# Efficient spatially-resolved multimode quantum memory

## Abstract

We propose a method that enables efficient storage and retrieval of a photonic excitation stored in an ensemble quantum memory consisting of Lambda-type absorbers with non-zero Stokes shift. We show that this can be used to implement a multimode quantum memory storing multiple frequency-encoded qubits in a single ensemble, and allowing their selective retrieval. The read-out scheme applies to memory setups based on both electromagnetically-induced transparency and stimulated Raman scattering, and spatially separates the output signal field from the control fields.

###### pacs:

42.50.Ex###### pacs:

42.50.Ct###### pacs:

42.50.-pOptical implementations of quantum information processing and transfer Quantum description of interaction of light and matter; related experiments Quantum optics

The distribution of entangled states over large distances is a basic requirement for realizing quantum networks and numerous quantum communication protocols [1, 2]. It is also of importance for fundamental experimental tests of quantum theory [3]. As recently pointed out [4] current entanglement distribution protocols using quantum repeater stations between two parties [5, 6] will only work over large distances if significantly improved quantum memories become available. Ideally these memories should also be able to store multiple qubits and allow their selective retrieval [7]. Here we propose a scheme to achieve these goals for photonic qubits stored in a quantum memory (QM) consisting of an ensemble of three level -type absorbers as shown in Fig. 1(a).

Our scheme is based on the approach
[8, 9, 10, 11, 12, 13, 14]
where the qubit is encoded in a *signal* light pulse, and
stored in the ensemble via a classical *control* field [see
Fig. 1(a)] propagating colinearly with the signal
in the positive -direction [as defined in
Fig. 1(b)]. Storage of the signal produces a
highly-asymmetric spin wave in the medium [15, 16].
After some time the pulse can be retrieved by applying another
control field. If this field also propagates in the positive
-direction, the spin wave of the medium has a low overlap with
the read-out mode, resulting in a low memory efficiency unless one
resorts to a much larger coupling strength for read-out. Instead
one can reverse the read-out control field, which maximizes this
overlap. However for non-zero Stokes shift [see
Fig. 1(a)] the memory efficiency suffers due to
momentum conservation. It is desirable to have
for independent addressability of the transitions
and
, as selection rules are rarely
stringent enough to ensure this. Furthermore, in colinear quantum
memories one also has to spectrally separate the retrieved signal
field from the stronger control field. For a weak signal this
filtering is difficult, and another method for resolving the
fields, such as orienting the control field slightly off-axis
[17, 18], would be advantageous.

The aim of this paper is to overcome these limitations in the
read-out. Our proposal applies to QMs based on both stimulated
Raman scattering [19, 20, 15] and
electromagnetically-induced transparency (EIT)
[8, 11, 13, 14]. We show that
using our method, a multimode QM can be realized whereby
frequency-encoded qubits are stored in and retrieved selectively
from a single ensemble. This would allow manipulation of stored
qubits, for example in an optical lattice QM [21],
without needing to use multiple ensembles for the different
logical states. The number of modes that can be stored in this way
is limited experimentally by angular resolution and the achievable
frequency of signal and control fields. In a conventional
single-mode memory the signal and control pulses are colinear and
the levels are chosen so that state is energetically
lower than . In this case , and the
stored spin wave has a momentum
(we call this a
*phase mismatch*), rendering backwards read-out of the signal
inefficient. To overcome this (i.e. to *phasematch* the
system) we choose the level configuration shown in
Fig. 1(a), and orient the read-in control
field at an angle of
to the signal
propagation direction [see Fig. 1(b)]. This
eliminates the longitudinal phase mismatch and allows efficient
retrieval of the signal field. Furthermore the output signal field
direction, determined by momentum conservation, is spatially
distinct from the read-out control field. The scheme allows
high-fidelity retrieval of signal pulses with duration
provided that , and , with the
length of the ensemble and the speed of light. In the
following we detail the application of this method to a general
QM, including a discussion of the constraints that arise and the
storage times that can be achieved. We then show how our scheme
could be used to implement a multimode Raman QM consisting of a
single ensemble, and derive an expression for the maximum number
of modes it is possible to store.

