Abstract
Quantum interfaces (QIs) that generate entanglement between photonic and spinwave (atomic memory) qubits are basic building block for quantum repeaters. Realizing ensemblebased repeaters in practice requires quantum memory providing long lifetimes and multimode capacity. Significant progress has been achieved on these separate goals. The remaining challenge is to combine the two attributes into a single QI. Here, by establishing spatial multimode, magneticfieldinsensitive and longwavelength spinwave storage in lasercooled atoms inside a phasepassivelystabilized polarization interferometer, we constructed a multiplexed QI that stores up to three longlived spinwave qubits. Using a feedforwardcontrolled system, we demonstrated that a multiplexed QI gives rise to a 3fold increase in the atom–photon (photon–photon) entanglementgeneration probability compared with singlemode QIs. For our multiplexed QI, the measured Bell parameter is 2.51±0.01 combined with a memory lifetime of up to 1 ms. This work represents a key step forward in realizing fiberbased longdistance quantum communications.
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Introduction
Quantum repeaters (QRs)^{1} hold promise for distributing entanglement over long distances (>1000 km) via optical fibers, thereby providing a feasible path to realize longdistance quantum communications^{2,3,4} and quantum networks^{5,6}. In QRs, long distances are divided into short elementary links, with each link comprising two nodes that store quantum states^{[1–4}. For each link, entanglement between the two nodes is required to be established in a “heralded” way^{2,7}. Various physical systems^{8} such as atomic ensembles^{2,4,9} and single quantum systems, including single atoms^{10,11}, ions^{12,13} and solidstate spins^{14,15} have been proposed as nodes. The atomicensemblebased nodes, which were initially proposed in the Duan–Lukin–Cirac–Zoller (DLCZ) protocol^{4}, are formed by quantum interface (QIs) that create quantum correlations between a spin wave (SW) and a photon via spontaneous Raman emissions (SREs) induced by a write pulse^{2,16,17,18,19,20,21,22,23,24,25,26,27,28,29}. The quantum correlations created via SREs form the basis of generating entanglement between a photonic qubit and a SW qubit^{2,30,31,32,33}. Ensemblebased QIs are attractive because a large number of atoms ensure an efficient quantum memory (QM)^{2,19,20}. In an improved DLCZ scheme^{34,35}, the QR uses SW–photon entanglement (SWPE) instead of quantum correlations as nodes, thereby removing the requirement for longdistance phase stability in the original DLCZ protocol^{36}. Over the past decade, QIs that generate SWphoton (atom–photon) entanglement through SREs^{30,31,32,33,37,38,39,40,41,42} or storage of photonic entanglement^{43,44,45} in atomic ensembles have been demonstrated. With atom–photon entanglement, quantum teleportation from photon^{46,47} or matter^{48} to matter and elementary entanglement generations^{36,49} have been demonstrated.
In QRs, QMs are required to have long lifetimes to store the generated entanglement in elementary links^{2,3,7,50,51}. To achieve longlived DLCZlike QMs, the decoherence of SWs in cold atomic ensembles has been widely studied^{21,22,23,24,25,26,27,38}. Atomic motions and inhomogeneous broadening of the spin transitions were shown to cause SW dephasing. Motioninduced decoherence was suppressed either using an approximately collinear configuration^{23,24} or confining the atoms in optical lattices^{25,26,27,38}, where the approximately collinear configuration requires the angle of the Stokes propagation direction relative to the write beam to be small and a longwavelength SW to be attained^{23}. Inhomogeneousbroadeninginduced decoherence may be reduced using magneticfieldinsensitive coherences in SW storage^{23,24,25,26,27,38}. Longlived (0.1 s) and nonmultiplexed atom–photon entanglement was demonstrated in opticallattice atoms^{38}, in which the memory qubit was stored as two spatiallydistinct SWs, both associated with the \(0 \leftrightarrow 0\) magneticfieldinsensitive coherence, and the corresponding photonic qubit encoded into two arms of a Mach–Zehnder interferometer. However, to maintain maximal entanglement, the relative phase between the arms was actively stabilized to zero by coupling an auxiliary laser beam into the interferometer^{38}.
DLCZtype QRs using singlemode memory have very slow repeater rates for practical use^{2,52,53,54,55,56}. To overcome this problem, multiplexed versions of the DLCZ schemes were proposed^{2,7,54,55,56,57,58}. Multimode quantum storages of single SWs have been mainly implemented with rareearthiondoped (REID) crystals^{59,60,61,62} or cold atoms^{58,63,64,65,66,67,68,69,70,71}. With REID crystals, spinwavephoton quantum correlations in more than 10 temporal modes have been demonstrated^{59,60} via the DLCZ approach. The longest quantumcorrelation storage time was pushed to 1 ms applying spinecho operations^{59}. However, the values of the crosscorrelation function at storage times 0.5 and 1 ms are 4.2 in that experiment^{59}, which is lower than the threshold of 6 that enables the violation of the Bell inequality^{2}. With temporal multimode DLCZlike QMs employing REID crystal^{60}, Kutluer and colleagues^{33} experimentally demonstrated singlemode timebin entanglement between a SW and a photon. With further development employing more modes in this experiment^{33}, multimode entanglement in time will be generated. We also note that multimode light storage with storage times well above 1 min have been demonstrated in a REID crystal^{72}. Continuousvariable entanglement between light and a crystal has been generated in two temporal modes^{73}. With cold atoms, multimode SWPE generations^{58,65,66,67,68,69} have been experimentally demonstrated. However, the lifetimes for preserving multimode entanglement in these experiments are below 50 μs so far. Limited to this lifetime, the entanglement between two multimode QMs linked by fibers of length longer than 10 km cannot be generated in a “heralded” way^{67} (Supplementary Note 1). Thus, integrating multiplexed and longlived qubit storages in single quantum memories remains challenging but a crucial task. For example, Pu et al.^{66} demonstrated a spatially multiplexed DLCZ memory with 225 individually memory cells (modes) by dividing a cold atomic ensemble into a twodimensional (2D) array of microensembles. By storing SW modes in magneticfieldinsensitive coherence and loading the cold atoms into an optical lattice, the lifetime, which is ~30 µs in that experiment^{66}, will be greatly prolonged. However, because of the relatively small size of the latticetrapped atoms, the number of stored modes is strongly limiting. In a reported experiment^{38}, there are actually a large number of magneticfieldinsensitive SW modes that are along different directions around zaxis and overlap on the center of the latticetrapped atoms. Each of these SW modes is nonclassically correlated with a Stokes photon mode. By encoding these Stokes modes into multiplexed qubits, multiplexed atom–photon entanglement with longlived lifetimes can be generated. Unfortunately, the reported scheme^{38} to encode the memory qubit is complex and very difficult to use for multiplexing. Very recently, a spatially multiplexed DLCZtype quantum memory generating Belltype entanglement was demonstrated in cold atoms^{74}. Using a spatially resolved singlephoton camera, the work achieved largescale multimode quantum correlations between Stokes and antiStokes photons, with lifetimes up to 50 µs. Yet, live feedback retrieval has not been demonstrated due to lacks of singlephoton spatial routing in that work.
