mathrm{pc}}$ , 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 the collinear situation (for a given incidence angle ${\theta }_{\mathrm{pc}}{)}_{:}$ , the two cones are tangent to one another exactly at the pump beam direction. If ${\theta }_{\mathrm{pc}}$ is increased, the two cones tilt towards the pump and intersect along two rays. Along the two directions (‘1’ and ‘2’), where the cones overlap, the light is described by a Bell entangled state:

$$|\psi ⟩=\frac{(|H_1,V_2⟩+{\mathrm{e}}^{\mathrm{i}\phi }|V_1,H_2⟩)}{\sqrt{2}},\text{\hspace{1em}(32)}$$

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 birefringent phase shifter (or slightly rotating the nonlinear crystal), giving the possibility of preparing any of the four EPR-Bell states (Shih 2003).

Figure 12 shows three examples of parametric sources based on type-I and type-II phase matching, 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). Collinear type-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 Bell states. This scheme has been used successfully to demonstrate many quantum entanglement phenomena (see section 3).

Phase-matching allows one to choose the wavelength and determine the bandwidth of the down-converted photons. The latter is in general rather broad and varies from a few nanometres up to some tens of nanometres. For the nondegenerate case one typically gets a bandwith of 5–10 nm, whereas in the degenerate case (where the central frequencies of both photons are equal), the bandwidth can be as large as 70 nm.

This photon-pair creation process is very inefficient (efficiency $10^{-7}–10^{-11})$ . The number of photon pairs per mode is thermally distributed within the coherence time of the photons and follows a Poissonian distribution for larger time windows (Walls and Milburn 1994).

A typical rate of pair generation into single-mode optical fibres, corresponding to 10 mW cw pump power, is $10^7$ pairs per second. The probability of having two pairs instead of one in a time slot of 1 ns is then about 1%. With a 10% detection yield on one of the fibres, the rate of single-photon production in the other fibre is $10^6$ per second. In order to obtain the same 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 dark counts 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 must be strongly attenuated to reduce the probability of obtaining two or more photons. The rate of emission of single photons is therefore limited. The working principles of the microscopic sources discussed in this section are profoundly different. These sources are built around a single emitting nanometric object (a nano-object), producing photon distributions which are very far from Poissonian. In most cases, for example, the probability density of emitting two photons at the same time can be completely neglected, whereas it is still high for an attenuated Poisson source with the same brightness. In most cases, the emission process is spontaneous and takes place after a rapid excitation of the emitter. In order to achieve emission at a high repetition rate, and in order to avoid excitation events in which no photons are emitted, two conditions have to be fulfilled:

(i) The emitting state must be prepared at the end of the excitation cycle with as high a probability as possible (ideally, with certainty). A number of excitation schemes have been proposed 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 optical pulse pumps an organic dye molecule from its electronic ground state to a vibrationally excited level of its excited electronic state. This short-lived state in turn quickly relaxes to the emitting state. Because it lives about 1000 times longer than the vibronic state, the latter state can accumulate a population from several pumping interactions. Repeated attempts at pumping by the many photons of the exciting pulse thus efficiently transfer the ground state population into a 100% population of the emitting state. A variant of this scheme is electrical pumping of a 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 hole are left to recombine, they will quickly relax to a single-exciton state, which will then emit a single photon.

Coherent preparation. It is also possible to prepare the emitting state with resonant light, but one then has to separate the emitted photon from the excited ones, either temporally, by a timedelayed detection, or spectrally, by detecting a red-shifted photon emitted on a vibrational level of the ground state. One possibility for resonant excitation is to apply a ‘pi-pulse’, i.e. a pulse whose duration and intensity are so chosen that the vector state exactly evolves from the ground state to the excited state during the excitation pulse (see, e.g. Shen (1984)). Another option is rapid adiabatic passage: the laser frequency is swept through resonance with the two-level system, quickly enough to avoid relaxation by spontaneous emission or dephasing but slowly enough 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 the excited state after one frequency sweep. Alternatively, the system’s resonance can be swept through resonance with a fixed laser, as was demonstrated for an organic molecule (Brunel et 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 a high-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. The emission yield primarily depends on the photophysics of the nano-object at hand, but it can be improved if spontaneous emission is enhanced, for example, by the Purcell effect of a resonant cavity (see section 4). In the present section, we shall discuss separately the emission quantum yield 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 and photophysical properties (quantum yield, spectrum, lifetime, polarization, radiation pattern, antibunching and/or multi-excitation shifts, etc), which determine the characteristics of the emitted photons and the properties of the source. Where applicable, we shall indicate current realizations, 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 in practice for cold atoms and trapped ions, the transitions are narrow and lifetime-limited. A big advantage of atoms is that their states are perfectly reproducible and well-known, including the hyperfine structure of the levels. With the right atom or ion, transitions can be found throughout 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 known levels. We already mentioned atomic cascades in section 5. In the STIRAP process discussed earlier (see section 4.2, Parkins et al (1993), Kuhn et al (2002), McKeever et al (2004)), an adiabatic passage between two hyperfine levels of the ground state is driven by a laser pulse close to resonance with an excited state. The emitted photon is stimulated into the mode of a resonant high-finesse cavity. Since there is only one atom in the cavity and since only one passage is performed at each laser pulse, only one photon is produced in the cavity (see figure 13). The incoherent pumping scheme does not apply to atoms. In contrast to emitters in condensed matter, they offer no possibility for fast relaxation between levels in the picosecond range.

Typical radiative lifetimes of allowed atomic transitions are about 30 ns, corresponding to a linewidth of a few megahertz. The whole oscillator strength is concentrated in one single transition, and because of the absence of non-radiative channels, the fluorescence quantum yield is unity.

One of the essential features of a single-photon emitter is that it should be strongly anharmonic, i.e. two- or multi-excitation states should have very different emission properties from the singly excited state. This condition is easily satisfied for few-electron systems such as atoms and ions. They never emit two or more photons (with the same wavelength) simultaneously and consequently present strong antibunching. Simultaneous emission of two photons in general indicates that two atoms are present in the detection volume (McKeever et al 2004). The polarization and radiation pattern of the emitted photon are determined by the 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 matter system (except for a possible recoil, which is usually negligible). Therefore, the emitted 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 vibrational sidebands should be negligible or eliminated.

