Abstract
Highrate generation of hybrid photonmatter entanglement remains a fundamental building block of quantum network architectures enabling protocols such as quantum secure communication or quantum distributed computing. While a tremendous effort has been made to overcome technological constraints limiting the efficiency and coherence times of current systems, an important complementary approach is to employ parallel and multiplexed architectures. Here we follow this approach experimentally demonstrating the generation of bipartite polarizationentangled photonic states across more than 500 modes, with a programmable delay for the second photon enabled by qubit storage in a wavevectormultiplexed coldatomic quantum memory. We demonstrate Clauser, Horne, Shimony, Holt inequality violation by over 3 standard deviations, lasting for at least 45 μs storage time for half of the modes. The ability to shape hybrid entanglement between the polarization and wavevector degrees of freedom provides not only multiplexing capabilities but also brings prospects for novel protocols.
Introduction
Efficient and robust entanglement generation between distant parties remains one of the most fundamental steps towards practical implementations of quantum enhanced protocols^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}, enabling amongst others quantum secure communication^{1,10,16,17,18,19,20} or distributed quantum computing^{21}. The optical domain remains a platform of choice for current stateoftheart experimental demonstrations of quantum communication protocols; however, a fundamental exponential transmission loss limits the attainable distance between the parties to several tens of km^{22,23}. While direct amplification of the quantum states is forbidden by the nocloning theorem, noisetolerant quantum repeaters have been proposed to distribute entanglement over extended distances by generating entanglement between intermediate stations (repeater nodes) and performing the entanglement swapping protocol^{24,25}. Quantum memories play a significant role in such a task facilitating adaptive strategies, and enabling hierarchical architecture of the network with neighbor pairs of nodes waiting for each other, to accomplish entanglement the generation step before trying the entanglement swap.
While there are proposals for memoryless errorcorrectionbased schemes^{26,27}, a significant technological advancement is still required, making the quantummemorybased repeaters the most promising solution for nearterm quantum networks. The performance of such repeaters heavily relies on the efficiency and lifetime of quantum memories, and a significant effort has been devoted to their improvement^{28,29}. Stateoftheart quantum memories with either high retrieval efficiency^{30,31,32,33,34} or long storage times^{35,36,37,38} have been demonstrated. Even though, practical quantum repeater networks will most likely require a combination of those results with each other in a parallelized or multiplexed architecture^{3,7,39,40,41,42}, in particular facilitated by multimode quantum memories^{42,43,44} and optical switching networks. Importantly, there has been a tremendous progress in developing either spatially^{44,45}, temporally^{40,46,47,48}, spectrally^{49} or hybrid^{42} multimode quantum memories. Recently demonstrated wavevector multiplexing constitutes a viable alternative offering both an unprecedented number of modes and versatile inmemory processing capabilities^{43,50}. In our group, we have recently proposed and analyzed theoretically a scheme to use the wavevector multiplexing paradigm with a coldatomic quantum memory as a feasible platform for a quantum repeater^{51}. Roughly speaking, during entanglement generation the large number of modes allows for successful completion of this first step in almost all cases.
Here, we experimentally demonstrate a highrate bipartite polarization entanglement generation employing ca. M ≈ 550 pairs of photonic modes of the wavevector multiplexed quantum memory (WVMUXQM), with the possibility of delaying a second photon in pair for up to tens of μs. The entanglement is verified by a moderesolved Bell state measurement certifying nonlocality of generated states. The wavevector multiplexing approach is compatible with modern twophoton quantum repeater protocols robust to global phase fluctuations, and particularly suitable with promising multimode quantum optical channels employing multicore fibers^{52,53} or freespace transmission^{54}. Our scheme constitutes an important step towards quasideterministic entanglement generation^{51}, which for the achieved parameters would improve the quantum communication rate roughly Mfold.
We note that the Bell state generation has been demonstrated before in fewmode quantum memories^{55,56}, and recently with a similar mode conversion setup and a long lifetime of 1 ms^{57}, albeit only in three modes.
Results
Multimode generation of Bell states
Robust quantum repeater protocols based on twophoton interference require highrate generation of photonatom entanglement^{58,59,60} of the form \(({S}_{{\rm{H}}}^{\dagger }{a}_{{\rm{H}}}^{\dagger }+{S}_{{\rm{V}}}^{\dagger }{a}_{{\rm{V}}}^{\dagger })/\sqrt{2}\), where \({a}_{{\rm{H}}}^{\dagger }\) (\({a}_{{\rm{V}}}^{\dagger }\)) corresponds to a photonic creation operator for horizontal (vertical) polarization and \({S}_{{\rm{H}}}^{\dagger }\) (\({S}_{{\rm{V}}}^{\dagger }\)) is the creation operator for an atomic excitation (spinwave) associated with the emission of an H (a V) photon. In a basic scenario, two atomic ensembles labeled H and V generate horizontally and vertically polarized photons, respectively, to further have them combined on a polarizing beamsplitter; however, such a scheme requires four atomic ensembles per repeater node (a twoensemble interface per each neighbor) and complicates any multiplexing or multimode platform. Alternatively, with a lightmatter interface supporting 2M singlepolarization modes, M photonic modes can have H polarization (V polarization) and be superimposed onto the other orthogonally polarized M modes, provided the modes are coherent with each other. While we introduce this idea in the context of a wavevector multiplexed quantum memory (WVMUXQM), such an approach could be suited to other multimode systems.
