1. Introduction

Storage and retrieval of single photons with preserved quantum states is of great importance for long-distance quantum communication and quantum computation [1–3]. A practical quantum memory is desirable with high storage efficiency, long coherence time, and low noise. In the past decade, many schemes have been proposed and demonstrated for optical storage based on coherent light-matter interactions, such as electromagnetically induced transparency (EIT) [4–6], off-resonance Raman interaction [7], and photon echo [8]. Of these techniques, photon echo has recently become attractive due to its promising storage efficiency (as high as 87% [9]), large mode capacity, and compatibility with solid state interfaces [10, 11]. However, these experimental demonstrations of high efficiency were all limited to coherent light pulses, and the recent implementations with entangled photons [12–14] only achieved a highest memory efficiency of 21% [12].

On the other hand, the EIT memory is compatible with quantum state operations of single-photon wave packets [15–18] and squeezed states [19, 20]. Recent progress includes storing narrow-band single photons generated from atomic systems [16–18] and spontaneous parametric down conversion [21]. Also, the EIT memory time has been pushed to milliseconds by prolonging the ground-state coherence [22]. However, EIT quantum memories have suffered from low efficiency so far, with the highest single-photon storage efficiency being only 17% [18,23], preventing the scheme from practical applications.

In this paper, we report an experimental demonstration of efficient storage and retrieval of narrow-band single-photon waveforms using EIT in a cold atomic ensemble. With the ability to control both single-photon wave packets and the memory bandwidth, we obtain a storage efficiency up to (49±3)% while the nonclassical property is maintained. To our knowledge, it represents the highest storage efficiency for a single-photon waveform to date. Because an efficiency above 50% is necessary to operate a memory for error correction protocols in one-way quantum computation [24], our result brings the atomic quantum light-matter interface closer to practical quantum information applications [10].

2. Experimental setup

Figure 1 illustrates the experimental configuration we use to generate, store, and retrieve narrow-band single photons. We make use of two two-dimensional (2D) ⁸⁵Rb magneto-optical traps (MOT1 and MOT2) with high optical depth (OD), which are described in details in [25]. The similar setup has been also used for producing and observing single-photon optical precursors [26]. Each cold atomic cloud, with a temperature of about 100 μK, has a length of 1.7 cm and transverse diameter of 0.7 mm. From MOT1, we produce Stokes (ω_s) and anti-Stokes (ω_as) paired photons [27], with the presence of counter-propagating pump (ω_p, 780nm) and coupling (ω_c, 795nm) beams aligned at a 3° angle with respect to the Stokes-anti-Stokes axis. The pump laser is blue detuned by 60 MHz from the |1〉 → |4〉 transition. The coupling laser is on resonance with the |2〉 → |3〉 transition. Both the pump and coupling lasers have the same collimated beam diameter of 1.6 mm and their linewidths are narrower than 1 MHz. The Stokes and anti-Stokes photons are coupled into two opposing single-mode fibers (SMF). When the Stokes photon is detected by the single-photon detector D₁, we send its paired anti-Stokes photon through an amplitude electro-optical modulator (EOM, 10 GHz, EOspace), which is driven by a triggered waveform generator. In this way, we are able to generate heralded single anti-Stokes photons with controllable waveforms [28]. We then store the anti-Stokes photons in the cold atoms at MOT2, controlled by a second coupling beam directed from the same coupling laser in MOT1. The anti-Stokes photon single mode is focused to the center of MOT2 along its longitudinal axis and has a 1/e² diameter of 245 μm at the waist. The coupling beam at MOT2, with a 1/e² diameter of 1.0 mm, is aligned at a 3° angle with respect to the anti-Stokes propagation. We run the experiment periodically with a MOT time of 4.5 ms followed by a photon generation window of 0.5 ms for each cycle. The MOT magnetic fields remain on all the time. In both MOTs, at end of the trapping time, we optically pump all the atoms to the ground level |1〉. Coincidence counts are recorded by a time-to-digital converter (Fast Comtec P7888) with 1 ns bin width.

 

Fig. 1 Schematics of the experimental setup for storage and retrieval of heralded single photons with controllable waveforms. The ⁸⁵Rb energy levels are chosen as |1〉 = |5S_1/2, F = 2〉, |2〉 = |5S_1/2, F = 3〉, |3〉 = |5P_1/2, F = 3〉 and |4〉 = |5P_3/2, F = 3〉. In MOT1, with the presence of pump (ω_p) and coupling (ω_c) lasers, we produce counter-propagating Stokes (ω_s) and anti-Stokes (ω_as) photon pairs. Conditioned on detection of the Stokes photon, its paired anti-Stokes photon passes through an electro-optical modulator (EOM) and is stored in MOT2. F₁ and F₂ are two narrow-band optical filters.

