Abstract
A device being a pinnacle of development of an optical quantum memory should combine the capabilities of storage, intercommunication, and processing of stored information. In particular, the ability to capture a train of optical pulses, interfere them in an arbitrary way and finally, perform ondemand release could realize arbitrary optical computation. Here we demonstrate the operation of a coherent optical memory being able to store optical pulses in the form of collective spinwave excitations in a twodimensional wavevector space. During storage, we perform complex beamsplitter operations and demonstrate a variety of protocols implemented at the processing stage, including realtime controlled interference of a pair of spinwave modes with 95% visibility. The highly multimode structure of the presented memory lends itself to enhancing classical optical telecommunication, as well as parallel processing of optical qubits at the singlephoton level.
Introduction
As optical quantum memory technologies mature, the range of their applications increases. Basic memories operating in a single temporal and spatial mode can store only a single optical pulse, which may interfere with another incoming pulse. This operation can only be performed in the presence of the coupling light.^{1} Such memories, based either on Raman scattering or electromagnetically induced transparency (EIT), can achieve high efficiencies,^{2} but offer very limited capacity as multiplexing is limited by the number of atomic magnetic sublevels employed.^{2,3,4} While a single atomic ensemble may be split into an array to offer parallel storage of light,^{5} such a scheme hinders manipulations within the memory as communication between memory cells must be inherently lightbased. It is thus highly desirable to independently store many optical pulses within the same group of atoms. Such a multiplexing scheme may utilize either the spatial^{6,7} or temporal degree of freedom.^{8,9,10} In the latter case, considered in the context of the atomicensemble based quantum memories, the Gradient Echo Memory (GEM)^{11,12,13,14,15} scheme stands out as an efficient way to engineer the phasematching at readout stage to achieve modeselective storage and retrieval. A similar feature is inherently offered by the atomic frequency comb (AFC) memories based on ensembles of ions in solids^{8,9,16} thanks to their large inhomogeneous broadening. In the spatial degree of freedom atomic ensembles allow storage of light in many angularemission modes through spinwave wavevector multiplexing.^{6,17} These schemes allow storage of hundreds of optical modes, also when used with nonclassical states of light.
Manipulation of stored optical pulses, however, remains a substantial challenge, both from technical and fundamental points of view. The AFC memory has been demonstrated to allow preprogrammed interference of two stored pulses with a single output port.^{18,19} Most recently similar beamsplitter operations in the AFC memory have also been implemented in a multidimensional Hilbert space.^{20} Within the GEM scheme a beamsplitter operation between preselected, stored pulses and an input pulse has been realized.^{21,22} Remarkably, these schemes allow basic spectral and temporal manipulations of stored light. More work is needed however to reach the regime of efficient and arbitrary manipulations. In particular, the acStark shift caused by an additional light field has been studies in opticallattice based quantum memories,^{23} as well as used and proposed as a versatile way to realize the GEM scheme^{24} and to control photonechoes.^{25} Recent theoretical proposals went beyond the simple gradient shape and suggested engineering the stored spinwave shape to realize Kapitsa–Dirac diffraction^{26} or a quantum memory protected with a disordered password.^{27} These schemes stand out in relation to an established alloptical technique of manipulating stationary light^{28}—since control light is not present during the manipulation process, decoherence is significantly reduced. Finally, a recent experiment used the acStark shift to realize a spinwave beamsplitter at the singleexcitation level demonstrating Hong–Ou–Mandel interference for stored light.^{29}
Here we present the realization of the acStarkbased spinwave multiport interferometric processor (SMIP) and join the advantages of the transverse wavevector and temporal multiplexing to realize a variety of operations on the stored coherent states of light. The previously untackled regime of complex light patterns used to engineer spin waves is explored, which allows us to tap into the full threedimensional potential of the wavevectormultiplexed coherent optical memory. We show that thanks to the engineering of the spatial profile of acStark modulation the stored pulses may be processed, interfered, and conditionally retrieved. The scheme features both reprogrammable reordering and interference of pulses within the multipleinput, multipleoutput paradigm, essential to realize true unitary operations. In the paper, we first introduce the protocol by deriving its theoretical principles and realizing a scheme reminiscent of the Gradient Echo Memory.^{12} Next, we realize a series of programmable beamsplitting experiments in spatial and temporal degrees of freedom. Highvisibility interference of a pair of modes is demonstrated. Finally, we propose potential further applications and give technical details of the experiment and lightatom propagation simulations involved.
