1. Introduction

Quantum networks (QWs) play an important role in distributed long-distance quantum communications [1–3] and quantum computations [4,5]. The quantum interfaces that generate entanglement between an atomic memory and a photon [6–16] can be used as quantum node and then are basic building blocks for QWs. The elementary step in QWs is to establish entanglement between two remote memories (nodes) [1–2]. With the quantum interfaces, one can establish entanglement between two remote memories by performing two-photon Hong-Ou-Mandel interference [17–22]. Also, the remote-memory entanglement can be established by mapping quantum state of a photon entangled with an atomic memory into another remote memory [23]. Hybrid quantum networks use disparate quantum memory (QM) systems as nodes and then can benefit from the advantages of the disparate systems. In hybrid QWs, an open question is wave-packet matching of the interfacing photons that are generated from or stored in disparate memory matters (nodes). For example, the pulse widths of the single photons generated from disparate memory matters are different. Specifically, the typical pulse widths of single photons are ∼3 ns [24] for single quantum dots, ∼0.4 [25,26] µs for solid-state atomic ensemble, ∼70-100 ns [8,9,12] for cold atoms, 1 ns [27] (40 µs [28]) for room-temperature atomic vapors via off-resonance (motional averaging) Duan–Lukin–Cirac–Zoller protocol, and ∼0.5 µs [23] or 12 µs [29] for single atoms or ions in QED cavities. Also, for achieving high-efficiency optical quantum storages in different memory matters via various storage schemes, the required pulse-widths of the single photons are different. For example, the required pulse-width is 200ns-3µs for cold atoms [30–34] via EIT storage scheme, ∼7 µs for crystals [35] via AFC storage scheme, 0.7µs for single atom via EIT storage scheme [36], 1 ns for hot atoms [37], 7 ns for cold atoms [38] and ∼260 fs for bulk diamond [39] via Raman storage scheme, respectively. However, HOM interference requires that the wave-packets of the two photons are perfectly overlapped, if not, visibility of HOM dip will be degraded [40]. So, the perfect wave-packet matching of the two photons is necessary for establishing hybrid QWs. Recently, two experiments have experimentally demonstrated the non-classical correlation between two different memories [41], and HOM interferometer between two photons from two different memory matters [42], where the wave-packets of the single photons are changed into appropriate values, and then the connection between the different memories are experimentally available. With various physical systems such as cavity-quantum-electrodynamics single atom [43], a single trapped ion [44], DLCZ quantum memory in cold atomic ensemble [45], cold atoms [46–49] and hot atomic vapour [50] with four-wave-mixing, the single photons with highly tunable waveform lengths have been experimentally demonstrated. However, in these experiments the single photon is only non-classically correlated with or stored in an atomic memory, it is not entangled with an atomic memory. Thus, the influences of changes of single-photon wave-packet on the atom-photon entanglement remain unexplored.

In this article, we demonstrated controls of temporal durations of waveform of the Stokes photon entangled with an atomic spin-wave memory in cold atoms via varying DLCZ-like light-atom interaction time. The retrieval efficiency R and Bell parameter S as a function of the width of Stokes-photon pulse are measured. The results show that in the width ranging from 40 ns to 50 µs, the retrieval efficiency is basically kept unchanged and the violation of the Bell inequality can be achieved. Specifically, when the pulse width is 40 ns, we achieved R= 20% and S= 2.64 ± 0.02, while when it is 50 µs, we achieved R= 15% and S= 2.26 ± 0.05, respectively. To our knowledge, the 50 µs long wave packet of a single photon represents the longest wave packet of a single photon entangled with an atomic spin-wave memory. By fitting the experimental data with theoretical calculations, we reveal the reason for the decrease in S parameter with the pulse duration, which results from that background noise per pulse increases with the pulse duration.

