Abstract
A quantum memory for light is a key element for the realization of future quantum information networks^{1,2,3}. Requirements for a good quantum memory are versatility (allowing a wide range of inputs) and preservation of quantum information in a way unattainable with any classical memory device. Here we demonstrate such a quantum memory for continuousvariable entangled states, which play a fundamental role in quantum information processing^{4,5,6}. We store an extensive alphabet of twomode 6.0 dB squeezed states obtained by varying the orientation of squeezing and the displacement of the states. The two components of the entangled state are stored in two roomtemperature cells separated by 0.5 m, one for each mode, with a memory time of 1 ms. The true quantum character of the memory is rigorously proved by showing that the experimental memory fidelity 0.52±0.02 significantly exceeds the benchmark of 0.45 for the best possible classical memory for a range of displacements.
Main
The continuousvariable regime represents one of the principal avenues towards the realization of quantum information processing and communication^{4,5,6}. In the optical domain it operates with wellknown optical modulation and detection techniques and allows for deterministic quantum operations. In the atomic domain it has been developed on the platform of atomic ensembles^{2,3,7}. Advances in the realization of continuousvariable quantum protocols include unconditional quantum teleportation involving light^{8} and atoms^{9}, a number of results on memory^{2,3,10,11} and quantum key distribution^{12}. Hybrid continuous/discretevariable operations^{13,14,15} paving the road towards continuousvariable quantum computation^{16,17} have also been reported. However, the ability to store nonclassical continuousvariable states of light is crucial to enable further progress, in particular, for continuousvariable linear optics quantum computing with offline resources^{17}, continuousvariable quantum repeaters^{18,19}, entanglementenhanced quantum metrology, iterative continuousvariable entanglement distillation^{20}, continuousvariable clusterstate quantum computation^{21}, communication/cryptography protocols involving several rounds^{22} and quantum illumination^{23}. Compared with a number of impressive results reporting discretevariable quantum memories at the singlephoton level (see reviews^{1,2,3} and references therein), there have been very few experiments towards quantum memory for continuousvariable nonclassical states. A fractional, 20 nsec, delay of 50 nsec pulsed continuousvariable entangled states in the atomic vapour has recently been demonstrated^{24}. Memory based on electromagnetically induced transparency for a squeezed vacuum state has been reported^{25,26}, albeit with the fidelity below the classical benchmark^{27}. Very recently, classical benchmarks for storing displaced squeezed states^{28,29} have been derived, which made experimental implementation of such storage a timely challenge. Such states form an alphabet for continuousvariable quantum information encoding^{16}. 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 twomode 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 continuousvariable 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_{±}=ω_{0}±ω_{L}, where ω_{0} is the carrier frequency of light. The entanglement condition for this Einstein–Podolsky–Rosen state^{6} 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(ω_{L}t) mode ; and the corresponding sin(ω_{L}t) mode. Before the input state of light undergoes various losses it is a 6 dB squeezed state. In the photonnumber representation for the two modes, the state is Ψ〉=0.80〉_{+}0〉_{−}+0.481〉_{+}1〉_{−}+0.292〉_{+}2〉_{−}+0.183〉_{+}3〉_{−}+⋯. The displaced squeezed states are produced (Fig. 1a) using an optical parametric amplifier^{31} (OPA) with the bandwidth of 8.3 MHz and two electrooptical modulators (EOMs; see the Methods section for details). The alphabet of quantum states Ψ_{i}, which we refer to as ‘initial pure states’ (see the inset in Fig. 2 and Table 1) is generated by displacing the twomode squeezed vacuum state by varying values [〈x_{L}〉;〈p_{L}〉]=[0,3.8,7.6;0,3.8,7.6] and by varying the orientation of the squeezed quadrature between and .
The two photonic modes are stored in two ensembles of caesium atoms contained in paraffincoated glass cells (Fig. 1a) with the groundstate coherence time around 30 ms (ref. 3). ω_{0} is bluedetuned 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 groundstate spins with the Larmor frequency ω_{L}=2π·322 kHz. This ensures that the atoms efficiently couple to the entangled ω_{±}=ω_{0}±ω_{L} sidebands of light^{3}. The two ensembles 1(2) are optically pumped in F=4,m_{F}=4(−4) states, respectively, which leads to the opposite orientation of their macroscopic spin components J_{x1}=−J_{x2}=J_{x}.
