Highly-efficient quantum memory for polarization qubits in a spatially-multiplexed cold atomic ensemble

Quantum memory for flying optical qubits is a key enabler for a wide range of applications in quantum information. A critical figure of merit is the overall storage and retrieval efficiency. So far, despite the recent achievements of efficient memories for light pulses, the storage of qubits has suffered from limited efficiency. Here we report on a quantum memory for polarization qubits that combines an average conditional fidelity above 99% and efficiency around 68%, thereby demonstrating a reversible qubit mapping where more information is retrieved than lost. The qubits are encoded with weak coherent states at the single-photon level and the memory is based on electromagnetically-induced transparency in an elongated laser-cooled ensemble of cesium atoms, spatially multiplexed for dual-rail storage. This implementation preserves high optical depth on both rails, without compromise between multiplexing and storage efficiency. Our work provides an efficient node for future tests of quantum network functionalities and advanced photonic circuits.

Q uantum memories enabling the storage of an input photonic qubit and its later retrieval with a fidelity beating any classical device constitute essential components in quantum communication networks and optical quantum information processing 1,2 . Over the past years, storage of optical qubits has been demonstrated in a variety of physical platforms, including individual atoms in high-finesse cavities 3 , ion-doped crystals, [4][5][6] and large ensembles of neutral atoms [7][8][9] . Quantum memories capable of storing qubits encoded into multiple degrees of freedom of light have also been achieved recently 10 .
The storage and retrieval efficiency of such devices is a stringent parameter for the envisioned applications 2,[11][12][13] and boosting this parameter has been a long-standing quest. This figure of merit is crucially important to reduce the entanglement distribution time in quantum repeater architectures and thereby develop scalable communication links 9,14,15 . It is also essential for increasing the success rate of gate operations 16 or for building up iterative quantum state engineering schemes in optical quantum circuits 17 . A memory efficiency exceeding the important 50% threshold would as well enable protocols to perform in the no-cloning regime without postselection 18 or error correction for qubit losses in linear optics quantum computation 19 . However, to date, the highest storage and retrieval efficiencies achieved for qubits, independently of the photonic degrees of freedom, are below 30% 10,[20][21][22][23][24] .
These limited values contrast with the recent progresses achieved in the demonstrations of optical memories. Ultra-high optical depths (OD) have indeed been obtained in laser-cooled elongated ensembles of neutral atoms 25,26 and then used to realize high-efficiency single-mode optical storage 27,28 based on long-lived collective excitations. However, the increase in OD, which is a strong prerequisite for large efficiency 29 , often comes at the expense of spatial multimode capacity as the atomic cloud is elongated along a direction and radially compressed. It results in a reduced transverse size that may render some strategies such as dual-rail storage arduous. Qubit storage with large efficiency remains thereby a challenging goal.
Here we demonstrate a faithful quantum memory for polarization qubits with a storage and retrieval efficiency close to 70%. Our realization is based on electromagnetically induced transparency (EIT) in a single spatially -multiplexed ensemble of cold cesium atoms featuring a large OD. The qubits are implemented using attenuated coherent states at the single-photon level. The reported efficiency approaches the maximal performance achievable on the D 2 line used here, as shown by a comprehensive model that includes all the involved atomic transitions. Relative to previous works, this advance has been made possible by combining a high OD medium, efficient spatial multiplexing and low -noise operation.

Results
Preparation of high-OD cold atomic ensemble. To obtain an ensemble with large OD, our experiment is based on an elongated 2D magneto-optical trap (MOT) of cesium atoms 25,26,30 . As sketched in Fig. 1a, the MOT relies on two pairs of rectangularshaped coils and three retro-reflected trapping beams with a twoinch diameter and a total power of 350 mW. The resulting cigarshaped ensemble has a length of 2.5 cm.
The experiment cycle is completed with a repetition rate of 20 Hz (see Supplementary Note 1). After a 37.5-millisecond-long MOT loading, the OD is further increased by linearly ramping up the magnetic field gradient in the transverse directions to radially compress the ensemble. For this purpose, the trapping coil current is increased from 4 to 16 A over 8 ms. After switching off the MOT coils, polarization gradient cooling is performed for 1.95 ms by ramping down the power of the trapping and repump beams following an exponential profile, while ramping up the detuning of the trapping beam from −17 to −107 MHz.
