Long-lived and multiplexed atom-photon entanglement interface with feed-forward-controlled readouts

The quantum interface (QI) that generates entanglement between photonic and spin-wave (atomic memory) qubits is a basic building block for quantum repeaters. Realizing ensemble-based repeaters in practice requires quantum memory providing long lifetime and multimode capacity. Significant progresses have been achieved on these separate goals. The remaining challenge is to combine long-lived and multimode memories into a single QI. Here, by establishing multimode, magnetic-field-insensitive and long-wavelength spin-wave storage in laser-cooled atoms that are placed inside a phase-passively-stabilized polarization interferometer, we constructed a multiplexed QI that stores up to three long-lived spin-wave qubits. Using a feed-forward-controlled system, we demonstrated that the multiplexed QI gives rise to a 3-fold increase in the atom-photon (photon-photon) entanglement-generation probability compared to single-mode QIs. The measured Bell parameter is 2.5+/-0.1 combined with a memory lifetime up to 1ms. The presented work represents a key step forward in realizing fiber-based long-distance quantum communications.

Kutluer et. al. experimentally demonstrated time-bin entanglement between a SW and a photon [32] . With more modes being used in the experiment, multimode entanglement in time will be generated with the crystal. Additionally, continuous-variable entanglement between light and a crystal has been generated in two temporal modes [67] . However, to date, the durations for preserving multimode entanglement are below 50 μs.
Limited to this lifetime, the entanglement creation between two multimode QMs linked by more than 10-km fibers can't be established in the "heralded" way (see Supplementary Material [68]). The realizations of QRs using multimode SWPE QIs require QMs to have long storage durations and multiplexed qubit storages [3,4,48,56] , more specifically, to store each of the multiplexed atomic qubits individually as a superposition of long-lived spin waves. However, that goal remained elusive.
In the present experiment, we overcome the difficulties by integrating multimode, MFI and long-wavelength spin-wave storage in a single QM system. The QM system was formed from an ensemble of laser-cooled 87 Rb atoms placed in a polarization interferometer (PI). The PI was formed using two identical beam displacers (BDs). Three optical channels (OCs) across the PI were built for multimode storages. The two arms of each OC, which correspond to the H-and V-polarization modes of the BDs, were used to encode photonic qubits. The relative phase between the paired arms was passively stable [69][70][71][72] . The six ( 32  ) spatial modes arranged in a two-dimensional array were focused at the center of the atoms with a lens. The atomic excitations, created by SREs, are stored as MFI spin waves of long wavelengths. We then realized an MQI that generated long-lived SWEP in the three channels.
The cold atomic ensemble was centered in a PI formed by BD1 and BD2 (see Fig.1a). The experiment relied on SREs induced by write pulses propagating along z-axis to create entangled pairs, each pair comprising a Stokes photon and a spin-wave excitation (cf. below). To realize the MQI, we set up three optical channels (spatial modes) that go through the PI to collect and detect both the Stokes and retrieved photons. The three channels (labeled by OCi =1,2,3 ) are arranged in a vertical plane with a separation of 4 mm. Each channel is pre-aligned with light beam. For example, the light beam in OCi emitted from the i-th single-mode fiber at  [21,73]   which represent the i-th atomic qubit. In the present experiment, the excitation probability for each OC is almost the same, i.e., 1 ≈ 2 ≈ 3 ≈ . For ≪ 1, the entangled state between the i-th atomic and photonic qubits is described as photon is + -polarized, the corresponding excitation in the 1 ( ) ( 2 ( ) ) mode is stored as the MFS spin wave and decays rapidly [75] . However, these photons are abandoned because they are excluded from the collections (see Supplementary Material [68]).
Returning to the entangled state -th and 2 ( ) before they overlap at BD1. Using the i-th phase compensator (labeled PC i in Fig. 1a), we set the phase difference + to zero. The generation of atom-photon (photon-photon) entanglement based on m=3 storage modes constitutes the MQI operation. To enable the MQI to be available for the multiplexed QR scheme [56] , we introduced an optical switch network (OSN) to route the retrieved qubits into a common single-mode-fiber [56] (CSMF). Passing through the CSMF and a /2  plate, the qubits () i T ( = 1 to 3) impinge on a polarization-beam splitter, PBS T . The two outputs of the PBS T are sent separately to detectors 1 and 2 . Then, the atoms are prepared in the initial state via optical pumping [74] . If no Stokes photon is detected during the write pulse, the atoms are pumped directly back into the initial state. Subsequently, the next trial starts.
To show that the MQI provides long-lived spin-wave storage, we examined the dependence of the retrieval efficiency on storage time t.
The retrieval efficiency of the m-mode MQI is measured as  The quality of the m-mode SWPE is described by the Clauser-Horne-Shimony-Holt (CHSH) Bell parameter ( ) [56] written as: where, for example, To demonstrate that our three-mode MQI preserves entanglement over a long duration, we measured the decay of the parameter ( =3) for various storage times t (blue squares in Fig. 3). At = 1 , ( =3) = 2.07 ± 0.02, which violates the CHSH inequality by 3.5 standard deviations. Material [68]). Their average fidelity ̄= 89.7 ± 1.2% is in agreement with the value of ( =3) . ). The blue squares (circles) dots in Fig. 4a (Fig. 4b) are the measured values of ( ) ( , ( ) ) as a function of m and show that the MQI gives rise to a three-fold increase in the atom-photon (photon-photon) entanglement-generation probability compared with single-mode QIs. Considering the imperfect OSN efficiency ≈ 0.8, which remains fixed regardless of m [55] , the MQI increases the photon-photon entanglement-generation probability by a factor of × = 2.4 compared with the single-mode QI without OSN.
We have demonstrated a three-mode MQI that preserves SWPE for 1 ms. This lifetime is 20 times longer than the best results among the multimode SWPE QIs reported to date. If the three-mode MQIs instead of single-mode MQIs are used as nodes of an elementary link, the probability of entanglement generation in the link will be increased 3-fold.
To apply the present MQI in QR applications, its performances needs to be further improved. Millisecond lifetimes are mainly limited by motional dephasing (see Supplementary Material [68]) but can be prolonged to 0.2 s by trapping the atoms in an optical lattice [25,26] . The multimode number can be increased by extending the apertures of the optical devices. The multimode capacity may be extended further using the multiplexing schemes with two or more degrees of freedom [76,77] , e.g., combining a temporal multiplexing scheme [65] with the present spatial approach.
Considering an MQI that stores 65 spatial and 10 temporal spin-wave qubits, the total number of memory qubits reaches To minimize transmission losses in the fibers, the Stokes photons at Rb transitions may be converted into photons in the telecommunications band [48,[78][79][80] . The lower retrieval efficiency (15%) can be increased using high optical-depth cold atoms [81] or coupling the atoms with an optical cavity [23,38] . We remark that the recent experiment demonstrates probabilistic entanglement generation in an elementary link over 22-km field-deployed fiber [48] , where the link uses single-mode SWPE QIs as nodes. The deterministic entanglement generation in this link [82] requires very long storage time (~150 s) [48] , which lead to very low rate (