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3. Single-photon storage

The physical mechanism of EIT memory has been well studied in terms of dark-state polaritons [5]. As the photon wave packet is spatially compressed inside the medium, we turn off the coupling laser to adiabatically convert the photon state into a long-lived atomic spin wave that involves only the two ground levels |1〉 and |2〉. After a controllable time delay, we turn on the coupling laser again to retrieve the photon wave packet. There are two important parameters characterizing the performance of a single-photon memory. The first is the storage efficiency, defined as the probability of storing and retrieving the single photon,

We first characterize the photon source. In the following experiments, we fix the pump and coupling laser Rabi frequencies during the biphoton generation in MOT1 at Ω_p = 0.4γ₁₃ and Ω_c1 = 5.1γ₁₃, where γ₁₃ = 2π × 3 MHz is the electric dipole relaxation rate between |1〉 and |3〉. The optical depth of MOT2 is maintained at OD₂=60. By varying the OD at MOT1 (OD₁), we produce paired photons with controllable temporal length [27]. At a low OD₁, the two-photon temporal correlation time length is determined by 1/(γ₁₃ +γ₁₂) ≃ 1/γ₁₃ = 53 ns, where γ₁₂ ≃ 0.03γ₁₃ is the ground-state dephasing rate [30]. At a high OD₁, we enter the group delay regime where the photon pair temporal length is determined by the relative group delay time between the paired anti-Stokes and Stokes photons [30]. The red curves in Fig. 2 show the two-photon coincidence counts between detectors D₁ and D₂ with 1 ns bin width, collected for 900 s. At OD₁=7, the heralded anti-Stokes photon has a temporal length of about 50 ns, as shown in Fig. 2(a), while at OD₁=35 we prolong the length to 200 ns as shown in Fig. 2(b). Excluding the uncorrelated accidental coincidences, there are a total of 3300 (or 18100) biphoton coincidence counts detected by D₁ and D₂ for OD₁=7 (or 35), which is consistent with the theoretical prediction at the group delay regime where the total rate of paired counts scales linearly as OD₁ [30]. This is because the on-resonance spectral brightness is proportional to ${\text{OD}}_1^2$ but the bandwidth reduces linearly with OD1. Including the coincidence counts between D₁ and D₃ (the measured BS splitting ratio is about 45%:55%), we detect a total of 7400 (or 42400) photon pairs in 900 s, corresponding to a photon pair detection rate of 8 (or 47) pair/s. Taking into account the detector quantum efficiencies (50% each), fiber-fiber coupling efficiencies (70% at MOT1 and 72% at MOT2), EOM transmission (50%), fiber connection efficiency (61%), filter transmissions (65% each), and the duty cycle (10%), this corresponds to a generation rate of about 4900 (or 28900) pair/s from MOT1. At OD₁=7 (or 35), for each click at D₁, the success probability of detecting its heralded photon at D₂ and D₃ is 2.8% (or 4.1%), which, accounting all the losses and efficiencies, corresponds to a pairing efficiency of 56% (or 82%) when they are produced from MOT1. The incident anti-Stokes photon rate in MOT2 is about 1000/s (or 6200/s). The nonclassical properties of the paired photons can be measured by violation of the Cauchy-Schwartz inequality ${\left[g_{i,j}^{(2)}(\tau )\right]}^2/\left[g_{i,i}^{(2)}(0)g_{j,j}^{(2)}(0)\right]\le 1$ [31]. Because the paired photons are generated through spontaneous four-wave mixing, there is no correlation between different pairs. Therefore, the correlation function $g_{s,as}^{(2)}(\tau )$ can be obtained by normalizing the two-photon coincidence counts to the background floor resulting from accidental coincidences between uncorrelated photons. We obtain $g_{s,as}^{(2)}(\tau )$ with maximum values of 150 and 95 for the input waveforms in Fig. 2(a) and (b), respectively. With $g_{s,s}^{(2)}(0)=g_{as,as}^{(2)}(0)=2.0$ measured using a fiber beam splitter, we obtain a violation of the inequality by a factor of 5625 and 2256, respectively. To characterize the single-photon nature of the heralded photons, we measure $g_c^{(2)}=0.10\pm 0.02$ for the short photon (with a coincidence window of 100 ns) and $g_c^{(2)}=0.17\pm 0.02$ for the long photon (with a coincidence window of 200 ns), each with a total time of 2100 s.

 

Fig. 2 Direct storage and retrieval of single photons without waveform shaping. Single photons with (a) a short waveform and (b) a long waveform are produced from MOT1 at OD₁=7 and 35, respectively. The coincidence counts are recorded by D₁ and D₂. Other parameters are OD₂=60, Ω_c2 = 11γ₁₃, and γ₁₂ = 0.03γ₁₃.