The atoms are prepared in the g j i ¼ 6S 1=2 ; F ¼ 3 state by turning off the repump light earlier than the trapping light, as well as sending an additional depumping light resonant on the  Fig. 1 Quantum memory for polarization qubits in a multiplexed large OD atomic cloud. a A polarization qubit is encoded via a quarter (QWP) and a half wave plate (HWP) and converted into a dual-rail qubit with a beam displacer (BD). The orthogonally polarized beams, separated by 4 mm, are then mapped into an elongated ensemble of laser-cooled cesium atoms prepared in a 2D magneto-optical trap in a glass chamber. The spatial multiplexing is realized by focusing the two parallel paths into the 2.5-centimeter-long ensemble with a small crossing angle of 0.5°in order to preserve a large OD for each mode, an essential but challenging feature. A single control beam propagates with an angle of 1°relative to the signal modes in the plane of symmetry. b A large OD of 300 is obtained. The blue points correspond to the experimental data while the red solid line gives the theoretical fit. c Energy levels of the Cs D 2 line involved in the EIT scheme. The atoms are prepared in F = 3 and populate all the Zeeman levels. Signal and control fields have the same circular polarization to avoid residual absorption. A comprehensive model is derived to take into account all the atomic levels, including the excited levels out of resonance. This model allows to understand the fundamental limits for storage and retrieval in such a setting, as described in the text. d Typical EIT spectrum as a function of the signal detuning when the control beam is kept on resonance. The red solid line corresponds to the full model transfer is required as no perfect EIT can be obtained in the absence of Zeeman optical pumping if s j i is used as the initial ground state. Finally, experiments are started 2 ms after the MOT coils are turned off to allow sufficient decay of the magnetic field. Three pairs of coils are used to compensate residual magnetic fields and to limit the inhomogeneous broadening to 50 kHz, as measured by microwave spectroscopy.
At the end of the preparation stage, the OD for a probe resonant to the g j i ! e j i transition reaches a value of about 300, as shown in Fig. 1b. The temperature of the atoms is measured to be 20 μK with a time-of-flight technique.
Realization of high-efficiency memory. Having prepared a large OD ensemble, we now turn to the memory protocol. The reversible mapping is based on EIT that enables the conversion of a signal photon into a long-lived collective excitation by dynamically changing the power of an auxiliary control field [31][32][33][34] . The signal and control beams, which are tightly phase locked (Supplementary Note 1), have waist diameters of 250 μm and 2 mm, respectively, and they intersect at the center of the MOT with a small angle of 1°. The control beam on the s j i ! e j i transition has the same circular polarization as the signal, as shown in Fig. 1c, to avoid absorption in an atomic system involving various EIT channels due to the presence of Zeeman sublevels. Proper alignment for the light polarization is especially important for a high OD medium, as only a small residual fraction of OD would lead to a significant absorption for the signal. Figure 1d gives a typical EIT spectrum. A transmission close to 80% is obtained at large OD.
To ensure negligible leakage during the storage process, the power of the control beam is chosen to provide a slow-light delay equal to twice the probe pulse duration when the control is continuously on. Figure 2a gives a single-photon level measurement of the slowed pulse and an example of a dynamic memory operation with a few microseconds storage. Before detection, the signal passes through a home-made atomic filter and a commercial lens-based cavity for spectral filtering (Quantaser FPE001A), with an overall rejection of 70 dB for the control field.
Thanks to the large OD achieved here, we could investigate the scaling behavior of the storage and retrieval efficiency. Figure 2b shows the efficiency as a function of the OD, which is varied by adjusting the power of the trapping beams during loading. At each value, the time delay is maintained constant by adapting the control beam power. As can be seen, the memory efficiency saturates at an OD of around 200 before decreasing. The maximal efficiency achieved here reaches (69 ± 1)%. This represents a record on the cesium D 2 line and, more importantly, the highest achievable value in this configuration.
To understand this scaling and OD tradeoff, the complex level structure has to be taken into account. In the alkali-metal atoms, hyperfine interaction in the excited state indeed introduces several levels and EIT features can differ from the usual threelevel Λ approximation, as previously studied in our group in various contexts [35][36][37][38] . Even for cold atoms, the off-resonance excitation of multiple excited levels can have a strong effect on the medium susceptibility. This is especially true for the D 2 line of cesium atoms, for which the levels are only separated by 30 to 50Γ, where Γ denotes the natural linewidth of the excited state. These off-resonant excitations result in AC Stark shifts and effective additional ground state decoherence proportional to the control power. This decoherence rate limits the achievable transparency and therefore the storage efficiency at large ODs 28 . In addition, the atoms can be distributed in the ground state over many Zeeman sublevels, as it is the case in our experiment, and this configuration can lead to further inefficiency.
In Fig. 2b, experimental results are compared with a full model based on the Maxwell-Bloch equations and that takes into account the interaction of the probe and control field not only with all the excited levels but also with the Zeeman states (Supplementary Note 3). The theoretical lines correspond to the case with a single Zeeman level of the ground state g j i, e.g. m = +3, and to the case with an equal population in all the Zeeman levels. As can be seen, the experimental results are in strong agreement with this second case. The efficiency would only be slightly increased to 75% by optical pumping, which is a challenging task for atomic ensembles with very large OD. The model also confirms that the intrinsic ground-state decoherence γ 0 = (1.0 ± 0.5) × 10 −3 Γ estimated in our experiment is not the limiting factor for the memory performance.
