The continuous-variable regime represents one of the principal avenues towards the realization of quantum information processing and communication4,5,6. In the optical domain it operates with well-known optical modulation and detection techniques and allows for deterministic quantum operations. In the atomic domain it has been developed on the platform of atomic ensembles2,3,7. Advances in the realization of continuous-variable quantum protocols include unconditional quantum teleportation involving light8 and atoms9, a number of results on memory2,3,10,11 and quantum key distribution12. Hybrid continuous/discrete-variable operations13,14,15 paving the road towards continuous-variable quantum computation16,17 have also been reported. However, the ability to store non-classical continuous-variable states of light is crucial to enable further progress, in particular, for continuous-variable linear optics quantum computing with offline resources17, continuous-variable quantum repeaters18,19, entanglement-enhanced quantum metrology, iterative continuous-variable entanglement distillation20, continuous-variable cluster-state quantum computation21, communication/cryptography protocols involving several rounds22 and quantum illumination23. Compared with a number of impressive results reporting discrete-variable quantum memories at the single-photon level (see reviews1,2,3 and references therein), there have been very few experiments towards quantum memory for continuous-variable non-classical states. A fractional, 20 nsec, delay of 50 nsec pulsed continuous-variable entangled states in the atomic vapour has recently been demonstrated24. Memory based on electromagnetically induced transparency for a squeezed vacuum state has been reported25,26, albeit with the fidelity below the classical benchmark27. Very recently, classical benchmarks for storing displaced squeezed states28,29 have been derived, which made experimental implementation of such storage a timely challenge. Such states form an alphabet for continuous-variable quantum information encoding16. An exciting feature of displaced squeezed states is that the ratio of the quantum to classical fidelity grows inversely proportionally to the degree of squeezing.

Here we report the realization of a quantum memory for a set of displaced two-mode squeezed states with an unconditionally high fidelity that exceeds the classical benchmark calculated on the basis of the method in ref. 28. The fidelity between the input state and the memory state is a sufficient condition for a memory or a teleportation protocol to be genuinely quantum. In fact, because continuous-variable protocols are deterministic, that is they have a unity efficiency, the fidelity becomes the preferred performance criterion. The experimental fidelity demonstrated here, which exceeds the classical benchmark fidelity, implies that our quantum memory cannot be mimicked by any classical device.

We store a displaced entangled state of two sideband modes of light and with the frequencies ω±=ω0±ωL, where ω0 is the carrier frequency of light. The entanglement condition for this Einstein–Podolsky–Rosen state6 is (ref. 30) where canonical quadrature operators obey . For a vacuum state . The entanglement of the and modes is equivalent to simultaneous squeezing of the cos(ωLt) mode ; and the corresponding sin(ωLt) mode. Before the input state of light undergoes various losses it is a 6 dB squeezed state. In the photon-number representation for the two modes, the state is |Ψ〉=0.8|0〉+|0〉+0.48|1〉+|1〉+0.29|2〉+|2〉+0.18|3〉+|3〉+. The displaced squeezed states are produced (Fig. 1a) using an optical parametric amplifier31 (OPA) with the bandwidth of 8.3 MHz and two electro-optical modulators (EOMs; see the Methods section for details). The alphabet of quantum states Ψi, which we refer to as ‘initial pure states’ (see the inset in Fig. 2 and Table 1) is generated by displacing the two-mode squeezed vacuum state by varying values [〈xL〉;〈pL〉]=[0,3.8,7.6;0,3.8,7.6] and by varying the orientation of the squeezed quadrature between and .