2. Experimental setup

The experimental setup is shown in Fig. 1(a). The atomic ensemble is a cloud of cold ⁸⁷Rb atoms, whose relevant atomic levels are shown in Fig. 1(b), where, $|g \rangle = |{{\textrm{5}^\textrm{2}}{S_{{\textrm{1} / \textrm{2}}}},F = \textrm{1}} \rangle$, $|s \rangle = |{{\textrm{5}^\textrm{2}}{S_{{\textrm{1} / \textrm{2}}}},F = \textrm{2}} \rangle$, $|{{e_\textrm{1}}} \rangle = |{{\textrm{5}^\textrm{2}}{P_{{\textrm{1} / \textrm{2}}}},F^{\prime} = \textrm{1}} \rangle$, and $|{{e_\textrm{2}}} \rangle = |{{\textrm{5}^\textrm{2}}{P_{{\textrm{1} / \textrm{2}}}},F^{\prime} = \textrm{2}} \rangle$. Each level includes Zeeman sublevels, for example, $|g \rangle$ is written as $|{g,m} \rangle$, where the magnetic quantum number $m ={\pm} 1,0$. After the atoms are released from the magneto-optical trap, we prepare the atoms into Zeeman levels $|{g,m ={\pm} \textrm{1,0}} \rangle$ via optical pumping with a cleaning laser. In the beginning of a spin-wave-photon entanglement (SWPE) generation trial [Fig. 1(c)], a write laser pulse with a tunable duration ${\tau _w}$ is applied onto the atoms. The value of the duration ${\tau _w}$ is controlled by an AOM (see Fig. 1(a)). The write pulse is $\Delta $=20 MHz blue-detuned to $|g \rangle \to |{{e_2}} \rangle$ transition, which induces spontaneous Raman scattering of ${\sigma ^ - }$-polarized (${\sigma ^ + }$-polarized) Stokes photons and simultaneously create the spin wave $|+ \rangle$($|- \rangle$), where the spin wave $|+ \rangle$($|- \rangle$) is associated with the coherence $|{g,m = 0} \rangle \leftrightarrow |{s,m = 0} \rangle$ and $|{g,m ={-} 1} \rangle \leftrightarrow |{s,m ={-} 1} \rangle$ ($|{g,m ={-} 1} \rangle \leftrightarrow |{s,m = 1} \rangle$ and $|{g,m = 0} \rangle \leftrightarrow |{s,m = 2} \rangle$), which is shown in Fig. 1(b). Also, the spin coherence $|{g,m ={-} 1} \rangle \leftrightarrow |{s,m ={-} 1} \rangle$ will be created in the spontaneous Raman scattering, but is neglected in the spin wave $|+ \rangle$ since it can’t be retrieved in the following retrieval step. In the Stokes detection channel (purple dash line in Fig. 1(a)), we transform the ${\sigma ^ + }$(${\sigma ^ - }$)-polarized Stokes photons into the horizontally- (H-) or vertically- (V-) polarized photon by a λ/4 plate. The joint state of the atom–photon system may be written as [7,51] $\rho _{ap}^{} = |\textrm{0} \rangle \left\langle \textrm{0} \right|+ \sqrt \chi |{\Phi _{\textrm{a - p}}^{}} \rangle \left\langle {\Phi _{\textrm{a - p}}^{}} \right|$, where $|\textrm{0} \rangle$ denotes the vacuum, $\chi ({ \ll \textrm{1}} )$ represents the probability of creating the $|{\Phi _{\textrm{a - p}}^{}} \rangle$ state per write pulse, $\Phi _{\textrm{a - p}}^{} = ({\cos \vartheta {{|+ \rangle }_a}{{|H \rangle }_S} + \sin \vartheta {{|- \rangle }_a}{{|V \rangle }_S}} )$ denotes the spin-wave-photon entanglement state, $|H \rangle _S^{}$ ($|V \rangle _S^{}$) denotes a H- (V-) polarized Stokes photon, $\cos \vartheta$ is the relevant Clebsch-Gordan coefficient with the asymmetric angle of $\vartheta \approx 0.81 \ast ({\pi /4} )$[6]. The Stokes photons are collected at about 1°relative to the direction of the write laser beam (blue line), which defines their wave vector ${\boldsymbol{k_S}}$. The collection is achieved by a single-mode fiber, which is labeled as SMF_S in Fig. 1(a). The wave-vector of the spin-wave excitation is given by ${\boldsymbol{k_{SWE}}} = {\boldsymbol{k_W}} - {\boldsymbol{k_S}}$, where, ${\boldsymbol{k_W}}$ is the wave-vector of the write pulse. After the SMF_S, the Stokes photons go through a phase compensator (PC) [52], which