To overcome this key difficulty, we developed a polarization interferometer that encodes multiple Stokes modes correlated with magneticfieldinsensitive SWs into multiplexed photonic qubits. We used an ensemble of lasercooled ^{87}Rb atoms as DLCZtype memory and the approximately collinear configuration to suppress the motioninduced decoherence^{23}. The polarization interferometer was formed from two identical beam displacers (BDs). Three optical channels (OCs) across the polarization interferometer were built for multimode storages. The two arms of each OC, which held the H and Vpolarization modes from the BDs, were used to encode photonic qubits. The relative phase between the paired arms was passively stable^{75,76,77,78}. The atomic excitations, created by SREs, were stored as magneticfieldinsensitive SWs. We then realized a threemode QI that preserves SWPE for 1 ms.
Results and discussion
Experimental setup and analysis
The cold atomic ensemble was centered in a polarization interferometer formed by BD1 and BD2 (see Fig. 1a). The experiment relied on SREs induced by write pulses propagating along the zaxis to create correlated pairs of Stokes photons and SW excitations. To realize the multiplexed QI (MQI), we set up three optical channels (spatial modes) that go through the polarization interferometer. The three channels (labeled OC_{i = 1,2,3}) are arranged in a vertical plane with a separation of 4 mm. Each channel is prealigned with light beam. For example, the light beam in OC_{i} emitted from the ith singlemode fiber at the left site (labeled by \({{{\mathrm{SMF}}}}_{AS}^{(i)}\)) enters BD1, which splits the H (horizontally) and V (vertically)polarized components of the beam into two arms (spatial modes) of the polarization interferometer. The two spatial modes are denoted by \(A_L^{(i)}\) and \(A_R^{(i)}\), respectively; here, “A” denotes the arm, subscripts L and R distinguish the two modes, and superscript i denotes the ith OC for the two modes. The use of L and R as subscripts is based on the following fact that the two modes passes through the Gsection on the left and at the right, respectively (see Fig. 1b). Exiting from BD1, the two components of the beam parallel propagate in a horizontal plane, with the same separation of 4 mm. Hence, there are six spatial modes (\(A_\alpha ^{(i)}\) with \(\alpha = R,L;i = {{{\mathrm{1}}}}\,{{{\mathrm{to}}}}\,{{{\mathrm{3}}}}\)), which are arranged in parallel in a twodimensional array (Gsection of the array in Fig. 1b). The optical elements (Fig. 1a) including two identical lenses (L1 and L2) and two beam transformation devices (BTD1 and BTD2), are inserted in the polarization interferometer, where BTD1 (BTD2) is formed by two lenses, which shrink (expand) the beam array by factor F_{BTD} (“Methods” section). The effective multimode storages rely on strong couplings of the Stokes and retrieved photons with the atoms. To this end, we use lens L1 to focus the six modes at the center of the atoms. To ensure the multimode storages have long lifetimes, we have to store longwavelength SWs, which in turn require the angles \(\vartheta _{A_\alpha }^{(i)}\) of the six modes \(A_\alpha ^{(i)}\) relative to the write beam (zaxis) to be reduced to very small values^{23,79}. The angles are calculated from \(\vartheta _{A_\alpha }^{(i)} = \left( {\sqrt {{{{\mathrm{4}}}}(i  {{{\mathrm{2}}}})^{{{\mathrm{2}}}} + {{{\mathrm{1}}}}} } \right)B_f/{{{\mathrm{2}}}}f\) (Supplementary Note 2), where \(B_f\) denotes the beam separation of the array on lens L1, and \(f\) the focal length of L1. To reduce the values of the angles significantly, we selected \(f = {{{\mathrm{1}}}}{{{\mathrm{.425 m}}}}\) and used BTD1 to reduce the array beam separation by factor F_{BTD} = 2. After BTD1, the array propagates parallel to L1 and has a separation of \(B_f \approx {{{\mathrm{2}}}}\,{{{\mathrm{mm}}}}\). We then obtain small angles \(\big\{ {\vartheta _{A_R}^{({{{\mathrm{1}}}})} = \vartheta _{A_L}^{({{{\mathrm{1}}}})} \approx {{{\mathrm{0}}}}{{{\mathrm{.09}}}}^\circ ,\vartheta _{A_R}^{({{{\mathrm{2}}}})} = \vartheta _{A_L}^{({{{\mathrm{2}}}})} \approx {{{\mathrm{0}}}}{{{\mathrm{.04}}}}^\circ ,\vartheta _{A_R}^{({{{\mathrm{3}}}})} = \vartheta _{A_L}^{({{{\mathrm{3}}}})} \approx {{{\mathrm{0}}}}{{{\mathrm{.09}}}}^\circ } \big\}\), which correspond to lifetimes limited by the atomic motion of \(\{ {{{\mathrm{840}}}}\,\upmu {{{\mathrm{s,}}}}\,{{{\mathrm{1850}}}}\,\upmu {{{\mathrm{s,}}}}\,{{{\mathrm{840}}}}\,\upmu {{{\mathrm{s}}}}\}\) for modes \(\left\{ {A_{R,L}^{({{{\mathrm{1}}}})},A_{R,L}^{({{{\mathrm{2}}}})},A_{R,L}^{({{{\mathrm{3}}}})}} \right\}\), respectively (Supplementary Note 3). Additionally, the 1/e spot size of each array mode at the atomic center is 0.55 mm, which is much less than the atomic transverse size (2 mm). After passing through the atoms, the six crossed beams are transformed to a parallel beam array by L2. Then, the array goes through BTD2 and is expanded by factor F_{BTD }= 2. After the transformation, this array has the same beam separations and sizes (see Fig. 1b). Next, the array passes through BD2, which combines the paired arm modes into single spatial modes; for example, \(A_R^{(i)}\) and \(A_L^{(i)}\) modes are combined into single light beam in OC_{i}. Finally, with high efficiency (“Methods” section), the light beam in OC_{i} is coupled to the ith singlemode fiber on the right site (labeled \({{{\mathrm{SMF}}}}_S^{(i)}\) in Fig. 1a).
