1. Introduction

Quantum networks rely on efficient transfer between flying qubits and stationary quantum nodes [1,2]. The synchronization between two distant nodes which facilitates the distribution of entanglement over a large-scale network is essential for many protocols, such as quantum repeaters [3] and linear optics quantum computation (LOQC) [4]. Quantum memories, devices that can store and retrieve flying qubits on demand, are crucial components for this purpose. Intensive efforts have been made to improve the storage-and-retrieval efficiency (SE) with promising results, including the $92\%$ efficiency with electromagnetically induced transparency (EIT) [5], $82\%$ with the off resonance Raman interaction [6], $87\%$ with the gradient echo in atomic vapor [7], and $69\%$ with the gradient echo in solid-state medium [8]. However, most of these works use attenuated or single-photon-level coherent light. It is essential to store the Fock-state single photons for a wider range of quantum information applications, such as LOQC [4] and entanglement transfer [2,9].

Based on spontaneous Raman transition or four-wave mixing, single photons can be generated from atomic systems, either atoms inside an optical cavity [10–12] or atomic ensembles [2,13–15]. The merit of using a atom-based single photon source is that the central frequencies match the atomic transitions and the photons are inherently narrow-band, which enables efficient interaction with atoms. To date, efforts towards efficient quantum storage of single photons generated by atomic systems have achieved an SE of $90.6\%$ [16] and of $> 85\%$ [2], both based on EIT protocol [2]. However, the setup of atom-based photon source is relatively complicated, which is more cumbersome to scale up for large-scale quantum networks.

On the other hand, one of the most widely used sources of photonic entanglement, spontaneous parametric downconversion (SPDC) with crystals, are suitable for large-scale protocols [17]. However, the drawback of using these SPDC-based photon sources is the broad bandwith ($\sim$THz) which hinder efficient interaction with atoms. This issue can be solved elegantly by placing the crystals into high-finesse optical cavities, thereby suppressing the spectral linewidth of single photons to MHz level [18,19]. Nevertheless, although quantum storage has been realized with SPDC-based photon sources [17,18,20–24], the SEs are typically low. The best SE reported in a previous work is $36 \%$ [17], which is not yet sufficient to overcome the $50\%$ no-cloning limit [25] and loss tolerance in the cluster state computation [26].

In this work, we demonstrate a quantum storage of $> 70\%$ efficiency for SPDC-generated single photons, which is the best record for such photon source to our knowledge. In addition, we demonstrate the quantum memory for polarization qubits with an average fidelity of > 96 %. The crystal-based photon source and atomic memory forms a basic quantum node in a quantum network. The compact SPDC-based photon source allows relatively easy scaling up for large-scale quantum networks.

2. Experimental setup

The experimental setup consists of a photon-pair source and a cold atom system, as shown in Fig. 1(a). Nondegenerate, narrowband photon pairs are generated from the cavity-enhanced SPDC setup with a single longitudinal mode. The design is similar to that of Ref. [19] but the wavelength of the photons is different (894.6 nm for cesium $D_1$ transition). A frequency-doubling laser (Toptica TA-SHG-PRO) with a wavelength of 447.3 nm pumps the type-II periodically poled KTiOPO4 (PPKTP) crystal in a linear cavity. The polling period and length of the crystal is 20.822 $\mu m$ and 2 cm, respectively. The crystal is operating around 30 $^{0}C$. Both sides of the crystal are coated but with different specifications. The pump input side is anti-reflective(AR)-coated for 447.3 nm with a reflectivity (R) of < 0.002 and high-reflective(HR)-coated for 894.6 nm with a reflectivity of 0.9982. The other side of the crystal is AR-coated for both 447.3 and 894.6 nm (R<0.002) with a distance of $\sim$ 3 mm to the output coupling mirror. This mirror is spherically concave with a radius of curvature of 75 mm and is HR-coated for both 447.3 and 894.6 nm with a reflectivity of 0.9995 and 0.9983, respectively. The finesse of the cavity for 894.6 nm is $\sim$ 1794. The 447.3 nm pump light double passes the crystal but the SPDC cavity does not act like a cavity for this wavelength. The mode spacing of the signal and idler photons, both around 894.6 nm, is 3.769 and 3.940 GHz, respectively. The cluster frequency is 86.7 GHz and the FWHM bandwidth of the SPDC is 66.4 GHz. With these parameters, the output of the photon pairs are mainly in the single longitudinal mode, similar to that of Ref. [19]. This important feature not only reduce the complexity of using an etalon filter to choose a selected mode but also increase the useful photon generation rate [19].

 

Fig. 1. (a) The experimental setup: cavity-enhanced SPDC photon source and EIT-based cold-atom quantum memory with a dual-rail setup for photonic polarization qubits are shown in the upper and lower parts, respectively. Inset includes the energy-level diagram of the Cs $D_1$ transition and the laser excitations. Signal photons as polarization-encoded qubits drive the $\vert F=3\rangle$ to $\vert F'=4\rangle$ transition. The strong control field drives the $\vert F=4\rangle$ to $\vert F'=4\rangle$ transition. (b) Timing sequence for the experiment. The pump power switches between a high-power and a low-power phase at 3.2 kHz, while the cold-atom system operates at a 32 Hz repetition rate. After being triggered by idler photons, the control beam power switches through a write-store-read sequence.