The WVMUXQM is a highopticaldensity coldatomicensemblebased quantum memory with several hundreds of wavevector or equivalently photonic emission angle modes. The modes are interfaced via spontaneous offresonant Raman scattering to probabilistically generate a twomode squeezed state of atomic excitations (spinwaves), and a writeout (Stokes) photons in each mode, as detailed in the ref. ^{43}. The probability of generating a single pair of a spinwave and a writeout photon per mode shall be denoted by χ ≪ 1. Importantly, the intermode coherence of WVMUXQM memory has been previously verified^{61}.
Bell state generation across many modes
Experimentally, we employ the wavevectors (or emission angles) of photons as the degree of freedom for the M modes. The memory is interfaced with spatially large write/read beams with a well defined and the fixed angle corresponding to wavevectors k_{W} and \(\frac{{{\bf{k}}}_{{\bf{R}}}}{ {{\bf{k}}}_{{\bf{R}}} }=\frac{{{\bf{k}}}_{{\bf{W}}}}{ {{\bf{k}}}_{{\bf{W}}} }\) for write and read beam, respectively. In the memory writing process, a spinwave–photon pairs across many angular modes are generated. The quantum state of a single pair of this kind can be described as a coherent superposition of planewave contributions:
where k_{W} − k_{w} = K represents a unique spinwave wavevector for each (further called writeout—w) generated photon’s wavevector k_{w} and A represents the rectangular field of view in the wavevector space. Additionally, the writeout photons coming from the memory have circular polarization that is further transformed to vertical one, which we explicitly denote by V subscript.
In a single write laser shot many pairs can be generated, and the full quantum state of the memory and writeout field is:
where χ_{M} is a total (across M modes) pair generation probability, which for small χ is just χ_{M} ≈ χM. The state \(\left{{\Psi }}\right\rangle\) describes many spinwave–photon pairs distributed across all possible modes including multiphoton contributions to a single mode. However, as the probability of such an event is χ^{2} (for a chosen mode), for small χ they are not likely to happen, even with relatively high average of total excitations per shot. Therefore, we can focus on a single photon–spinwave state \(\left\psi \right\rangle\) from which we generate the WVMUX Bell state.
To generate the polarization degree of freedom (DoF) Bell state, we split the field of view containing 2M modes in half and superimpose the two resulting regions. Prior to the superimposition, one half is sent through halfwaveplate to change polarization of the writeout photons form vertical (V) to horizontal (H). After the superimposition we end up with a WVMUX polarization DoF Belllike state:
where the optical phase between generated writeout H and V photons with k_{w} wavevector is φ_{w}(k_{w}). The spin waves (atomic excitations) can be readout after a programmable delay:
where X ∈ {H, V} and \({b}_{{\rm{X}}}^{\dagger }({{\bf{k}}}_{{\bf{r}}})\) denotes a creation operator for the readout photon with polarization X and wavevector k_{r}, and ϕ(k_{r}, k_{w}) characterizes the correlation between writeout and readout photons. The operation brings our state to:
where we introduced a WVMUX Bell state creation operator \({\mathfrak{{B}^{\dagger }}}({{\bf{k}}}_{{\bf{r}}},{{\bf{k}}}_{{\bf{w}}})\), with φ(k_{r}, k_{w}) = φ_{w}(k_{w}) + φ_{r}(k_{r}) and φ_{r}(k_{r}) being an additional phase difference for orthogonally polarized readout photons.
We note that with anticollinear write and read beam configuration (\(\frac{{{\bf{k}}}_{{\bf{R}}}}{ {{\bf{k}}}_{{\bf{R}}} }=\frac{{{\bf{k}}}_{{\bf{W}}}}{ {{\bf{k}}}_{{\bf{W}}} }\)) the writeout–readout wavevectors on average satisfy \(\frac{{{\bf{k}}}_{{\bf{w}}}}{ {{\bf{k}}}_{{\bf{w}}} }=\frac{{{\bf{k}}}_{{\bf{r}}}}{ {{\bf{k}}}_{{\bf{r}}} }\). Since the readout of a spinwave is facilitated by lightatom interaction in a finite ensemble, the angular spread of readout photons around specific k_{r} corresponding to a registered k_{w} is inversely proportional to the transverse size of the atomic cloud. Assuming Gaussian profile for the transverse atomic density, we get a Gaussian mode function (conditional on the choice of the writeout wavevector k_{w}) for the readout photon with k_{r} wavevector, described by the correlation function:
with the covariance matrix \({\mathcal{C}}\), that determines the correlation strength between writeout and readout photons in the kspace. The correlation size gives the number of memory modes in terms of Schmidt decomposition:
where α is a geometric factor, which for a rectangular area with transversallyGaussian correlation is α = 0.565^{43}, and A = L_{x} × L_{y} denotes the observed rectangular area in the wavevector space (note that in total we observe 2A which is divided into H and V part) and λ_{1}, λ_{2} correspond to eigenvalues of the \({\mathcal{C}}\) matrix (e.g., λ = σ^{2} for one dimensional Gaussian parameter σ).
Finally, we can rewrite \(\left{\psi }_{{\rm{B}}}\right\rangle\) in terms of the twophoton (writeout–readout pair) subspace and separate the wavevector and polarization photonic DoFs, obtaining:
with a polarization DoF state:
We note that the state given by Eq. (8) has a nontrivial interdependence between the wavevector and polarization DoF via the wavevectordependent phase φ_{w}(k_{w}) + φ_{r}(k_{r}) (see “Methods” section), which is experimentally feasible to be arbitrarily shaped e.g., by placing a spatial light modulator (SLM) in the farfield of the atomic cloud.
Throughout this work we choose the coordinates so that the photons are correlated as follows:
where we denote k_{i} = (x_{i}, y_{i}); i ∈ {r, w}.