is used to eliminate the phase shift between the H and V polarizations caused by SMF_S. Next, the Stokes photons pass through an optical-spectrum-filter set (OSFS) which consists of three Fabry-Perot etalons. Blocking by OSFS, the leakage of the write laser beam in the Stokes channel was attenuated by a factor of ${10^{ - 9}}$. We then measured the background noise probability resulting from the write laser beam in the Stokes channel, which is not more than $2.5 \times {10^{ - 4}}$/write pulse in our presented experiment. The OSFS transmit the Stokes field with a transmission efficiency of ∼70%. Subsequently, the Stokes photon passes through a λ/2 plate. By rotating it, one can change the polarization angle θ_S of the Stokes fields. Then, the Stokes photon is guided to a polarizing beam splitter PBS_S, which transmits the horizontal (H) polarization to a detector D_S1 and reflects the vertical (V) polarization to a detector D_S2 for θ_S =0. The detection events at the detectors $D_{S1}^{}$ and ${D_{S2}}$ are analyzed with a field programmable gate array (FPGA). As soon as a Stokes photon is detected by $D_{S1}^{}$ or $D_{S2}^{}$, the storage of a spin-wave in $|+ \rangle$ or $|- \rangle$ mode is herald. Thus, the FPGA will send out a feed-forward signal to stop the write process. After a time delay ${t_D}$, a ${\sigma ^ - }$-polarized read laser pulse that is resonant on the $|s \rangle \to |{{e_1}} \rangle$ transition and counter-propagates with the write beam is applied to convert the spin wave $|+ \rangle$ ($|- \rangle$) into a ${\sigma ^ - }$(${\sigma ^ + }$) -polarized anti-Stokes photon. According to the phase-matching condition ${\boldsymbol{k_W}} - {\boldsymbol{k_S}} = {\boldsymbol{k_{AS}}} - {\boldsymbol{k_R}}$, where ${\boldsymbol{k_R}}$ is the wave vector of the read laser pulse, the retrieved (anti-Stokes) photon directs into the spatial mode determined by the wave-vector $\boldsymbol{k_{AS}} \approx \textrm{ - }{k_S}$, i.e., it propagates in the opposite direction to the Stokes photon. The ${\sigma ^ - }$(${\sigma ^ + }$) -polarized anti-Stokes fields are transformed into H (V) -polarized field by a λ/4 plate labelled as QW_AS in Fig. 1(a). Thus, the atom–photon state $\Phi _{\textrm{a - p}}^{}$ is transformed into the two-photon entangled state $\Phi _{\textrm{pp}}^{} = \cos \vartheta |H \rangle _S^{}|H \rangle _{AS}^{} + \sin \vartheta |V \rangle _S^{}|V \rangle _{AS}^{}$, where, $|H \rangle _{AS}^{}$ ($|V \rangle _{AS}^{}$) denotes a H- (V-) polarized anti-Stokes (readout) photon. As shown in Fig. 1(a), the channel for collecting and detecting anti-Stokes (readout) photon is similar to that for the Stokes (write-out) photon. The two outputs of PBS_AS are sent to single-photon detectors ${D_{AS1}}$ (${D_{AS2}}$). The polarization angle ${\theta _{AS}}$ of the anti-Stokes field is changed by rotating a λ/2 plate before PBS_AS. After the retrieval, a cleaning pulse with a duration of 200 ns is applied to pump the atoms into the initial level $|g \rangle$. Then, the next trial starts. If no Stokes photon is detected during the write process, the atoms will be directly pumped back into the initial level by the read and cleaning pulses. Subsequently the next trial starts. The time sequence of the above-mentioned SWPE is shown in Fig. 1(c). While, in the measurement for the cross-correlation function g_S,AS which is defined in the following Eq. (5), we will apply a write pulse following by a cleaning laser pulse and then apply a read pulse. Such cycle is repeated by a large number in the measurement. The time sequence of the measurement for g_S,AS is shown in Fig. 1(d).

