## Abstract

We generated narrowband pairs of nearly identical photons at a rate of 5 × 10^{4} pairs per second from a laser-cooled atomic ensemble inside an optical cavity. A two-photon interference experiment demonstrated that the photons could be made 90% indistinguishable, a key requirement for quantum information-processing protocols. Used as a conditional single-photon source, the system operated near the fundamental limits on recovery efficiency (57%), Fourier transform–limited bandwidth, and pair-generation-rate–limited suppression of two-photon events (factor of 33 below the Poisson limit). Each photon had a spectral width of 1.1 megahertz, ideal for interacting with atomic ensembles that form the basis of proposed quantum memories and logic.

The generation of photon pairs is useful for a broad range of applications, from the fundamental [exclusion of hidden-variable formulations of quantum mechanics (*1*)] to the more practical [quantum cryptography (*2*) and quantum computation (*3*)]. A key parameter determining the usefulness of a particular source is its brightness, i.e., how many photon pairs per second are generated into a particular electromagnetic mode and frequency bandwidth. Parametric down-converters based on nonlinear crystals are excellent sources of photon pairs, but they are comparatively dim because their photon bandwidths range up to hundreds of GHz. However, new applications are emerging that demand large pair-generation rates into the narrow bandwidths (5 MHz) suitable for strong interaction of the photons with atoms and molecules (*2*, *4*–*7*).

We report the development of a source of photon pairs with spectral brightness near fundamental physical limitations and approximately three orders of magnitude greater than the best current devices based on nonlinear crystals (*8*). Unlike parametric downconverters, however, the atomic ensemble can additionally act as a quantum memory and store the second photon, allowing triggered (i.e., deterministic) generation of the second photon. Triggered delays of up to 20 μs have been demonstrated (*9*–*15*), and it is expected that optical lattices hold the potential to extend the lifetime of these quantum memories to seconds (*9*). Lastly, proposed applications in quantum information (*2*, *3*) rely on joint measurements of single photons for which indistinguishability is crucial for high fidelity. We observe large degrees of indistinguishability in the time-resolved interference between the two generated photons (*16*–*19*).

A range of approaches using atomic ensembles to strongly couple matter and light are actively being pursued. These include room-temperature atomic vapors (*10*, *11*) and laser-cooled atomic ensembles both in free space (*12*–*15*) and in optical cavities (*9*). Simultaneous generation of pairs of strongly correlated photons has been reported (*20*) with a 7% success rate for generation of the second photon and large violations of a Cauchy-Schwartz inequality (*21*), *G* = 400 > 1, that indicates the quantum nature of the observed correlations. Three- to fivefold suppressions of undesired two-photon events have been reported (*22*–*24*). Single photons have been captured in and released from atomic ensembles (*22*, *24*). Measurement-induced entanglement of independent ensembles of atoms has been demonstrated (*23*, *25*). The two-photon Hong-Ou-Mandel interference used here (*16*) has also been used to demonstrate the degree of indistinguishability of single photons emitted from quantum dots (*17*), and from a single atom in a high-finesse cavity (*18*, *19*).

We concentrate on the regime of minimum delay time between the generation of the photons within a pair in order to characterize the source, while keeping in mind that the present results should straightforwardly extend to the regime of delayed photon generation explored in previous work (*9*). The experimental setup consisted of a laser-cooled ensemble of *N* = 10^{4} Cs atoms in the TEM_{00} mode of a low-finesse *F* = 250, single-mode optical cavity (Fig. 1 and supporting online text). Photon pairs were generated by a four-wave mixing process that relies on quantum interference in the emission from an entangled atomic ensemble (*2*) to enhance the probability of scattering a second “read” photon into the cavity to near unity given the initial scattering of a “write” photon into the cavity (Fig. 1). Without collective enhancement, the maximum probability that the read photon would be scattered into the cavity was only 7.3 × 10^{–4}, set by the cavity cooperativity parameter, and was nearly three orders of magnitude lower than the observed value of 0.57(9).

To first verify that the light emitted in one polarization was correlated in time with the light in the other polarization, we measured second-order correlation functions *g*_{wr}(τ)|_{T}. These are simply the measured coincidence count rate between the detectors D1 and D2 normalized by the rate one would expect for two completely uncorrelated beams of the same average intensities. The label *T* specifies the size of the coincidence windows (and will hereafter be made implicit), and τ specifies a time offset between the write and read windows. At fixed *T* = 8 ns (Fig. 2, inset), the time-resolved cross-correlation has peak coincidence rates 100(10) times as high as those for uncorrelated beams.

To normalize out possible classical contributions to the cross-correlation data *g*_{wr}(20 ns) of Fig. 2, we also accurately measured the autocorrelations *g*_{ww}(0) and *g*_{rr}(0) using two detectors for each of the write and read beams. For a bin size *T* = 60 ns, the normalized cross-correlation is , representing a large violation of the Cauchy-Schwarz inequality *G* ≤ 1 that purely classically correlated beams must satisfy (*21*).

To assess the usefulness of the system as a source of single photons heralded by the detection of a write photon, we examined two relevant quantities: (i) the recovery efficiency, defined as the probability of generating a read photon conditioned on having detected a write photon; and (ii) the degree to which two–read photon events are suppressed below that of a Poisson distribution with the same average intensity.

