Single-photon sources

For an ideal single-photon source, the probability of observing n counts for M emitted photons is p_x(n) = binomial(M, n) (xT)^n (1-xT)^(M-n).

“The probability of observing n counts for M emitted photons is therefore p x (n) = n M (xT ) n (1 -xT ) M-n . (17”

For the coherent source, adding the variances of the signal and reference measurements gives the variance for the transmission coefficient T^2_C = (1/(xM T))(T+1).

“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: T 2 C = 1 xM T (T + 1).”

For the transmission measurement with an ideal single-photon source, the variance is T^2_SP = (1/(xM T))(T+1-2xT).

“For the transmission measurement, the variance becomes T 2 SP = 1 xM T (T + 1 -2xT ).”

The ratio of the variances for the single-photon source to the coherent source is T^2_SP/T^2_C = (1-2xT)/(1+T).

“With a similar result for the reference measurement, the ratio of the variances for the squeezed to coherent cases is T 2 SP T 2 C = 1 -2x T 1 + T , (18”

For a coherent source with large average number M of photons transmitted by an absorbing medium with intensity transmission T and measured by a photon-counting detector with quantum yield x, the distribution of detected counts is Poissonian with average S_C = xMT and shot noise N_C = √(xMT).

“The distribution of the detected counts is also Poissonian, with the average number of detected counts (supposed to be large) S C = xMT and with the shot noise (standard deviation) N C = √ xMT .”

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 = μ 1.

“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 = µ 1.”

Even with resonant-cavity enhancement, the spontaneous emission lifetimes of the emitters discussed in this paper are longer than 100 ps.

“Even with resonant-cavity enhancement, the spontaneous emission lifetimes of the emitters discussed in this paper are longer than 100 ps.”

Pairs of correlated photons (twin photons) were obtained at high rates by parametric downconversion.

“At about the same time, pairs of correlated photons (twin photons) were obtained at high rates by parametric downconversion (Burnham andWeinberg 1970, Hong andMandel 1986).”

• cites Burnham andWeinberg 1970Not checked yet

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• cites Hong andMandel 1986Not checked yet

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If a one-photon state is sent on a beam splitter with photon-counting detectors on the reflected and transmitted beams, no coincidence between counts measured by the two detectors is observed.

“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).”

From the mid-1980s, single ions in traps provided long observation times with one and the same ion.

“From the mid-1980s, single ions in traps provided long observation times with one and the same ion (Diedrich and Walther 1987).”

• cites Diedrich and Walther 1987Not checked yet

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Single-photon sources are crucial components in recent proposals for quantum computation by means of linear optics.

“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.”

• cites Knill et al 2001Not checked yet

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Sub-Poissonian statistics were first demonstrated in the fluorescence of a very faint atomic beam excited by a continuous laser.

“Sub-Poissonian statistics were first demonstrated in the fluorescence of a very faint atomic beam excited by a continuous laser (Kimble et al 1977).”

• cites Kimble et al 1977Not checked yet

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Since only emission between the lowest two levels can be detected, complex systems behave as two-level emitters.

“As a consequence, since only emission between the lowest two levels can be detected, the complex system behaves as a two-level emitter.”

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.

“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.”

One-photon states present a peculiar anticorrelation effect, which does not exist for a classical wave.

“One-photon states present a peculiar anticorrelation effect, which does not exist for a classical wave.”

For perfectly antibunched light, the second-order correlation function cancels for zero time.

“For perfectly antibunched light, the correlation function therefore cancels for zero time: g (2) (τ = 0) = 0.”

Two photons incident at the same time on the two input ports of a 50%/50% beam splitter interfere in such a way that they both exit from one of the output ports.

“Fearn and Loudon (1987) and Hong et al (1987) pointed out that two photons incident at the same time on the two input ports of a 50%/50% beam splitter interfere in such a way that they both exit from one of the output ports, as illustrated in figure 4.”