 

Fig. 1. Overview of the experiment. (a) Experimental setup. ${\lambda / 4}$(${\lambda / 2}$) quarter-wave (half-wave) plate; AOM: acousto-optic modulator; PC: phase compensation module; FC: fiber collimator; PBS: polarizing beam splitter; OSFS: optical-spectrum-filter set; SMF: Single-mode fiber; D_S1, D_S2, D_AS1, D_AS2: Single photon detectors (SPCM-NIR, Excelitas Technologies); FPGA: programmable gate array; (b) Relevant atomic levels. $\circlearrowright(\circlearrowleft)$ represents ${\sigma ^-{-}-}$(${\sigma ^ + }$) –polarized write (read) laser beam; (c) Time sequence of the experimental trials for measuring the Bell parameter S ; (d) Time sequence of the experimental trials for measuring cross-correlation function g_S,AS and retrieval efficiency . W, C, R: write, cleaning and read laser pulses (all the lasers are DL Pro made in TOPTICA Photonics), ${t_D}$ denotes the time delay between the write and read pulses.

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3. Experimental results

By changing the write-laser pulse duration ${\tau _w}$, we varied the atom-light interaction time and then demonstrated the control of Stokes-photon wave-packet length. With the increase in the write-pulse duration ${\tau _w}$, we decreased write laser power and fixed the Stokes excitation probability at $\chi \approx 1\%$ in this demonstration. The Stokes photon counts ${C_S}$ are measured according to ${C_S} = {C_{{D_{S1}}}} + {C_{{D_{S2}}}}$, where ${C_{{D_{S1}}}}$(${C_{{D_{S2}}}}$) is the photon counts at ${D_{S1}}$ (${D_{S2}}$) for the polarization angle ${\theta _S} = 0$. Figure 2(a), (b) and (c) show the three examples of the measured Stokes-photon histograms when the widths of the write pulses are ${\tau _w} = 40\textrm{ }ns$, ${\tau _w} = 5\textrm{ }\mu s$ and ${\tau _w} = 50\textrm{ }\mu s$. The red lines in these figures are the temporal wave shapes of the Stokes photon without any background subtraction. The blue lines represent background noise levels. The wave-shape durations of the Stokes-photon in (a), (b) and (c) are ${\sim} 40ns$, ${\sim} 5\textrm{ }\mu s$ and ${\sim} 50\textrm{ }\mu s$, respectively, which show that the durations ${\tau _S}$ of the Stokes-photon pulses are the same as that of the applied write-laser pulses, i.e., ${\tau _S} \approx {\tau _w}$. It should be pointed out that since the background noise probability resulting from the write laser is not more than $2.5 \times {10^{ - 4}}$/write pulse, it doesn’t influence on the fixing of the excitation probability 1%.

 

Fig. 2. Three examples of the Stokes-photon temporal wave shapes. The Stokes photons are generated by applying write pulses with (a) 40 ns, (b) 5µs and (c) 50µs durations (full-width at half-maximum, FWHM). All detection events were binned into 10 ns.

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The quantum correlations between the Stokes and anti-Stokes photons can be characterized by the cross-correlation function

The Stokes, anti-Stokes and coincidence detection probabilities can be expressed as:

