Experimental realization of entanglement in multiple degrees of freedom between two quantum memories

Entanglement in multiple degrees of freedom has many benefits over entanglement in a single one. The former enables quantum communication with higher channel capacity and more efficient quantum information processing and is compatible with diverse quantum networks. Establishing multi-degree-of-freedom entangled memories is not only vital for high-capacity quantum communication and computing, but also promising for enhanced violations of nonlocality in quantum systems. However, there have been yet no reports of the experimental realization of multi-degree-of-freedom entangled memories. Here we experimentally established hyper- and hybrid entanglement in multiple degrees of freedom, including path (K-vector) and orbital angular momentum, between two separated atomic ensembles by using quantum storage. The results are promising for achieving quantum communication and computing with many degrees of freedom.


Supplementary Figure 2: Interference curves for polarization entanglement in OAM bases.
Interference curves before (a, c, e, g, i, k, m, o) and after (b, d, f, h, j, l, n, p) storage for polarization entanglement when SLMs are projected in the bases of OAM as illustrated. According to the interference curves, we calculate the average visibility in OAM bases accordingly. Before storage, the average visibility is 92.6%, 96.0%, 94.3%, 94.9%, 91.8%, 92.9%, 91%, 93.1% for interference curves in Supplementary Figure 2     '  represents the retrieved photon-photon state after storage.

Supplementary Note 2: Hybrid entanglement
Supplementary Figure 3 is detailed setup for storage of hybrid entanglement. The generated entanglement between Signal 1 and spin wave through SRS process is represented below, 1 A S1 A S1 where |D A > and |U A > refer to the spin wave related to the path U and D in MOT A accordingly, |H S1 >and |V S1 > represent the generated horizontal and vertical polarization of Signal-1 photon respectively. During this process, the power of SLM In order to prepare the hybrid entangled state we wanted, the polarization information of signal 1 photons is transferred from polarization DOF into OAM DOF. We input the Signal 1 photons into a special designed Sagnac interferometer, which is inserted with a vortex phase plate with a topological charge of ±1 for oppsite incidenting direction. The entangled state evolves as, A S1 A S1 S1 S1 where, the states |L> and |R> are the OAM states with topological charge of ±1 . After Signal 1 photons passing through a half-wave plate with 22.5°of the optical axis with respect to the vertical axis, the vertical and horizontal states are changed to be diagonal and anti-diagonal polarized state, the output state filtered by a polarization beam splitter can be denoted as, 1 A S1 A S1 Thus the preparation of hybrid entanglement is achieved.

Supplementary Note 3: Fidelity computation
We use the formular

Supplementary Note 4: Experimental details including coupling efficiencies and counts rate
The coupling efficiency (for both paths M and N) of Signal 1 from space to fibre is 75%.
Signal-1 photons are filtered using three homemade cavities (with temperature control) with 45% transmittance and 70 dB isolation. Signal-2 photons are filtered using two homemade cavities with 65% transmittance and 40 dB isolation. The reflectivity of both SLMs is ~75%. The efficiency of every single photon detector is ~50%.
During our experiment, the dark-count rate is ~180/s for both detectors while the photon counts rate (including dark-count rate) is ~800/s. The overall rate of successful events is 2.5/s accounting for ~500 coincidence counts (in the bases of H-H or L-L) in total 200 s. So, on each 500-ns run, the probability (corrected for loss) of generating entanglement between two atomic ensembles is estimated as 2.5/(2800×100×75%×65%×45%×50%×50%×75%×75%)~2.9×10 -4 , in which 2800×100 is the trial number per second, the first 75% is coupling efficiency for Signal 1 from space to fibre, 65%×45% is for cavity transmission efficiency, the 50%×50% is the total detecting efficiencies of two detectors, 75%×75% is the total reflectivity of two SLMs.

Supplementary Note 5: Memory performance
Because our magnetic field for trapping can't be shut down completely within 1.4 ms due to big value of inductance, our memory time is rather limited to 1.4 s as discussed before in previous work 1 , the 100-ns memory efficiency in MOT B is ~25% for photon carrying no OAM information as depicted in Supplementary Figure 4 and is ~20% for photon carrying OAM with topological value 1 l  . In general, memory time can be improved by compensating the magnetic field or by using magnetic field-insensitive states and reducing atomic motion by using optical lattice, a millisecond and even hundred millisecond storage time could be achieved [2][3][4][5] . In addition, the dynamic decoupling method can also be used to improve the storage time 6 . Memory efficiency can be increased by using waveform modulation 7 .