• cites Fearn and Loudon (1987)Not checked yet

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• cites Hong et al (1987)Not checked yet

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Coincidences between two detectors in a split detection beam are insensitive to dead-times and afterpulses.

“Coincidences between the two detectors are then insensitive to dead-times and afterpulses.”

Antibunching alone is not sufficient for obtaining single photons on demand.

“Yet, antibunching alone is by no means sufficient for obtaining single photons on demand.”

Provided the probability of generating two pairs at the same time remains negligible, correlated photon pairs from parametric downconversion can be used as sources of heralded single photons.

“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.”

Dephasing and spectral diffusion, which arise from fast and slow fluctuations of the transition frequency, complicate the indistinguishability of photons from a single-photon source.

“However, dephasing and spectral diffusion, which arise from fast and slow fluctuations of the transition frequency, complicate this picture.”

Early demonstrations of single-photon emission have been extended to many different systems that could become useful emitters of single photons, including semiconductor nanocrystals, colour centres in diamond, and single quantum dots.

“Those early demonstrations have now been extended to many different systems which could become useful emitters of single photons.”

• Also stated in: § 1. Introduction · hedged

  “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.”

• cites Lounis et al 2000Not checked yet

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• cites Brouri et al 2000 sourceNot checked yet

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• cites Michler et al 2000bNot checked yet

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The first successful generation of single photons was based on a cascade transition of calcium atoms.

“The first successful generation of single photons (Clauser 1974) was based on a cascade transition of calcium atoms.”

• cites Clauser 1974 sourceNot checked yet

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A pulsed excitation gives rise to periodic emission of single photons.

“A pulsed excitation gives rise to periodic emission of single photons (d).”

Single emitters usually deliver photons one at a time.

“Single emitters usually deliver photons one at a time.”

• Also stated in: § 2.3. Antibunching and second-order correlation function · hedged

  “The photons have a tendency to be emitted one by one, i.e. they are 'antibunched'”

• cites Kimble et al 1977Not checked yet

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In order to give rise to fully destructive interference, the two photons must be completely indistinguishable, i.e. they must be in exactly the same mode.

“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.”

In the absence of slow spectral diffusion, the frequency linewidth (full-width at half-maximum) is ν = 1/(2πT1) + 1/(πT2*).

“In the absence of slow spectral diffusion, the resulting frequency linewidth (full-width at half-maximum),”

If the dipole lies in the focal plane of the collection objective, the polarization is close to linear, parallel to the dipole, for all photons across the whole spatial mode.

“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.”

The depth of the two-photon interference effect will no longer be 100% but only T2/2T1 when dephasing is present.

“The depth of the two-photon interference effect will no longer be 100% but only T 2 /2T 1 (Bylander et al 2003).”

• cites Bylander et al 2003Not checked yet

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The noise with the single-photon source is always less than with the laser.

“which shows that, as expected, the noise with the single-photon source is always less than with the laser.”

In complex geometries, the local field correction depends on details of the structure and must be calculated numerically.

“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.”

• cites Lukosz and Kunz 1977Not checked yet

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An additional loss of indistinguishability arises from incoherent pumping.

“An additional loss of indistinguishability arises from incoherent pumping.”

In many cases, the spectrum of a single-photon source is broader than the Fourier-transform of the time-profile of the emitted pulse.

“In many cases, however, the spectrum of a source is broader than the Fourier-transform of the time-profile of the emitted pulse.”

The polarization state of a single-photon source is determined by the microscopic nature of the emitter and by the way it is coupled to the emission mode.

“It is determined by the microscopic nature of the emitter and by the way it is coupled to the emission mode.”

The noise reduction is nearly total for high detection quantum yields (x ≈ 1) and for weakly absorbing samples (T ≈ 1).

“The noise reduction is nearly total for high detection quantum yields (x ≈ 1) and for weakly absorbing samples (T ≈ 1).”

If an emitter is placed in a large bulk dielectric with refractive index n, its emission lifetime is significantly modified.