We then measured the background noise probability B per Stokes-photon (write-laser) pulse when the duration of the write pulse is changed from 40 ns to 50 µs. Since the duration of the Stokes pulse ${\tau _S}$ equal to that of the write-pulse duration ${\tau _w}$, we set the detection time interval of the Stokes photon to be equal to ${\tau _w}$. The background noise probability B includes the dark counts of single-photon detectors, leakages of the laser beams in the experimental system due to imperfect switch off, as well as the photon counts from the write-laser leakage. It doesn’t include the emission from the atoms. So, we measured it without the trapped atoms. When measuring the background noise probability, the write laser power was decreased with increasing the write-pulse duration. For each duration value, by setting the write laser power to an appropriate value, the excitation probability was fixed at $\chi \approx 1\%$, which is the same as the measurement in Fig. 2. The measured results are shown in Fig. 3. We find that the noise probability B increases linearly with the width ${\tau _w}$, i.e., it can be written as $B = k{\tau _w}$ with $k = \textrm{4}\textrm{.84} \times \textrm{1}{\textrm{0}^{ - 4}}/1\mu s$.

 

Fig. 3. The measured background noise probability B as a function of the width ${\tau _w}$. The fitting (red line) to the experimental data is based on $B = k{\tau _w}$ with $k = \textrm{4}\textrm{.84} \times \textrm{1}{\textrm{0}^{ - 4}}/1\mu s$.

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The retrieval efficiency is measured as $\gamma \approx {{({{P_{S,AS}} - {P_S}{P_{AS}}} )} / {[{{\eta_{AS}} \cdot ({{P_S} - B{\eta_S}} )} ]}}$. The time sequence of measurement for the retrieval efficiency is the same as that for g_S,AS in Fig. 1(d). Figure 4(a) plots the measured retrieval efficiency as a function of time delay ${t_D}$ of the read laser pulse for the case that 100 ns write pulse is applied, where the time interval for detecting Stokes photon is set to be 100 ns. In this measurement since the write-pulse duration ${\tau _w}$ is very small, the time delay ${t_D}$ corresponds to the storage time t, i.e., $t \approx {t_D}$. From Fig. 4(a), one can see that the measured retrieval efficiency decreases with the increase in storage time t due to spin-wave decoherence. The solid (blue) line in Fig. 4(a) is the fitting to the measured data according to $\gamma (t) \approx {\gamma _0}{e^{ - {{(t/{\tau _0})}^2}}}$ with ${\gamma _0} = 22\%$, which yields a memory lifetime of ${\tau _0} \approx 53\mu s$. Figure 4(b) plots the measured retrieval efficiency as a function of the width ${\tau _w}$ for a fixed storage time t of ∼1µs, where the time interval for detecting Stokes photon is set to be the width of ${\tau _w}$. As shown in Fig. 4(b), the measured retrieval efficiencies decrease with the increase in the width ${\tau _w}$. The main physical reason for this decrease explained below. Actually, when the write pulse with duration ${\tau _w}$ is applied onto the atoms, the spin wave may be generated at any time during the range of [${t_D},\textrm{ }{t_D} + {\tau _w}/\textrm{2}$]. So, the average storage time of the spin wave can be written as $t = {t_D} + {\tau _w}/\textrm{2}$. For the case that ${\tau _w} \approx {\tau _0}$, for example, ${\tau _w} \approx \mathrm{50\mu s}$, the average spin-wave storage time in Fib.4(b) is $t = {t_D} + {\tau _w}/\textrm{2} \approx \mathrm{26\mu s}$. After such an average storage time, the decoherence of the spin wave will be distinctly, which will lead to a significant decrease in the retrieval efficiency. Such a decrease can be evaluated according to the measured result in Fig. 4(a), where the retrieval efficiency will decrease by 22% after a storage time of 26µs. So, the spin-wave decay with increasing storage time t is a main reason for that the retrieval efficiency decreases with the excitation pulse duration ${\tau _w}$. Another physical reason for the decrease in Fig. 4(b) probably results from the spin-wave decay with increasing the interaction time between the write-laser pulse and the atoms. Driving by the weak write-laser pulse, the atoms, which are initially prepared in the state $|{5{S_{1/2}},F = \textrm{1}} \rangle$, will be excited to the state $|{5{P_{1/2}},F^{\prime} = \textrm{2}} \rangle$ with a small probability of ${P_e} \sim {({{\Omega _w}{\tau_w}} )^2}$ [53,54], where, ${\Omega _w}$ denotes the Rabi frequency of the write-laser beam. The population in the excited state will decay to $|{5{S_{1/2}},F = \textrm{1}} \rangle$ via spontaneous emissions, which will induce atomic random motion and then lead to a spin-wave decay with increasing the duration ${\tau _w}$. Due to such spin-wave decay, the retrieval efficiency decreases with the duration ${\tau _w}$. To precisely understand the relationship between the retrieval efficiency and the excitation pulse duration, and the possible decoherence factor, further studies are needed.