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Due to the similar reflectivities of the two cavity ends, the down-converted photons travel nearly 50/50 % forward and backward. The pump power alternates between a high-power ($\sim 18 mW$) and a low-power ($\sim 50 \mu W$) phase with a switching rate of 3.2 kHz. The duty cycle for the high-power phase is $70\%$. It is controlled by an acousto-optic modulator (AOM1) (see Fig. 1 a). During the high-power phase, which corresponds to the optical parametric oscillator (OPO) output, the backward beam provides the signal for locking the cavity length to maintain the double resonances condition. The forward beam is chopped by AOM2 with the same driving frequency as AOM1 but is out of phase with AOM1. Its duty cycle is $20\%$. This allows the cavity output beam to pass only during the low-power phase (which corresponds to the SPDC photon-pair output), preventing the single photon counter (SPCM1) from being saturated by the OPO output during the high-power phase. The signal and idler photons are separated by a polarizing beam splitter (PBS). The signal photons are sent to the cold-atom system while the idler photons are detected as triggers. The spectral linewidth of the heralded single photons is 2.2 MHz, as determined by the cross correlation function [19,27]. The OPO output light beats with a frequency-stabilized reference laser (not shown in Fig. 1(a)). The central frequency of the signal beam is locked at $^{133}Cs$ $D_1$ transition ($\sim 894.6$ nm) by locking the beat frequency to a certain value.

The quantum memory is implemented with the electromagnetically induced transparency (EIT) protocol [5,28] using a magneto-optical trap (MOT) of cesium with cigar-shaped atomic clouds [28]. The EIT memory is operated at the $D_1$ line, with the signal photons driving the $\vert F=3\rangle$ to $\vert F'=4\rangle$ transition and the strong control field driving the $\vert F=4\rangle$ to $\vert F'=4\rangle$ transition, as shown in Fig. 1(a). The choice of $D_1$ line allows a reduced control-intensity-dependent ground-state decoherence rate due to the off-resonant excitation of the control field to the nearby transitions [5,17]. The temporally dark and compressed MOT along with Zeeman-state optical pumping act to increase the optical depth of the atomic media [28]. The achieved optical depth is $\approx \{235, 306, 334\}$ when the MOT repetition rate is set to $\{32, 16, 8\}$Hz, respectively. The optical depth is determined by EIT spectral fitting by setting the decay rate of the optical coherence ($\gamma _{ge}$) to be $\sim 0.7\Gamma$, where $\Gamma = 2\pi \times 4.56$ MHz is the natural linewidth of the $D_1$ transition. This decay rate is larger than the ideal value of 0.5$\Gamma$ (solely due to the spontaneous decay) due to the additional decay rate from the finite laser linewidth and laser frequency fluctuations [5].

The idler photons are detected by single-photon detector (SPCM1) for heralding, while the signal photons are sent to the atomic memory setup through a $400$m optical fiber (Thorlabs 780HP), which is used to induce an $\approx 2 \mu$s temporal delay. This delay is technically essential so that the optical switch for the control beam can respond before the signal photons passing through the MOT cell. However, the fiber also causes a photon loss of $\sim$ 27%. It is due to the relatively slow respond time ($\sim 1.6 \mu$s) of the function generator used for controlling the intensity of the control field. If we could find a faster solution in the electronics, the length of this fiber and thus the loss could be reduced. The signal photons then pass through the polarization displacer to map the polarization of the photons into two spatial modes [16,29]. A half-wave plate is added in one of the spatial mode to ensure that both modes have the same polarization before interacting with the cold atomic ensemble. The atomic population is optically pumped towards the rightmost Zeeman state to increase the optical depth. Such a dual-rail setup allows us to store any polarization of the flying qubits without losing much of the optical depth [16,29]. Before entering the MOT cell, the signal beam is focused by a lens to an intensity $e^{-2}$ the diameter of $\approx 90 \mu m$ around the atomic clouds while the control beam is collimated to a diameter of $\approx 520 \mu m$. The angle between the signal and control beam is $\approx 4^{0}$, which is selected to minimize noise from the control beam leakage while still suppressing the ground state decoherence rate at $\approx 5 \times 10^{-3} \Gamma$ due to the residual Doppler broadening.

After passing through the MOT cell, the signal beam pass through a reverse dual-rail setup, to combine and collimate both spatial modes. The signal photons are then collected by a single-photon detector (SPCM2) before passing through four irises, some mirrors and lens, two elatons (Quantaser FPE001), one optical isolator, a bandpass filter (Semrock FF01-900/25) and a fiber for directing the photons into the detector, all of which (not completely shown in Fig. 1(a)) are used to filter out the noise. The reduction of external noise is essential to preserve the quantum nature of single photons, as discussed in Appendix 7.3 and 7.4. The overall collection efficiency, defined as the power ratio between that at the photon counter and the output right after the SPDC source, is $\approx 2.8 \%$. The significant photon loss is certainly an issue for quantum memory applications but is a common challenge. There is a trade off between getting a high signal to noise ratio and a high transmission of the signal photons. To achieve a higher overall collection efficiency, one has to pay more efforts to reduce the loss on each component along the signal beam path.