We consider the setup shown in Fig. 1. The classical control field for read-in is oriented in the - plane at an angle of to the signal. If is small, the walk-off between the two fields can be neglected. The control field envelope can then be represented at time and longitudinal position by the slowly varying Rabi frequency , where . The frequency of the transition is . We introduce dimensionless coordinates for the longitudinal position; , with the integrated Rabi frequency, for the time; and , where (analogously for ) for the transverse position. We define the slowly-varying operators for the signal field and for the spin wave, where and are the slowly-varying signal and spin-wave amplitudes respectively, and , which describes a Stark shift due to the control field and a modification of the signal group velocity. We decompose the phase mismatch as , with the unit vector oriented along the -axis. The coupling of the signal field to the ensemble is , with the resonant optical depth [16] and , where accounts for dephasing and loss of the optical polarization.

We assume that either or are much larger than the signal bandwidth, the maximum control Rabi frequency, and , and adiabatically eliminate the excited state . Within this limit our theory describes the interaction for arbitrary signal and control pulse shapes in both the EIT and Raman regimes. The absorbers are all initially optically pumped into state , and the population of state is assumed to remain negligible for each absorber. In the slowly-varying and paraxial approximations the Maxwell-Bloch equations are

(1) | |||||

(2) |

where , , and the coupling . Note that these equations are similar to those in [15], except we also include the transverse Laplacian [22].

The dynamics of Eqs. (1) and (2) decomposes into a linear mapping between signal and spin wave modes. The optimally-efficient mapping excites an asymmetric spin wave approximated by exponential decay in . The symmetry of the interaction [15, 16] dictates that the optimal spin wave for forwards retrieval, , is the mirror image in of this, i.e. , where a superscript identifies a quantity associated with the read-out. The efficiency of forward readout then depends on the overlap of with , which is generally low. Switching the propagation direction of the control field, to readout in the backward direction, flips around, so that its overlap with is perfect. However, due to the residual momentum of the spin-wave coherence after storage, we then have , where

The linear -dependence of the real terms in reduces the overlap of the stored spin wave and the output mode, leading to a reduction in the memory efficiency. This is shown in Fig. 2(a) for a cesium QM, where levels and are the and hyperfine levels of the state respectively (i.e. is energetically lower than ). Our strategy is to find a configuration for which is independent of . To do this we choose to be energetically higher than and set , where the angle of the read-out control field is as defined in Fig. 1(b). This phasematches the memory in the Raman and EIT configurations – far from resonance the functions in are real but negligibly small, and on resonance they are imaginary, so do not contribute to the mismatch. Figure 2(b) shows that this results in a high-fidelity read-out in Cs, where now and are the and hyperfine levels respectively. Note that a transverse phase mismatch does not necessarily degrade the readout efficiency; any residual simply modifies the angle at which the retrieved signal field is emitted. However, having requires a modification of the frequency of the control field to ensure that momentum and energy are conserved.

We have assumed that the walk-off along the -axis between the signal and control fields is negligible, i.e. small compared to the pulse durations, so that . A more stringent condition is that the transverse walk-off of the beams is small compared to their beam waists, which requires , with the cross-sectional area of the control pulse. As well as setting the angle , the Stokes shift imposes an upper limit to the control field bandwidth, since there should be no control photons at the signal frequency. Diffraction limits the length of the ensemble according to the condition , where is the control wavelength. These conditions can be combined into , with the maximum storage bandwidth given by . The upper bound on is relaxed if the control field focussing is loosened, but the control pulse energy should then be increased. However, in general, the above considerations imply that the number density required for high-efficiency (90%) phasematched operation of a -ensemble QM is given approximately by , with , in both EIT and Raman configurations.

In a Cs QM the state is split by the hyperfine interaction with the nuclear spin (), to give the (our ), and states (our ), with . The state ( line), which lies () above state , would be used for the excited state . If , then the phasematching angle is . This would enable high-efficiency storage and retrieval of a single photon with in an ensemble with . This bandwidth requires a number density of around , which can be achieved in a cesium vapor at a temperature of . For a thermal vapor we must consider motional dephasing of the memory, as the phase introduced by the read-in control field will average out [23, 24]. The stored spin wave oscillates at the frequency , and so the entanglement fidelity of the memory [25] is given by , with the storage time of the memory, Boltzmann’s constant, and the atomic mass. For the above parameters the maximum storage time with is , i.e. three orders of magnitude larger than the signal pulse duration. Longer storage times could be achieved by using a storage unit consisting of an ultra-cold gas, but this requires a larger sample than has currently been achieved experimentally.