The relevant Rb atomic levels (Fig. 1c) are \( a \rangle = {{{{\mathrm{5}}}}^{{{\mathrm{2}}}}S_{1/2},F = {{{\mathrm{1}}}}} \rangle\), \( b \rangle =  {{{{\mathrm{5}}}}^{{{\mathrm{2}}}}S_{1/2},F = {{{\mathrm{2}}}}} \rangle\), \( {e_{{{\mathrm{1}}}}} \rangle =  {{{{\mathrm{5}}}}^{{{\mathrm{2}}}}P_{1/2},F^{\prime} = {{{\mathrm{1}}}}} \rangle\), and \( {e_{{{\mathrm{2}}}}} \rangle =  {{{{\mathrm{5}}}}^{{{\mathrm{2}}}}P_{1/2},F^{\prime} = {{{\mathrm{2}}}}} \rangle\). After the atoms are prepared in the Zeeman state \(\left {a,m_a =  {{{\mathrm{1}}}}} \right\rangle\) via optical pumping^{80}, we start the SWPE generation (“Methods” section). At the beginning of a trail, a write pulse of 20 MHz bluedetuned to the \( a \rangle \to  {e_{{{\mathrm{2}}}}} \rangle\) transition is applied to the atoms. This write pulse induces the Raman transition \( {a,m_a =  {{{\mathrm{1}}}}} \rangle \to  {b,m_b = {{{\mathrm{1}}}}} \rangle\) (\( {a,m_a =  {{{\mathrm{1}}}}} \rangle \to  {b,m_b =  {{{\mathrm{1}}}}} \rangle\)) via \( {e_{{{\mathrm{2}}}},m\prime = {{{\mathrm{0}}}}} \rangle\), which may emit \(\sigma ^ \)polarized (\(\sigma ^ +\)polarized) Stokes photons and create simultaneously SW excitations associated with the coherence \( {m_a =  {{{\mathrm{1}}}}} \rangle \leftrightarrow  {m_b = {{{\mathrm{1}}}}} \rangle\) (\( {m_a =  {{{\mathrm{1}}}}} \rangle \leftrightarrow  {m_b =  {{{\mathrm{1}}}}} \rangle\)) (Fig. 1c), where \( {m_a =  {{{\mathrm{1}}}}} \rangle \leftrightarrow  {m_b = {{{\mathrm{1}}}}} \rangle\) and \( {m_a =  {{{\mathrm{1}}}}} \rangle \leftrightarrow  {m_b =  {{{\mathrm{1}}}}} \rangle\) are the magneticfieldinsensitive and the magneticfieldsensitive coherences, respectively. A Stokes photon coupled to the \(A_R^{(i)}\) (\(A_L^{(i)}\)) mode and moving towards the right is denoted by \(S_R^{(i)}\) (\(S_L^{(i)}\)). In this case, one excitation is created in the SW mode \(M_R^{(i)}\) (\(M_L^{(i)}\)) defined by the wavevector \({{{\mathbf{k}}}}_{M_R}^{(i)} = {{{\mathbf{k}}}}_{{{\mathrm{w}}}}  {{{\mathbf{k}}}}_{S_R}^{(i)}\) (\({{{\mathbf{k}}}}_{M_L}^{(i)} = {{{\mathbf{k}}}}_{{{\mathrm{w}}}}  {{{\mathbf{k}}}}_{S_L}^{(i)}\)), where \({{{\mathbf{k}}}}_{{{\mathrm{w}}}}\)denotes the wavevector of the write pulse, and \({{{\mathbf{k}}}}_{S_R}^{(i)}\) (\({{{\mathbf{k}}}}_{S_L}^{(i)}\)) that of the Stokes photon \(S_R^{(i)}\) (\(S_L^{(i)}\)). In Fig. 1a, the \(\sigma ^ \)polarized \(S_R^{(i)}\) (\(S_L^{(i)}\)) photons are transformed into H (V)polarized photons by the λ/4 plate labeled QW_{1S} (QW_{2S}). After BD2, the H (V)polarized \(S_R^{(i)}\) and \(S_L^{(i)}\) modes are combined to form a Stokes qubit \(S^{(i)}\) and then are coupled to \({{{\mathrm{SMF}}}}_S^{(i)}\). In addition, the corresponding excitations in the \(M_R^{(i)}\) and \(M_L^{(i)}\)modes are stored as magneticfieldinsensitive SWs, which represent the ith atomic qubit. In a single channel, for example, in the ith channel, the joint state of the atom–photon system may be written as^{31} \(\rho _{ap}^{i{{{\mathrm{  th}}}}} =  {{{\mathrm{0}}}} \rangle \langle {{{\mathrm{0}}}}  + \chi _i {{\Phi}_{{{{\mathrm{a  p}}}}}^{i{{{\mathrm{  th}}}}}} \rangle \langle {{\Phi}_{{{{\mathrm{a  p}}}}}^{i{{{\mathrm{  th}}}}}} \), where \(\left {{{\mathrm{0}}}} \right\rangle\) denotes the vacuum, \(\chi _i\)(\(\ll 1\)) represents the probability of creating the \( {{\Phi}_{{{{\mathrm{a  p}}}}}^{i{{{\mathrm{  th}}}}}} \rangle\) state in each trial, \({\Phi}_{{{{\mathrm{a  p}}}}}^{i{{{\mathrm{  th}}}}}\)=\(\left H \right\rangle _S^{(i)}\)\(\left {M_R} \right\rangle _{{{{\mathrm{MFI}}}}}^{(i)}\) + e^{iφi}\(\left V \right\rangle _S^{(i)}\)\(\left {M_L} \right\rangle _{{{{\mathrm{MFI}}}}}^{(i)}\) denotes entanglement between the ith atomic and photonic qubits. Here, \(\left H \right\rangle _S^{(i)}\) (\(\left V \right\rangle _S^{(i)}\)) denotes the H (V)polarized Stokes photon of the qubit \(S^{(i)}\), \(\left {M_R} \right\rangle _{{{{\mathrm{MFI}}}}}^{(i)}\) (\(\left {M_L} \right\rangle _{{{{\mathrm{MFI}}}}}^{(i)}\)) a single SW excitation associated with the magneticfieldinsensitive (MFI) coherence and stored in the modes \(M_R^{(i)}\) (\(M_L^{(i)}\)), and \(\varphi _i\) the phase difference between the \(S_R^{(i)}\) and \(S_L^{(i)}\) fields. If the \(S_R^{(i)}\) (\(S_L^{(i)}\)) photon is \(\sigma ^ +\)polarized, the corresponding excitation in the \(M_R^{(i)}\) (\(M_L^{(i)}\)) mode is stored as the magneticfieldsensitive SW and decays rapidly^{81}. However, these photons are abandoned because they are excluded from collections (“Methods” section).