Figure 1(b) depicts the time sequence of our experiment. The repetition rates for the photon pair source and MOT are 3.2 kHz and 32 (or 16) Hz, respectively. Each time period includes $\approx$ 30 (or 60) ms for atom loading and state preparation and then the MOT is turned off for 0.72 ms. During the MOT off time, the photo source is in the photon-pair phase for $\approx 94 \mu s$. The $34 \mu$s wait time allows the photons leaking from the OPO phase to decay to a negligible level [17]; the latter $60 \mu s$ time window is left for single-photon storage. Once SPCM1 detects an idler photon, an electronic signal is sent to an optical switch to ramp down the intensity of the control beam to the optimal intensity ($3$ mW) for storage [5]. Before this, the control beam is set to a stronger intensity ($10$ mW) in order to clear out the atomic population at the $\vert F=4\rangle$ state. After the signal photons entering the atomic ensemble, the control beam is ramped off to convert the signal photons into atomic spin-waves. After a given storage time, the control beam is ramped up to retrieve the signal photons. The intensity of this reading control field could be different from that of the writing control field. We denote the intensity ratio of reading to writing control field as $\xi _{R/W}$ for simplicity and later use. The signal photons are then detected by SPCM2 before being recorded by an oscilloscope (Rohde & Schwarz RTO2014).

3. Characterization of single-photon source

We first characterize the nonclassical property of the heralded single photons by measuring the normalized cross-correlation function $g^{(2)}_{si}(t;\tau ) = G^{(2)}_{si}(t;\tau )/(G^{(1)}_{s}(t) G^{(1)}_{i}(t+\tau ))$, where $G^{(2)}_{si}(t;\tau ) = \langle {\hat {a}^{\dagger }_i(t) \hat {a}^{\dagger }_s(t + \tau ) \hat {a}_s(t + \tau ) \hat {a}_i(t)} \rangle {}$ and $G^{(1)}_{s(i)}(t) = \langle {\hat {a}^{\dagger }_{s(i)}(t) \hat {a}_{s(i)}(t) }\rangle {}$. In the experiment, the normalized cross-correlation function can be determined by $g^{(2)}_{si} = p_{s,i}/(p_s p_i)$, where $p_{s} (p_i)$ is the probability of detecting the signal (idler) photons, and $p_{s,i}$ is the probability of detecting the coincidence events of both the signal and idler photons [17]. A time-bin of 10 ns is used to determine the photon count. $g^{(2)}_{si}$ is determined by the photon count at the peak of the biphoton waveform and the average count far away from the biphoton waveform. If one assumes the second-order auto-correlations of signal and idler to be $1 \leq [g_{ss}, g_{ii}] \leq 2$, then $g^{(2)}_{si} > 2$ manifests violation of the classical property [19,21,30]. It is desirable to pursue a better $g^{(2)}_{si}$ to enlarge the violation factor [19]. Therefore, we vary the pump power to determine the best working point, as shown in Fig. 2(b). It is evidently to see a slight degradation of $g^{(2)}_{si}$ due to the leakage of control beam photons. Details of the theory used to model the data can be referred to Appendix 7.3. At the best working point, the maximum $g^{(2)}_{si}$ right after the photon source and after arriving at the quantum memory setup are $\approx 35$ and $\approx 10$, respectively, as shown in Fig. 2(a). The data of Fig. 2(a) for both cases correspond to an accumulation of idler trigger events for $1.6\times 10^{5}$ and $2.7 \times 10^{6}$ with a data collection time of $\sim$ 10 and 160 minutes, respectively.

The anti-bunching behavior of single photons is confirmed by measuring the conditional second order autocorrelation function for the signal photons $g^{(2)}_{ss|i}$, which should be $0$ for an ideal single-photon Fock state and $1$ for coherent light. The minimum value of the $g^{(2)}_{ss|i}$ measured after arriving at the quantum memory setup is $0.17(0.07)$, as shown in Fig. 2(c).

 

Fig. 2. (a) The normalized cross-correlation functions of the photon pairs after arriving at the quantum memory setup and right after the photon source are indicated by the black and yellow points, respectively. (b) Peak $g^{(2)}_{si}$ for the photon pairs measured after arriving at the quantum memory setup. The solid lines are the fitted curves obtained using Eq. (9) with $\kappa ^{2} = A \times (\text {Pumping Power})$. The fitting parameters are $\{\Gamma _{cav}, n_s, n_i, A\} = \{22.94 , 0.46, 0.21, 0.068\}$ for the case with the control beam and $\{22.94, 0.35, 0.21, 0.068\}$ for the case without the control beam. (c) Normalized autocorrelation function of the photon pairs measured after arriving at the quantum memory setup. The black dashed line is a fit curve with a formula $g^{(2)}_{ss|i}(\tau)=1-(1-G_{min})e^{-(\gamma(\tau-\tau_{0}))^2/2}$ [31]. The fitting parameters $G_{min}, \tau_{0}, \gamma$ are $0.17(0.07), 0.057(0.011), 15.80(1.65)$, respectively.

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4. Storage of heralded single photons