Experimental Bellstate preparation
To experimentally generate a state given by Eq. (8) across M ≈ 550 modes, we employ a WVMUXQM with a specifically designed Mach–Zehnder interferometers (MZI) placed in the farfield of the atomic cloud (see “Methods” section). A simplified experimental setup is depicted in Fig. 1. Each MZI allows dividing the emission cone (wavevector range) of writeout (readout) photons to H and V modes and to superimpose the two parts. Furthermore, the wavevectordependent phase φ_{w}(k_{w}), φ_{r}(k_{r}) can be shaped by tilting one of the MZI mirrors. The MZI consists of three mirrors, a halfwave plate (HWP) and a polarizing beamsplitter (PBS). All modes have initially H polarization which is not reflected at PBS. The first mirror reflects half of the emission cone (half of the wavevector modes), which is then routed by the second mirror into a port of PBS and transmitted through. The other half of the emission cone passes through HWP, which rotates the polarization to V, and is reflected by the third mirror into the second port of the PBS. Vpolarized half is reflected at the PBS, hence the two parts are superimposed in wavevector (spatial) DoF. Importantly, both parts undergo the same number of reflections. Effectively, for each pair of modes MZI connects two wavevector modes into a product of a single wavevector mode and two polarization modes.
Each MZI works as a mode converter between the wavevector and polarization DoF. The wavevector DoF enables parallelization or multiplexing for efficient quantum repeater protocols^{51}, while the polarization DoF facilitates Bell state generation and enables robust twophoton protocols for entanglement creation and connection between repeater nodes^{58,59,60}.
We note that the conversion between wavevector (or spatial) and polarization DoFs has been demonstrated before, also in the context of Bell state generation^{41,48,57,62}, albeit with a fewmode system.
Bellstate measurement
To quantify the entanglement of experimentally generated states we perform a wavevectorresolved Bellstate measurement (BSM) with the writeout and readout arm corresponding to the two parties—Alice and Bob—usually considered in the Bell setting. With a linear MZI phase, we select measurement bases for Alice and Bob lying on the equator of the Bloch sphere. Such a choice ensures the BSM visibility remains constant regardless of the wavevectordependent phase. The local Alice and Bob generalized measurement operators {Θ(k)}_{k}(ξ) are given by
with \({{{\Pi }}}_{\xi }={\hat{\sigma }}_{{\rm{x}}}\cos \xi +{\hat{\sigma }}_{{\rm{y}}}\sin \xi\), where \({\hat{\sigma }}_{{\rm{x}}}\), \({\hat{\sigma }}_{{\rm{y}}}\) are the Pauli operators and ξ parametrizes the measurement (e.g., ξ = 0, ξ = π/2 correspond to a measurement in diagonal and circular polarization bases, respectively).
Writeout and readout paths undergo a projective polarization measurement on a beamdisplacer (BD) with ξ adjusted by halfwave plates (HWP). The two output ports ± of the BD are observed with a singlephoton sensitive IsCMOS camera^{63}, which resolve individual wavevector modes with 1 px corresponding to 2.38 rad/mm (see “Methods” section). Importantly, some of the events registered with the camera correspond to dark counts, photons from the multiexcitation component \({\mathcal{O}}(\chi )\) or misalignment of the experimental setup. To model those imperfections we assume a depolarizing channel^{51} over the polarization DoF, which transforms:
where the visibility \({\mathcal{V}}\) in general depends on the wavevectors of writeout and readout photons.
Expected value and Bell parameter
In the experimental demonstration, we select a linear MZI phase
which for maximally correlated writeout and readout wavevectors k_{w} = k_{r} = k allows us to observe a family of Belllike states \(\left{{\Phi }}({\bf{a}}\cdot {\bf{k}}+{\varphi }_{0})\right\rangle\) with a = a_{w} + a_{r}.
Let us calculate the average outcome of local Alice’s and Bob’s measurements for a single pair of writeout and readout wavevectors Θ(k_{w}, ξ_{w}) ⊗ Θ(k_{r}, ξ_{r}). Postselecting outcomes with a registered pair of photons at k_{r} and k_{w}, we get the expected value to be
where the cosine is given by the trace over polarization DoF: \({\rm{Tr}}\left[({{{\Pi }}}_{{\xi }_{{\rm{w}}}}\otimes {{{\Pi }}}_{{\xi }_{{\rm{r}}}})\left{{\Phi }}\left(\varphi \right.({{\bf{k}}}_{{\bf{r}}},{{\bf{k}}}_{{\bf{w}}})\right\rangle \left\langle {{\Phi }}\left(\varphi \right.({{\bf{k}}}_{{\bf{r}}},{{\bf{k}}}_{{\bf{w}}})\right\right]\). Selecting two bases \({\mathcal{A}}=\{{\xi }_{{\rm{w}}}^{(1)},{\xi }_{{\rm{w}}}^{(2)}\}\), \({\mathcal{B}}=\{{\xi }_{{\rm{r}}}^{(1)},{\xi }_{{\rm{r}}}^{(2)}\}\) for Alice and Bob each, respectively, we can formulate the Bell parameter in the wavevector space:
For an optimal selection of bases
the Bell parameter
depends only on the visibility
To violate Clauser, Horne, Shimony, Holt (CHSH) inequality ? and certify nonclassical correlations one needs \({\mathcal{V}}({{\bf{k}}}_{{\bf{r}}},{{\bf{k}}}_{{\bf{w}}})> 1/\sqrt{2}\).