Returning to the entangled state \({\Phi}_{{{{\mathrm{a  p}}}}}^{i{{{\mathrm{  th}}}}}\), the qubit \(S^{(i)}\) is guided into the ith polarizationbeam splitter (\({{{\mathrm{PBS}}}}_S^{(i)}\)) after the \({{{\mathrm{SMF}}}}_S^{(i)}\). The two outputs of the \({{{\mathrm{PBS}}}}_S^{(i)}\) are sent to singlephoton detectors \(D_{S_{{{\mathrm{1}}}}}^{(i)}\) and \(D_{S_{{{\mathrm{2}}}}}^{(i)}\). The polarization angle of qubit \(S^{(i)}\), denoted by \(\theta _S^{(i)}\) may be changed by rotating the λ/2plate before the \({{{\mathrm{PBS}}}}_S^{(i)}\). Here, we set \(\theta _S^{({{{\mathrm{1}}}})} = \theta _S^{({{{\mathrm{2}}}})} = \theta _S^{({{{\mathrm{3}}}})} = \theta _S\) Once a photon is detected by \(D_{S_R}^{(i)}\) (\(D_{S_L}^{(i)}\)), a magneticfieldinsensitive excitation, which is stored in the mode \(M_R^{(i)}\) or \(M_L^{(i)}\) (\(M_R^{(i)}\)or \(M_L^{(i)}\)), is heralded. After a storage time t, we apply a read pulse that counterpropagates with the write beam to convert the magneticfieldinsensitive SW excitation \( {M_R} \rangle _{}^{(i)}\) (\( {M_L} \rangle _{}^{(i)}\)) into an antiStokes photon \(AS_R^{(i)}\)(\(AS_L^{(i)}\)). The retrieved photon \(AS_R^{(i)}\) (\(AS_L^{(i)}\)) is emitted into the spatial mode determined by the wavevector constraint \({{{\mathbf{k}}}}_{AS_R}^{(i)} \approx  {{{\mathbf{k}}}}_{S_R}^{(i)}\) (\({{{\mathbf{k}}}}_{AS_L}^{(i)} \approx  {{{\mathbf{k}}}}_{S_L}^{(i)}\)); i.e., it propagates in arm \(A_R^{(i)}\) (\(A_L^{(i)}\)) in the opposite direction to the \(S_R^{(i)}\) (\(S_L^{(i)}\)) photon. The \(AS_R^{(i)}\) (\(AS_L^{(i)}\)) photon is \(\sigma ^ +\)polarized and transformed into the H (V)polarized photon by a λ/4 plate labeled QW_{1AS} (QW_{2AS}). After BD1, the \(AS_R^{(i)}\) and \(AS_L^{(i)}\) fields are combined to form a polarization qubit (\(AS^{(i)}\)). Thus, the atom–photon state \({\Phi}_{{{{\mathrm{a  p}}}}}^{i{{{\mathrm{  th}}}}}\) is transformed into the twophoton entangled state, \({\Phi}_{{{{\mathrm{pp}}}}}^{i{{{\mathrm{  th}}}}} = \left H \right\rangle _S^{(i)}\left H \right\rangle _{AS}^{(i)} + e^{i(\varphi _i + \psi _i)}\left V \right\rangle _S^{(i)}\left V \right\rangle _{AS}^{(i)}\), where \(\psi _i\) denotes the phase difference between the antiStokes fields in arms \(A_R^{(i)}\) and \(A_L^{(i)}\) before they overlap at BD1. Using the phase compensator in the OC_{i} (see left site of Fig. 1), we set the phase difference \(\varphi _i + \psi _i\) to zero. We next describe how the spin waves in our experiment are independently stored, i.e., the orthogonality of the two adjacent spinwave modes, that enables us to avoid entanglement degradations arising from cross talk between the adjacent spin waves. As pointed out in a previous paper^{82}, independent storage requires the angular separation \(\theta _d\) of adjacent spinwave modes to be of order ~λ/w, where \(\lambda\)(\(w\)) is the wavelength (spot size) of the Stokes (antiStokes) modes. The actual angular separation in our experiment is >0.08°, i.e., \({{{\mathrm{1}}}}{{{\mathrm{.4}}}} \times {{{\mathrm{10}}}}^{{{{\mathrm{  3}}}}}\) (Supplementary note 2), which is in agreement with the calculated result of \(\theta _d = \lambda /w \approx {{{\mathrm{1}}}}{{{\mathrm{.4}}}} \times {{{\mathrm{10}}}}^{{{{\mathrm{  3}}}}}\) using \(\lambda = {{{\mathrm{795 nm}}}}\) and \(w \approx {{{\mathrm{550}}}}\,\mu {{{\mathrm{m}}}}\). Therefore, the adjacent spinwave states (e.g., \(M_R^{(i)}\) and \(M_L^{(i)}\) or \(M_R^{(i + {{{\mathrm{1}}}})}\) states) are orthogonal in our experiment. The generation of atom–photon (photon–photon) entanglement using m = 3 storage modes, constitutes the MQI. To enable the MQI to be available for the multiplexed QR scheme^{58}, we introduced an optical switch network (OSN), which routes the retrieved photons into a specific mode (common singlemode fiber). Passing through this fiber and a λ/2plate, the qubits \(AS^{(i)}\) (i = 1 to 3) impinge on a polarizationbeam splitter, PBS_{AS}.
Experimental results
To show that the MQI provides longlived SW storage, we examined the dependence of retrieval efficiency on the storage time t. The retrieval efficiency of the mmode MQI is measured as \(\gamma ^{(m)} = \mathop {\sum}\nolimits_{i = {{{\mathrm{1}}}}}^m {P_{S,{{{\mathrm{ }}}}AS}^{i{{{\mathrm{  th}}}}}} /\left( {\eta _{AS}\mathop {\sum}\nolimits_{i = {{{\mathrm{1}}}}}^m {P_S^{i{{{\mathrm{  th}}}}}} } \right)\), where \(\eta _{AS}\) denotes the detection efficiency in the antiStokes channel, \(P_{S,{{{\mathrm{ }}}}AS}^{i{{{\mathrm{  th}}}}} = P_{D_{S_{{{\mathrm{1}}}}},{{{\mathrm{ }}}}D_{AS_{{{\mathrm{1}}}}}}^{i{{{\mathrm{  th}}}}} + P_{D_{S_{{{\mathrm{2}}}}},{{{\mathrm{ }}}}D_{AS_{{{\mathrm{2}}}}}}^{i{{{\mathrm{  th}}}}}\)the Stokes–antiStokes coincidence probability, \(P_{D_{S_{{{\mathrm{1}}}}},{{{\mathrm{ }}}}D_{AS_{{{\mathrm{1}}}}}}^{i{{{\mathrm{  th}}}}}\) (\(P_{D_{S_{{{\mathrm{2}}}}},{{{\mathrm{ }}}}D_{AS_{{{\mathrm{2}}}}}}^{i{{{\mathrm{  th}}}}}\)) the probability of detecting a coincidence between the detectors \(D_{S_{{{\mathrm{1}}}}}^{(i)}\) (\(D_{S_{{{\mathrm{2}}}}}^{(i)}\)) and \(D_{AS_{{{\mathrm{1}}}}}\) (\(D_{AS_{{{\mathrm{2}}}}}\)),\(P_S^{i{{{\mathrm{  th}}}}} = P_{D_{S_{{{\mathrm{1}}}}}}^{i{{{\mathrm{  th}}}}} + P_{D_{S_{{{\mathrm{2}}}}}}^{i{{{\mathrm{  th}}}}}\)the Stokesdetection probability, and \(P_{D_{S_{{{\mathrm{1}}}}}}^{i{{{\mathrm{  