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

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

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

6.3. Organic molecules

Some organic dyes and aromatic molecules emit fluorescence with a high quantum yield and were the first condensed-matter systems for which antibunching was demonstrated. The photophysical properties of molecules make them interesting models for understanding the behaviour of strongly confined electron systems such as colour centres and semiconductor nanoparticles, to be discussed later. In the present section, we discuss separately the photophysical properties of organic fluorophores at cryogenic and ambient temperatures and single-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 present optical transitions in the visible or near-IR. As compared with those of atoms, the eigenstates of 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 the ground 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 ground and excited electronic states is a very narrow line, called the zero-phonon line (ZPL) (Sild and Haller 1988). Zero-phonon linewidths are often lifetime-limited at low temperatures. The intensity of the ZPL compared with the total intensity of the absorption spectrum is called the Debye–Waller factor. A typical value is 0.1, and it decreases quickly for temperatures higher than that of liquid helium, while the width of the ZPL increases steeply because of phonon-induced dephasing processes. A molecule also absorbs at a higher frequency than the ZPL, towards vibrational excited states of the electronic excited state.

Fluorescence is emitted from the lowest vibration state of the excited electronic state towards vibrational levels of the ground state, giving rise to vibrational fluorescence lines (shifted to the red with respect to the ZPL by vibration frequencies, $100-3000{\mathrm{cm}}^{-1})$ and to phonon sidebands (width about 100 cm−¹). Only the ZPL, the bluest emission at low temperatures, is narrow because its width is determined by the long lifetime of the excited state, nanoseconds. The width of the other vibronic lines is limited by the lifetime of the vibrations, about a few picoseconds.

The molecular transition is a linear dipole with an oscillator strength of about unity, which corresponds to a spontaneous emission lifetime of a few nanoseconds and to a ZPL linewidth of about 30 MHz. The fluorescence quantum yield is very close to unity (typically better than 95% 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 reached by absorption of a second or third photon. However, highly-excited singlet states relax very rapidly towards the lowest excited singlet state in about a picosecond or less. This observational fact is summarized by Kasha’s rule: only the lowest excited state of a given spin multiplicity can efficiently emit light (Birks 1970). Multiple excitations therefore cannot lead to emission of two or more photons. However, many photochemical channels open for highly excited molecular states, which promote the irreversible photodestruction of the molecule, specially under ambient conditions.

The natural molecular fluorescence spectrum presents a number of lines and bands, spread over 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 to stimulate the ZPL emission by coupling to a resonant cavity (section 4), which is not easy for molecular 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 liquid helium temperatures. Sometimes, even in crystals, defects can lead to frequency variations by spectral diffusion. Depending on the molecular compound and host crystal, excursions to a triplet lifetime can in some cases limit the molecular brightness (Bernard et al 1993).

Under low-temperature conditions, the fluorescence of single molecules shows deep antibunching and damped Rabi oscillations (Basche´ et al 1992). Aromatic molecules were used for an early demonstration of a single-photon source in condensed matter (Brunel et al 1999), as shown in figure 14.

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

Only very broad absorption and emission bands appear at room temperature. Their shape and width depend on the matrix. A typical value for the linewidth is $300{\mathrm{cm}}^{-1}$ and is mainly due to thermal fluctuations. Therefore, even in crystalline environments, lines of single organic molecules at room temperature are never much narrower. Thermal fluctuations, i.e. vibrations and phonons in solids, cause fast dephasing of the electronic oscillations. In soft molecular matrices, the sharp transitions (ZPL) disappear altogether above 50 K. As a consequence of fast dephasing, the emission is very far from lifetime-limited, by a factor of at least 10 000!

Figure 14. Left: experimental measurements of the second-order correlation of photons delivered by 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 delay stem from two successive emissions by the molecule, due to a double excitation (the passage was not rapid enough). This is confirmed by the antibunching dip at zero delay, more visible on the simulations.

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

Molecular photostability is a much more serious issue. It is often severely limited at room temperature because of the many photochemical processes that can be activated from the excited state by thermal fluctuations. A chemically stable dye in a polymer matrix does not survive more than seconds the typical excitation intensities of single-molecule observations. Furthermore, the complexity and flexibility of the environment at room temperature allows the existence of metastable dark states, leading to extended dark periods, during which emission stops (Zondervan et al 2003, 2004). However, if the molecules can be protected from atmospheric oxygen and are rigidly held in a crystalline matrix, the stability can sometimes be extended to minutes or hours (Fleury et al 1998, Kulzer et al 1999), even under intense pulsed illumination, as has been shown for terrylene in p-terphenyl crystals (Lounis and Moerner 2000).

The fluorescence of single molecules at room temperature also shows very strong antibunching (Ambrose et al 1997, Fleury et al 2000), due to the very short lifetime of higher excited levels. For systems presenting a high photo-stability, for example terrylene in a p-terphenyl crystal (see figure 15), single molecules can operate as single-photon sources (Lounis and Moerner 2000, Treussart et al 2002).

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

6.3.3. Multichromophoric and conjugated systems. Here, we only discuss the roomtemperature 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 each chromophore is transferred by hopping and relaxation towards sites with lower excitation energy. At higher intensity, when two excitations meet, one of them is efficiently annihilated and 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 very efficiently prevents emission of two photons at the same time by a coupled system and explains 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 all subject to the same processes of photochemical destruction that limit the operation time of single fluorophores at room temperature. For such large assemblies, however, the operation time 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 quencher for neighbouring sites, a new fluorophore becomes the lowest-energy trap and can resume emission. Single polymer molecules have been in operation for as long as hours on a glass surface 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 electronic states around interstitial or substitutional impurities, vacancies, inserted charge carriers or combinations thereof. Although much is known from conventional ensemble spectroscopy about a huge number of colour centres in a variety of hosts (Crawford et al 1972, Gellermann 1991), only a handful of systems have led to observations of individual defects, and even fewer have been shown to possess the photophysical properties suitable for single-photon generation. Defects in low-band-gap semiconductors have been proposed as sources 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 been proposed as single-photon sources. A major advantage of inorganic materials, and in particular of diamond, is their mechanical stiffness and stability. Although the electronic wavefunctions and 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 and because of the protection it affords against small aggressive molecules such as oxygen.