Using the off-axis QM scheme described above, it is possible to store multiple signal fields of different frequencies in a single ensemble, and retrieve them selectively, as demonstrated in Fig. 3. Hence one can implement a multimode QM [7] in which frequency-encoded qubits are stored, and retrieved either one after the other, or at the same time but spatially separated. To see this we consider the same setup as in Fig. 1, but now the signal field consists of two components, labeled 1 and 2, of frequency and respectively. Signal component () is Raman resonant with control field , which has frequency , detuning , and is angled at to the signal for phasematching. Both signal components can be stored in the ensemble, and the resulting spin wave has two components separated in (where is the momentum component in the -direction), due to the different transverse momenta of the control fields [see Fig. 3(a)]. Backwards read-out of each mode can be performed using a control pulse with the appropriate orientation and frequency, in analogy with the single mode case. These control fields could be applied at different times to read out the modes one after the other, or the angles can be changed so that the modes are emitted in different directions [as shown in Fig. 3(b)]. We ensure that , with the bandwidth of signal component , so that signal field cannot initially interact with control field (). This condition requires that is large, i.e. in the Raman regime. However, once a material excitation is created by the absorption of signal , this excitation may interact with control field .

To calculate the form of the stored spin wave and the retrieved signal field, we again assume that the angles relative to the signal field of both control fields are small. Control field has wavevector in the -direction with rescaled magnitude . We assume that control field has Rabi frequency , with the maximum Rabi frequency, an envelope function normalized so that , and the integrated Rabi frequency . Also due to the large number of absorbers in our ensemble, the probability of a single absorber being excited by both signal components is assumed to be negligible. The coupling constant is now , where . The dimensionless operator for signal component is written as , and the spin wave operator is now , where is the slowly-varying amplitude of signal field , and the spin-wave momentum mismatch is now . The complex exponents are , and respectively. The Maxwell-Bloch equations are thus

(3) | |||

(4) |

with , and ; subscripts indicate the signal components to which the quantities refer. The solution for the two-component signal field, which can be achieved in the same way as for the single-field case, is the convolution of the integral kernels for the single-field interaction [15] with terms that arise due to the mixing of the two components. This mixing occurs, for example, when during storage signal component interacts with both control fields by a second-order process, and could potentially degrade the read-in. However, our calculations have shown that the mixing only becomes significant when the read-in coupling is – much larger than the coupling sufficient for signal storage with near-unit efficiency (). For we see that the storage process for the two-component signal field is almost identical to the results obtained when the two signal components are assumed to interact independently of each other. The read-out process also obeys Eqs. (3) and (4), and a similar lack of mixing ensures that the modes can be addressed independently. Note that if the ’s have a relative phase of between them, then the two absorption processes interfere destructively, and the storage efficiency is low. For effective storage of the signal components this phase must be zero, which can be ensured either by changing the sign of the ’s, or by phase-shifting the ’s.

Since one can in principle have arbitrarily-far blue-detuned fields (avoiding any higher levels), the main limitations on the number of modes that can be stored are experimental – the achievable optical frequencies and the possible angular resolution . The maximum control field angle allowed is given by and the number of different control fields allowed is . The smaller angles considered here correspond to the fields being far blue-detuned, which may be outside the optical regime. Taking this into account, the number of modes that can be stored is given by , with the largest allowed blue detuning from . For the Cs QM discussed previously, taking the angular resolution to be , setting , and restricting so that no other transitions are excited, this corresponds to modes. As with the single-mode case the phase introduced by control field is . Hence the limitation on storage time of the memory due to dephasing, as calculated earlier for a single mode, is given by the entanglement fidelity of the mode that requires the smallest control field angle for phasematching.

In summary, we have demonstrated a QM scheme using an ensemble of three-level -type absorbers that allows efficient storage and phase-matched spatially-resolved read-out of the signal field. We use this method to demonstrate how a multimode QM could be implemented using a single atomic ensemble. Multimode memories will enable the storage of multiple qubits in one memory setup, significantly decreasing the resources required for entanglement distribution protocols. Furthermore, storing multiple qubits in a single ensemble opens up new possibilities for scalable entanglement manipulation.

This work was supported by the EPSRC through the QIP IRC (GR/S82716/01) and project EP/C51933/01, and in part by the National Science Foundation under Grant No. NSF PHY05-51164. JN thanks Hewlett-Packard and FCW thanks Toshiba for support. IAW was supported in part by the European Commission under the Integrated Project Qubit Applications (QAP) funded by the IST directorate as Contract Number 015848.

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