Implementation of a dual-rail memory. Next we present the extension of the setup to the qubit storage. The polarization mapping is implemented by using a dual-rail strategy in a single ensemble 39,40 . This method has been used in various implementations 10,41-43 , but the challenge was here to maintain a large OD and a strong noise cancellation despite this multiplexing in an elongated ensemble with a very reduced cross-section.
For this purpose, as illustrated in Fig. 1a, the signal beam passes through a beam displacer based on a birefringent calcite crystal (Thorlabs BD40) that provides a large 4-mm separation between the two orthogonally polarized beams. The two paths are then focused on the center of the MOT 44 with a 500-mm focal length lens. Their transverse separation at the MOT edge is estimated to be 100 μm, which is much smaller than the 1-mm transverse size of the MOT. In contrast to the usual parallel scheme, i.e., without focusing the signal beam, here both beams cross the center of the MOT, enabling the attainment of a large and similar OD for the two paths. In the experiment, the OD is chosen to be around 200. At the memory output, the two paths are recombined into a single spatial mode with a second beam displacer. The position of the second lens is optimized to obtain a visibility over 99% between these two paths, which is a critical step to achieve high fidelity. The two displacers form a passively stable Mach-Zehnder interferometer, where the relative phase is set to zero by adjusting the tilt of the second beam displacer.
Highly efficient and faithful polarization qubit storage. We now proceed to the qubit storage in this dual-rail setting. Polarization qubits are implemented with weak coherent states with a mean photon number per pulse n = 0.5 and subsequently stored into the memory. The retrieved states are then characterized by usual quantum state tomography 45 . Figure 3a gives the reconstructed density matrices in the H j i; V j i f glogical basis. From the measured matrices, one can estimate the conditional fidelity of the output states with the initially encoded state. The values for the complete set of inputs are listed in Fig. 3b. The average fidelity is 97.7 ± 0.8% and raises up to 99.5 ± 0.5% after correction for background noise that mainly comes from residual control leakage and detector dark counts. In the absence of an input signal, the background floor corresponds to 5 × 10 −4 events per detection window.
In order to conclude about the quantum performance of the storage, we need to compare our fidelity with the maximal one achievable using a classical memory device, based for instance on the so-called measure and prepare strategy. It can be shown that the classical benchmark is given by a fidelity equal to (N + 1)/(N + 2) for a state containing N photons 46 , which is equal to 2/3 for the particular case of a single photon. In our case, the bound has to be modified to take into account the Poissonian statistics of the probe state and the finite memory efficiency, as done in refs. 3,4 . Figure 4 presents the achieved fidelities as a function of the mean photon number per pulse n. The measured fidelities are largely above the classical benchmark for n as low as 0.02. As shown in the inset of Fig. 4, the memory time for maintaining the quantum nature of the storage reaches more than 20 μs. In our experiment, the lifetime is mainly limited by the residual magnetic field along the elongated atomic cloud (Supplementary Note 2).
For these measurements, the average storage and retrieval efficiency reaches (68.5 ± 2)%, as detailed in Fig. 3b. This value is the highest efficiency reported so far for a reversible memory that demonstrates the quantum storage of photonic qubits. Moreover, we have shown that, because of the multiple level structure of the cesium D 2 line, the achieved efficiency is the highest attainable in this configuration. Direct extension of our setup to the D 1 line where excited levels are much more separated should enable to reach an efficiency above 90% 28 , although this has yet to be demonstrated in the quantum regime.
In our realization, the pulse duration of the qubits has been chosen around 400 ns to obtain the highest efficiency after one pulse-width delay. It corresponds to a memory bandwidth of a few MHz, as expected from EIT storage 2 . Single-photon sources with sub-MHz bandwidths have been demonstrated [47][48][49] and can be adapted to our reported quantum memory.

Discussion
In summary, we have demonstrated a highly efficient memory for optical qubits by successfully operating a large OD elongated atomic ensemble in a dual-rail configuration. This combination enables the reversible mapping of arbitrary polarization states not only with fidelities well above the classical benchmark but also with an overall storage and retrieval efficiency close to 70%. This value represents the highest efficiency to date for the storage and readout of optical qubits in any physical platform and is more than double of the previously -reported values. It also  outperforms the important 50% threshold required to beat the no-cloning limit without post-selection. Besides the aforementioned network architecture scalability and potential loss-tolerant schemes, the achieved efficiency opens the way to tests of advanced quantum networking tasks where the storage node efficiency plays a critical role, such as in certification protocols or unforgeable quantum money 50,51 . Moreover, the designed platform is directly compatible with recent works based on spatially -structured photons and multiple degree of freedom storage 10 and can now yield very efficient realizations to boost high-capacity network channels.
Data availability. The data that support the findings of this study are available from the corresponding authors on request.