Visibility
BSM visibility indirectly quantifies the quality of the generated entangled state for further entanglement distillation protocols. Entanglement of distillation, i.e., the number of maximally entangled states that can be distilled per a copy of the generated state, would provide a resourceoriented characterization; however, it is difficult to calculate in a generic case. Optimistically, one may use another entanglement monotone such as entanglement of formation, concurrence or negativity to upper bound the entanglement of distillation. For a Werner state, given by RHS of Eq. (12), all those monotones can be calculated analytically^{64} and depend only on the visibility. Furthermore, concurrence and negativity are in this case linear functions of \({\mathcal{V}}\), hence we will further focus on the visibility as a figure of merit quantifying the entanglement of generated states.
In addition to experimental setup imperfections, the influence of multiphoton excitations and noise either from dark counts or stray photons constitutes a fundamental factor limiting the visibility. Importantly, these factors also affect the Glauber secondorder intensity crosscorrelation between writeout and readout photons g^{(2)}(k_{r}, k_{w}). As shown in “Methods” section, the visibility can be written as:
Interestingly, this result facilitates estimating BSM visibility via g^{(2)} function measurements which do not require the BSM setup. Importantly, with the BSM setup inplace, the g^{(2)} function lets us observe wavevectorresolved interference in the correlation patterns, facilitating observation of the phase profile at MZIs (see the Supplementary Material Sec. S1 and S2 for more details about the wavevector space correlations).
Experimental BSM
To observe the moderesolved Bell parameter \({\mathcal{S}}({\mathcal{A}},{\mathcal{B}},{{\bf{k}}}_{{\bf{r}}},{{\bf{k}}}_{{\bf{w}}})\) we shall look at maximally correlated pairs of writeout and readout wavevectors \(\arg \max  \phi ({{\bf{k}}}_{{\bf{w}}},{{\bf{k}}}_{{\bf{r}}}){ }^{2}\). As detailed in the Supplementary Material Sec. S3, for each point in sum coordinates [y_{s} ≡ (y_{w} + y_{r})/2, x_{s} ≡ (x_{w} + x_{r})/2], we bin the coincidences in a rectangular region 2nσ_{x} × 2nσ_{y} around (0, 0) point in difference (y_{w} − y_{r}, x_{w} − x_{r}) coordinates. Importantly, this affects the secondorder correlation function and thus also the BSM visibility which reads:
where the binning factor is
and where \({\tilde{B}}_{{\rm{w}}}={B}_{{\rm{w}}}/M\) (\({\tilde{B}}_{{\rm{r}}}={B}_{{\rm{r}}}/M\)) denotes the noise probability per mode in the writeout (readout) arm, before detection. The visibility is expected to be uniform in sum coordinates (x_{s}, y_{s}). For further analysis we choose n = 1 (F(n) ≈ 0.825) as a tradeoff between decreasing the visibility (as n grows) and gathering a larger statistics for each point. The choice of Alice and Bob bases \({\mathcal{A}},{\mathcal{B}}\) is optimal for a subset of wavevector modes. While, we note that a constant MZI phase would enable a choice of bases simultaneously optimal for all modes, linear phase provides experimentally feasible characterization in terms of the BSM visibility. Figure 2 depicts subsequent analysis stages leading to the Bell parameters in sum coordinates \({\mathcal{S}}({x}_{{\rm{s}}},{y}_{{\rm{s}}})\). For Fig. 2a–d we choose a single pair of bases \({\xi }_{{\rm{w}}}^{(2)},{\xi }_{{\rm{r}}}^{(2)}\) and illustrate coincidence maps with different combinations of polarization measurement outcomes in writeout/readout arms. Figure 2e–h depict expected values for the BSM measurement with each combination of Alice and Bob bases e.g., Fig. 2e corresponds to the following operation on maps depicted in Fig. 2a–d: e = (a − b − c + d)/(a + b + c + d). Wavevectorresolved Bell parameter, depicted in Fig. 2i, is obtained according to Eq. (16) by taking a combination of BSM expected values. The linear MZI phase is clearly visible in all maps, modulating the results along y_{s}. Hence, we take an average of the Bell parameter along x_{s}. As depicted in Fig. 2j, the averaged Bell parameter \({\langle {\mathcal{S}}({x}_{{\rm{s}}},{y}_{{\rm{s}}})\rangle }_{{x}_{{\rm{s}}}}\) is sinusoidal with a fitted amplitude of 2.60 ± 0.19 yielding CHSH violation by over three standard deviations.
BSM with memory
BSM visibility
The Bell parameter under a choice of optimal bases is directly proportional to the BSM visibility, making the visibility a viable figure of merit for the generated writeout–readout bipartite states. Furthermore, from the perspective of entanglement distillation protocols, the BSM visibility determines the entanglement monotones measuring the ebit content of a single generated state. Importantly, entanglement distribution protocols such as quantum repeater nets can greatly benefit from the delayed release of the readout photon, which allows improving the protocol success rate via, among others, hierarchical architectures^{25,59} or multiplexing of the readout photonic mode^{51}. Hence, we perform a series of measurements with increasing memory time t ∈ [0.3, 60.3] μs and with a very quickly changing linear MZI phase, which allows us to retrieve wavevectorresolved BSM visibility from the sinusoidal patterns in coincidence maps.