We now discuss the experiment conducted for the storage of heralded single photons. The repetition rate is set to 16 Hz in this experiment. We first operate the photon source in the OPO regime such that the signal is weak coherent light and take EIT transmission spectrum. We then determine the interested parameters, e.g. optical depth, Rabi frequency of the control field, ground-state decoherence rate and the FWHM bandwidth of the EIT transparent window, by spectral fitting of the EIT spectrum. With an optical depth of $\sim$ 300, we typically operate the EIT memory at a FWHM transparent bandwidth of $\sim$ 6 MHz, which is wider than the photon bandwidth of 2.2 MHz. Under such a condition, the group delay of the waveform of signal photons is $\sim$ 400 ns, which is roughly an optimum value to compress the signal waveform into atomic memory without losing the tails. The waveform and the non-classical correlation of the retrieved signal photons can be manipulated by varying the control intensity during the retrieval process. The data obtained using various ratios between the reading to writing control intensity ($\xi _{R/W}$ from 1 to 5.6) with a storage time of $400$ns are shown in Fig. 3(a). It is evident that the peak of the retrieved signal pulse is higher for a larger $\xi _{R/W}$. We emphasize that the implementation of EIT memory in the $D_1$ line allows us to increase the $g^{(2)}_{si}$ of the signal photons by increasing the writing control beam intensity without sacrificing the storage efficiency, in comparison to the case of the $D_2$ transition [17]. This is due to the significant reduction in the off-resonant coupling of the control field to the nearby transition for the $D_1$ line as compared to that of the $D_2$ case [5,17]. The peak $g^{(2)}_{si}$ and storage-and-retrieval efficiency (SE) versus $\xi _{R/W}$ are depicted in Fig. 3(b) and (c), respectively. The way to determine the efficiency is described in Appendix 7.2. The efficiency is around $\sim 70\%$ for different $\xi _{R/W}$. A slight deviation from the average efficiency for some data points may due to the drifts in system parameters (e.g. optical depth) since it takes $\sim (6-8)$ hours to collect each data set.

 

Fig. 3. (a) Raw data for the input single-photon waveform and the retrieved waveform with $\xi _{R/W} = \{1, 1.2, 3.3, 5.6\}$, respectively. $N_{s,i}$ indicates the coincidence counts of the signal and idler photons; $N_i$ indicates the idler photon count. (b) Peak $g^{(2)}_{si}$ of the retrieved photons versus the $\xi _{R/W}$. The dashed curve denotes the classical limit ($g^{(2)}_{si}=2$). (c) The efficiency of the retrieved photons versus $\xi _{R/W}$. The dashed line denotes an efficiency of 0.7.

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The efficiency of 70% is better than the 36 % of our previous work on storage of single photons generated by a SPDC source due to multiple reasons [17]. First, we operate the EIT memory using the $D_1$-line of cesium in this work, instead of the $D_2$-line in the previous work. As already discussed in Ref. [5], the off-resonant excitation of the control field to the nearby transitions introduce a control-intensity-dependent ground-state decoherence rate, which degrades the storage efficiency at larger optical depths. This issue is much serious for the $D_2$-line EIT system, compared to the $D_1$-line system, due to the much smaller hyperfine splitting in the excited states and the existence of one cycling transition. Second, we have a higher optical depth of $\sim$300 in this work, compared to $\sim$50 in the previous work. The memory efficiency is higher for a larger optical depth as shown in Ref. [5]. Third, the photon bandwidth of our previous work is 6.2 MHz while the optical depth is only up to 50. Refer to Ref. [32], it is impossible to operate the memory in the adiabatic EIT regime under such a condition. Instead, the memory is operated in the Aulter-Townes splitting regime [33]. Under such a situation, there is a significant fraction of the photon waveform that is non-slowed and can’t be stored. Besides, the control intensity is relatively high with such a high photon bandwidth. Due to the first reason mentioned above, the control-induced ground-state decoherence rate is relatively high which also leads to a degradation in the efficiency.

Although the peak height of the signal pulse is greater for a stronger $\xi _{R/W}$, as shown in Fig. 3(a), the peak $g^{(2)}_{si}$ for the retrieved signal photons saturates at higher $\xi _{R/W}$, as shown in Fig. 3(b). The peak $g^{(2)}_{si}$ increases by a factor of $\sim 2$ for $\xi _{R/W}$ from 1 to 4.2. This means that one can manipulate the waveform (or bandwidth) and the non-classical correlation of the retrieved signal photons by varying the intensity of the writing control field [17]. The saturation of the peak $g^{(2)}_{si}$ at high $\xi _{R/W}$ is due to the increase in the background noise which comes from the control beam leakage and Raman-induced noise. More discussions of the noise are described in Appendix 7.4. In all cases, the peak $g^{(2)}_{si}$ are all larger than 2. Therefore, the non-classical property of the heralded single photons is preserved by the EIT memory. The highest retrieved $g^{(2)}_{si}$ peak reaches $12.54 (1.27)$ at $\xi _{R/W} = 3.3$.

Next, we measure the retrieved signal photons for various storage times, as shown in Fig. 4(a). The peak $g^{(2)}_{si}$ and efficiency versus the storage time are shown in Fig. 4(b) and (c), respectively. The efficiency can be fitted by a curve $SE_0e^{-t^{2}/\tau ^{2}}$ with $SE_0=0.69$ and $\tau = 18.4 \mu$s. The $e^{-1}$ storage time ($\tau$) is significantly shorter than that obtained in our previous work which was $325 \mu$s [5]. This is mainly due to the larger angle ($4^{0}$) between the signal and control beams, which results in a relatively large residual Doppler broadening. There is a trade off between getting a higher signal to noise ratio and a longer storage time, which correspond to a larger and a smaller angle between the signal and control beam, respectively. At a smaller angle, the control leakage into the single photon detector degrades the signal to noise ratio. The $4^{0}$ is nearly the smallest angle we could achieve while maintaining an acceptable signal to noise ratio. One possibility to achieve a longer storage time is to cool the atoms to a colder temperature. The typical atomic temperature is $\sim 150 \mu$K in this experiment. It is possible to use the $\Lambda$-enhanced grey molasses cooling to cool down atoms to $\sim 3\mu$K [34]. At such a temperature, the residual Doppler broadening can be reduced by a factor of $\sim$7 and the storage time can be lengthened by the same factor [35]. For an even longer storage time (e. g. on the second scale), one needs to trap the atoms with an optical dipole trap with a magic-valued magnetic field [36]. However, the experimental setup will be much more complicated.