Figure 1: Set-up and pulse sequence.
figure 1

a, At the sender station two-mode entangled (squeezed) light is generated by the OPA. A variable displacement of the state is achieved by injecting a coherent input into the OPA modulated by EOMs. The output of the OPA is shaped by a chopper, and combined on a polarizing beam splitter with the local oscillator (LO) beam, such that the squeezed light is on only during the ‘input pulse’. A beam shaper and a telescope create an expanded flat-top intensity profile. The light is then sent to the receiver (memory) consisting of two oppositely oriented ensembles of spin-polarized caesium vapour in paraffin-coated cells and a homodyne detector. The detector signal is processed electronically and used as feedback onto the spins obtained using radiofrequency magnetic field pulses. b, Pulse sequence for the initiation of the memory, storage and verification. Radiofrequency ‘feedback’ pulses are 0.15 ms long. QND: quantum non-demolition. c, Atomic-level structure illustrating interaction of quantum (dashed lines) and classical (solid lines) modes with the memory.

Figure 2: Fidelities.
figure 2

The graph shows the values of the experimental fidelity (circles) and the theoretical benchmark values (squares) as a function of the size of the set of states dmax with one vacuum unit of displacement corresponding to . The inset illustrates the alphabet of states used in the experiment. The three sets of states with dmax=0;3.8;7.6 used for the determination of the experimental values of the fidelity plotted in the graph are shown in the inset within dashed, dotted and dashed–dotted squares, respectively. The panel in the bottom shows schematically the propagation channels used in the calculations of the fidelity of the quantum memory (left part) and of the benchmark fidelity of the classical memory (right part). The error bars on the experimental data represent the standard deviations of the results, where all statistical and systematic errors have been included.

Table 1 Initial states of light and stored memory states.

The two photonic modes are stored in two ensembles of caesium atoms contained in paraffin-coated glass cells (Fig. 1a) with the ground-state coherence time around 30 ms (ref. 3). ω0 is blue-detuned by Δ=855 MHz from the F=4F′=5 of the D2 transition (Fig. 1c). Atoms are placed in a magnetic field that leads to the precession of the ground-state spins with the Larmor frequency ωL=2π·322 kHz. This ensures that the atoms efficiently couple to the entangled ω±=ω0±ωL sidebands of light3. The two ensembles 1(2) are optically pumped in F=4,mF=4(−4) states, respectively, which leads to the opposite orientation of their macroscopic spin components Jx1=−Jx2=Jx.

The atomic memory is conveniently described by two sets c,s of non-local, that is, joint for the two separate memory cells, canonical atomic operators , where the superscript rot denotes spin operators in a frame rotating at ωL. It can be shown that the cosine and sine light modes couple only to the atomic c and s modes, respectively32. As a consequence, in the protocol described below the upper (lower) entangled sideband mode of light is stored in the 1 (2) memory cell, respectively. As the equations describing the interaction are the same, we omit the indices c,s from now on.

Light emitted from the sender station to the receiver (memory) station consists of quantum x-polarized modes and a strong y-polarized part that serves as the driving field for interaction with atoms and as the local oscillator for the subsequent homodyne measurement (Fig. 1). The interaction of light and a gas of spin-polarized atoms under our experimental conditions can be described by the equations32:

where the coupling constant κ is a function of light intensity, density of atoms and interaction time, and Z2=6.4 is a function of the detuning alone. In the limit κZ these equations describe a swap of operators for light and atoms, that is, a perfect memory followed by squeezing by a factor Z2. However, in our experiment the swapping time is too long compared with the atomic decoherence time. To speed up the memory process we add a quantum measurement and feedback steps to this swap operation.

The sequence of operations of the quantum memory protocol is shown in Fig. 1b. We start the memory protocol with initializing the atomic memory state in a spin-squeezed state (SSS). The spin-squeezed state with Var(xA)=0.43(3) and Var(pA)=1.07(5) is generated3,7 starting from a (nearly) coherent spin state with Var(xA)=0.55(4) by the sequence (Fig. 1b—preparation of initial state) of the ‘quantum non-demolition spin-squeezing pulse’ and a feedback into both the c and s modes. The feedback is achieved with pulses of the magnetic field at the frequency ωL applied to the two cells. The memory input light pulse is then sent, followed by the measurement of the output light operator xL by the polarization homodyne detection. The measurement result is fed back onto the pA with a gain g. The resulting xAfin and pAfin for the optimized g and κ=1 can be found from equation (1)

In the absence of decoherence the operator xL is perfectly mapped on the memory operator pAfin. The operator pL is stored in xAfin with the correct mean value 〈pL〉=〈xAfin〉 (as 〈xA〉=0) but with an additional noise due to xA.