Spinwave decoherence
Fundamentally, the BSM visibility at larger memory times is limited by the spinwave decoherence. For WVMUXQM’s lightmatter interface, we select the socalled clock transitions insensitive to the first order to stray magnetic fields and fix the quantization axis by introducing external constant magnetic field along the cloud (zaxis); hence, the decoherence mechanism in our case mainly due to thethermal atomic motion. Inevitably, random displacement of atoms distorts the spatial structure of a spinwave^{43}. The decoherence quantified as an average overlap with the initial spinwave state
is Gaussian with the characteristic time τ depending on the spinwave wavevector and given by:
with γ depending on the atomic mass of ^{87}Rbm, cloud temperature T and Boltzmann constant k_{B}. Spinwave decoherence can be accounted for by plugging into Eq. (21) a timedependent retrieval efficiency
Visibility model
In our experimental setup the MZI used to divide the atomic emission cone into H and V polarized parts superimposes a mode with the lowest modulus wavevector \(\min  {{\bf{k}}}_{{\bf{H}}}\) from the H part onto the highest modulus wavevector \(\max  {{\bf{k}}}_{{\bf{V}}}\) from the V part and vice versa. As a consequence, spinwaves corresponding to different polarization parts decohere with a different rate τ(∣k_{H}∣) ≠ τ(∣k_{V}∣) effectively further deteriorating BSM visibility. Importantly, it could be amended by an improved MZI setup.
Let us denote \({k}_{\min }=\min ( {{\bf{k}}}_{{\bf{H}}} , {{\bf{k}}}_{{\bf{V}}} )\), \({k}_{\max }=\max ( {{\bf{k}}}_{{\bf{H}}} , {{\bf{k}}}_{{\bf{V}}} )\). Since longer wavevectors are associated with faster decoherence, let as also denote \({\tau }_{\min }=\tau ({k}_{\max })\), \({\tau }_{\max }=\tau ({k}_{\min })\) and \({{\Delta }}={\tau }_{\min }^{1}+{\tau }_{\max }^{1}\). As demonstrated in “Methods” section, the additional visibility reduction is:
With the increasing storage time the decoherence deteriorates the visibility. As detailed in Methods, the storagetime–dependent version of Eq. (21) can be written as:
The total visibility is given by the product of Eq. (28) and Eq. (27).
BSM visibility maps
Experimentally, the BSM visibility can be retrieved from coincidence measurements with a quickly changing linear MachZehnder inteferometer phase. The coincidence map for each combinations of measurement ports (s_{r}, s_{w}) = {(+, +), (+, −), (−, +), (−, −)} is smoothed with onedimensional Gaussian filters (σ = 1 px for y_{i} direction and σ = 10 px for x_{s} direction) and for each x_{s} a row of data along y_{s} is selected. Each row is divided into overlapping 50 px segments with an 1 px step and to each segment we fit a model given by \(a\cos (2\pi {f}_{{\rm{y}}}{y}_{{\rm{s}}})+b\) and obtain visibility as a/b, which we assign to the y_{s} coordinate of the visibility map given by the center position of the 50 px segment. Figure 3 depicts experimental data along a fitted visibility model given by the product of Eq. (28) and Eq. (27), assuming τ(k) = γ/∣k∣. Hatched regions correspond to visibility above \(1/\sqrt{2}\), which predicts CHSH violation while additional isolines are drawn at levels from 0.6 downward with a step of 0.1. The model fit yielded
with γ corresponding to a temperature of \(T=4{7}_{4}^{+5}\ \mu \,\text{K}\,\). For completeness, we note that an independent measurement, described in Supplementary Material Sec. S4, yielded a consistent temperature of \(T=4{8}_{5}^{+6}\ \mu \,\text{K}\,\), the χdependent noise \({\tilde{B}}_{{\,}}\rm{r}^{(\chi )}\,=\,0.131\,\pm\, 0.015\) and a readout efficiency of η_{r}(0) = 0.405 ± 0.015.
Modeaveraged visibility
Obtained visibility maps let us study the properties of an average memory mode. As depicted in Fig. 4a, up to ca. 30 μs nearly all modes would violate CHSH, while for 45 μs it remains true for half of the modes. The average visibility amongst the CHSH violating modes remains fairly constant with increasing memory time. Figure 4b depicts very good agreement of the visibility averaged over all modes \(\left\langle {\mathcal{V}}\right\rangle\) with the modeaveraged prediction from the measured Glauber crosscorrelation \(\left\langle ({g}^{(2)}(t)1)/({g}^{(2)}(t)+1)\right\rangle\). Visibly, above 45 μs the performance of a significant number of modes has severely deteriorated. Nevertheless, modes with a lower modulus wavevector can have significantly longer lifetimes. Unfortunately, the number of modes with a given modulus wavevector ∣k∣ is roughly proportional to ∣k∣, hence there are few modes with a long lifetime.
Wavevectordependent decoherence rate
For quantum memories utilizing a dense cold ensemble of atoms, it has been demonstrated that thermal motion constitutes the main source of spinwave decoherence^{38}, provided the socalled clock transitions—in the first order robust to external magnetic field fluctuations—are utilized for lightmatter interface. Characteristically, decoherence due to thermal motion leads to Gaussian temporal profile of the retrieval efficiency, as given by Eq. (26), with a characteristic time of τ(∣k∣) = γ/∣k∣. A vast range of wavevector modes simultaneously utilized in the WVMUXQM provides a unique opportunity to precisely observe experimentally the decoherence time τ(∣k∣) dependence on the wavevector length ∣k∣. As depicted in Fig. 5, experimental results are in a good agreement with the predictions of thermalmotioninduced decoherence.