th}}}}}\) (\(P_{D_{S_{{{\mathrm{2}}}}}}^{i{{{\mathrm{  th}}}}}\)) the probability of detecting a photon at \(D_{S_{{{\mathrm{1}}}}}^{(i)}\) (\(D_{S_{{{\mathrm{2}}}}}^{(i)}\)). Both \(P_{S,{{{\mathrm{ }}}}AS}^{i{{{\mathrm{  th}}}}}\) and \(P_S^{i{{{\mathrm{  th}}}}}\)are measured for \(\theta _S = \theta _{AS} = {{{\mathrm{0}}}}^\circ\), and \(\theta _{AS}\) is the polarization angle of the \(AS^{(i)}\) qubits, which is set by the λ/2 plate before PBS_{AS}. The measured retrieval efficiency of the MQI based on storages of the three SW qubits are shown in Fig. 2 (black circles). The fitting function (solid red curve) based on \(\gamma ^{(m = {{{\mathrm{3}}}})}(t) = \gamma _{{{\mathrm{0}}}}e^{  t/\tau _{{{\mathrm{0}}}}}\) yields a zerodelay retrieval efficiency \(\gamma _{{{\mathrm{0}}}} \approx {{{\mathrm{15}}}}\% {{{\mathrm{ }}}}\)and 1/e storage time \(\tau _{{{\mathrm{0}}}} \approx {{{\mathrm{870}}}}\,{{{\mathrm{\mu}}}} {{{\mathrm{s}}}}\). This lifetime is consistent with the average lifetime over the three SW qubits (Supplementary note 4). The double excitations lead to errors in the Stokes–antiStokes coincidences in a write trial^{2}. To avoid double excitations, we kept the excitation probabilities \(\chi _i\) (\(i = 1\,{{{\mathrm{to}}}}\,{{{\mathrm{3}}}}\)) to a low level, which is achieved by manipulating the write laser pulse. We evaluated the value of the excitation probability\(\chi _i\) from the Stokesphoton generation rate \(C_S^{i  {{{\mathrm{th}}}}} = rP_{_S}^{i  {{{\mathrm{t}}}}h} = r\chi _i\eta _S^{i  {{{\mathrm{t}}}}h}\), where \(r = 8 \times 10^4\) is the repetition rate for storage time \(t = 1\,\mu {{{\mathrm{s}}}}\) (see “Methods” section for details), and \(\eta _{_S}^{i  {{{\mathrm{th}}}}}\) is the detection efficiency of the ith Stokes channel. In the experiment described, the detection efficiencies for the 1st, 2nd, and 3rd channels are approximately identical, i.e., \(\eta _{_S}^{1  th} \approx \eta _{_S}^{2  th} \approx \eta _{_S}^{3  th} = \eta _{_S}^{} \approx 0.19\). The measured Stokesphoton generation rates of the three channels are all \(\sim 150{{{\mathrm{ s}}}}^{{{{\mathrm{  }}}}1}\). With these data, we evaluated the Stoke detection probabilities \(P_S^{1  {{{\mathrm{th}}}}} \approx P_S^{2  {{{\mathrm{th}}}}} \approx P_S^{3  {{{\mathrm{th}}}}} \approx P_S \approx 1.9 \times 10^{  3}\) and the excitation probabilities \(\chi _1 \approx \chi _2 \approx \chi _3 \approx \chi \approx 0.01\). In addition, the detected antiStokesphoton generation rate \(C_{AS}\) for each channel is ~84 s^{−1}. Using \(C_{AS} = rP_{AS}\), we obtained the antiStokesphoton probability \(P_{AS} \approx 1 \times 10^{  3}\). The probability of detecting a coincidence between the Stokes and antiStokes detectors is \(P_{S,AS} \approx 5 \times 10^{  5}\) for each channel. The quantum correlation g^{(2)} between the Stokes and antiStokes photons can be calculated using the relation^{2} \(g^{(2)} = P_{S,AS}/P{}_SP_{AS}\) and yields \(g^{(2)} \approx 26\). Decreasing the excitation probability\(\chi\) further increases significantly the quantum correlation g^{(2)}^{19,83,84}.
The quality of the mmode SWPE described by the Clauser–Horne–Shimony–Holt Bell parameter \(S^{(m)}\)^{58} is written
with the correlation function \(E^{(m)}(\theta _S,\theta _{AS})\) defined by
where, for example, \(C_{D_{S_{{{\mathrm{1}}}}},D_{AS_{{{\mathrm{1}}}}}}^{i{{{\mathrm{  th}}}}}(\theta _S,\theta _{AS})\) (\(C_{D_{S_{{{\mathrm{2}}}}},D_{AS_{{{\mathrm{2}}}}}}^{i{{{\mathrm{  th}}}}}(\theta _S,\theta _{AS})\)) denotes the coincidence counts between detectors \(D_{S_{{{\mathrm{1}}}}}^{(i)}\) (\(D_{S_{{{\mathrm{2}}}}}^{(i)}\)) and \(D_{AS_{{{\mathrm{1}}}}}\) (\(D_{AS_{{{\mathrm{2}}}}}\)) for the polarization angles \(\theta _S\)and \(\theta _{AS}\). In the \(S^{(m)}\)measurement, we used the canonical settings \(\theta _S = {{{\mathrm{0}}}}^\circ\), \(\theta {\prime}_S = {{{\mathrm{45}}}}^\circ\), \(\theta _{AS} = {{{\mathrm{22}}}}{{{\mathrm{.5}}}}^\circ\), and \(\theta {\prime}_{AS} = {{{\mathrm{67}}}}{{{\mathrm{.5}}}}^\circ\). To demonstrate that our threemode MQI preserves entanglement over a long duration, we measured the decay of the parameter \(S^{(m = {{{\mathrm{3}}}})}\) for various storage times t (blue squares in Fig. 3). At t = 1 ms, \(S^{(m = {{{\mathrm{3}}}})} = {{{\mathrm{2}}}}{{{\mathrm{.07}}}} \pm {{{\mathrm{0}}}}{{{\mathrm{.02}}}}\), which violates the Bell inequality by 3.5 standard deviations.
The quality of the photon–photon (atom–photon) entanglement generated from the mmode MQI can also be characterized by the fidelity, given by \(F^{(m)} = {{{\mathrm{Tr}}}}\left( {\sqrt {\sqrt {\rho _r^{(m)}} \rho _d\sqrt {\rho _r^{(m)}} } } \right)^2\), where \(\rho _r^{(m)}\) (\(\rho _d\)) denotes the reconstructed (ideal) density matrix of the twophoton entangled state. From measurements of the Stokes–antiStokes coincidences for \(t = 1\,\mu {{{\mathrm{s}}}}\) and \(\chi = 1\%\), we reconstructed \(\rho _r^{(m = 3)}\) (Supplementary Note 6), which yields \(F^{(m = 3)} = 90.4 \pm 1.6\%\). We also reconstructed the density matrices \(\rho _r^{i  {{{\mathrm{th}}}}}\) of the entangled states \({\Phi}_{{{{\mathrm{pp}}}}}^{i  {{{\mathrm{th}}}}}\), which yields fidelities \(F^{(1  {{{\mathrm{th}}}})} = 88.6 \pm 1.13\%\), \(F^{(2  {{{\mathrm{th}}}})} = 92.0 \pm 1.5\%\), and \(F^{(3  {{{\mathrm{th}}}})} = 88.4 \pm 0.85\%\) (Supplementary Note 6).