“If the emitter is placed in a large bulk of dielectric with refractive index n, the emission lifetime is significantly modified.”

• cites Nienhuis and Alkemade (1976)Not checked yet

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If the spectrum of a single-photon source is Fourier-transform-limited, two photons will be indistinguishable.

“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.”

With squeezed light, fluctuations of a given quadrature are reduced with respect to those of coherent light.

“With squeezed light, fluctuations of a given quadrature are reduced with respect to those of coherent light (Kimble andWalls 1987, Drummond 1999).”

• cites Kimble andWalls 1987Not checked yet

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• cites Drummond 1999Not checked yet

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Photon-counting devices based on III-V semiconductors are now available in the near-infrared window.

“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).”

• cites Hiskett et al 2000 sourceNot checked yet

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• cites Rarity et al 2000 sourceNot checked yet

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• cites Stucki et al 2001Not checked yet

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If the dipole is perpendicular to the focal plane, the polarization in the collected mode is radial.

“If the dipole is perpendicular to the focal plane, the polarization in the collected mode is radial (Fourkas 2001).”

• cites Fourkas 2001 sourceNot checked yet

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A first correction factor from the different density of photon modes in the dielectric yields the relation τ_f(n) = (1/n) τ_f(1).

“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”

• cites Nienhuis and Alkemade (1976)Not checked yet

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In the spherical Lorentz cavity approximation, and if the polarizability of the object is not too different from that of the surrounding matrix, the lifetime in a dielectric with index n becomes τ_f(n) = (1/n) (3/(n^2+2))^2 τ_f(1).

“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, Vallée et al 2005), the lifetime in a dielectric with index n becomes (Imbusch and Kopelman 1981)”

• cites Schuurmans et al 2000Not checked yet

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• cites Vallée et al 2005 sourceNot checked yet

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• cites Imbusch and Kopelman 1981Not checked yet

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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.

“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).”

• cites Fourkas 2001 sourceNot checked yet

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• cites Novotny et al 2001Not checked yet

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The only way for Eve to remain unnoticed is to restrict her measurement rate to less than the dark count rate p dark of Bob's detectors.

“The only way for her to remain unnoticed is to restrict her measurement rate to less than the dark count rate, p dark , of Bob's detectors.”

For low dark count probabilities, the single-photon source permits more attenuation, i.e. a longer fibre, than an attenuated laser source.

“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.”

In the PNS attack, Eve obtains full information on Bob's bits without introducing any error.

“In this way, Eve has obtained full information on Bob's bits without introducing any error.”

• AssumptionThe authors state this as an assumption: Eve has unlimited technological power within the laws of physics.

The condition for security of the communication is that the rate of bits received by Bob, T AB η det, be larger than his detectors' dark counts probability p dark, which may hide Eve's measurements: T AB > p dark / η det.

“The condition for security of the communication will be that the rate of bits received by Bob, T AB η det , be larger than his detectors' dark counts probability, p dark , which may hide Eve's measurements (Brassard et al 2000):”

• cites Brassard et al 2000Not checked yet

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For three-photon GHZ states, the contradiction between the Einstein-Podolsky-Rosen assumptions and quantum mechanics arises even for an individual event.

“There, the contradiction between the Einstein-Podolsky-Rosen assumptions and quantum mechanics arises even for an individual event.”

• cites Greenberger et al 1989 sourceNot checked yet

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Rarity and collaborators proposed a physical quantum-mechanical random generator based on the 50:50 splitting of a beam of single photons.

“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.”

• cites Rarity et al 1994 sourceNot checked yet

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Quantum theory predicts that, for each photon, the which-way decision is truly random and independent of history or other experimental parameters.

“Quantum theory predicts that, for each photon, the 'which-way' decision is truly random and independent of history or other experimental parameters.”

Each individual photon going through a 50:50 beam splitter has an equal probability of being reflected or transmitted.

“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.”