Results
Operation of the lightatom interface
The coherent atomic optical memory based on an elongated (σ_{x} × σ_{y} × σ_{z} ≈ 0.15 × 0.15 × 2.5 mm^{3}) ensemble of Rb87 atoms employs a strong control field \({\cal E}_C\) to map a weak signal field \({\cal E}_{in}\) onto the atomic coherence ρ_{hg} between the two metastable ground states, for which we take g〉 → F = 1, m_{F} = −1, and h〉 → F = 2, m_{F} = 1 (see Fig. 1c for the atomic level scheme). In the experiment the atoms are first optically pumped to the g〉 state and control and signal fields (both λ = 780 nm) operate with opposite circular polarizations. Typically we use 300 ns long pulses for storage and retrieval of atomic coherence. While the interaction is well characterized by a set of coupled MaxwellBloch equations (see Methods), first we rather choose to describe the atomlight coupling qualitatively. In particular, Fig. 1b illustrates the geometry in which the coupling and signal fields copropagate through an elongated atomic ensemble. Assuming that the coupling beam diameter is significantly larger than the transverse size of the ensemble, we may actually solve the coupled equations within the first order in the coupling strength and obtain a simple result by which a signal \({\cal E}(k_x,k_y)\) couples to an atomic coherence
where \(K_{z0} = \sqrt {\omega ^2{\mathrm{/}}c^2  k_x^2  k_y^2}  \omega _C{\mathrm{/}}c\), δ is a Dirac delta function and ω and ω_{C} are frequencies of signal and coupling fields, respectively, and c is the speed of light. For k_{x} = k_{y} = 0 the longitudinal wavevector simplifies to a constant component K_{z0} = Δ_{0}/c ≈ 0.14 rad mm^{−1}, where Δ_{0} ≈ 2π × 6.8 GHz is the nominal frequency splitting between levels g〉 and h〉. To exclude these trivial dependencies from further consideration we will define the stored spinwave excitation as
where \({\cal F}\) stands for the Fourier transform in the spatial domain, N(x, y, z) is the atom number density and \({\ast}\) denotes convolution (here in the wavevector space). Importantly, after mapping the optical field we obtain a spinwave excitation at K_{z} = 0 in terms of S.
The process of reverse mapping or retrieval driven by the same coupling field occurs in a symmetric way. Essentially, an atomic spinwave excitation will be mapped onto an optical field proportional to S in terms of the transverse wavevector dependence only if K_{z} = 0. This requirement arises due to the phasematching condition. In particular, the allowed spread along K_{z} is inversely proportional to the atomic cloud length σ_{z} and most importantly spin waves with large K_{z} component (K_{z}σ_{z} ≫ 1) will remain stored in the memory. This remains true unless we change the frame of reference significantly by selecting much different K_{x}, K_{y} and change them substantially by δK_{x}, δK_{y}, as the actual phase matching is satisfied on a K_{z} = 0 plane only for original (unmodified) spin waves. When we modify the transverse wavevector component the plane tilts and for large K_{x} (K_{y}) the spin wave becomes phasemismatched. However, for small K_{x} (K_{y}) and δK_{x} (δK_{y}) the phasemismatch \({\mathrm{\Delta }}K_z \approx \frac{\lambda }{{4\pi }}\left( {\left( {K_x + \delta K_x} \right)^2  K_x^2} \right)\) is small and thus we may use the phasematching planar approximation to learn which spin waves are retrievable.