 

Fig. 4. (a) Raw data for the retrieved signal photons for various storage times of 0.4, 10, and 15$\mu s$. The symbols indicate: $t_{store}$: storage time; $t_s$: signal photon arrival time; $t_i$: idler photon arrival time. (b) The $g^{(2)}_{si}$ of the retrieved pulses (blue squares) versus the storage time. The red solid line is a fitted curve with the expression of $g^{(2)}_{si,0} e^{-t/\tau }$ [5], where the fitting parameters are $g^{(2)}_{si,0} = 12.1$ and $\tau = 12.1 \mu s$. The dashed line for $g^{(2)}_{si} = 2$ depicts the classical bound. (c) Efficiency (blue squares) versus the storage time. The red solid line is a fitted curve with the expression $\text {SE}_0 e^{-t^{2}/\tau ^{2}}$, where the fitting parameters are $\text {SE}_0 = 0.69$ and $\tau = 18.4 \mu s$. (d) Comparison of the DBP versus efficiency for the storage of heralded single photons generated by SPDC sources where AFC stands for the quantum memory protocol based on atomic frequency comb. The definition of DBP is explained in the main text. A: [20], B: [18], C: [17]; D: [21];E: [23]; F: [22]; G: [24]

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The efficiency is below $50\%$ when the storage time is longer than $\sim 10 \mu s$, while the $g^{(2)}_{si}$ drops to half of the value without storage at $\sim 8 \mu s$. The delay-bandwidth product (DBP), usually defined as the ratio of the storage time when the efficiency is 50% to the FWHM duration of the input signal pulses, is an important figure of merit for quantifying the ability of a memory to store the quantum nature of the signal. However, our work is the only one so far to report an efficiency of > 50 % for the storage of single photons generated by SPDC sources. Remember that there do exist some works with an efficiency of > 50% for storage of single photons generated by spontaneous Raman transition or spontaneous four-wave mixing in atomic ensembles [2,16], but we only compare the works with SPDC photon sources here. To be able to make a comparison with other works using SPDC photon sources, we adopt a definition of DBP as the product of the storage time when the retrieved $g^{(2)}_{si}$ is 50% of its initial value with the null storage time. The comparison are shown in Fig. 4(d) [5,32,37]. Although there is still a lot of room for improvement, the DBP of $\approx 138$ and the efficiency of 70% we have achieved are the best among these works. The DBP value for the original and the new definitions could be different but should be comparable. As a reference, the DBP value of the original definition is 161 in our experiment.

5. Storage of polarization qubits

The quantum memory should be able to store arbitrary qubit states. We encode the polarization qubits into the heralded single photons by the dual-rail scheme [16,29], as shown in Fig. 1(a). The polarizations of the signal photons are mapped into two spatial modes. For example, for a qubit with a quantum state $\frac {1}{\sqrt {2}}(|{1}\rangle _H + e^{i \phi }|{1}\rangle _V)$, the dual-rail setup converts it into

Figures 5(a-d) depict the reconstructed density matrices of the retrieved signal photons for which the initial polarization states are $|{D}\rangle =\frac {1}{\sqrt {2}}(|{H}\rangle +|{V}\rangle )$, $|{R}\rangle =\frac {1}{\sqrt {2}}(|{H}\rangle +i|{V}\rangle )$, $|{H}\rangle$, and $|{V}\rangle$, respectively [38]. The average raw fidelity reaches $89.06(2.31)\%$. The performance is mainly limited by the low heralding efficiency ($\sim 0.38 \%$) and the background noise [29]. After subtracting the background noise, the average corrected fidelity is $96.88(3.22)\%$ (Fig. 5(e)). The average storage efficiency for these data is $55.8 (3.8)\%$. The major reason for the reduced efficiency compared to the single-rail setup is due to a reduction in the optical depth from $306$ to $235$ due to the increase in the repetition rate from 16 to 32 Hz for faster data collection. Besides, the two spatial modes of the signal photons are separated by $\sim$3 mm before focusing into the MOT cell. The distance between the two modes is $\sim$ 127 $\mu$m inside the atomic ensembles. The optical depth and the control intensity experienced by the two spatial modes could be slightly different and reduced.

 

Fig. 5. (a)-(d) The reconstructed density matrices of the retrieved signal photons with the input photon state prepared as $|{D}\rangle, |{R}\rangle, |{H}\rangle,$ and $|{V}\rangle$, respectively. The colored bar on the right side indicates the phase of the components. The raw fidelity of $|{D}\rangle, |{R}\rangle, |{H}\rangle, |{V}\rangle$ are $88.23(2.12) \%$, $90.42(2.34) \%$, $90.16(1.81) \%$, $87.44(2.86) \%$. The corrected fidelity of $|{D}\rangle, |{R}\rangle, |{H}\rangle, |{V}\rangle$ are $95.96(2.81) \%$, $96.64(3.12) \%$, $96.47( 2.55) \%$, $98.45(4.17) \%$. The corrected fidelity is estimated by deducting the average background noise. (f) The fidelity of $|{D}\rangle$ versus the storage time. There is a drop in the raw fidelity due to the declining of the photon signal, while the corrected fidelity remains at the same level but with a larger uncertainty. The dashed lines in (e, f) indicate the $95\%$ fidelity.