As shown by Brassard et al (2000), to eschew a PNS attack in the presence of dark counts, the transmission efficiency T AB of the line from Alice to Bob should obey T AB > p dark /(η det μ) + μ/(2η det).

“As shown by Brassard et al (2000), in order to eschew a PNS-attack in the presence of dark counts, the transmission efficiency, T AB , of the line from Alice to Bob should obey the following criterion:”

• cites Brassard et al (2000)Not checked yet

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A single measurement of a GHZ state will discriminate with certainty between quantum mechanics and a local hidden-variable theory.

“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.”

If Eve has unlimited technological power within the laws of physics, a photon number splitting (PNS) attack is in principle possible.

“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, Lütkenhaus 2000):”

• cites Brassard et al 2000Not checked yet

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• cites Lütkenhaus 2000Not checked yet

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• AssumptionThe authors state this as an assumption: Eve has unlimited technological power within the laws of physics.

A continuous stream of random numbers at a rate of 1 Mbit s^-1 could be produced with no correlations between the generated numbers using attenuated LED light sources.

“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.”

• cites Jennewein et al 2000Not checked yet

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• cites Stefanov et al 2000Not checked yet

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All these experiments have confirmed the predictions of quantum mechanics.

“All these experiments have confirmed the predictions of quantum mechanics.”

The violation of Bell's inequality was demonstrated with pairs of independent photons.

“In such a way, the violation of Bell's inequality was demonstrated with pairs of independent photons (Fattal et al 2004b).”

• cites Fattal et al 2004bNot checked yet

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The reliability and secrecy of the key transmission can be improved by applying error-correcting and privacy-amplifying algorithms to the transmitted bits.

“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).”

• cites Bennett et al 1995Not checked yet

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A perfect single-photon source associated with perfect detection would give access to arbitrarily small absorptions, impossible to measure with a laser source because of photon noise.

“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.”

The intercept-resend attack necessarily induces an additional error rate.

“She thus necessarily induces an additional error rate.”

Detector after-pulses and dead-times may introduce strong correlations or anticorrelations between successive bits.

“The detector's after-pulses and dead-times may introduce strong correlations or anticorrelations between successive bits.”

The recent generation of polarization-entangled photon pairs with a quantum-dot source is a first step toward closing the detection loophole.

“A first step in that direction is the recent generation of polarization-entangled photon pairs with a quantum-dot source (section 6.6).”

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.

“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.”

If the Purcell factor is much greater than 1, the emitter will radiate much faster in the cavity than in free space.

“If the Purcell factor is much greater than 1, the emitter will radiate much faster in the cavity than in free space.”

Michler et al embedded self-assembled InAs and InGaAs quantum dots in microdisc cavities to generate single photons.

“Michler et al (2000) embedded self-assembled InAs and InGaAs quantum dots in microdisc cavities to generate single photons.”

• cites Michler et al 2000Not checked yet

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Assuming a normalized Lorentzian mode density with a width of ω 0 /Q for the cavity, the spontaneous emission enhancement is γ/γ 0 = 3/(4π 2) λ 3 /V Q + f.

“Assuming a normalized Lorentzian mode density with a width of ω 0 /Q for the cavity, ρ cav (ω) ≈ 2Q/π ω 0 , we deduce the following spontaneous emission enhancement:”

• AssumptionThe authors state this as an assumption: The cavity mode density is a normalized Lorentzian with a width of ω 0 /Q.

Both experiments show that single-photon devices can provide completely secure communication in conditions under which this would be impossible with attenuated laser pulses.

“Both experiments show that single-photon devices can provide completely secure communication in conditions under which this would be impossible with attenuated laser pulses.”

• cites Waks et al 2002Not checked yet

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• cites Beveratos et al 2002Not checked yet

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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).

“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).”

To date, two groups have used a single-photon source in a QKD link which includes sifting, error-correction and privacy amplification.