The atomic memory is conveniently described by two sets c,s of nonlocal, that is, joint for the two separate memory cells, canonical atomic operators , where the superscript rot denotes spin operators in a frame rotating at ω_{L}. It can be shown that the cosine and sine light modes couple only to the atomic c and s modes, respectively^{32}. 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 xpolarized modes and a strong ypolarized 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 spinpolarized atoms under our experimental conditions can be described by the equations^{32}:
where the coupling constant κ is a function of light intensity, density of atoms and interaction time, and Z^{2}=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 Z^{2}. 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 spinsqueezed state (SSS). The spinsqueezed state with Var(x_{A})=0.43(3) and Var(p_{A})=1.07(5) is generated^{3,7} starting from a (nearly) coherent spin state with Var(x_{A})=0.55(4) by the sequence (Fig. 1b—preparation of initial state) of the ‘quantum nondemolition spinsqueezing pulse’ and a feedback into both the c and s modes. The feedback is achieved with pulses of the magnetic field at the frequency ω_{L} applied to the two cells. The memory input light pulse is then sent, followed by the measurement of the output light operator x_{L}^{′} by the polarization homodyne detection. The measurement result is fed back onto the p_{A}^{′} with a gain g. The resulting x_{A}^{fin} and p_{A}^{fin} for the optimized g and κ=1 can be found from equation (1)
In the absence of decoherence the operator x_{L} is perfectly mapped on the memory operator p_{A}^{fin}. The operator p_{L} is stored in x_{A}^{fin} with the correct mean value 〈p_{L}〉=〈x_{A}^{fin}〉 (as 〈x_{A}〉=0) but with an additional noise due to x_{A}.
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 photonecho approaches where such an adjustment, to the best of our knowledge, has not been demonstrated.
The deleterious noise of the initial state of atoms x_{A} is suppressed by the initial spinsqueezing 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 =0^{0} and =90^{0}, 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 η_{tr}=0.80(4) (which includes the OPA output coupling efficiency 0.97) resulting in the ‘memory input state’, ρ_{in}, with Var(x_{L}·cos()−p_{L}·sin())=0.20(2) and Var(x_{L}·sin()+p_{L}·cos())=1.68(9). The entrance (reflection) losses at the windows of the memory cells lead to further attenuation by the factor η_{ent}=0.90(1). Between the interaction and detection light experiences losses described by the detection efficiency η_{det}=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 x_{A}^{fin} and p_{A}^{fin} 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(x_{A}^{fin}) and 0.38 (11) to Var(p_{A}^{fin}), 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 x_{A}) and 0, for the two quadratures, respectively. This added noise can be due to atomic decoherence, uncancelled noise from the initial antisqueezed p_{A} 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 d_{max}=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 calculated^{28} from the overlap of displaced Ψ_{i} 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 ρ_{in} as the quantum memory that is Ψ_{i} propagated through the transmission channel with η_{t r}=0.80. The benchmark fidelity is found as an average overlap for an input distribution within a square {〈x_{L}〉,〈p_{L}〉≤d_{max}} 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 finitedimensional optimization employing semidefinite programming. The result of the truncation of the Hilbert space is the constant upper bound of 0.45 of the benchmark for d_{max}>3.5 (see Supplementary Information), whereas the actual benchmark decreases further for a larger d_{max} (the benchmark for an infinite Gaussian distribution^{28} of displacements is 0.38).
For the experimental alphabet with d_{max}=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 narrowband optical cavity.
We have experimentally demonstrated the deterministic quantum memory for continuousvariable 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 nonGaussian continuousvariable multiphoton states. Stored states can be directly processed in the atomic memory, as in the repeater scheme^{18}, measured, as in multiround protocols^{22}, or transferred onto another processor by means of teleportation^{3}.
Methods
Verification.
Atomic memory state tomography is carried out by measuring the x_{L}^{′} of the verification pulse equation (1). We run a series of measurements of p_{A}^{fin} and x_{A}^{fin}, the latter carried out after applying a magnetic πpulse 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 x_{A}^{fin} and p_{A}^{fin} are sufficient for a complete description of the atomic state.
Calibrations.
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 〈p_{L}^{input}〉 in the input pulse. The mean is stored in the atomic x_{A}^{′}, which is read out after the magnetic πpulse with the probe pulse. The measured mean of the probe pulse is then 〈x_{L}^{′probe}〉=κ^{2}〈p_{L}^{input}〉, from which κ^{2} 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 ω_{0}, 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 ω_{0} (Fig. 1). With such a modulated seed, the output of the OPA is a displaced twomode 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.
Losses.
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 η_{tot} can be divided into three parts, the channel propagation transmission η_{tr} from the OPA to the front of the memory cells (including the OPA output efficiency 0.97), the entrance transmission η_{ent} and the detection efficiency η_{det}, such that η_{tot}=η_{tr}·η_{ent}·η_{det} (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 η_{tot}=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 onehalf of the losses through the cells to the entrance losses and the other half to the detection losses we find , and η_{tr}=η_{tot}/(η_{ent}η_{det})=0.80(4).
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Acknowledgements
This work was supported by EU projects QESSENCE, HIDEAS, CORNER, COMPAS, EMALI and COQUIT.
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Experimental group: K.J., W.W., H.K., T.F., B.M.N. and E.S.P. Calculation of the classical benchmark: M.O., M.B.P., A.S. and M.M.W.
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Jensen, K., Wasilewski, W., Krauter, H. et al. Quantum memory for entangled continuousvariable states. Nature Phys 7, 13–16 (2011). https://doi.org/10.1038/nphys1819
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DOI: https://doi.org/10.1038/nphys1819
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