The ability to reproduce the correct mean values of the input state in the memory by adjusting g and κ is a characteristic feature of our protocol that arguably makes it better suited for storage of multiphoton states compared with, for example, electromagnetically induced transparency and photon-echo approaches where such an adjustment, to the best of our knowledge, has not been demonstrated.

The deleterious noise of the initial state of atoms xA is suppressed by the initial spin-squeezing sequence and by the extra factor due to the swap interaction. In the absence of passive (reflection) losses for light and atomic decoherence we find from equation (2) the expected fidelity of 0.95 and 0.61 for the states squeezed with the =00 and =900, respectively, with the mean fidelity of 0.78.

The true quantum character of our memory is preserved despite substantial transmission losses schematically shown in the bottom of Fig. 2. Entangled states of light are sent to the receiver memory station through a channel with the transmission coefficient ηtr=0.80(4) (which includes the OPA output coupling efficiency 0.97) resulting in the ‘memory input state’, ρin, with Var(xL·cos()−pL·sin())=0.20(2) and Var(xL·sin()+pL·cos())=1.68(9). The entrance (reflection) losses at the windows of the memory cells lead to further attenuation by the factor ηent=0.90(1). Between the interaction and detection light experiences losses described by the detection efficiency ηdet=0.79(2) (see the Methods section).

Following the storage time of 1 ms between the end of the input pulse and the beginning of the verifying pulse (see Fig. 1b for the time sequence) we measure the atomic operators with a verifying probe pulse. The measured mean values and variances of the atomic operators xAfin and pAfin are summarized in Table 1 (see the Methods section for calibration of the atomic operators). From these values and the loss parameters, we can calculate the noise added during the storage process that comes on top of the noise added by transmission and entrance losses. We find that the memory adds 0.47 (6) to Var(xAfin) and 0.38 (11) to Var(pAfin), whereas for the ideal memory, according to equation (2), we expect the additional noise to be 0.36 (5) (due to the finite squeezing of the initial atomic operator xA) and 0, for the two quadratures, respectively. This added noise can be due to atomic decoherence, uncancelled noise from the initial antisqueezed pA quadrature and technical noise from the EOMs.

The overlap integrals between the stored states and the initial pure states are given in the Table 1. The average fidelities calculated from the overlap values for square input distributions with the size dmax=0,3.8 and 7.6 are plotted in Fig. 2. The choice of the interaction strength κ=1 minimizes the added noise but leads to the mismatch between the mean values of the stored atomic state and of the initial pure state of light by the factor . This mismatch is the reason for the reduction of the experimental fidelity for states with larger displacements.

The classical benchmark memory fidelity is calculated28 from the overlap of displaced Ψi states with the states stored in the (hypothetical) classical memory positioned in place of the quantum memory (see the panel in the bottom of Fig. 2). This ensures that the classical memory has the same input state ρin as the quantum memory that is Ψi propagated through the transmission channel with ηt r=0.80. The benchmark fidelity is found as an average overlap for an input distribution within a square {|〈xL〉|,|〈pL〉|≤dmax} with all input states with the mean values within the square having equal probability. The squeezing of the pure input states is fixed to the experimental value of 6 dB, and all phases of squeezing are allowed. The upper bound values on the classical benchmark fidelity are plotted in Fig. 2. The benchmark values have been obtained by first truncating the Hilbert space to a finite photon number and then solving the finite-dimensional optimization employing semi-definite programming. The result of the truncation of the Hilbert space is the constant upper bound of 0.45 of the benchmark for dmax>3.5 (see Supplementary Information), whereas the actual benchmark decreases further for a larger dmax (the benchmark for an infinite Gaussian distribution28 of displacements is 0.38).