“To date, two groups have used a single-photon source in a QKD link which includes sifting, error-correction and privacy amplification.”

• cites Waks et al 2002Not checked yet

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• cites Beveratos et al 2002Not checked yet

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The radiative transition rate γ of an atom from an excited initial state |e to a lower energy final state |g depends on the density of available photon states ρ(ω) at the transition frequency ω and on the atom-vacuum field dipole interaction Hamiltonian.

“The radiative transition rate, γ , of an atom from an excited, initial state |e to a lower energy, final state |g depends on the density of available photon states, ρ(ω), at the transition frequency, ω, and on the atom-vacuum field dipole interaction Hamiltonian.”

Efficient cavity-coupled single-photon sources were demonstrated with single quantum dots embedded in microcavities.

“Efficient cavity-coupled single-photon sources were demonstrated with single quantum dots embedded in microcavities (Ohnesorge et al 1997, Gérard et al 1998, Solomon et al 2001).”

• cites Ohnesorge et al 1997Not checked yet

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• cites Gérard et al 1998Not checked yet

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• cites Solomon et al 2001Not checked yet

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The Purcell factor increases with Q/V only as long as the coupling parameter g is less than the decay rates of the system (γ and γ cav).

“The Purcell factor increases with Q/V only as long as the coupling parameter, g, is less than the decay rates of the system (γ and γ cav ).”

In this case β ≈ 80% of the emitted light was coupled into a single cavity mode.

“In this case β ≈ 80% of the emitted light was coupled into a single cavity mode.”

• cites Pelton et al 2002Not checked yet

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• cites Vuckovic et al 2003Not checked yet

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By altering ρ(ω), for example by placing the atom in a cavity or near an interface, the spontaneous emission rate can be enhanced or suppressed.

“Thus, by altering ρ(ω) (placing the atom in a cavity or near an interface), the spontaneous emission rate can be enhanced or suppressed (Purcell 1946, Drexhage 1974).”

• cites Purcell 1946Not checked yet

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• cites Drexhage 1974Not checked yet

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The atom-cavity system must be able to transfer quantum information between atoms and photons in a coherent manner.

“The atom-cavity system, in particular, must be able to transfer quantum information between atoms and photons in a coherent manner (Maître et al 1997, Brattke et al 2001).”

• cites Maître et al 1997Not checked yet

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• cites Brattke et al 2001Not checked yet

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The photon-pair creation process is very inefficient (efficiency ≈ 10 -7 -10 -11).

“This photon-pair creation process is very inefficient (efficiency ≈ 10 -7 -10 -11 ).”

Kimble's group employed a single Cs atom trapped in the optical cavity.

“To circumvent this problem, Kimble's group employed a single Cs atom trapped in the optical cavity.”

• cites McKeever et al 2004Not checked yet

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The stimulated emission process requires a strong coupling to a single mode of a high-finesse optical cavity.

“The stimulated emission process requires a strong coupling to a single mode of a high-finesse optical cavity (Law and Kimble 1997).”

• cites Law and Kimble 1997Not checked yet

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Photon pairs can be generated by spontaneous parametric down-conversion (SPDC) in non-centrosymmetric crystals such as KDP, BBO, LBO or LiNbO 3.

“Photon pairs can be generated by SPDC in non-centrosymmetric crystals such as KDP, BBO, LBO or LiNbO 3 .”

• cites Shen 1984Not checked yet

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The probability p multi that a non-empty weak coherent pulse contains more than one photon is approximately n 2.

“p multi = p n 2 p n 1 = 1 -p(0) -p(1) 1 -p(0) ≈ n 2 , (30”

In contrast to solid-state single-photon sources, the 85Rb atom-cavity source operates under non-stationary conditions because atoms enter and leave the cavity randomly.

“In contrast to solid-state single-photon sources, this source operates under non-stationary conditions because atoms enter and leave the cavity randomly.”

The reversible transfer of quantum information between the emitter and the photons opens the way for quantum networks.