A lower bound on the read recovery efficiency *R* is obtained from the measured detection losses *q*_{r}, combined with the measured probability of detecting a read photon given the detection of a write photon *R*^{cond}_{det}. The inset to Fig. 3 shows the conditional detection probability versus bin size. The integrated conditional detection probability *R*^{cond}_{det} = 0.031(2) is estimated from the *T* = 0 intercept of a linear fit to the data at large bin size *T*. The read photon detection efficiency *q*_{r} = 0.053(8) includes contributions from cavity mirror losses (0.45), fiber coupling (0.75), and detector quantum efficiency (0.40). Extrapolated to just outside the cavity output mirror, the recovery efficiency is R^{cond}_{cav} = 0.26(4). The physical recovery efficiency for a cavity of the same linewidth, but with losses completely dominated by transmission of one of the two mirrors, is *R*^{cond} = *R*^{cond}_{det}/*q*_{r} = 0.57(9). Given the low-finesse *F* = 250 of the present cavity, this ideal regime could be easily achieved with current technologies.

The conditional suppression of two-photon events was measured using one detector to herald the arrival of a write photon at time *t* and two detectors to measure the autocorrelation of the subset of the read photons that fall within a time bin of size ±*T*/2 centered at *t* + τ. With τ = 20 ns, the conditional autocorrelation starts near unity (no suppression) at large bin sizes *T* (Fig. 3) due to backgrounds not correlated with the registered write photon, and it decreases monotonically until the bin size becomes comparable to the read photon emission time near *T* ∼ 100 ns. The largest measured suppression of two-photon events = 0.03(3) occurs at *T* = 60 ns.

A conditional single-photon source can only suppress two-photon events to the fundamental limit = ϵ_{w}*g*_{ww} associated with the random emission of two pairs of photons within the same time bin *T*. In the ideal case, the autocorrelation of the write photons is *g*_{ww} = 2. The write photon generation probability ϵ_{w} can be extracted from the measured write photon detection probability , thus predicting the fundamental limit for a conditional photon source at this pair-generation rate of ≥ 0.026(3). The agreement with the measured value = 0.03(3) indicates that spurious background counts are not a serious limitation to the performance of the single-photon source at the present photon-generation rate.

The identicalness of the write and read photons was examined via two-photon interference at the polarizing beam splitter (Fig. 1A) (*16*–*19*). This was accomplished by analyzing the write and read photons in a polarization basis rotated by 45° with respect to the usual basis used to deterministically separate the photons. Neglecting interference between the two photons, one expects that in half the cases, the photons register a coincidence count on opposite detectors. However, if the write and read photons perfectly overlap in time and frequency, there is a complete destructive interference for the probability of a coincidence count [a so-called Hong-Ou-Mandel interference (*16*)]. The fractional reduction of the coincidence count rate below ½ of its original value is a direct measure of the degree of indistinguishability of the photons.

Figure 4 shows the measured coincidence rate expressed as a cross-correlation between detectors *D*_{1} and *D*_{2}, when the write and read photons are polarization separated *g*_{wr}(τ) (Fig. 4A) and are allowed to interfere *g*_{45}(τ) (Fig. 4B). The destructive interference is most pronounced near τ = 0, and it decreases as |τ| increases because the finite time separation allows one to infer with increasing reliability which detection event corresponds to the write photon and which to the read photon. For data sets at larger π-pump intensities for which the read photons are emitted more promptly, we observe suppressions of the two-photon coincidence rate below ½ by as much as 90(20)% integrated over *T* = 5 μs, indicating that the photons can be made nearly identical.

As a model of the expected two-photon interference for the data of Fig. 4B, we assume that the photons differ by at most a fixed frequency offset Δϵ (*19*). The quantum probabilities *C*_{wr} and *C*_{45} that a given photon pair will register as a coincidence event at time separation τ are related by . Two predictions of *g*_{wr}(τ) are obtained from the measured *g*_{wr}(τ) and assuming Δω/2π = 0 and 2.5 MHz (Fig. 4B). The second prediction accurately describes the observed data, indicating that we are observing a quantum beat between the photons. However, the frequency difference is somewhat larger than the measured Zeeman and calculated light shifts that might give rise to Δω ≠ 0.

The two-photon interference results above can be directly mapped onto a *gedanken* version of the experiments in (*26*–*29*), wherein polarization entanglement is generated via post selection (see supporting online text and Fig. 4C). The mapping is performed assuming that quantum mechanics is correct (*1*, *27*–*29*). At τ = 0, the predicted CHSH Bell's parameter is *S* = 2.68(2), a violation of the Bell's inequality |*S*| ≤ 2. The predicted violation is not closer to the theoretical maximum *S*_{max} (dashed line of Fig. 4C), largely due to backgrounds set by the two-photon generation rate.

The frequency bandwidths of the write and read photons are 1.1(2) MHz, making them ideal for interacting with narrowband systems such as atoms, molecules, and optical cavities. By separately heterodyning the write and read photons with laser light derived from the π-pump laser (measured linewidth of 50 kHz), we obtained the power spectral density of the photons from the Fourier transform of the measured second-order autocorrelation function (Fig. 4D). The photons are nearly Fourier-transform limited, as can be seen from the 2-MHz full width at half-maximum power spectrum (Fig. 4E) of the measured cross-correlation function *g*_{wr}(τ) taken at slightly different parameters.

These measurements show that pairs of nearly identical photons are generated at an approximate rate of 5 × 10^{4} pairs/s into a single Gaussian transverse mode. The spectral brightness of 5 × 10^{4} pairs/s per MHz^{–1} is ∼10^{3} times as bright as the best sources based on parametric downconversion with nonlinear crystals (*8*). The system can operate very near fundamental limits on recovery efficiency, photon bandwidth, and two-photon suppression for a conditional single-photon source. In addition, identical photon pairs are necessary for certain quantum information protocols such as quantum computation with linear optics (*3*). The identical photon pairs also have potential applications for sub-shotnoise spectroscopy of atomic ensembles.