For the experimental alphabet with dmax=3.8 the experimental fidelity (the average overlap integral for the top eight representative states in Table 1) is higher than the classical benchmark value, which proves the true quantum nature of the memory.

Outperforming the classical benchmark means that our memory is capable of preserving entanglement in the case when one of the two entangled modes is stored whereas the other is left propagating, which is the case, for example, in the quantum repeater. Using experimentally obtained values of the added noise we evaluate the performance of our memory for the protocol where the upper sideband mode is stored in one of the memory cells whereas the other entangled mode is left as a propagating light mode. We find the Einstein–Podolsky–Rosen variance between the stored mode and the propagating mode to be 1.52 (−1.2 dB below the separability criterion), which corresponds to the lower bound on the entanglement of formation of 1/7ebit (see Supplementary Information for details of the calculation). This version of the memory can be implemented by splitting the and modes, which can be accomplished by a narrow-band optical cavity.

We have experimentally demonstrated the deterministic quantum memory for continuous-variable multiphoton entangled states. The present memory lifetime is limited by collisions and residual magnetic fields leading to the decoherence time of 30 ms. The fidelity can be further improved by means of reduction of the reflection losses, increasing the initial atomic spin squeezing and reduction of the atomic decoherence. Our approach is feasible also for non-Gaussian continuous-variable multiphoton states. Stored states can be directly processed in the atomic memory, as in the repeater scheme18, measured, as in multiround protocols22, or transferred onto another processor by means of teleportation3.



Atomic memory state tomography is carried out by measuring the xL of the verification pulse equation (1). We run a series of measurements of pAfin and xAfin, the latter carried out after applying a magnetic π-pulse before the verification pulse, for many copies of the same input state of light. Owing to the Gaussian statistics of the states, the mean values and the variances of xAfin and pAfin are sufficient for a complete description of the atomic state.


Before carrying out the storage, we calibrate the interaction strength κ and the feedback gain g, such that the mean values of the light state inside the memory (that is, after the entrance loss) are transferred faithfully. κ is calibrated by creating a mean value 〈pLinput〉 in the input pulse. The mean is stored in the atomic xA, which is read out after the magnetic π-pulse with the probe pulse. The measured mean of the probe pulse is then 〈xL′probe〉=κ2pLinput〉, from which κ2 is determined. Using similar methods we can calibrate the electronic feedback gain g.

Generation of the displaced squeezed input states.

The OPA, which is pumped by the second harmonic of the master laser and generates the entangled squeezed vacuum states, is seeded with a few microwatts of the master laser light with the carrier frequency ω0, which is amplitude and phase modulated by two EOMs at a frequency of 322 kHz, thus creating coherent states in the ±322 kHz sidebands around ω0 (Fig. 1). With such a modulated seed, the output of the OPA is a displaced two-mode squeezed state. The output of the OPA is mixed at a polarizing beam splitter with the strong local oscillator (driving) field from the master laser.


To calculate the mean values and variances of the stored state and the input states, we need to know the optical losses. The total losses ηtot can be divided into three parts, the channel propagation transmission ηtr from the OPA to the front of the memory cells (including the OPA output efficiency 0.97), the entrance transmission ηent and the detection efficiency ηdet, such that ηtot=ηtr·ηent·ηdet (all of the η terms are intensity transmission coefficients). From the measurement of the quadratures of the squeezed light (with variances 0.29 (1) and 1.34 (6)), we find the total losses ηtot=0.567(35). We measure the transmission through the cells of 0.817 (20), the transmission through the optics after the cells of 0.889 (10) and estimate the efficiency of the photodiodes to be 0.98 (1). Assigning one-half of the losses through the cells to the entrance losses and the other half to the detection losses we find , and ηtr=ηtot/(ηentηdet)=0.80(4).