 

Fig. 4. (a) Measured retrieval efficiency $\gamma$ as a function of storage time t (temporal delay between the read and write laser pulses), where the write-pulse duration is set to a fixed value of ${\tau _w} = 100ns$. (b) Measured retrieval efficiency $\gamma$ as a function of write-pulse width ${\tau _w}$, where the time delay is set to a fixed value ${t_D}$=1µs. In both measurements, the excitation probability is set to the fixed value $\chi \approx {1\%}$.

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According to the Eq. (1), we measured the cross-correlation function g_S,AS as a function of the width ${\tau _w}$ for a fixed excitation probability $\chi \approx 1\%$ per trial (write pulse) and the time delay between the read and write pulses ${t_D}\textrm{ = 1}\mu \textrm{s}$. In the measurement, the time interval for detecting Stokes photon is set to be ${\tau _w}$ . As shown in Fig. 5, the value of g_S,AS decrease with the increase in the width ${\tau _w}$. We attribute such decrease to the increase in the background noise B and the decrease in the retrieval efficiency $\gamma$ with the width ${\tau _w}$. This view point can be explained by the following theoretical calculations. According to the expressions of ${P_{S,AS}}$, ${P_S}$ and ${P_{AS}}$, we rewrite the quantum correlation g_S,AS as:

 

Fig. 5. The measured cross-correlation function g_S,AS as a function of the width ${\tau _w}$. In the measurement, the time interval for detecting Stokes photon is set to be the corresponding value of ${\tau _w}$. The black solid curve is the fitting to the measured data based on Eq. (5) with the experimental parameters of $\chi \approx {1\%}$ and ${t_D}$=1µs. In the fitting, the branching ratio is set to be $\xi = 0.27$.

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The quality of the spin-wave-photon entanglement can be described by the Clauser–Horne–Shimony–Holt Bell parameter S, which is written as:

 

Fig. 6. The measured Bell parameter S as a function of the width ${\tau _w}$. The red solid curve is the fitting to the measured data based on Eq. (7) with the parameter ${V_0} \approx 9\textrm{5}.7\%$ and the values of $g_{S,\textrm{AS}}^{}({\tau _w})$ taken from the data in Fig. 5.

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4. Conclusion

Based on atom-photon entanglement generation via DLCZ scheme, we demonstrated the control of waveform length of the Stokes photon by varying light-atom interaction time in cold atoms. The Bell parameter S as the function of the duration of the Stokes photon pulse (write-pulse width) is measured, which shows that violations of the Bell inequality can be achieved for the Stokes photon pulse in the duration ranging from 40 ns to 50 µs with S= 2.64${\pm} $0.02 and S= 2.26${\pm} 0.05$ for 40 ns and 50 µs durations, respectively. To our knowledge, the 50 µs long wave packet of a single photon represents the longest wave packet of single photon non-classically correlated with or entangled with an atomic spin-wave memory. In the next works, the bandwidth of the Stokes-photon should be measured, which is important quantity in Hong-Ou-Mandel interference [55]. Another question in hybrid QWs is the wavelength matching of the interfacing photons from two disparate atomic memories. By using cascaded quantum frequency conversion [41], one can overcome this question. In our presented experiment, the dark state in the readout, which is mentioned in the experimental setup, leads to a loss in the retrieval efficiency. In the next works, different energy level without any dark states should be selected and then the loss in the retrieval efficiency will be avoided. In summary, our presented work provides a road to achieve entanglement between a single photon with highly tunable wave shape and a spin-wave memory and then benefits for hybrid quantum networks.