Discussion
We experimentally demonstrated the generation of polarizationentangled bipartite Bell states in ca. 550 photonic modes, with an inherent programmable delay for the second photon in a pair. Our approach harnesses hybrid atomphoton entanglement generation in a single cold highdensity ensemble of Rb87 atoms, which is the central part of the wavevectormultiplexed quantum memory (WVMUXQM). Ca. 1100 wavevector (angular emission) modes of the WVMUXQM are divided into H and V polarized photonic modes, which are superimposed pairwise forming 550 modes in a polarization superposition. Importantly, a writeout photon in one of those combined modes may have been created in an H or a V part, in each case having a different initial wavevector and thus being entangled with a collective atomic excitation—a spinwave—in a different memory mode. The wavevectors of writeout and corresponding readout photons are correlated with each other; hence, a writeout photon created in the H(V) part of writeout emission cone is accompanied by a readout photon also in the H(V) part, albeit of the readout emission cone. Hence, the readout modes need to be superimposed in the same way as writeout modes. This way, a writeout photon and a correlated readout photon may be either both H polarized or both V polarized, constituting a polarization entangled state. Fundamentally, this concept realizes a conversion between wavevector and polarization degrees of freedom.
We have demonstrated CHSH inequality violation by more than 3 standard deviations with the Bell parameter reaching \({\mathcal{S}}=2.60\pm 0.19\). Via wavevectorresolved Bell state measurement (BSM) with varying storage times, we established experimentally and with a theoretical model the BSM visibility—quantity proportional to the Bell parameter under the optimal choice of bases—behavior with wavevector and temporal resolution. After 45 μs storage time, ca. 50% of modes still indicate CHSH violation, while after 60 μs it amounts to around 10%. Importantly, with singlecopy distillation protocols^{65}, even slightly entangled states can be probabilistically distilled into Bell states, hence the BSM visibility averaged over the modes—which reflects the average ebit content of a generated state—is a significant figure of merit. In our experiment, it has fallen merely to around 50% at 60 μs storage time, as compared to the initial ca. 80% for immediate readout. With the current experimental parameters, we estimate the probability to generate and detect at least one atomphoton Bell state across all pairs of modes to be 1 − (1 − ηχ)^{M} ≈ 35% (89% when only the detection and not the filtering system efficiency is contained in η).
Quantum repeater networks constitute the most versatile application of entanglement amending the fundamental exponential loss of photonic channels. While the performance and feasibility of WVMUXQM platform for quantum repeater networks has been discussed elsewhere^{51}, we note here that obtaining hybrid entanglement is essential for twophoton protocols^{58,59,60} which solve the phase stability issues inherent to the DLCZ protocol^{25}. From the perspective of feasible quantum repeaters, the 800 nm photons are suboptimal having an order of magnitude higher fibertransmission losses than telecom; however, conversion to telecom photons as well as lightatom interfaces at telecom wavelengths have been demonstrated^{2,66,67,68}. Furthermore, quantum repeater protocols generally do not require phase stability between different modes, rendering freespace transmission a viable alternative.
Importantly, WVMUXQM platform offers versatility beyond entanglement generation, such as intramemory processing of stored spinwave states^{69}—including operations on nonclassical singleexcitation states^{50}, continuous variable and temporal processing^{70,71} or ultranarrow band temporal imaging^{72}. Furthermore, the rich atomic structure of alkali atoms may be employed to design various lightmatter interfaces suited for a particular application^{73}.
Finally, we note that our experimental demonstration is far from the ultimate capabilities of wavevectormultiplexed memories. Cold atomic memories with higher coherence times^{37} and readout efficiencies^{31} has been demonstrated. The wavevector multiplexing principle could be applied with atomic clouds captured in a dipole trap or an optical lattice, which can prolong the memory lifetime to the subsecond regime^{36,62}. However, the simplest improvement would be to observe a larger angular range of photonic emission from the memory. As estimated in our theoretical work^{51}, the number of modes attainable with standard optical components and commercial CMOS image sensors would reach over 5000 enabling feasible highrate entanglement generation.
Methods
Wavevectordependent phase
The ability to shape the wavevector dependent phase φ_{w}(k_{w}), φ_{r}(k_{r}) opens vast possibilities. For instance, consider the moststrongly correlated writeout and readout wavevectors k_{w} = k_{r} = k. By selecting φ(k) ≡ φ_{w}(k) + φ_{r}(k) periodically equal to either π or 0 on a rectangular grid, we generate either a Φ_{−} or Φ_{+} Bellstate, respectively, depending on the wavevector k. Interestingly, the wavevector of the scattered writeout photon k is inherently random and selected by the process of spontaneous Raman scattering. Such a quantumrandom sampling of generated states is equivalent to a quantumrandom choice of measurement bases for Alice and Bob in the Bell setting (since either the states or the bases may be equivalently altered). After the measurement, the central WVMUXQM can reveal the exact phase profile φ(k) that was employed, allowing Alice and Bob to interpret their results.