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We then measure the storage efficiency without shaping the waveform of anti-Stokes photons by leaving the EOM at its maximum transmission. To store the anti-Stokes photon, we switch off the coupling laser at MOT2 (Ω_c2 = 11γ₁₃) for a period of 100 ns after detecting its paired Stokes photon. The retrieved photon waveforms are displayed as the green curves in Fig. 2(a) and (b). The coupling laser has switch-on and -off times of 50 ns. For both waveforms, we obtain the same storage efficiency of (20±2)%. The measured $g_c^{(2)}=0.24\pm 0.17$ and 0.44 ± 0.15 confirm that we indeed retrieve single photons. However, in both cases, the waveform profiles are not preserved after retrieval.

Previous work for coherent pulse storage suggests that the storage efficiency can be substantially improved by optimizing the pulse shape to match the EIT bandwidth. We follow the optimization procedure described in [32, 33]. We work with the initial waveform in Fig. 2(b). The oscillatory structure on top of the leading edge is an optical precursor at the single-photon level [26, 27, 34], which is produced from the optical frequency components far away from the atomic resonance and whose wave front always travels at the speed of light in vacuum. This optical precursor component cannot be slowed and stored because its frequency components do not fall inside the EIT slow light window [26], thus we remove it from the single-photon waveform by switching on the EOM 100 ns after the leading edge. At MOT2, we fix the coupling laser Rabi frequency as Ω_c2 = 11γ₁₃ and OD₂=60. We feed back the time-reversed waveform of the retrieved photon to shape its input waveform using the EOM. The optimal input-output waveforms for the maximum storage efficiency are obtained after three iterations, as shown in Fig. 3(a), which becomes identical for more than three iterations and the storage efficiency saturates [32]. The input single-photon waveform has a full width at half maximum (FWHM) of 50 ns and a peak $g_{s,as}^{(2)}$ value of 23. The output single-photon waveform is retrieved after a storage time of 2 pulse length (100 ns). The optimal storage efficiency is (36±3)%. To confirm that at this optimal condition, the retrieved photon waveform is the time-reversal of the input, we plot the time-reversed retrieved photon waveform together with the input waveform with a proper rescaling in the inset [Fig. 3(b)], and they match each other very well. This time-reversal relationship between the input and output waveforms agrees with the theoretical prediction and indicates that the phase coherence information of the single-photon wave packet is preserved during storage [35]. For a quantitative estimation, we calculate the temporal waveform likeness L = |∫ ψ_in(τ)ψ_out (−τ)dτ|²/[∫ |ψ_in(τ)|²dτ ∫ |ψ_out (τ)|²dτ]. Since we work at the group delay regime, the single-photon waveform from MOT1 is nearly transform-limited [26, 30]. After the amplitude modulation by the EOM, we obtain the optimal waveform envelope without time-varying phase. Therefore, we have ${\psi }_{in}(\tau )=\sqrt{G_{\mathit{in}}^{(2)}(\tau )}$ and ${\psi }_{\mathit{out}}(\tau )=\sqrt{G_{\mathit{out}}^{(2)}(\tau )}$, where $G_{\mathit{in}}^{(2)}(\tau )={〈{\widehat{a}}_s^{†}(t){\widehat{a}}_{\mathit{as}}^{†}\left(t+\tau \right){\widehat{a}}_{\mathit{as}}\left(t+\tau \right){\widehat{a}}_s(t)〉}_{\mathit{in}}$ and $G_{\mathit{out}}^{(2)}(\tau )={〈{\widehat{a}}_s^{†}{\widehat{a}}_{\mathit{as}}^{†}\left(t+\tau \right){\widehat{a}}_{\mathit{as}}\left(t+\tau \right){\widehat{a}}_s(t)〉}_{\mathit{out}}$ are Glauber correlation functions before and after storage that can be obtained from the coincidence counts in Fig. 3. Here â^† and â are photon creation and annihilation operators. For this optimal storage, we obtain L=93%. We measure $g_c^{(2)}=0.30\pm 0.06$ for the input photon and $g_c^{(2)}=0.26\pm 0.13$ for the retrieved photon with a coincidence window of 100 ns and a total time of 3300 s. They are below the two-photon threshold of 0.5.

 

Fig. 3 Storage and retrieval of a single photon with optimal waveform. (a) The optimal input (red curve) and output (retrieval, green curve) heralded single-photon waveforms are measured as coincidence counts between D₁ and D₂. (b) The inset shows the time-reversed retrieved photon waveform matches the input photon waveform after normalization. The operating parameters at MOT2 are OD₂=60, Ω_c2 = 11γ₁₃, and γ₁₂ = 0.03γ₁₃. The measured storage efficiency is (36±3)%.