“The reversible transfer of quantum information between the emitter and the photons opens the way for quantum networks.”

Entangled photon pairs can be obtained in certain atomic cascades because of the polarization correlation between the successive emitted photons.

“Entangled photon pairs can be obtained in certain atomic cascades (j = 0 → j = 1 → j = 0, j being the total angular momentum of the atomic states) because of the polarization correlation between the successive emitted photons.”

Along the two directions where the cones overlap, the light is described by a Bell entangled state.

“Along the two directions ('1' and '2'), where the cones overlap, the light is described by a Bell entangled state:”

Aspect's experiment used the 4p 2 1 S 0 → 4s 4p 1 P 1 → 4s 2 1 S 0 cascade in a calcium atom beam, which yields two polarization-correlated photons (λ 1 = 551 nm and λ 2 = 423 nm) upon two-photon excitation.

“For example, Aspect's experiment used the 4p 2 1 S 0 → 4s 4p 1 P 1 → 4s 2 1 S 0 cascade in a calcium atom beam, which yields two polarization-correlated photons (λ 1 = 551 nm and λ 2 = 423 nm) upon two-photon excitation from 4s 2 1 S 0 to 4p 2 1 S 0 .”

• cites Aspect et al 1982bNot checked yet

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Single-photon number-states may be approximated by coherent states with a very low average photon number n.

“Single-photon number-states may be approximated by coherent states with a very low average photon number n .”

When n is small, most pulses are empty: p(n = 0) ≈ 1-n.

“When n is small, most pulses are empty: p(n = 0) ≈ 1n .”

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 was varied.

“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.”

• cites Grangier et al 1986 sourceNot checked yet

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Nonlinear interaction of a pump laser pulse with the crystal occasionally splits one high-frequency pump photon into two lower-frequency signal and idler photons.

“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).”

• cites Shen 1984Not checked yet

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For quantum cryptography, the reliability of the source is related to the probability p multi that a non-empty weak coherent pulse contains more than one photon.

“For quantum cryptography, the reliability of the source is related to the probability, p multi , that a non-empty weak coherent pulse contains more than one photon.”

In 1986, Aspect and co-workers used this atomic cascade source to demonstrate the wave-particle duality of a photon by sending single photons on a beam splitter.

“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.”

• cites Grangier et al 1986 sourceNot checked yet

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The photon production step is based on a stimulated Raman process driven by an adiabatic passage (STIRAP) between two ground states of a single atom.

“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.”

• cites Parkins et al 1993Not checked yet

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• cites Vitanov et al 2001 sourceNot checked yet

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Recently, single-photon sources using single atoms strongly interacting with cavities have been demonstrated by the groups of Rempe and Kimble.

“Recently, single-photon sources using single atoms strongly interacting with cavities have been demonstrated by the groups of Rempe and Kimble (Kuhn et al 2002, McKeever et al 2004).”

• cites Kuhn et al 2002Not checked yet

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• cites McKeever et al 2004Not checked yet

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The value of p multi can be made arbitrarily small.

“The value of p multi can be made arbitrarily small.”

Sources based on incoherent excitation of the emitting state of a quantum emitter do not meet the requirement that single-photon states 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 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.

“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(Zondervan et al , 2004)).”

• cites Zondervan et al 2003 sourceNot checked yet

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• cites Zondervan et al 2004Not checked yet

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For systems presenting high photostability, for example terrylene in a p-terphenyl crystal, single molecules can operate as single-photon sources.

“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 andMoerner 2000, Treussart et al 2002).”

• cites Lounis and Moerner 2000Not checked yet

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• cites Treussart et al 2002Not checked yet

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Depending on the molecular compound and host crystal, excursions to a triplet lifetime can in some cases limit the molecular brightness.

“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).”

• cites Bernard et al 1993Not checked yet

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As a consequence of fast dephasing, the emission of organic molecules at room temperature is very far from lifetime-limited, by a factor of at least 10 000.