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The above measurements are also repeated for longer storage times, as depicted in Fig. 5(f). There is a decrease in the raw fidelity as the storage time increases, while the corrected fidelity stays at nearly the same level, although the uncertainty rises. The reason for the reduction in the raw fidelity is twofold. First, memory efficiency drops when the storage time increases due to the decay of the spin waves resulting from the finite ground-state decoherence rate and the atomic motions. Second, there is an increase in the Raman-induced noise for longer storage times. This noise increase is due to the excitation of the atoms in the $|F=4\rangle$ ground state by the control field after it is turned back on for retrieval. The atoms in the $|F=4\rangle$ ground state may come from either cold atoms after collisional relaxation or the background hot atoms in that state travelling through the control beam. At the atom density in the MOT, the former reason is small and the latter reason should dominate. More discussions are described in Appendix 7.4.

Finally, it should be noted that there is a strong spectral correlation between the signal and idler photons for the continuous-wave pumped cavity-enhanced SPDC source. This may cause some problems in certain quantum information applications [39] but can be resolved by exploiting the pulsed pumping scheme [18].

6. Conclusions

In summary, we demonstrate the quantum storage of heralded single photons and polarization qubits generated from a cavity-enhanced SPDC source. The non-classical correlation is preserved and can be manipulated by the EIT memories. Both the efficiency and delay-bandwidth product are the highest achieved to date in experiments using SPDC-generated single photons. The corrected average fidelity of the polarization qubits is $96.88 (3.22)\%$. The fidelity is limited by the low heralding efficiency (due to the optical loss) and the noise, but this could be improved with more technical efforts. Our work paves the way towards the development of a large-scale quantum network.

7. Appendix

7.1. Measurement of the autocorrelation function

In the measurement process, a 50/50 beam splitter is used to split the signal photon channel into two and then detect photons on these two arms. The conditional second-order autocorrelation function reads,

(2)$$\label{} g_{ss|i}^{(2)} = \frac{N_{s_1, s_2, i} N_i}{N_{s_1, i} N_{s_2, i}}$$

, where $N_{s_1, s_2, i}$ is the three-fold coincidence counts of the heralded idler and signal photons on both arms. $N_{s_{1(2)}, i}$ is the two-fold coincidence counts for the heralded idler and signal photons on one of the two arms; $N_i$ indicates the idler photon count [16].

7.2. Determination of the efficiency

The coincidence counts with and without storage are first subtracted from the corresponding average background count, which is determined for a time window away from the biphoton profile. The coincidence counts left, with the profile of a nearly two-sided exponential decay curve, are those due to the true two-photon events. The total counts within a time window, centered around the peak count of three times FWHM of the biphoton profile, are calculated for both the data with and without storage [19,27]. The ratio of the count with storage to that without storage is the efficiency.

7.3. Noise model for the photon pairs

The noise model for the photon pair detection is illustrated in Fig. 6. The photon pairs generated from the "photon source" box are then sent to the two detection modules. There are inevitable losses along the propagation channel. The total arrival efficiency for the signal (idler) photons is denoted by $R_{s(i)}$. In addition, both channels may suffer from the problem of noise. The number of noise counts entering the two channels are denoted by $N_i$ and $N_s$, respectively.

 

Fig. 6. Illustration of the noise model discussed in Appendix 7.3.

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We follow Ref. [27] to consider the theory of a cavity-enhanced SPDC photon-pair source. The total cavity decay rates for the signal and idler photons ($\Gamma _{s,cav}, \Gamma _{i,cav}$) contain the cavity output coupling rate ($\gamma _{s,cav}, \gamma _{i,cav}$) and other decay rates, such as the crystal loss. In our experiment, the total cavity decay rates for the signal and idler photons are nearly the same, due to their having nearly the same wavelengths, as evidenced by the symmetry of the decay time of the cross-correlation function of the biphotons [27]. For simplicity, we assume $\Gamma _{s,cav} = \Gamma _{i,cav} \equiv \Gamma _{cav}$. Based on the cavity-enhanced property of the SPDC, the coincidence count of the photon pairs is

(3)$$\begin{aligned} & \langle{Vac}| a^{\dagger}_{i}(t_i)a^{\dagger}_{s}(t_{s}) a_{s}(t_{s}) a_{i}(t_i)|{Vac}\rangle= \\ & (\kappa^{2}\times e^{-\Gamma_{cav} |t_s - t_i|} + \frac{4}{\Gamma_{cav}^{2}}\kappa^{4}) R_i R_s \end{aligned}$$

where $|{Vac}\rangle$ is the vacuum state; and $\kappa$ is the coupling constant between the field and the crystal, which is proportional to the field strength and the nonlinear coefficient of the crystal [27]. During data fitting, we denote $\kappa ^{2}=AP$, where $P$ is the pump power and $A$ is a constant related to property of the crystal. The generation rates for the signal and idler photons are

(4)$$\langle{Vac}| a^{\dagger}_{s}(t) a_{s}(t) |{Vac}\rangle = \ \frac{2}{\Gamma_{cav}} \kappa^{2} R_s,$$

(5)$$\langle{Vac}| a^{\dagger}_{i}(t) a_{i}(t) |{Vac}\rangle = \ \frac{2}{\Gamma_{cav}} \kappa^{2} R_i,$$

respectively. Since there should be no correlation between the noise (of a given channel) and the photons from the photon source or the noise from the other channel, its field operator commutes with the field operator of the photon source and that of the noise of the other channel. The field operator of photons (or noise) of the same channel follows the standard commutation relation, e.g. $[a_s, a_s^{\dagger }]=1$. The coincidence count contains both photons from the source and photons from the noise channels. Using the commutation relations, the coincidence count is,