Figure 16. Atomic structure of an NV-centre in a diamond crystal. One of the carbon atoms is replaced by a nitrogen (N), and one of its neighbours is missing (vacancy V). The electronic structure 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/).

6.4.1. Nitrogen-vacancy centre in diamond. The first single colour centre ever detected was a nitrogen-vacancy (NV) centre in diamond (Gruber et al 1997), at room temperature. This defect is a carbon vacancy situated next to a nitrogen atom, with an electron trapped (see a scheme of the structure in figure 16). The ground and optically excited states are triplets. The luminescence spectrum of this defect is a broad band (more than $1000{\mathrm{cm}}^{-1}$ , extending from 640 to 720 nm), with a weak and rather narrow ZPL (at 640 nm, width about $50{\mathrm{cm}}^{-1})$ , still visible 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 being lifetime-limited and therefore are not indistinguishable.

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

Diamond has a high refractive index, which seriously limits the fraction of luminescence extracted in microscopy on a flat interface. An elegant solution to this collection problem is the use of nanocrystals (size 50–100 nm) containing one or a few centres for which no totalreflection effect takes place. This improves the collection efficiency by more than a factor of 2 and additionally reduces the background. Because of the reduction in the surrounding index of refraction, the luminescence lifetime is lengthened to 20–30 ns, depending on the individua nanocrystal. Nanocrystals offer the additional advantage of the ease of manipulation and sample preparation.

Figure 17. Antibunching of a single CdSe nanocrystal, under cw excitation. Note the depth of the dip for zero delay, indicating the very low probability of multiphoton emission (after Lounis et al (2000)).

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

6.4.2. Nickel–nitrogen complex. Another colour centre of diamond was recently proposed as 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 found at low concentrations in natural diamonds or can be created by implantation. The fluorescence spectrum displays an intense and narrow ZPL at 802 nm, a convenient wavelength for fibre communications. The linewidth, $20{\mathrm{cm}}^{-1}$ at room temperature, is much lower than that of the NV-centre, and could be further narrowed by coupling to a cavity (section 4). The ZPL bears 70% of the total emitted intensity (the Debye–Waller factor is 0.7). The fluorescence lifetime is 11.5 ns, comparable with that of the NV-centre. However, a dark metastable state limits the fluorescence intensity. Its lifetime is 170 ns, and it has a rather high production yield of 20% from the excited state.

6.5. Semiconductor nanocrystals

Nanocrystals of II–VI semiconductors such as ZnS have been studied as individual emitters for several years (Empedocles et al 1996), but antibunching in their luminescence was discovered only recently (Lounis et al 2000, Michler et al 2000, Messin et al 2001). An example of an antibunching curve is shown in figure 17. Among the many possible materials and structures with 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. Their size-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 excited with a variety of sources. The role of the ZnS shell, or capping layer, is to protect the core exciton from surface defects and dangling bonds, which dramatically reduce the luminescence efficiency (Hines and Guyot-Sionnest 1996, Dabbousi et al 1997).

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

The luminescence of nanocrystals arises from radiative recombination of the lowest-lying exciton state, with a lifetime of 20–30 ns at room temperature. This emission time lengthens considerably at low temperatures (Nirmal et al 1995, Labeau et al 2003). The spectrum is a single line (ZPL), with a weak phonon sideband. At room temperature, the ZPL is strongly broadened to several hundreds of ${\mathrm{cm}}^{-1}$ (50 meV) by dephasing and spectral diffusion and is thus very far from lifetime-limited. The emission line narrows significantly at low temperatures, down to about $1{\mathrm{cm}}^{-1}$ (0.1 meV) in the best cases (Empedocles et al 1996), but never reaches the lifetime limit, probably because of spectral diffusion. Photoinduced charge rearrangements in the passivation layer and the environment have been shown to lead to spectral shifts of the exciton line (Empedocles and Bawendi 1997). Spectral diffusion and the very long luminescence lifetime are two weak points of nanocrystals for low-temperature applications 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 al 1999, Chung et al 2003). The emission quantum yield was recently shown to be very close to unity, but only during the bright periods of their blinking (Brokmann et al 2004b).

Being formed of thousands of atoms, semiconductor nanocrystals in principle support multiple excitations. However, all multi-excited states are nearly dark because of very efficient exciton–exciton annihilation (also called Auger processes (Klimov et al 2000)) similar to the singlet–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 can very efficiently dissipate the exciton energy by non-radiative relaxation. In a neutral, singlyexcited 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 matrix element for such a multi-phonon transition is exceedingly small. Radiative recombination is the main relaxation channel for mono-exciton states and therefore occurs with near-unity quantum yield. As soon as an additional charge carrier (or exciton) is present, however, its interaction with the emitter exciton leads to highly excited electron or hole states, for which many onephonon relaxation channels exist. The electronic energy is thus very efficiently dissipated non-radiatively. The efficient annihilation of multi-excitations is the key mechanism of the strong 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 other words the strong fluctuations of their luminescence with time. The occasional transfer of a 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 (Kuno et 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 nanocrystals are efficient emitters at ambient conditions. They are easy to manipulate and to couple to efficient collecting optics in a room-temperature microscope and have better stability than single organic chromophores. However, blinking limits their practical applications as sources of single photons.

6.6. Self-assembled quantum dots and other heterostructures

The core–shell nanocrystals wejust discussed are grown in solution by the methods of chemica synthesis. Self-assembled quantum dots, on the other hand, are also islands of a low-band-gap semiconductor 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/or reactive gases, by chemical vapour deposition. If one tries to grow a given material epitaxially on a (slightly) mismatched substrate, the elastic energy arising from the mismatch is at first compensated by surface forces for the first few monolayers but is soon relaxed by the growth of three-dimensional domains when more material is added. This transition, called Stranski–Krastanov instability, often appears after the first few ‘wetting’ monolayers have been deposited. With pure materials, high-quality substrates, ultra-high vacuum and wellcontrolled temperature conditions, the Stranski–Krastanov growth leads to uniformly sized, regularly spaced, three-dimensional islands of the deposited material (Moriarty 2001). The resulting quantum dots are usually protected by a thick layer of the high-band-gap substrate material, and they can support localized excitons. The well-controlled growth and environment ensure that the surface of the dots is very clean and that the radiative recombination rates and the photostability are excellent. Quantum dots are often excited in the electron–hole continuum. The excitons and/or charge carriers are trapped in the dots, where they recombine radiatively. For InAs in GaAs, the spectral range is typically 850–1000 nm, for InAs in InP it 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 single quantum dot gives a narrow, close to Fourier-limited line, in a spectral domain depending on the material (Gammon et al 1996, Bayer and Forchel 2002, Zwiller et al 2004). The spontaneous emission rate is of the order of few nanoseconds but can be significantly reduced by the Purcell effect if the dots are placed inside a resonant cavity (see section 4). The emission quantum yield is thought to be very close to unity.