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The storage efficiency is also affected by the dephasing rate γ₁₂ between the two ground levels |1〉 and |2〉 which causes decoherence loss and absorption at high OD. We obtain the effective dephasing rate by best-fitting it to the EIT transmission spectrum. The finite dephasing rate is caused by stray magnetic fields, atomic motion, and the coupling beam profile. In the above measurements, the coupling beam at MOT2 has a diameter of 1.0 mm, and we obtain γ₁₂ = 0.03γ₁₃. To reduce the dephasing rate, we increase the coupling beam diameter to 1.6 mm and obtain γ₁₂ = 0.01γ₁₃. The coupling laser power, which is limited by the maximum power allowed by the fiber EOM, remains the same as that in the previous measurements and thus we have Ω_c2 = 6.88γ₁₃. Figure 4(a) shows the optimal storage and retrieval of single photon waveforms under the new conditions. Due to the narrower EIT bandwidth, the optimal waveform has a longer temporal length (FWHM=100 ns) and a peak $g_{s,\mathit{as}}^{(2)}$ value of 12. For a delay of two pulse length (200 ns), we obtain a storage efficiency of (49±3)%. The inset shows that the suitably rescaled time-reversed output waveform matches the input waveform well [Fig. 4(b)]. We obtain a temporal waveform likeness of L=96%. For the conditional correlation measurement, we take the FWHM (100 ns) as the coincidence window and obtain $g_c^{(2)}=0.10\pm 0.06$ for the input waveform and $g_c^{(2)}=0.14\pm 0.14$ for the retrieved photon. The higher $g_c^{(2)}$ after retrieval is caused by the single photon storage loss in presence of detector dark counts and multiphoton events from the background noise photons. As we increase the coincidence window to 200 ns to contain more accidental counts from noise photons and dark counts, the $g_c^{(2)}$ of the input and retrieved photon become 0.28 ± 0.07 and 0.66 ± 0.22 respectively, which are still below the classical limit. The measured memory lifetime is about 1.6 μs, which is mainly determined by the inhomogeneous MOT magnetic field.

 

Fig. 4 Optimal storage and retrieval of single photons with reduced ground-state dephasing rate. (a) The optimal input (red curve) and output (retrieval, green curve) heralded single-photon waveforms. (b) The inset shows the time-reversed retrieved photon waveform matches the input photon waveform after normalization. The operating parameters at MOT2 are OD₂=60, Ω_c2 = 6.88γ₁₃, and γ₁₂ = 0.01γ₁₃. The measured storage efficiency is (49±3)%.

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In our configuration, we find that the connection between $g_c^{(2)}$ and the normalized cross correlation function ${\overline{g}}_{s,\mathit{as}}^{(2)}$ (averaged over the same coincidence window) can be expressed as $g_c^{(2)}\simeq \left(2g_{s,\mathit{as}}^{-(2)}+1\right)/\left[{\left(g_{s,\mathit{as}}^{-(2)}+1\right)}^2\right]$. In the case ${\overline{g}}_{s,\mathit{as}}^{(2)}>>1$, this reduces to $g_c^{(2)}\simeq 2/g_{s,\mathit{as}}^{-(2)}$, which agrees with our experimentally measured values within their statistical errors. As we increase the coincidence window length, $g_c^{(2)}$ increases because of an increasing probability of detecting multiphoton events from uncorrelated noise photons and dark counts. Oppositely, ${\overline{g}}_{s,\mathit{as}}^{(2)}$ drops as the coincidence window length increases. As a measure of the ratio of correlated photons (signal) to uncorrelated photons (noise), ${\overline{g}}_{s,\mathit{as}}^{(2)}$ provides a quick estimate of the quality of heralded single photons. It is clear that both $g_c^{(2)}<1$ and the violation of Cauchy-Schwartz inequality require ${\overline{g}}_{s,\mathit{as}}^{(2)}>2$ for beating the classical limit. ${\overline{g}}_{s,\mathit{as}}^{(2)}>4$ indicates the near-single-photon character of the heralded anti-Stokes photon ( $g_c^{(2)}<0.5$).

4. Conclusion

In summary, we have demonstrated optimal storage and retrieval of heralded single-photon wave packets using EIT in cold atoms. At a high OD of 60, we obtain a storage efficiency of close to 50% for the optimal single-photon waveform with a temporal likeness of 96%. In our experiment, the storage efficiency might be limited by the inhomogeneous stray magnetic field, the coupling beam profile, atomic motion, and possibly some nonlinear processes at high optical depth. The storage efficiency in this work refer only to the capability of the EIT atomic medium at MOT2 in storing and retrieving optimal single-photon waveforms, and it does not include the fiber connection loss and the EOM insertion loss. The EOM amplitude modulation loss of about 50% in our system can be counted into the heralded single-photon generation efficiency. This modulation loss can be eliminated, in principle, using other waveform shaping techniques, such as chirp (using phase-frequency modulation) and compression [36, 37].