The WVMUXQM platform
The WVMUXQM is based on an elongated (0.5 mm × 0.5 mm × 7 mm), highopticaldepth ensemble of rubidium87 atoms prepared in magnetooptical trap (MOT). The optical depth (OD) amounts to 150 at the F = 1, m_{F} = 1 → F = 2, m_{F} = 2 transition of D2 line, which corresponds to OD = 25 at the Write laser transition (in resonance). To define the quantization axis, the cloud is kept in a constant (1 Gauss) magnetic field oriented along the propagation axis (zaxis). The MOT loading time is set to 2 ms which includes compression stage (700 μs) and polarization gradient cooling, with magnetic gradient switched off (300 μs). After MOT loading stage the atoms are optically pumped to the F = 1, m_{F} = −1 state using 70 μslong hyperfine pump pulse (resonant with the F = 2 → F = 2 transition of D1 line), that illuminates the cloud from four sides (along cooling beams) and has 15 mW power in total. The Zeeman sublevel pumping is achieved by illuminating the atoms along the zaxis by a circularly polarized laser beam (resonant with F = 1 → F = 1 transition of D2 line) for 55 μs from the beginning of the hyperfine pump pulse. We estimate the efficiency of the Zeeman pumping to be about 70%^{43}. Longer hyperfine pump pulse duration combined with additional (5 mW, 75 μs) “clearing” pulse of the Read laser guarantees, that the storage level (F = 2, m_{F} = 1) is emptied before the quantum memory protocol. To generate the twomode squeezed state of spinwave and a writeout photon, we use Raman interaction in the Λ scheme. The writeout photon and spinwave pairs are generated using a 30 MHz reddetuned σ_{+}polarized Write laser pulse (300 ns) on \(F=1\to F^{\prime} =2\) transition of D2 line. The Write laser power is chosen to provide desired pair generation probability χ ≈ 0.01, which in our case corresponds to about 10 μW. Since we filter the writeout photons to have orthogonal polarization to the laser photons, the spin waves of our interest are created between the F = 1, m_{F} = − 1 and F = 2, m_{F} = 1 states. To convert (readout) the spin waves to readout photons we use σ_{−}polarized Read laser pulse (300 ns) tuned to the \(F=2\to F^{\prime} =2\) transition of the D1 line. The Read laser power is chosen to give the readout pulse duration of approx. 200 ns, which corresponds to about 100 μW. The doubleΛ configuration (consisting of D1 and D2 lines of rubidium87) allows us to efficiently filter writeout and readout photons from backemissions, that may lead to uncorrelated detection events and spoil the measurements. Finally, to provide the best signaltonoise ratio, the writeout and readout photons pass through narrowband atomic filters. The filters are glass cells containing optically pumped ^{87}Rb that absorb residual laser light (which is separated from photons by 6.8 GHz), while being transparent to the signal (writeout or readout) photons (see Parniak et al. for details of the filtering system^{43}).
Singlephoton sensitive IsCMOS camera
Generation of Bell states across many modes requires efficient detection of singlephotons with a high spatial resolution. High number of modes—several hundreds—render arrays of singlemode singlephoton detectors practically unfeasible; however, recent development in singlephoton cameras shows the detection technology is sufficient for nearterm applications^{63,74}. Among singlephoton camera solution, the most commonly employed are intensified CCD (ICCD) and electronmultiplying CCD (EMCCD). Neither of those solutions achieves acquisition times on the order of a few μs, which would allow realtime response to detected photons required in many protocols such as quantum repeater operation. Additionally, EMCCD cameras have a high readout noise resulting in high darkcount rates and consequently deteriorating the fidelity of the generated entanglement. Recently commercialized, intensified scientific CMOS (IsCMOS) cameras solve those issues to a great extent. Additionally, socalled quanta image sensors^{74} offer further improvements, especially in terms of quantum efficiency.
In our experiment, we employ a custom IsCMOS camera which involves a 2stage image intensifier (luminous gain of 5 × 10^{6}) imaged by a relay lens (fnumber of 1.1, magnification of −0.44) onto a 5.5 Mpx (mega pixel), 2560 × 2160 px CMOS sensor (effective pixel pitch ca. 15 μm accounting for relay lens magnification). Image intensifier involves a GaAs photocathode which around 800 nm offers the overall detection efficiency of ca. 20%. Active multiplexing, required in many applications^{51} would inherently need realtime feedback from the camera, which has not been implemented in the current experiment; however, it would be possible with one of the modern fast CMOS sensors.
Mode detection crosstalk
With the high resolution of singlephoton cameras, the probability for two or more generated writeout photons to share the same (up to the camera resolution) central wavevector is given by ≈1 − n!/(n − ⌈χM⌉)! × 1/n^{⌈χM⌉}, where n = 130 × 160 is the number of observed pixels (px). In our case (χ ≈ 0.01) the probability is 4.8 × 10^{−4} and we will neglect such cases. Similarly, if we focus on a single jth mode with a detected writeout photon, the probability of registering a readout photon coming from another mode in the 3σ radius around the mostlikely position of jth readout can be approximated as \(1{[1\chi {\eta }_{{\rm{r}}}\eta \times \pi {(3\sigma )}^{2}/n]}^{{\rm{M}}1}\), where σ is the mode size in pixels, η_{r} ≈ 40% corresponds to the readout efficiency and η ≈ 8% to the filtering and detection system efficiency. In our case the probability amounts to around 0.17%. Therefore, in most cases, we can consider a single isolated jth mode with the results being valid for any j.
The used formulae have been derived in the Supplementary Material Sec. S6.
Visibility model
Assuming low average number of detected photons per experiment \(\bar{n}\ll 1\), we can approximate the photon number crosscorrelation function as:
where p_{w,r} ≡ p_{w,r}(k_{r}, k_{w}) is the singleexperiment probability of observing a coincidence between a writeout and readout photon and p_{w} ≡ p_{w}(k_{w}) (p_{r} ≡ p_{r}(k_{r})) denotes the marginal probability of observing a writeout (readout) photon. The coincidence probability can be written as p_{w,r} = g^{(2)}p_{w}p_{r}, with g^{(2)} ≡ g^{(2)}(k_{r}, k_{w}). For a given point (k_{r}, k_{w}) in the wavevector space, we measure the BSM visibility by comparing the number of coincidences during measurement set up for maximally constructive (+) or destructive (−) interference. The numbers of coincidences serve as probability estimates and hence give the visibility as
In (−) settings only noise coincidences are registered i.e. \({p}_{{\rm{w,r}}}^{()}={p}_{{\rm{w}}}{p}_{{\rm{r}}}\) while \({p}_{{\rm{w,r}}}^{(+)}={p}_{{\rm{w,r}}}={g}^{(2)}{p}_{{\rm{w}}}{p}_{{\rm{r}}}\). This directly gives us Eq. (20).