In contrast to nanocrystals, multiple excitations can coexist in self-assembled quantum dots and can also coexist with charge carriers, leading to characteristic bi- or multi-exciton lines and 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 quantum dots can be traced back to two effects: (1˚) the higher purity and crystalline quality of the selfassembled dots, as less defects sites are available to promote non-radiative recombinations and (2˚) the smaller size and the larger quantum confinement of nanocrystals, since Coulomb interactions between charge carriers and excitons are enhanced by confinement. Another important difference is that self-assembled quantum dots can also be excited via relaxation of excitons and charge carriers from the wetting layer and the substrate, which can replenish the dot with excitations at a high rate.

Multi-excitations give rise to photoluminescence lines which, at low temperatures, are red-shifted with respect to the single-exciton line. For a single-photon source, it is important to make sure that multi-excitation lines are eliminated and that only photons from the singleexciton transition are selected. The biexciton line could also be selected as a source of single photons as it too shows antibunching (Thompson et al 2001, Zwiller et al 2002), but it requires a higher exciting intensity than the single exciton. An interesting alternative would be to use the bi-exciton and exciton lines to generate correlated pairs of single photons (Benson et al 2000).

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 with permission).

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

One ofthe most attractive features ofself-assembled quantum dots is their narrow emission line at very low temperatures (lower than 5 K), close to the Fourier-limit. Yamamoto’s group in Stanford has demonstrated the large degree of indistinguishability (more than 80% for a particular 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 photostable and show no long-lived dark state.

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

In conclusion, self-assembled quantum dots present many advantages as potential sources of single photons. They are bright, stable and nearly Fourier-limited at sufficiently low temperatures. The epitaxial growth and lithography methods facilitate the incorporation of the dots 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 helium temperatures and on the other hand spectral filtering by a monochromator or filter, which entails 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 were introduced to recombine in a quantum well (Kim et al 1999). In that case, multi-excitations were avoided because Coulomb blockade prevented more than one electron and more than one hole at the same time to enter the recombination region. However, Coulomb blockade in the comparatively large structures necessitated temperatures of some tens of millikelvins, making light collection inefficient and the whole experiment difficult. The more recent demonstration of an electrically driven single-photon source (Yuan et al 2002) based on a single quantum dot in a p–i–n junction (similar to a conventional light-emitting diode) only requires temperatures around 5 K. Again, a spectral selection makes it possible to isolate the single-exciton line of a single quantum dot. Antibunching was demonstrated under pumping by short voltage pulses. A special kind of quantum dot can be created around a single impurity in a quantum well (Strauf et al 2002), for example a nitrogen impurity in a ZnSe well. Such states are intermediate between colour centres and semiconductor quantum dots because the exciton wavefunction largely extends in the surrounding semiconductor host. Here also, cryogenic temperatures were required for the isolation of single defects, but no spectral filtering was needed 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 the spectroscopy 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 as single-photon sources. The numbers given in table 2 are only estimates providing a rough comparison of the different systems. The choice of a given system will depend on the intended uses of the single-photon source. For example, the indistinguishability of the emitted photons requires a Fourier-transform limited emission, which can only be achieved under conditions where the system is thoroughly protected from fluctuations of the environment. Only two possibilities exist so far, either atoms or ions in ultra-high vacuum, and localized electronic states in condensed matter at liquid-helium temperatures, in molecules or self-assembled quantum dots. Even at very low temperatures, semiconductor nanocrystals appear to be too sensitive to charge rearrangements and spectral diffusion to reach the Fourier-limit.

A single-photon source operated at room-temperature has the advantage of compactness and convenience, together with a high efficiency of collection, thanks to efficient microscope objectives. Up to 20% of the total emission can be detected, while this efficiency usually does not 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 resonant cavity. Although any small object can in principle be coupled to a high-finesse cavity, solidstate semiconductor devices have the advantage that the cavity can be grown along with the emitting structures, which simplifies the whole production process and does not require later manipulations.

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

	Cold atom	Ion in trap	Molecule low T	Molecule RT	Col. cent. RT	III-V Q-dot	II-VI nanocryst.
Emission spectrum	Sharp line	Sharp line	ZPL + broad lines	Broad band	Broad line, band	Sharp line	Broad line
Typical linewidth	10 MHz	10 MHz	30 MHz, 30 GHz	10 THz	1-30 THz	1 GHz	30 GHz
Emission lifetime	15 ns	15 ns	4 ns	4 ns	10 ns	300 ps	30 ns
Fourier-limited	Yes	Yes	Yes, no	No	No	Yes	No
Max. rate of emiss.	100 kHz	100 kHz	100 MHz	100 MHz	10 MHz	1 GHz	10 MHz
Dark states	No	No	Yes	Yes	Yes	No	Yes
Multi-excit.	No	No	No	No	No	Yes	No
Long-term stability	No, but reproduc.	Yes	Yes	No, in general	Yes	Yes	No, blink
Operat. temper.	μK	mK	1-10 K	300 K	300 K	1-30 K	300 K
Requirements	UHV, lasers	UHV, trap	Liq. He	Easy	Easy	MBE, liq. He	Easy

7. Conclusion and outlook

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

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

The third class of potential uses for single photons exploits entanglement, a key feature of quantum mechanics. In the last 15 years, a more and more precise roadmap for all-optica quantum 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 being contained in the measurement process itself. Single-photon states play a central part in these linear optical quantum computers. Bright and reliable sources of single photons would greatly help to explore and demonstrate their working principles and, more generally, those of quantum teleportation, quantum networks and other fascinating quantum phenomena.