(6)$$\begin{aligned} G^{(2)}_{si}(t_{s}, t_{i}) = & \langle{Vac}| (a^{\dagger}_i(t_i)+ a^{\dagger}_{N_i}(t_i) )(a^{\dagger}_{s}(t_{s})+ \\ & a^{\dagger}_{N_s}(t_{s})) (a_{s}(t_{s}) + a_{N_s}(t_{s})) (a_i(t_i)+ a_{N_i}(t_{i}))|{Vac}\rangle\\ = & \langle{Vac}| a^{\dagger}_i(t_i)a^{\dagger}_{s}(t_{s}) a_{s}(t_{s}) a_i(t_i)|{Vac}\rangle + \\ & \langle{Vac}| a^{\dagger}_{N_i}(t_i)a^{\dagger}_{s}(t_{s}) a_{s}(t_{s}) a_{N_i}(t_i)|{Vac}\rangle + \\ & \langle{Vac}| a^{\dagger}_{i}(t_i)a^{\dagger}_{N_s}(t_{s}) a_{N_s}(t_{s}) a_{i}(t_i)|{Vac}\rangle + \\ & \langle{Vac}| a^{\dagger}_{N_i}(t_i)a^{\dagger}_{N_s}(t_{s}) a_{N_s}(t_{s}) a_{N_i}(t_i)|{Vac}\rangle. \end{aligned}$$

The noises operators follow

(7)$$\begin{aligned} \langle{Vac}|a^{\dagger}_{N_i}(t_i) a_{N_i}(t_i)|{Vac}\rangle = N_i\\ \langle{Vac}|a^{\dagger}_{N_s}(t_s) a_{N_s}(t_s)|{Vac}\rangle = N_s. \end{aligned}$$

With some reductions, the last three terms in Eq. (6) can be reduced to,

(8)$$\begin{aligned} & \langle{Vac}| a^{\dagger}_{N_i}(t_i)a^{\dagger}_{N_s}(t_{s}) a_{N_s}(t_{s}) a_{N_i}(t_i)|{Vac}\rangle =\\ & \langle{Vac}| a^{\dagger}_{N_i}(t_i) a_{N_i}(t_i) a^{\dagger}_{N_s}(t_{s}) a_{N_s}(t_{s}) |{Vac}\rangle=N_i N_s,\\ & \langle{Vac}| a^{\dagger}_{N_i}(t_i)a^{\dagger}_{s}(t_{s}) a_{s}(t_{s}) a_{N_i}(t_i)|{Vac}\rangle =\\ & \langle{Vac}| a^{\dagger}_{N_i}(t_i) a_{N_i}(t_i) a^{\dagger}_{s}(t_{s}) a_{s}(t_{s}) |{Vac}\rangle \\ = & N_i \langle{Vac}|a^{\dagger}_{s}(t_{s}) a_{s}(t_{s}) |{Vac}\rangle = N_i R_s \frac{2}{\Gamma_{cav}}\kappa^{2},\\\\ & \langle{Vac}| a^{\dagger}_{N_s}(t_s)a^{\dagger}_{i}(t_{i}) a_{i}(t_{i}) a_{N_s}(t_s)|{Vac}\rangle =\\ & \langle{Vac}| a^{\dagger}_{N_s}(t_s) a_{N_s}(t_s) a^{\dagger}_{i}(t_{i}) a_{i}(t_{i}) |{Vac}\rangle \\ = & N_s \langle{Vac}|a^{\dagger}_{i}(t_{i}) a_{i}(t_{i}) |{Vac}\rangle = N_s R_i \frac{2}{\Gamma_{cav}}\kappa^{2}. \end{aligned}$$

Using the aforementioned relations, the normalized cross correlation function for $t_s=t_i$ is

(9)$$\begin{aligned} g^{(2)}_{si}(t_s=t_i) = 1 + \frac{\kappa^{2}}{ (\frac{2}{\Gamma_{cav}} \kappa^{2} + n_i)(\frac{2}{\Gamma_{cav}} \kappa^{2} + n_s)} \\ =1 + \frac{1}{(\frac{2}{\Gamma_{cav}} (n_s + n_i) + \frac{4}{\Gamma_{cav}^{2}} \kappa^{2} + \frac{n_i n_s}{\kappa^{2}})}. \end{aligned}$$

The data in Fig. 2(b) and Fig. 7 are fitted by this relation with $\kappa ^{2}=AP$, where $P$ is the pumping power and $A$ is a constant.

 

Fig. 7. $g_{si}^{(2)}$ versus the pump power measured in a 20 $\mu s$ time window with its starting time delayed by multiples of 20 $\mu s$ plus 34 $\mu s$ after switching the photon source output from the OPO to the SPDC phase. There is degradation in the $g_{si}^{(2)}$ value of the earlier intervals due to the noise from the leakage of photons in the OPO phase. The fitting parameters, $\{\Gamma, n_{s}, n_{i}, A\}$, for $\{0-20, 20-40, 40-60\}\mu s$ measurement time windows are $\{26., 0.105, 0.105, 0.059\}$, $\{26., 0.085, 0.085, 0.059\}$, and $\{26., 0.079, 0.079, 0.059\}$, respectively.