Spinwave manipulation with the acStark effect
As discussed above, only a limited space, or more precisely a thin threedimensional volume around K_{z} = 0 plane in the wavevector space may be populated by spin waves by means of Raman interaction. To manipulate the spin waves within and beyond this volume we use an additional far offresonant beam (marked in Fig. 1c as acS) that induces an additional differential acStark shift between levels g〉 and h〉 of Δ_{acS} that adds to Δ_{0}. The acStark beam propagating along the ydirection is zpolarized and reddetuned by ~1 GHz from the h〉 → e〉 transition induces Δ_{acS} ~ 1 MHz acStark shift with ~100 mW beam power. This shift causes the atomic coherence ρ_{hg}, and thus the spin wave, to accumulate an additional phase φ_{acS} = Δ_{acS}T over the interaction time T. Typically we use acStark pulses of approx. T ~ 2 μs duration. By spatially shaping the acStark beam intensity I_{acS}(x, z) (see Methods and refs. ^{29,30} for details) we induce a spatiallydependent phase shift φ_{acS}(x, z) ∝ I_{acS}(x, z), which due to the geometry of the experiment is limited to two dimensions (see Fig. 1b). Any spin wave is then reshaped as:
A basic example is an acStark analog of the GEM, in which a phase shift linear in z (φ_{acS} = βz) shifts the spin wave in the K_{z} direction by β.
Here we work with nearly planewave optical modes characterized by small wavevector (angular) spread and therefore the spin waves are welllocalized in the wavevector space. Discrete mode transformation in such a space are most conveniently performed by a spatially periodic acStark modulation. Taking a spatial period 2π/k_{acS} of the modulation in the form \(\varphi _{{\mathrm{acS}}}(x,z) = \varphi _{{\mathrm{acS}}}^{{\mathrm{periodic}}}({\bf{k}}_{{\mathrm{acS}}} \cdot (x,z))\) we may rewrite the spinwave transformation (3) using Fourier series as:
with Fourier coefficients c_{n} defined as:
Figure 1a presents a simplified simulation (assuming perfect writein and readout) of a protocol operation within this paradigm. In this exemplary protocol three pulses \({\cal E}_0\), \({\cal E}_{  k}\), \({\cal E}_k\) with transverse optical wavevector component k_{x} equal 0, −k and k, respectively, are stored, processed and released from the memory. When the first pulse is mapped to the ensemble, the created spin wave is phasemodulated using a sawtoothshaped (saw) (periodic) modulation \(\varphi _{{\mathrm{acS}}}(x,z) = \varphi _{{\mathrm{acS}}}^{{\mathrm{saw}}}(k_{{\mathrm{saw}}}z)\). Such a modulation with amplitude equal 2π shifts the spin wave in the K_{z} direction by k_{saw}, making the spin wave unreadable, as k_{saw}σ_{z} ≫ 1. Next, two pulses with k_{x} components separated by 2k arriving at the same time are written to the ensemble. Then, squareshaped (sq) modulation \(\varphi _{{\mathrm{acS}}}(x,z) = \varphi _{{\mathrm{acS}}}^{{\mathrm{sq}}}(k_{{\mathrm{sq}}}x)\) with k_{sq} = 2k is applied. The amplitude and phase of this modulation are chosen so that the first three Fourier coefficients are in following relations: \(c_{  1}^{{\mathrm{sq}}} = c_0^{{\mathrm{sq}}} =  c_1^{{\mathrm{sq}}}\). This way the two pulses with nonzero k_{x} component are combined in a way reminiscent of a two mode beamsplitter transformation^{29} resulting in constructive interference in the mode with k_{x} = −k. Note that the spin wave component corresponding to the first pulse is transformed (split) as well, but because of shifted K_{x} and most importantly K_{z} component it does not take part in the interference. At this stage the first readout is performed during which all readable (zero K_{z}) spin wave components are converted to light pulses. To read out the first pulse, all the previous transformations are undone by applying versions of previous modulations shifted by half a period in reversed order. At the very end of the protocol, the first pulse is retrieved.
Reconfigurable acStark Echo Memory
For the experimental demonstration we begin by moving the spin waves outside the zero K_{z} to allow storage of subsequent incoming optical pulses. This configuration, most reminiscent of the GEM^{12} and already proposed as a way to reverse pulses order in the twolevel GEM,^{26} here operates best with a triangleshaped grating (pattern A), which can be conventionally written in a closed form:
with k_{tri} = 9.6 rad/mm \(\hat e_{z}\) for which most essentially the zeroth order \(c_{0}^{{\mathrm{tri}}} \propto \left {{\mathrm{sinc}}({\cal A}{\mathrm{/}}2)} \right\) disappears periodically with modulation strength \({\cal A}^{{\mathrm{tri}}}\) (with period equal 2π) except for \({\cal A}^{{\mathrm{tri}}} = 0\). With this scheme, we may thus apply a grating with \({\cal {A}}^{\mathrm{tri}} = 2{\pi}\) and move the pulse out of the K_{z} = 0 plane. Due to the periodicity of \(c_{0}^{\mathrm{tri}}\) in the modulation strength \({\cal A}^{{\mathrm{tri}}}\), if a subsequent pulse is stored, the first and any previous pulse remains phasemismatched at consecutive grating operations with amplitude \({\cal A}^{{\mathrm{tri}}} = 2\pi\). To retrieve the pulses we apply a pattern with the same amplitude shifted by a half of period in the spatial domain (pattern B), that restores the spin waves to the K_{z} = 0 plane.