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

However, for all its advantages, parametric down-conversion only produces a Poisson distribution of photons, just as attenuated laser pulses would. To obtain single photons, one has to either strongly attenuate a Poisson source, which strongly limits its brightness, or to rely on a completely different architecture. By collecting the fluorescence photons from a single 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 on the emitters considered. Coupling the emitter to a resonant optical cavity makes it possible to 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 the leak modes. The strong-coupling regime makes other schemes possible, for example, the STIRAP process, in which a single photon is produced in the cavity mode upon an adiabatic passage.

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 their photophysical 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 (with single molecules, colour centres or semiconductor nanocrystals) are sufficient. If, on the other hand, the application requires coherence between the single photons, as most quantum computation protocols do, the emitter must be extremely well shielded from fluctuations in its environment. This condition leaves only two possibilities, either isolated atoms or ions in traps in ultra-vacuum, or such emitters as molecules or self-assembled quantum dots in rigid matrices at cryogenic temperatures. In the latter class of solid-state sources, III–V semiconductor heterostructures benefit from all the knowhow developed for electronic microlithography, by which whole emitting structures, including the resonant cavity, can be produced in one microfabrication process.

Many technical problems must still be solved before single-photon sources become as standard and user-friendly as lasers are. The simplest version of a photon gun, based on a single 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 in principle be built around single atoms or single quantum dots. However, the current setups must 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 quantum process must be considered as open to superposition and interference. Progress in the control of quantum processes has been very fast lately on the sources’ side, but it cannot be separated from progress on the detectors’ side. Higher collection and transmission efficiencies and higher 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, efficient and dedicated detectors must be built for near-infrared communication wavelengths. The manipulation of single photons has been made possible by the existence of single-photon detectors. The manipulation of higher number-states would benefit from detectors capable of distinguishing between multi-photon states, for example superconducting bolometers (Cabrera et al 1998). Quantum cryptography over large fibre distances will also require progress in fibre transmission.

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

Even with resonant-cavity enhancement, the spontaneous emission lifetimes of the emitters discussed in this paper are longer than 100 ps. Higher emission rates require shorter lifetimes. One could think of employing more efficient emitters. Large aggregates of molecules, for example J-aggregates, present enhanced fluorescence rates due to cooperative interference of the molecules in a coherence domain. However, the single-photon emission rate would be limited by the relaxation of higher excited states, with lifetimes of the order of 1 ps. Higher spontaneous emission rates would mean that two excitations could produce two photons simultaneously. Inefficient emitters with high non-radiative relaxation and therefore very short emission times, such as metal particles, would present the same problem. The easiest solution is therefore to rely on cavity enhancement of the current emitters to decrease their emission lifetimes down to some tens of picoseconds.

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

Acknowledgments

The authors thank Professor G Nienhuis for critical reading of the manuscript and suggestions, 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.

Brassard G, Lutkenhaus N, Mor T and Sander¨ s B C 2000 Phys. Rev. Lett. 85 1330

References

Alivisatos A P 1996 Science 271 933

Alspector J, Gupta B and Allen R B 1989 Adv. Neutral Inform. Process. Syst. 1 741

Ambrose W P, Goodwin P M, Enderlein J, Semin D J, Martin J C and Keller R A 1997 Chem. Phys. Lett. 269 365

Anderson P W 1954 J. Phys. Soc. Japan 9 316

Aspect A, Grangier P and Roger G 1981 Phys. Rev. Lett. 47 460

Aspect A, Dalibard J and Roger G 1982a Phys. Rev. Lett. 49 1804

Aspect A, Grangier P and Roger G 1982b Phys. Rev. Lett. 49 91

Bachor H A 1998 A Guide to Experiments in Quantum Optics (Weinheim: Wiley)

Basche T, Moerner W E, Orrit M and Talon H 1992´ Phys. Rev. Lett. 69 1516

Bayer M and Forchel A 2002 Phys. Rev. B 65 041308

Bell J S 1965 Physics (NY) 1 195

Bennett C H and Brassard G 1984 Proc. IEEE Int. Conf. on Computers, Systems and Signal Processing (Bangalore, India) (New York: IEEE)

Bennett C H and Brassard G 1985 IBM Tech. Discl. Bull. 28 3153

Bennett C H, Brassard G and Robert J-M 1988 SIAM J. Comput. 17 210

Bennett C H, Bessette F, Brassard G, Salvail L and Smolin J 1992 J. Cryptol. 5 3

Bennett C H, Brassard G, Crepeau C, Jozsa R, Peres A and Wootters W K 1993´ Phys. Rev. Lett. 70 1895

Bennett C H, Brassard G, Crepeau C and Maurer U M 1995´ IEEE Trans. Inform. Theory 41 1915

Benson O, Santori C, Pelton M and Yamamoto Y 2000 Phys. Rev. Lett. 84 2513

Bernard J, Fleury L, Talon H and Orrit M 1993 J. Chem. Phys. 98 850

Beveratos A, Brouri R, Gacoin T, Poizat J-P and Grangier P 2001 Phys. Rev. A 64 061802(R)

Beveratos A, Brouri R, Gacoin T, Villing A, Poizat J-P and Grangier P 2002 Phys. Rev. Lett. 89 187901

Birks J B 1970 Photophysics of Aromatic Molecules (London: Wiley-Interscience)

Birotheau L, Izrael A, Marzin J Y, Azoulay R, Thierry-Mieg V and Lada ¨ n F R 1992 Appl. Phys. Lett. 61 3023

Born M and Wolf E 1980 Principles of Optics (Oxford: Pergamon)

Bouwmeester D, Pan J, Mattle K, Eibl M, Weinfurter H and Zeilinger A 1997 Nature 390 575

Bouwmeester D, Ekert A and Zeilinger A 2000 The Physics ofQuantum Information (Berlin: Springer)

Brattke S, Varcoe B T H and Walther H 2001 Phys. Rev. Lett. 86 3534

Brokmann X, Giacobino E, Dahan M and Hermier J P 2004a Appl. Phys. Lett. 85 712

Brokmann X, Coolen L, Dahan M and Hermier J P 2004b Phys. Rev. Lett. 93 107403

Brouri R, Beveratos A, Poizat J P and Grangier P 2000 Opt. Lett. 25 1294

Brune M, Schmidt-Kaler F, Maali A, Dreyer J, Hagley E, Raimond J M and Haroche S 1996 Phys. Rev. Lett. 76 1800