Storagetime dependence
Let us first consider the visibility model as given by Eq. (21) and under experimentally verified assumptions of negligible writeout noise \({\tilde{B}}_{w}\ll \chi\) :
We shall further employ a model for the readout noise:
which has been verified in an additional measurement, as described in the Supplementary Material Sec. S4. Furthermore, with the calibrated values of τ_{B} ≈ 13 μs and \({\tilde{B}}_{{\rm{r}}}(\infty )\approx 5\times {\tilde{B}}_{{\rm{r}}}(0)\) we can approximate
with little error. Denoting
and using the decohrence model of Eq. (26) we get the Eq. (28).
Different wavevectors for superimposed H, V modes
Here we quantify the effects of different decoherence rates on the generated states. Let us denote
and
where \({y}_{\max }\) corresponds to the maximal y wavevector component entering the MZI. We shall focus on a single mode and modify the state given by Eq. (9) so that H and V parts have different (modulus) coefficients:
with
Assuming ∣k_{H}∣ > ∣k_{V}∣ the temporal evolution of the state in Eq. (41) brings it closer to \({\leftH\right\rangle }_{{\rm{r}}}{\leftH\right\rangle }_{{\rm{w}}}\). If we consider a reduced state e.g., of writeout only, the evolution moves the state from the equator of the Bloch sphere to the pole. Intuitively, as the polarization measurement projects the state on some axis in the equator plane, the visibility will be reduced. Direct calculation yields
with
Importantly, no assumption on the actual form of τ(k) need to be made to give an expression for \(\tilde{{\mathcal{V}}}(t)\), as long as Eq. (26) holds. Let us denote \({k}_{\min }=\min ( {{\bf{k}}}_{{\bf{H}}} , {{\bf{k}}}_{{\bf{V}}} )\), \({k}_{\max }=\max ( {{\bf{k}}}_{{\bf{H}}} , {{\bf{k}}}_{{\bf{V}}} )\). Since longer wavevectors are associated with faster decoherence, let as also denote \({\tau }_{\min }=\tau ({k}_{\max })\), \({\tau }_{\max }=\tau ({k}_{\min })\) and \({{\Delta }}={\tau }_{\min }^{1}+{\tau }_{\max }^{1}\). This way, we arrive at Eq. (27).
Interestingly, MZI configured to give different c_{H}(t) and c_{V}(t) could be used to demonstrate a violation of the socalled tilted Bell inequalities^{75} by generating a family of states with a varying degree of entanglement, across the memory modes. The only required modification in the experimental setup would be to remove quarter waveplates, depicted in Fig. 1, which would amount to performing BSM with polarization operators of the form \({\hat{\sigma }}_{{\rm{z}}}\cos \xi +{\hat{\sigma }}_{{\rm{x}}}\sin \xi\) corresponding to the projection on the Bloch sphere’s meridian.
Data availability
Data presented in Figs. 2–5 have been deposited in RepOD Repository for Open Data at https://doi.org/10.18150/RJBMOD. Any other data that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
Data was analyzed using custom photon analysis code available at https://github.com/Michuu/photonpacket. Other custom code used in data analysis is available from the corresponding author upon reasonable request.
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Acknowledgements
This work has been funded by the Ministry of Education and Science (Poland) grants no. DI2016 014846, DI2018 010848, by the National Science Centre (Poland) grants no. 2016/21/B/ST2/02559, 2017/25/N/ST2/01163, 2017/25/N/ST2/00713, by the Foundation for Polish Science MAB/2018/4 “Quantum Optical Technologies” project and by the Office of Naval Research (USA) grant no. N629091912127. The “Quantum Optical Technologies” project is carried out within the International Research Agendas programme of the Foundation for Polish Science cofinanced by the European Union under the European Regional Development Fund. M.P. has been also supported by the Foundation for Polish Science via the START scholarship. We thank K. Banaszek for the generous support and J. Kołodyński for insightful discussions.
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M.M. and A.L. performed the measurements assisted by M.L.; M.L. analysed the data assisted by M.M.; M.L. performed the theoretical analysis and wrote the manuscript assisted by M.M. and M.P.; M.L., M.M., A.L., M.P., and W.W. contributed to building and calibration of the experimental setup; M.L., M.M., and M.P. conceived the scheme and designed the experiment; W.W. and M.P. supervised the project.
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The University of Warsaw has filed a related Polish Patent Application P.434142 entitled “System for generating entangled photon pairs of multimode quantum memory for regeneration of a quantum signal at a long distance, a method for generating entangled photon pairs of multimode quantum memory for regeneration of a quantum signal at a long distance” invented by W. Wasilewski, M. Lipka, M. Mazelanik, M. Parniak, K. Zdanowski, A. Ostasiuk, and A. Leszczyński. The unpublished patent application is pending. The authors declare no other competing interests.
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Lipka, M., Mazelanik, M., Leszczyński, A. et al. Massivelymultiplexed generation of Belltype entanglement using a quantum memory. Commun Phys 4, 46 (2021). https://doi.org/10.1038/s42005021005511
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DOI: https://doi.org/10.1038/s42005021005511
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