The scheme lends itself to both firstin, firstout (FIFO) and lastin, firstout (LIFO) operation, as shown in Fig. 2. For the FIFO operation on two pulses, after storage of a second pulse, we apply a shifted pattern B to simultaneously transfer the first pulse back to the K_{z} = 0 plane and phasemismatch the second pulse. After the first retrieval operation, the phase matching is restored for the second pulse with pattern A.
The efficiency of our memory is currently limited by the optical depth of the ensemble as well as available coupling power. By comparing the intensity of light at the input and output of the memory we obtain writein efficiency for the first pulse of about 59 and 44% for the second pulse. For immediate retrieval (as for the second pulse in LIFO scheme) we achieve 35% efficiency, while net storage and retrieval efficiency is equal 44% × 35% = 15%. For the pulses that are manipulated the efficiency is diminished by dephasing due to the acStark light intensity inhomogeneities.^{30}
Programmable beamsplitting of stored optical pulses
To demonstrate the beamsplitting capability for pulses arriving at different times we use again the trianglewave modulation (tri) in the zdirection, with k_{tri} = 22 rad/mm. After storage of two subsequent pulses (which is done the same way as in FIFO and LIFO demonstration using pattern A) we apply the shifted pattern B for a half of period T, modulating the spin wave with amplitude π instead of 2π (see Fig. 2d). This way the two pulses are combined and K_{z} = 0 component of resulting spin wave becomes the first output port of the temporalmode beamsplitter. Then, after the first readout, we modulate the remaining part again with pattern B with amplitude \({\cal {A}} \approx {2.25}{\pi}\) to transfer a part of the second port to readable K_{z} = 0 plane, then the second readout is performed. Note that it is crucial to always perform the first readout as otherwise the unread spin wave will interfere and spoil the operation of the second output port. It is thus necessary to simulate the operation of this scheme to a full extent, including possibly imperfect first readout which can affect the second output port.
To characterize the interference of two pulses we change the relative phase between the pulses by varying the twophoton detuning δ. Fringes in the intensity registered in the two output ports appear accordingly. Essentially, the phase difference between the two interfering spin waves is the product of the twophoton detuning δ and the time between two storage operations τ. Furthermore, as we move outside the twophoton resonance the interaction becomes inefficient. This behavior is reminiscent of the wellknown Ramsey interference. In Fig. 2f we plot the total number of photons collected after the first (port a) and second (port b) readout as a function of the twophoton detuning δ. The observed behavior is properly predicted by the simulation described in detail in Methods. The maximal efficiency achievable in this case (triangleshaped grating) calculated using the provided Fourier formalism assuming perfect readout is 81% and 60% for port a and b respectively.