Brunel C, Lounis B, Tamarat P and Orrit M 1999 Phys. Rev. Lett. 83 2722

Brunner K, Bockelmann U, Abstreiter G, Walther M, Bohm G, Tr¨ ankle G and Weimann G 1992¨ Phys. Rev. Lett. 69 3216

Burnham D C and Weinberg D L 1970 Phys. Rev. Lett. 25 84

Buttler W T, Hughes R J, Kwiat P G, Lamoreaux S K, Luther G G, Morgan G L, Nordholt J E, Peterson C G and Simmons C 1998 Phys. Rev. Lett. 81 3283–6

Buttler W T, Hughes R J, Lamoreaux S K, Morgan G L, Nordholt J E and Peterson C G 2000 Phys. Rev. Lett. 84 5652

Bylander J, Robert-Philip I and Abram I 2003 Eur. Phys. J. D 22 295

Cabrera B, Clarke R M, Colling P, Miller A J, Nam S and Romani R W 1998 Appl. Phys. Lett. 73 735

Carmichael H J 1993 An Open Systems Approach to Quantum Optics (Lecture Notes in Physics) (Berlin: Springer)

Chance R R, Prock A and Silbey R J 1974 J. Chem. Phys. 60 2744

Chuang I L and Yamamoto Y 1995 Phys. Rev. A 52 3489

Chung I, Shimizu K T and Bawendi M G 2003 Proc. Natl Acad. Sci. USA 100 405

Cirac J I, Zoller P, Kimble H J and Mabuchi H 1997 Phys. Rev. Lett. 78 3221

Clauser J F 1974 Phys. Rev. D 9 853

Clauser J F and Shimony A 1978 Rep. Prog. Phys. 41 1881

Cohen-Tannoudji C, Dupont-Roc J and Grynberg G 1992 Atom–Photon Interactions (New York: Wiley)

Cramer J G 1986 Rev. Mod. Phys. 58 647

Crawford J H and Slifkin L M 1972 Point Defects in Solids (New York: Plenum)

Cresser J D, Hager J, Leuchs G, Rateike F-M and Walther H 1982¨ Dissipative Systems in Quantum Optics (Topics in Current Physics vol 27) ed R Bonifacio and L Lugiato (Berlin: Springer)

Dabbousi B O, Rodriguez-Viejo J, Mikulec F V, Heine J R, Mattoussi H, Ober R, Jensen K F and Bawendi M G 1997 J. Phys. Chem. B 101 9463

Dalibard J, Castin Y and Molmer K 1992 Phys. Rev. Lett. 68 580

De Martini F, Innocenti G, Jacobowitz G R and Mataloni P 1987 Phys. Rev. Lett. 59 2955

De Martini F, Di Giuseppe G and Marrocco M 1996 Phys. Rev. Lett. 76 900

Di Vincenzo D 2000 Fortschr. Phys. 48 771

Diedrich F and Walther H 1987 Phys. Rev. Lett. 58 203

Dieks D 1982 Phys. Lett. A 92 271

Drexhage K H 1974 Progress in Optics vol XII, ed E Wolfe (Amsterdam: North-Holland) p 165

Drexhage K H, Kuhn H and Schafer F P 1968¨ Ber. Bunsenges. Phys. Chem. 72 329

Drummond P D 1999 Quantum Squeezing (Berlin: Springer)

Dumeige Y, Treussart F, Alleaume R, Gacoin T, Roc´ h J F and Grangier P 2004 J. Lumin. 109 61

Einstein A 1905 Ann. Phys. 17 132

Einstein A, Rosen N and Podolsky B 1935 Phys. Rev. 47 777

Empedocles S A and Bawendi M G 1997 Science 278 2114

Empedocles S A, Norris D J and Bawendi M G 1996 Phys. Rev. Lett. 77 3873

Empedocles S A, Neuhauser R and Bawendi M G 1999 Nature 399 126

Fattal D, Diamanti E, Inoue K and Yamamoto Y 2004a Phys. Rev. Lett. 92 037904

Fattal D, Inoue K, Vuckovic J, Santori C, Solomon G S and Yamamoto Y 2004b Phys. Rev. Lett. 92 037903

Fearn H and Loudon R 1987 Opt. Commun. 64 485

Felix S, Gisin N, Stefanov S and Zbinden H 2001 ´ J. Mod. Opt. 48 2009

Fleury L, Sick B, Zumofen G, Hecht B and Wild U P 1998 Mol. Phys. 95 1333

Fleury L, Segura J M, Zumofen G, Hecht B and Wild U P 2000 Phys. Rev. Lett. 84 1148

Fourkas J T 2001 Opt. Lett. 26 211

Gabrielse G and Dehmelt H 1985 Phys. Rev. Lett. 55 67

Gaebel T, Popa I, Gruber A, Domhan M, Jelezko F and Wrachtrup J 2004 New J. Phys. 6 98

Gammon D, Snow E S, Shanabrook B V, Katzer D S and Park D 1996 Science 87 273

Gardiner C W, Parkins A S and Zoller P 1992 Phys. Rev. A 46 4363

Gasparoni S, Pan J W, Walther P, Rudolph T and Zeilinger A 2004 Phys. Rev. Lett. 93 020504

Gellermann W 1991 J. Phys. Chem. Solids 52 249

Gerard J M, Sermage B, Gayral B, Legrand B, Costard E and Thierry-Mieg V 1998´ Phys. Rev. Lett. 81 1110

Giacomini S, Sciarrino F, Lombardi E and De Martini F 2002 Phys. Rev. A 66 030302

Gisin N, Ribordy G, Tittel W and Zbinden H 2002 Rev. Mod. Phys. 74 145

Glauber R J 1965 Optical coherence and photon statistics Quantum Optics and Electronics ed C DeWitt et al (New York: Gordon and Breach) p 63

Gorman P M, Tapster P R and Rarity J G 2001 J. Mod. Opt. 48 1887–901

Gottesman D and Chuang I L 1999 Nature 402 390

Goy P, Raimond J M, Gross M and Haroche S 1983 Phys. Rev. Lett. 50 1903

Grangier P, Roger G and Aspect A 1986 Europhys. Lett. 1 173

Greenberger D M, Horne M A and Zeilinger A 1989 Bell’s Theorem, Quantum Theory, and Conceptions of the Universe ed M Kafatos (Dordrecht: Kluwer) p 69