For a multiplexed atom–photon entanglement interface, demonstrating that, compared with the nonmultiplexed interface, this interface has the capacity to enhance the probability of generating entangled atom–photon (photon–photon) pairs is important. The probability of generating an atom–photon (photon–photon) entangled pair corresponds to the total Stokes detection (Stokes–antiStokes coincidence) probability \(P_S^{(m)} = \mathop {\sum}\nolimits_{i = {{{\mathrm{1}}}}}^m {P_S^{i{{{\mathrm{  th}}}}}}\) (\(P_{S,AS}^{(m)} = \mathop {\sum}\nolimits_{i = {{{\mathrm{1}}}}}^m {P_{S,AS}^{i{{{\mathrm{  th}}}}}}\)). The blue squares in Fig. 4a (Fig. 4b) are the measured values of \(P_S^{(m)}\) (\(P_{S,AS}^{(m)}\)) as a function of m and show that the MQI gives rise to a threefold increase in the atom–photon (photon–photon) entanglementgeneration probability compared with singlemode QIs. The average OSN efficiency is written as \(\eta _{{{{\mathrm{OSN}}}}}^{} = {\Gamma}_A\left( {1  (m  1)\Upsilon _A/2} \right)\) (“Methods” section), where \({\Gamma}_A\) denotes the diffraction efficiency of the acoustooptic modulator, and \(\Upsilon _A\) the transmission loss for the case that modulator is not driven. For our experiment, \({\Gamma}_A \approx 82\%\) and \(\Upsilon _A \approx 0.02\) is small; with m = 3, we have \(\eta _{{{{\mathrm{OSN}}}}}^{} \approx 0.8\). Considering the efficiency of OSN, the MQI increases this probability by a factor of \(m \times \eta _{{{{\mathrm{SW}}}}} = {{{\mathrm{2}}}}{{{\mathrm{.4}}}}\) compared with the singlemode QI without OSN. In our experiment, the limitation in scaling up to more than three modes is because transverse sizes of the optical elements are small, specifically, beam displacers in the polarization interferometer. When the transverse sizes of the beam displacers and other optical elements are increased, the mode number may be scaled up. For example, when the transverse sizes of the optical elements are increased to \(26 \times 24{{{\mathrm{ mm}}}}^2\), the number of stored qubits (modes) may be extended to 12. With an increased mode number, the larger angles between the zaxis (write beam direction) and the Stokes fields lead to decreases in lifetimes. The evaluated average lifetime of 12 storage modes is about 510 μs. However, this decrease is not a fundamental limitation and may be overcome by trapping the atoms in an optical lattice.
Conclusion
The current work exploited the spatially multiplexed correlations between Stokes photon and SWs and the feedforward controlled retrieval, which were used in our previous work^{58}, to achieve a threefold increase in the generation rate of Stokes–antiStokes pairs, compared with the nonmultiplexed case. In contrast to the previous work, in which each memory qubit is stored as two spin waves with one being associated with the magneticfieldinsensitive Zeeman coherence and the another with magneticfieldsensitive coherence, the current work used a polarization interferometer to encode multiplexed photonic qubits onto multiple Stokes modes that are only correlated with magneticfield insensitive SWs of long wavelength. Therefore, the entanglement storage lifetimes reach up to 1 ms, which is 20 times higher than our previous result of 50 µs^{58}. However, to apply the present MQI in QR applications, its performance needs to be improved. Millisecond lifetimes are mainly limited by motional dephasing (Supplementary Note 5) but can be prolonged to 0.2 s by trapping the atoms in an optical lattice^{26,27}. In the experiment described, the number of modes is six, which can be increased by extending the apertures of the optical devices. The multimode number may be extended using multiplexing schemes with two degrees of freedom, e.g., combining a temporal multiplexing scheme^{68} with the present spatial approach. Considering an MQI that stores 12 spatial and 10 temporal SW qubits, the total number of memory qubits is \(N_m = 120\). To minimize transmission losses in fibers, the Stokes photons (795 nm) has to be converted into photons in the telecommunications band^{49,85,86,87}. The lower retrieval efficiency (15%) can be increased using high opticaldepth cold atoms^{75,76,88,89} or coupling the atoms with an optical cavity^{24,39} to enhance the collective interference (Supplementary Note 7). The measured average entanglement fidelity of the three channels is ~90%. We attributed the 10% decrease in entanglement fidelity to imperfect phase compensations of the OSN and the relatively lowquantum correlations (\(g^{(2)} \approx 26\)) between the Stokes and antiStokes photons (Supplementary Note 8). In future work, we need to enhance the entanglement fidelity by improving phase compensations of the OSN and increasing g^{(2)} (ref. ^{31}). Our present experiment shows a promising way to store multiplexed memory qubits as magneticfieldinsensitive SWs, thereby allowing a realization of an entanglement QI capable of storing a large number of longlived memory qubits in latticetrapped atoms. Such capabilities would be of benefit in QRbased longdistance quantum communications.
Methods
Experimental method
The experiment is performed in cycles. In each circle, the duration for the preparation of cold atoms and that for the experimental run of the SWPE generation are 42 ms and 8 ms, respectively, corresponding to a 20Hz cycle frequency. During preparations, >10^{8} atoms of ^{87}Rb are trapped in a twodimension magnetooptical trap (MOT) for 41.5 ms and further cooled by Sisyphus cooling for 0.5 ms. The cloud of cold atoms has a size of ~5 × 2 × 2 mm^{3}, a temperature of ~100 μK and an optical density of about 14. At the end of each preparation stage, a bias magnetic field of B_{0} = 4 G is applied along the zaxis (see Fig. 1a) and the atoms are prepared into the initial level \(\left {5^2S_{1/2},F = 1,m =  1} \right\rangle\) via optical pumping^{79}. After the preparation stage, the 8ms experimental run containing a large number of SWPEgeneration trials starts. At the beginning of a trial, a write pulse of 70 ns duration is applied on the atomic ensemble to generate correlated pairs of Stokes photons and SW excitations. The detection events at the Stokes detectors \(D_S^{(i = 1,2,3)}\) (i.e., \(D_{S_1}^{(i = 1,2,3)}\) and \(D_{S_2}^{(i = 1,2,3)}\) in Fig. 1a) are analyzed using a field programmable gate array (FPGA). As soon as a Stokes photon qubit is detected by any of these detectors, for example, \(D_S^{(i)}\) (\(D_{S_1}^{(i)}\) or \(D_{S_2}^{(i)}\)), the atom–photon entanglement is generated in the ith channel and the FPGA sends out a feedforward signal to stop the write processes. After a storage time t, a read laser pulse of 70 ns duration is applied to the atoms to convert the SW qubit into an antiStokes photon qubit \(AS^{(i)}\). At the same time, the FPGA delivers a feedforward signal to switch the optical switching network (OSN) and then the antiStokes photon qubit is routed into the common singlemode fiber (CSMF)^{58}. After a 1300 ns interval, a cleaning pulse of 200 ns duration is applied to pump the atoms into the initial level \(\left {5^2S_{1/2},F = 1,m =  1} \right\rangle\). Then, the next SWPEgeneration trial starts. If no Stokes photon is detected during the write pulse, the atoms are pumped directly back into the initial level by the read and cleaning pulses for the subsequent trial to start. The delay between the two adjacent write pulses for a storage time \(t \approx {{{\mathrm{1}}}}\,\upmu {{{\mathrm{s}}}}\) is 2000 ns. Therefore, the 8 ms experimental run contains ~4000 experimental trials. Considering that 1s experiment contains 20 cycles, the repetition rate of the SWPEgeneration trail is \(r = 8 \times 10^4\).