“As a consequence of fast dephasing, the emission is very far from lifetime-limited, by a factor of at least 10 000!”

A transition to an excited electronic state in a molecule in condensed matter is distributed over a broad range of frequencies, corresponding to the creation of additional vibrations and phonons.

“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.”

Aromatic molecules were used for an early demonstration of a single-photon source in condensed matter.

“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.”

• cites Brunel et al 1999Not checked yet

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Atom- or ion-based sources are nearly ideal for the demonstration of quantum-optics experiments.

“They are nearly ideal for the demonstration of quantum-optics experiments.”

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).

“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).”

Kasha's rule states that only the lowest excited state of a given spin multiplicity can efficiently emit light.

“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).”

• cites Birks 1970Not checked yet

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Even in crystalline environments, lines of single organic molecules at room temperature are never much narrower than about 300 cm-1.

“Therefore, even in crystalline environments, lines of single organic molecules at room temperature are never much narrower.”

Atom- or ion-based sources of single photons present many advantages as standard sources of pure and indistinguishable photons.

“In summary, atom-or ion-based sources of single photons present many advantages as standard sources of pure and indistinguishable photons.”

Singlet-singlet annihilation is a consequence of Kasha's rule and of the fast relaxation of high excited states.

“This process, called singlet-singlet annihilation, is a consequence of Kasha's rule and of the fast relaxation of high excited states.”

At very low temperatures, 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).

“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).”

• cites Sild and Haller 1988Not checked yet

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Single polymer molecules have been in operation for as long as hours on a glass surface before photobleaching eventually stopped the 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).”

• cites Lee et al 2004Not checked yet

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The main disadvantage of atom- or ion-based sources is their complexity of operation.

“Their main disadvantage is their complexity of operation.”

Multiple excitations in molecules cannot lead to emission of two or more photons.

“Multiple excitations therefore cannot lead to emission of two or more photons.”

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.

“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.”

If 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, even under intense pulsed illumination, as shown for terrylene in p-terphenyl crystals.

“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).”

• cites Fleury et al 1998Not checked yet

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• cites Kulzer et al 1999Not checked yet

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• cites Lounis and Moerner 2000Not checked yet

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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.

“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.”

For large multichromophoric assemblies, the operation time is lengthened because of the large number of available fluorophores in the assembly.

“For such large assemblies, however, the operation time is lengthened because of the large number of available fluorophores in the assembly.”

Simultaneous emission of two photons in general indicates that two atoms are present in the detection volume.

“Simultaneous emission of two photons in general indicates that two atoms are present in the detection volume (McKeever et al 2004).”

• cites McKeever et al 2004Not checked yet

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Blinking limits the practical applications of semiconductor nanocrystals as sources of single photons.

“However, blinking limits their practical applications as sources of single photons.”

All multi-excited states of nanocrystals are nearly dark because of very efficient exciton-exciton annihilation (Auger processes) similar to the singlet-singlet and singlet-triplet annihilation in multi-chromophoric molecules and polymers.

“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.”

• cites Klimov et al 2000Not checked yet

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The emission time of nanocrystals lengthens considerably at low temperatures.

“This emission time lengthens considerably at low temperatures (Nirmal et al 1995, Labeau et al 2003).”

• cites Nirmal et al 1995Not checked yet

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• cites Labeau et al 2003Not checked yet

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If only a well-defined number of photons is important, as in squeezed-light measurements or quantum cryptographic applications, simple devices operated at room temperature (with single molecules, colour centres or semiconductor nanocrystals) are sufficient.

“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.”

• AssumptionRests on an assumption: A simple photon gun based on a single diamond nanocrystal with a single colour centre at the focus of a microscope objective already exists.
• AssumptionRests on an assumption: Such simple devices were already employed in early demonstrations of QKD.

The condition of requiring coherence leaves only two possibilities: isolated atoms or ions in traps in ultra-vacuum, or molecules or self-assembled quantum dots in rigid matrices at cryogenic temperatures.