Gruber A, Drabenstedt A, Tietz C, Fleury L, Wrachtrup J and von Borczyskowski C 1997 ¨ Science 276 2012

Hanbury Brown R and Twiss R Q 1956 Nature 177 27

Haroche S 1992 Cavity quantum electrodynamics Les Houches session LIII, ‘Fundamental Systems in Quantum Optics’ ed J Dalibard et al (Amsterdam: Elsevier)

Hegerfeldt G C 1993 Phys. Rev. A 47 449

Heinzen D J, Childs J J, Thomas J F and Feld M S 1987 Phys. Rev. Lett. 58 1320

Hines M A and Guyot-Sionnest P 1996 J. Phys. Chem. 100 468

Hiskett P A, Buller G S, Loudon A Y, Smith J M, Gontijo I, Walker A C, Townsend P D and Robertson M J 2000 Appl. Opt. 39 6818

Hofkens J et al 2003 Proc. Natl Acad. Sci. USA 100 13146

Hong C K and Mandel L 1986 Phys. Rev. Lett. 56 58

Hong C K, Ou Z Y and Mandel L 1987 Phys. Rev. Lett. 59 2044

Imbusch G F and Kopelman R 1981 Laser Spectroscopy of Solids ed W M Yen and P M Selzer (Berlin: Springer) p 3

Jacobs B and Franson J 1996 Opt. Lett. 21 1854

Jaynes E T and Cummings F W 1963 Proc. IEEE 51 89

Jelezko F, Popa I, Gruber A, Tietz C and Wrachtrup J 2002 Appl. Phys. Lett. 81 2160

Jennewein T, Achleitner U, Weihs G, Weinfurter H and Zeilinger A 2000 Rev. Sci. Instrum. 71 1675

Jewell J L, Harbison J P, Scherer A, Lee Y H and Florez L T 1991 IEEE J. Quantum Electron. 27 1332

John S 1987 Phys. Rev. Lett. 58 2486

Kim J, Benson O, Kan H and Yamamoto Y 1999 Nature 397 500

Kimble H J 2003 Phys. Rev. Lett. 90 249801

Kimble H J and Walls D F 1987 J. Opt. Soc. Am. B 4 1450

Kimble H J, Dagenais M and Mandel L 1977 Phys. Rev. Lett. 39 691

Kiraz A, Atature M and Imamoglu A 2004¨ Phys. Rev. A 69 032305

Kitson S C, Jonsson P, Rarity J G and Tapster P R 1998 Phys. Rev. A 58 620

Kleppner D 1981 Phys. Rev. Lett. 47 233

Klimov V I, Mikhailovsky A A, McBranch D W, Leatherdale C A and Bawendi M G 2000 Science 287 1011

Knill E, Laflamme R and Milburn G J 2001 Nature 409 46

Kubo R 1954 J. Phys. Soc. Japan 9 935

Kuhn H 1970 J. Chem. Phys. 53 101

Kuhn A, Hennrich M and Rempe G 2002 Phys. Rev. Lett. 89 067901

Kulzer F, Koberling F, Christ T, Mews A and Basche T 1999 Chem. Phys. 247 23

Kuno M, Fromm D P, Hamann H F, Gallagher A and Nesbitt D J 2001 J. Chem. Phys. 115 1028

Kurtsiefer C, Mayer S, Zarda P and Weinfurter H 2000 Phys. Rev. Lett. 85 290

Kwiat P G, Mattle K, Weinfurter H and Zeilinger A 1995 Phys. Rev. Lett. 75 4337

Labeau O, Tamarat P and Lounis B 2003 Phys. Rev. Lett. 90 257404

Law C K and Kimble H J 1997 J. Mod. Opt. 44 2067

Lee T H, Kumar P, Mehta A, Xu K, Dickson R M and Barnes M D 2004 Appl. Phys. Lett. 85 100

Lombardi E, Sciarrino F, Popescu S and De Martini F 2002 Phys. Rev. Lett. 88 070402

Loudon R 1983 The Quantum Theory of Light (New York: Oxford University Press)

Lounis B and Moerner W E 2000 Nature 407 491

Lounis B, Bechtel H A, Gerion D, Alivisatos P and Moerner W E 2000 Chem. Phys. Lett. 329 399

Lukosz W and Kunz R E 1977 J. Opt. Soc. Am. 67 1607

Lutkenhaus N 2000¨ Phys. Rev. A 61 052304

Maˆıtre X, Hagley E, Nogues G, Wunderlich C, Goy P, Brune M, Raimond J M and Haroche S 1997 Phys. Rev. Lett. 79 769

Mandel L and Wolf E 1995 Optical Coherence and Quantum Optics (Cambridge: Cambridge University Press)

Maurer U M and Wolf S 1999 IEEE Trans. Inform. Theory 45 499

McCall S L, Levi A F J, Slusher R E, Pearton S J and Logan R A 1992 Appl. Phys. Lett. 60 289

McKeever J, Boca A, Boozer A D, Miller R, Buck J R, Kuzmich A and Kimble H J 2004 Science 303 1992

Messin G, Hermier J P, Giacobino E, Desbiolles P and Dahan M 2001 Opt. Lett. 26 1891

Michler P 2003 Single quantum dots fundamentals, applications and new concepts Top. Appl. Phys. 90 352

Michler P, Imamoglu A, Mason M D, Carson P J, Strouse G F and Buratto S K 2000a Nature 406 968

Michler P, Kiraz A, Becher C, Schoenfeld W V, Petroff P M, Zhang L, Hu E and Imamoglu A 2000b Science 290 2282

Milburn G J 1988 Phys. Rev. Lett. 62 2124

Monroe C 2002 Nature 416 238

Moriarty P 2001 Rep. Prog. Phys. 64 297

Muller A, Breguet J and Gisin N 1993 Europhys. Lett. 23 383

Nielsen M A and Chuang I L 2000 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press)

Nienhuis G and Alkemade C T J 1976 Physica B + C 81 181

Nirmal M, Norris D J, Kuno M, Bawendi M G, Efros A L and Rosen M 1995 Phys. Rev. Lett. 75 3728

Novotny L, Beversluis M R, Youngworth K S and Brown T G 2001 Phys. Rev. Lett. 86 5251