In the center of the atoms, the diameters of the write and read light beams are both ~1.1 mm; however, the power of the beams are ~100 μW and ~1 mW, respectively. The read light field is on resonance with transition \(\left b \right\rangle \to \left {e_1} \right\rangle\). At the right site in Fig. 1a, the BD2 perfectly combines both \(A_R^{(i)}\) (Hpolarization) and \(A_L^{(i)}\) (Vpolarization) modes into a single light beam propagating in channel OC_{i} and then the light beam is effectively coupled into the singlemode fiber \({{{\mathrm{SMF}}}}_S^{(i)}\). The measured coupling efficiencies for modes \(\left\{ {A_R^{(1)},A_L^{(1)}} \right\}\), \(\left\{ {A_R^{(2)},A_L^{(2)}} \right\}\), and \(\left\{ {A_R^{(3)},A_L^{(3)}} \right\}\) are {70.5%, 71.0%}, {70.6%, 71.5%}, and {70.8%, 70%}, respectively. For blocking the write (read) beam in the Stokes and antiStokes channels, we placed an opticalspectrumfilter set (OSFS) before each polarizationbeam splitter (PBS). Each OSFS comprises four Fabry–Pérot etalons, which attenuate the write (read) beam by factor \(\sim 2.7 \times 10^{  9}\) (\(\sim 3.7 \times 10^{  9}\)) and transmit the Stokes ((antiStokes) fields with a transmission efficiency of ~65%. Additionally, in the Stokes (antiStokes) detection channel, the spatial separation of the Stokes (antiStokes fields) from the strong write (read) beam provides an attenuation of \(\sim 10^{  4}\) for the write (read) beam. In the present experiment, we measured the uncorrelated noise probability in the antiStokes mode, which is \(p_n = 9 \times 10^{  5}\) per read pulse (70 ns). This noise mainly results from the leakage of the read beam into the antiStokes detection channel.
From the SRE induced by the write pulse, if the Stokes photon \(S_R^{(i)}\)(\(S_L^{(i)}\)) is \(\sigma ^ +\)polarization, it is transformed into a V (H)polarized photon by the λ/4 waveplate QW_{1S} (QW_{2S}) (see Fig. 1a) and then removed from OC_{i} by BD2.
The total detection efficiency for detecting the Stokes photons at the detector \(D_{S_1}\) (\(D_{S_2}\)), denoted \(\eta _S\), includes the transmission of the BS2 (η_{BS} = 90%), the coupling efficiency of the singlemode fiber SMF_{S} ~70.7%, the transmission of ~65% for the OSFS, the transmission of an multimode fiber (MMF) η_{MMF} = 92%, and the quantum efficiency 0.5 of the detector \(D_{S_1}\)(\(D_{S_2}\)), respectively. The overall detection efficiency is \(\eta _S = \eta _{{{{\mathrm{BS}}}}} \times \eta _{{{{\mathrm{SMF}}}}_S} \times \eta _{{{{\mathrm{MMF}}}}} \times \eta _D \approx 19\%\).
Similarly, the total detection efficiency for detecting the antiStokes photons at the detector \(D_{{{{\mathrm{AS}}}}_1}\)(\(D_{{{{\mathrm{AS}}}}_2}\)), denoted by \(\eta _{{{{\mathrm{AS}}}}}\), includes the transmissions of the BS1 (η_{BS }= 90%), the coupling efficiency ~70.7% of the singlemode fiber SMF_{AS}, the optical switching network (OSN) efficiency of 80%, the coupling efficiency of the CSMF (η_{CSMF} = 80%), the transmission of the OSFS η_{OSFS} = 65%, and the transmission of the MMF η_{MMF} = 92%, the quantum efficiency 0.5 of the detector \(D_{{{{\mathrm{AS}}}}_1}\)(\(D_{{{{\mathrm{AS}}}}_2}\)), respectively. The total detection efficiency is \(\eta _{{{{\mathrm{AS}}}}} = \eta _{{{{\mathrm{BS}}}}} \times \eta _{{{{\mathrm{SMF}}}}_{{{{\mathrm{AS}}}}}} \times \eta _{{{{\mathrm{OSN}}}}} \times \eta _{{{{\mathrm{CSMF}}}}} \times \eta _{{{{\mathrm{OSFS}}}}} \times \eta _{{{{\mathrm{MMF}}}}} \times \eta _D \approx 12\%\).
All error bars in the experimental data represent a ± 1 standard deviation, which is estimated from the Poissonian detection statistics using Monte Carlo simulations.
Propagating directions of the retrieved photons
Because of the collective interference of atoms, the retrieved photon \(AS_L^{(i)}\) (\(AS_R^{(i)}\)) is emitted into a welldefined spatial mode given by the phase matching condition, \({{{\mathbf{k}}}}_{{{{\mathrm{AS}}}}_R}^{(i)} = {{{\mathbf{k}}}}_{M_R}^{(i)} + {{{\mathbf{k}}}}_{{{\mathrm{r}}}} = {{{\mathbf{k}}}}_{{{\mathrm{w}}}}  {{{\mathbf{k}}}}_{S_R}^{(i)} + {{{\mathbf{k}}}}_{{{\mathrm{r}}}}\) (\({{{\mathbf{k}}}}_{{{{\mathrm{AS}}}}_L}^{(i)} = {{{\mathbf{k}}}}_{M_L}^{(i)} + {{{\mathbf{k}}}}_{{{\mathrm{r}}}} = {{{\mathbf{k}}}}_{{{\mathrm{w}}}}  {{{\mathbf{k}}}}_{S_L}^{(i)} + {{{\mathbf{k}}}}_{{{\mathrm{r}}}}\)), where \({{{\mathbf{k}}}}_{{{\mathrm{r}}}}\) (\({{{\mathbf{k}}}}_{{{\mathrm{w}}}}\)) is the wavevector of the read (write) beam. As the write and read light beams counterpropagate through the atoms, we have \({{{\mathbf{k}}}}_{{{\mathrm{r}}}} \approx  {{{\mathbf{k}}}}_{{{\mathrm{w}}}}\) and hence \({{{\mathbf{k}}}}_{{{{\mathrm{AS}}}}_R}^{(i)} \approx  {{{\mathbf{k}}}}_{S_R}^{(i)}\) (\({{{\mathbf{k}}}}_{AS_L}^{(i)} \approx  {{{\mathbf{k}}}}_{S_L}^{(i)}\)).
Phase compensators (PCs)
When the H and Vpolarized light fields propagate in the two paired arms in the BDs, the difference in the refractive indices of the two arms leads to a phase shift between the two light fields. We use PCs to overcome this problem. For example, to eliminate the phase shift due to the BDs in the ith channel (\({{{\mathrm{OC}}}}_i\)), we place the phase compensator \({{{\mathrm{PC}}}}_i\) between BD1 and the ith singlemode fiber at the left site. Each phase compensator is a combination of the λ/4, λ/2, and λ/4 waveplates^{58}. By rotating the λ/2 waveplate in the \({{{\mathrm{PC}}}}_i\), we eliminate the phase shift caused by the BDs \(\left( {\varphi _i + {\it{\uppsi }}_i} \right)\). We also insert a PC before each PBS (see Fig. 1a) to eliminate phase shifts caused by optical elements such as the singlemode fibers and acousticoptical modulators (AOMs).