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7.4. Characterization of the noise

The noise comes from multiple sources. The dark count of the photon detector which is $\sim 100 \text {Hz}$, contributes $\sim 5\%$ of the noise background. Careful blockage of the photon detection unit was carried out to minimize the leakage of stray light into the detector from the surroundings. We estimate that less than 10 % of the background noise is contributed by the stray light. Based on our previous work [17], we know that the leakage of residual photons from the OPO phase into the SPDC phase is also a source of noise. However, this contribution can be minimized by waiting for a certain time before data collection (34 $\mu s$ in our experiment, as shown in Fig. 1(b) to allow the residual leakage to decay to a negligible level. To investigate this issue, we split the detection time period ($60 \mu s$ as shown in Fig. 1(b)) into three equal intervals, then measured the $g_{si}^{(2)}$ of each interval right after the photon source, as shown in Fig. 7. It is evident that in the earliest time interval ($0-20 \mu s$), the residual photons still make a considerable contribution to the background noise. The data discussed in this paper were collected for the whole 60 $\mu s$ period, starting at 34 $\mu s$, after switching from the OPO to the SPDC phase. We estimate that the residual OPO photon leakage contributes an average of $\sim$ 5% to the noise background.

Next, we address the noise due to the leakage of the control beam. In the earlier experiment, we found that the reflection of the control beam from the etalon filter back to the SPDC photon source causes serious false triggering on the idler detector (SPCM1 in Fig. 1(a)). Figure 8(a) shows an example to demonstrate this problem with three different control beam powers applied during the optical pumping period (the 10 mW period shown in Fig. 1(b)). A higher control beam power during this period is used to pump the population to the $|F=3\rangle$ ground state to minimize the Raman-induced noise which will be described later. With a stronger control power, there is evidently a significant degradation in the biphoton coincidence. To solve this problem, one additional optical isolator is added to the path between the MOT cell and the etalon filter. This reduces the false triggering to a negligible level, similar to the case with the lowest control power shown in Fig. 8(a). The isolator is kept there for all subsequent experiments. To understand the contribution of leakage control photons to the probe detector (SPCM2), we can check the raw data shown in Fig. 2(b), with and without the presence of the control beam. It is determined that leakage of photons from the control beam contributes $\sim$ 14% of the background noise.

 

Fig. 8. (a) The coincidence data depict the effect of false triggering due to the reflection of the control beam on the idler detector. The coincidence counts deteriorate when the power of the control beam is increased during the pumping stage (the period with a control field of 10 mW illustrated in Fig. 1(b), its power is denoted as $\text {Pow}_{pp, cp}$). This is because the control beam is reflected back to SPCM1, which results in the increase of false triggers. In all three conditions as well as the case 3 in (b), the control beam is switched off during the storage phase (2.2 - 2.8 $\mu s$) and back on for the reading stage (after $2.8 \mu s$). The higher noise level after $2.8 \mu s$ is due to Raman-induced noise explained below. (b) The coincidence counts versus time for three cases to depict Raman-induced noise. For case 1, the control beam is constantly on, without hot cesium vapor in the cell (blue). For case 2, the control beam is constantly on with hot cesium vapor (red). It is evident that the noise background of case 2 is higher than that of case 1. We attribute this additional noise to Raman-induced noise, which comes from Raman transition for hot atoms in the $|{F=4}\rangle$ ground state entering the control beam region, absorbing the control photons and then emitting photons with the same frequency as signal photons. For case 3, the control beam is switched off for storage and back on for retrieval (green), while the hot vapor fills the cell. The background noise level is even higher than that of case 2 when the control beam is turned back on at time > $2.8 \mu s$. Again, this can be explained by Raman-induced noise. The higher noise level is due to more hot atoms in the $|{F=4}\rangle$ ground state entering the control beam region during its off period.

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Finally, we address the problem of Raman-induced noise, which is found to be the largest source of noise in our experiments. The coincidence data for cases 1 (blue) and 2 (red) are shown in Fig. 8(b). In the first case the cesium dispenser source is off and in the second it is heated to a typical level for the MOT operation. The control beam is continuously on in both cases. It is evident that the background noise is higher for case 2 than case 1 by $\sim$ 25% on average. In case 3 (green), the control beam is switched off for the storage phase (during 2.2 - 2.8 $\mu$s) and then switched on for the reading stage (after 2.8 $\mu$s), while the hot vapor is the same as case 2. It is evident that the background noise level increases to a level even higher than that for case 2 (by $\sim 40\%$) when the control field is turned back on (after 2.8 $\mu$s). The higher noise level in cases 2 and 3, compared to case 1 can be attributed to Raman-induced noise, which are generated when some hot atoms in the 6$S_{1/2}, F=4$ ground state, moving into the control beam region, absorb the control photons and emit photons at the signal frequency. The higher noise level for case 3 compared to case 2 after 2.8 $\mu$s is because that more hot atoms in the 6$S_{1/2}, F=4$ ground state move into the control beam region when it is off during the storage period. The data in Fig. 3 and Fig. 4 all suffer from this noise. One way to reduce such noise is to apply an optical pumping beam, driving either the $D_1$ or $D_2$ $|F=3\rangle \rightarrow |F=4\rangle$ transition, to pump the hot vapor in the glass cell to the 6$S_{1/2}, F=3$ ground state but with a shadow in its beam to avoid perturbing the trapped cold atoms. Another way is to to utilize the double MOT system to reduce the background pressure of the hot vapor while still maintain a large number of cold atoms for a high optical depth.

Funding

Ministry of Science and Technology, Taiwan (108-2112-M-001-030-MY3, 109-2639-M-007-002-ASP, 110-2639-M-007-001-ASP).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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