Supplementary Material
Entanglement establishment between two remote nodes in a ''heralded'' fashion.
As explained in the main text, a DLCZ-like elementary link comprises two nodes, each being formed by a QI. One essential requirement for the QR protocols is to establish entanglement between the two nodes [2,47] .
The m-fold compared with single-mode links [56] . We emphasize, however, that this increase can be achieved only when the entanglement in the multiplexed link is established in the "heralded" manner. The reason for this is that the multimode memory at each node needs to know that successful BSMs have been achieved and in which photonic modes so that it can retrieve the excitations in the corresponding spin-wave modes and send them to a common channel via feed-forward-controlled read outs [56] . According to the above discussions, one cannot realize a probability increase in the entanglement generation between two remote multimode QMs linked by more than 10-km fibers because of the short storage lifetimes (~50 µs).
Our present three-mode QM has a lifetime of τ 0~1 for preserving entanglement. If one want to establish entanglement between the two QMs in the "heralded" manner, the fiber distance used for connecting well-defined spatial mode given by the phase matching condition,  ).

Phase compensators (PCs). When H-and V-polarization light fields
respectively propagates in the two paired arms in the BDs, the refractive index difference between the two arms will lead to a phase shift between the two light fields. We use PCs to overcome the problems. For example, for eliminating the phase shift due to the BDs in the i-th channel ( i OC ) ,we place the phase compensator PC i between BD1 and the i-th single-mode fiber at the left site. Each phase compensators is a combination of λ/4, λ/2 and λ/4 wave-plates [56] . By rotating the λ/2 wave-plate in the PC i , we can compensate the phase shift due to the BDs ( ii  + ) to zero. We also insert a PC before each PBS in the Fig.1a to eliminate phase shifts due to the optical elements such as single-mode fibers and AOMs.

The increase in the multimode number
To The ABCD matrix for the BTD1 and BTD2 can be calculated by: Thus, when a beam array goes through the BTD1 (BTD2) from the left to right side, it can be shrunk (expand) by a factor F. In our current experiment, we use the lenses at hand, whose focus lengths are

The lifetimes limited by the atomic motion.
For the storage of a single-mode spin wave, it has been pointed out the decoerhence due to the atomic motions give a limit to the storage lifetimes, which may be described by /    [2,73] , where,