Optical switching network
The optical switching network is composed of m AOMs, which are placed in a straight line. To understand the optical circuit of OSN, Fig. 1a shows a network with three AOMs. The ith AOM is used to route the retrieved photon from the ith channel into the line. Then, the photon goes through (mi) AOMs that do not work and is coupled to the CSMF. The OSN efficiency of the ith channel, which corresponds to the transmission of the photon from ith channel to the fore of the CSMF, is \(\eta _{{{{\mathrm{OSN}}}}}^{i  th} = {\it{{\Gamma}}}_A\left( {1  {\it{\Upsilon }}_A} \right)^{m  i}\), where \({\it{{\Gamma}}}_A\) is the diffraction efficiency of the AOM, and \({\it{\Upsilon }}_A\) the transmission loss for the case that modulator is not driven. For a small \({\it{\Upsilon }}_A\), the OSN transmission efficiency for the ith channel is written \(\eta _{{{{\mathrm{OSN}}}}}^{i  th} = {\it{{\Gamma}}}_A\left( {1  (m  i){\it{\Upsilon }}_A} \right)\). The average OSN efficiency is \(\eta _{{{{\mathrm{OSN}}}}} = \frac{1}{m}{\sum} {_{i = 1}^m} {{{\mathrm{ }}}}\eta _{{{{\mathrm{OSN}}}}}^{i  th} = {\it{{\Gamma}}}_A\left( {1  \frac{1}{2}(m  1){\it{\Upsilon }}_A} \right)\), showing a linear decrease with m.
Configuration of the PI (polarization interferometer) (see Fig. 5a) and cross sections of the array at different sites (see Fig. 5b). Beam transformation devices (BTDs)
In the experiment, we used two BTDs (BTD1 and BTD2) to transform the beam array. The BTD1 (BTD2) is formed by a convex and a concave lens, denoted by L_{11} (L_{22}) and L_{12} (L_{21}), respectively, with L_{11} (L_{21}) placed on the lefthand side of L_{12} (L_{22}) (see Fig. 6). The focal lengths of the lenses L_{11} (L_{22}) and L_{12} (L_{21}) were chosen so that \(F_{{{{\mathrm{L}}}}11} = f_{{{\mathrm{L}}}}(F_{{{{\mathrm{L}}}}22} = f_{{{\mathrm{L}}}})\)) and \(F_{{{{\mathrm{L}}}}_{12}} =  f_{{{\mathrm{L}}}}/F_{{{{\mathrm{BTD}}}}}\)(\(F_{{{{\mathrm{L}}}}_{21}} =  f_{{{\mathrm{L}}}}/F_{{{{\mathrm{BTD}}}}}\)), with \(f_{{{\mathrm{L}}}} \; > \; 0\). The separation between L_{11} (L_{22}) and L_{12} (L_{21}) is \(\left( {1  \frac{1}{{F_{{{{\mathrm{BTD}}}}}}}} \right)f_{{{\mathrm{L}}}}\). The ABCD matrix for BTD1 and BTD2 were calculated from
\(\left(\begin{array}{cc}1 & 0 \\ F_{{{\mathrm{BTD}}}}/f_{{{\mathrm{L}}}} & 1\end{array}\right)\left(\begin{array}{cc} 1 & (1  1/F_{{{\mathrm{BTD}}}})f_{{{\mathrm{L}}}} \\ 0 & 1\end{array}\right)\left(\begin{array}{cc} 1 & 0 \\  1/f_{{{\mathrm{L}}}} & 1\end{array} \right) = \left(\begin{array}{cc} 1/F_{{{\mathrm{BTD}}}} & (1  1/F_{{{\mathrm{BTD}}}})f_{{{\mathrm{L}}}} \\ 0 & F_{{{\mathrm{BTD}}}}\end{array}\right)\) and \(\left(\begin{array}{cc} 1 & 0 \\  1/f_{{{\mathrm{L}}}} & 1\end{array} \right)\left(\begin{array}{cc} 1 & \left(1  1/F_{{{\mathrm{BTD}}}} \right)f_{{{\mathrm{L}}}} \\ 0 & 1\end{array} \right)\!\left(\begin{array}{cc} 1 & 0 \\ F_{{{\mathrm{BTD}}}}/f_{{{\mathrm{L}}}} & 1\end{array} \right) \!=\! \left( \begin{array}{*{20}{c}} F_{{{\mathrm{BTD}}}} & \left(1  1/F_{{{\mathrm{BTD}}}}\right)f_{{{\mathrm{L}}}} \\ 0 & 1/F_{{{\mathrm{BTD}}}}\end{array}\right).\)
Therefore, when a beam array goes through the BTD1 (BTD2) from left to righthand side, it is shrunk (expanded) by factor F_{BTD}. In our current experiment, we used the lenses at hand, for which the focus lengths are \(F_{{{{\mathrm{L}}}}_{{{{\mathrm{11}}}}}}\) = 2000 mm \(F_{{{{\mathrm{L}}}}_{{{{\mathrm{22}}}}}}\) = 2000 mm, \(F_{{{{\mathrm{L}}}}_{{{{\mathrm{21}}}}}}\) = −1000 mm, and \(F_{{{{\mathrm{L}}}}_{{{{\mathrm{21}}}}}}\) = −1000 mm, which correspond to factor F_{BTD} = 2. In future work, one may increase (decrease) the factor F_{BTD} (1/F_{BTD}) by increasing (decreasing) the lens ratio \( {F_{{{{\mathrm{L}}}}_{{{{\mathrm{11}}}}}}/F_{{{{\mathrm{L}}}}_{{{{\mathrm{12}}}}}}} \) \(( {F_{{{{\mathrm{L}}}}_{{{{\mathrm{22}}}}}}/F_{{{{\mathrm{L}}}}_{{{{\mathrm{21}}}}}}} )\).
Data availability
All data from this work are available through the corresponding author upon request.
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Acknowledgements
We acknowledge funding support from Key Project of the Ministry of Science and Technology of China (Grant No. 2016YFA0301402); The National Natural Science Foundation of China (Grants: No. 11475109, No. 11974228), Program for Shanxi Province; Fund for Shanxi “1331 Project” Key Subjects Construction.
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H.W. conceived the research. H.W., Z.X.X., S.J.L., and S.Z.W. designed the experiment. S.Z.W., M.J. W., Y.F.W., T.F.M. setup the experiment with assistances from all other authors. S.Z.W., M.J.W. took the data. S.Z.W. analyzed the data. H.W. wrote the paper.
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Wang, Sz., Wang, Mj., Wen, Yf. et al. Longlived and multiplexed atomphoton entanglement interface with feedforwardcontrolled readouts. Commun Phys 4, 168 (2021). https://doi.org/10.1038/s42005021006709
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DOI: https://doi.org/10.1038/s42005021006709
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