Ohnesorge B, Bayer M, Forchel A, Reithmaier J P, Gippius N A and Tikhodeev S G 1997 Phys. Rev. B 56 R4367–70

Orrit M and Bernard J 1990 Phys. Rev. Lett. 65 2716

Ou Z Y and Mandel L 1988 Phys. Rev. Lett. 61 50

Pan J W, Bouwmeester D, Daniell M, Weinfurter H and Zeilinger A 2000 Nature 403 515

Parkins A S, Marte P, Zoller P and Kimble H J 1993 Phys. Rev. Lett. 71 3095–8

Pelton M, Santori C, Vuckovic J, Zhang B, Solomon G S, Plant J and Yamamoto Y 2002 Phys. Rev. Lett. 89 233602

Pittman T B, Jacobs B C and Franson J D 2001 Phys. Rev. A 64 062311

Pittman T B, Jacobs B C and Franson J D 2002 Phys. Rev. A 66 052305

Planck M 1900 Verh. Deutsch. Phys. Ges. 2 202

Polzik E S, Carri J and Kimble H J 1992 Phys. Rev. Lett. 68 3020

Purcell E M 1946 Phys. Rev. 69 681

Ralph T C, White A G, Munro W J and Milburn G J 2001 Phys. Rev. A 65 012314

Rarity J G, Owens P C M and Tapster P R 1994 J. Mod. Opt. 41 2435

Rarity J G, Wall T E, Ridley K D, Owens P C M and Tapster P R 2000 Appl. Opt. 39 6746

Reynaud S 1983 Ann. Phys. 8 315

Santori C, Pelton M, Solomon G, Dale Y and Yamamoto Y 2001 Phys. Rev. Lett. 86 1502

Santori C, Fattal D, Vuckovic J, Solomon G S and Yamamoto Y 2002 Nature 419 594

Santori C, Fattal D, Vuckovic J, Solomon G S and Yamamoto Y 2004 New J. Phys. 6 89

Santos E 1992 Phys. Rev. A 46 3646–56

Schuurmans F J P, de Vries P and Lagendijk A 2000 Phys. Lett. A 264 472

Shen Y R 1984 The Principles ofNonlinear Optics (New York: Wiley)

Shih Y H 2003 Rep. Prog. Phys. 66 1009

Shih Y H and Alley C O 1988 Phys. Rev. Lett. 61 2921

Shih Y H and Sergienko A V 1994 Phys. Rev. A 50 2564

Short R and Mandel L 1983 Phys. Rev. Lett. 51 384

Sild O and Haller K 1988 Zero-Phonon Lines (Berlin: Springer)

Skinner J L and Reilly P D 1994 J. Chem. Phys. 101 959

Sleator T and Weinfurter H 1995 Phys. Rev. Lett. 74 4087

Solomon G S, Pelton M and Yamamoto Y 2001 Phys. Rev. Lett. 86 3903

Stefanov A, Guinnard O, Guinnard L, Zbinden H and Gisin N 2000 J. Mod. Opt. 47 595

Strauf S, Michler P, Klude M, Hommel D, Bacher G and Forchel A 2002 Phys. Rev. Lett. 89 177403

Stucki D, Ribordy G, Stefanov A, Zbinden H, Rarity J and Wall T 2001 J. Mod. Opt. 48 1967

Tapster P R, Rarity J G and Owens P C M 1994 Phys. Rev. Lett. 73 1923

Thompson R M, Stevenson R M, Shields A J, Farrer I, Lobo C J, Ritchie D A and Leadbeater M L 2001 Phys. Rev. B 64 201302

Tittel W, Brendel J, Zbinden H and Gisin N 1998 Phys. Rev. Lett. 81 3563

Treussart F, Alleaume R, Le Floc’h V, Xiao L T, Courty J M and Roc´ h J F 2002 Phys. Rev. Lett. 89 093601

Turchette Q A, Hood C J, Lange W, Mabuchi H and Kimble H J 1995 Phys. Rev. Lett. 75 4710

Twiss R Q, Little A G and Hanbury Brown R 1957 Nature 180 324

Vallee R, Van Der Auweraer M, De Schryver F C, Beljonne D and Orrit M 2005´ Chem. Phys. Chem. 6 81

Verberk R and Orrit M 2003 J. Chem. Phys. 119 2214

Vitanov N V, Fleischhauer M, Shore B W and Bergmann K 2001 Adv. At. Mol. Opt. Phys. 46 55

Vuckovic J, Pelton M, Scherer A and Yamamoto Y 2002 Phys. Rev. A 66 023808

Vuckovic J, Fattal D, Santori C, Solomon G and Yamamoto Y 2003 Appl. Phys. Lett. 82 3596

Waks E, Inoue K, Santori C, Fattal D, Vuckovic J, Solomon G S and Yamamoto Y 2002 Nature 420 762

Walls D F 1979 Nature 280 451

Walls D F and Milburn G J 1994 Quantum Optics (Berlin: Springer)

Weihs G, Jennewein T, Simon C, Weinfurter H and Zeilinger A 1998 Phys. Rev. Lett. 81 5039

Wootters W K and Zurek W H 1982 Nature 299 802

Xiao M, Wu L A and Kimble H J 1987 Phys. Rev. Lett. 59 278

Yablonovitch E 1987 Phys. Rev. Lett. 58 2059

Yamamoto Y and Imamoglu A 1999 Mesoscopic Quantum Optics (New York: Wiley)

Yamamoto Y, Machida S, Igeta K and Horikashi Y 1989 Coherence and Quantum Optics ed J H Eberly et al (New York: Plenum) p 1249

Yuan Z, Kardynal B E, Stevenson R M, Shields A J, Lobo C J, Cooper K, Beattie N S, Ritchie D A and Pepper M 2002 Science 295 102

Zondervan R, Kulzer F, Orlinskii S B and Orrit M 2003 J. Phys. Chem. A 107 6770

Zondervan R, Kulzer F, Kol’chenko M A and Orrit M 2004 J. Phys. Chem. A 108 1657

Zwiller V, Aichele A and Benson O 2004 New J. Phys. 6 96

Zwiller V, Jonsson P, Blom H, Jeppesen S, Pistol M E, Samuelson L, Katznelson A A, Kotelnikov E Y, Evtikhiev V and Bjork G 2002¨ Phys. Rev. A 66 053814