“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.”

The advantage of nanocrystal photostability is partly offset by a major disadvantage, their blinking, i.e. strong fluctuations of their luminescence with time.

“This advantage, however, is partly offset by a major disadvantage, their blinking, in other words the strong fluctuations of their luminescence with time.”

Photoinduced charge rearrangements in the passivation layer and the environment have been shown to lead to spectral shifts of the exciton line in nanocrystals.

“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).”

• cites Empedocles and Bawendi 1997Not checked yet

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The high peak intensities of a pulsed excitation can lead to complex dark states in NV centres, possibly through charge transfer processes and photochemical changes.

“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).”

• cites Dumeige et al 2004Not checked yet

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Because of the reduction in the surrounding index of refraction, the luminescence lifetime of NV centres in nanocrystals is lengthened to 20-30 ns, depending on the individual nanocrystal.

“Because of the reduction in the surrounding index of refraction, the luminescence lifetime is lengthened to 20-30 ns, depending on the individual nanocrystal.”

With pure materials, high-quality substrates, ultra-high vacuum and well-controlled temperature conditions, the Stranski-Krastanov growth leads to uniformly sized, regularly spaced, three-dimensional islands of the deposited material.

“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).”

• cites Moriarty 2001 sourceNot checked yet

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At cryogenic temperatures, a single quantum dot gives a narrow, close to Fourier-limited line, in a spectral domain depending on the material.

“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).”

• cites Gammon et al 1996Not checked yet

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• cites Bayer and Forchel 2002Not checked yet

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• cites Zwiller et al 2004 sourceNot checked yet

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If an application requires coherence between single photons, as most quantum computation protocols do, the emitter must be extremely well shielded from fluctuations in its environment.

“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.”

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.

“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 andGuyot-Sionnest 1996, Dabbousi et al 1997).”

• cites Hines and Guyot-Sionnest 1996Not checked yet

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• cites Dabbousi et al 1997 sourceNot checked yet

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There is a dark state, presumably a singlet, responsible for bunching in the correlation function of the NV centre.

“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).”

• cites Kurtsiefer et al 2000Not checked yet

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• cites Beveratos et al 2001Not checked yet

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• cites Jelezko et al 2002Not checked yet

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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 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.”

The single-photon emission rate of J-aggregates would be limited by the relaxation of higher excited states, with lifetimes of the order of 1 ps.

“However, the single-photon emission rate would be limited by the relaxation of higher excited states, with lifetimes of the order of 1 ps.”

• AssumptionRests on an assumption: Large aggregates of molecules, for example J-aggregates, present enhanced fluorescence rates due to cooperative interference of the molecules in a coherence domain.

Higher emission rates require shorter lifetimes.

“Higher emission rates require shorter lifetimes.”

Inefficient emitters with high non-radiative relaxation and therefore very short emission times, such as metal particles, would present the same problem.

“Inefficient emitters with high non-radiative relaxation and therefore very short emission times, such as metal particles, would present the same problem.”

On longer timescales, as the atoms themselves enter and leave the excitation beam according to Poisson statistics, coincidences of two photons emitted by two different atoms may be observed.

“On longer timescales, as the atoms themselves enter and leave the excitation beam according to Poisson statistics, coincidences of two photons emitted by two different atoms may be observed.”

The sub-Poissonian character of this emission only applies during the time an isolated atom crosses the excitation beam.

“The sub-Poissonian character of this emission only applies during the time an isolated atom crosses the excitation beam.”

Higher spontaneous emission rates would mean that two excitations could produce two photons simultaneously.

“Higher spontaneous emission rates would mean that two excitations could produce two photons simultaneously.”

The antibunching effect only manifests itself by an increase in the correlation function with delay, starting from unity for short delays.

“Therefore, the antibunching effect only manifests itself by an increase in the correlation function with delay, starting from unity for short delays.”