The relative phase between the pulses can be also modified within the spinwave domain. To demonstrate this, we implement another interference protocol; instead of splitting the first pulse into many orders we simply shift its K_{z} component by k_{saw} using sawtooth wave modulation:
in the zdirection. Then, the second pulse is written to the memory and the resulting spin wave is modulated using a triangleshaped grating of depth \({\cal {A}}^{{\mathrm{tri}}} \approx 1.16{\pi}\) satisfying the equation \(\left {c_0^{{\mathrm{tri}}}} \right = \left {c_1^{{\mathrm{tri}}}} \right = \left {c_{  1}^{{\mathrm{tri}}}} \right \approx 0.53\). The spatial period of the tri modulation is chosen to satisfy \({\bf{k}}_{{\mathrm{tri}}} = {\bf{k}}_{{\mathrm{saw}}} = 22\,{\mathrm{rad/mm}}\,\hat e_z\), thus the pulses are combined in such a manner that the zeroth order of the first pulse overlaps with first diffraction order of the second pulse and vice versa. The first interferometer port is again a resulting K_{z} = 0 spin wave component, so it can be completely readout without any additional manipulations. The second port this time is well defined and lies at a plane with K_{z} = k_{tri}. Thus, in principle the second port could be restored completely by applying reversed sawtooth pattern shifting back the spin wave by −k_{saw} to the readable region in wavevector space. Due to our setup limitations (see Methods) we probe the second port by applying the tri modulation with an amplitude equal π and subsequently the phasematched component (K_{z} = 0) is released. The relative phase between interfering components can be manipulated by changing the phase of one of the gratings (saw or tri), as for any shifted periodic modulation \(\varphi _{{\mathrm{acS}}}^{{\mathrm{periodic}}}(\xi  \zeta )\) the complex amplitudes of subsequent orders change as c_{n} ~ e^{inζ}. We directly witness this behavior by shifting the sawtooth grating portrayed in Fig. 3a in the zdirection and measuring interference fringes in the total energy of the released pulses. In Fig. 3c we plot the resulting interference pattern, accompanied by a proper simulation, showing the interference in wavevector space (Fig. 3b). Theoretical maximal efficiency taken as the ratio of cumulative energy of two output ports to total input energy under perfect readout conditions is however limited in this case by \(2\left {c_0^{{\mathrm{tri}}}} \right^2 \approx 56{\mathrm{\% }}\). Yet, the rest (unread part) of the spin wave remains in the memory and could be in principle retrieved using proper manipulation.
Transverse space interference and manipulation
To go beyond a single transverse mode we now add the K_{x} dimension to the scheme. In a simple yet highly robust scenario, we map two equally bright pulses arriving at the same time yet into two different spin waves with K_{x} = ±k, where k = 75.4 rad/mm. We then apply a sinusoidal grating modulation (pattern presented in Fig. 4a):
with \({\bf k}_{\mathrm{sin}} = 2k\hat e_x\) and \({\cal {A}}^{{\mathrm{sin}}} \approx 0.92\pi\) that again satisfies \(\left {c_0^{{\mathrm{sin}}}} \right = \left {c_1^{{\mathrm{sin}}}} \right = \left {c_{  1}^{{\mathrm{sin}}}} \right = {\cal C} \approx 0.55\). In this way the output ports at K_{x} = ±k are mixtures of both input ports in the 50:50 ratio. We then again use the fact that shifting the grating position ζ changes the phase at orders ±1 by ±ζ. We may thus write the (lossy) beamsplitter transformation as:
We scan the phase using a piezoactuated mirror mount in the far field of the ensemble (see Methods for details of the imaging setup) and observe highvisibility interference fringes. Notably, we obtain average visibility of 95% by comparing maximum and minimum intensities observed at each port, as portrayed in Fig. 4b. As in the previous section, the theoretical maximal efficiency in this case (assuming perfect readout) is limited by \(2{\cal C}^2 \approx 60{\mathrm{\% }}\).
Simultaneous spinwave processing in two dimensions
Finally, we combine the longitudinal and transverse manipulations to exhibit the timespace interference of two sequentially stored pulses. To access the timespace beamsplitting we design the squarewave grating
in the x − z direction \(\varphi _{{\mathrm{acS}}}^{{\mathrm{sq}}}({\bf{k}}_{{\mathrm{sq}}} \cdot (x,z))\), where \({\bf{k}}_{{\mathrm{sq}}} = 12\,{\mathrm{rad/mm}}\,\hat e_x + 5\,{\mathrm{rad/mm}}\,\hat e_z\). The periodic collapserevival behavior of a \(c_0^{{\mathrm{sq}}} = \left {{\mathrm{cos}}({\cal {A}}^{{\mathrm{sq}}}{\mathrm{/}}2)} \right\) allows us to use this very grating for both subsequent storage and interference of two coherent pulses. By applying the squarewave modulation (sq, pattern A in Fig. 5a) of amplitude \({\cal {A}}^{{\mathrm{sq}}} = \pi\) after arrival of the first pulse and \({\cal {A}}^{{\mathrm{sq}}} = \pi {\mathrm{/}}2\) after the second pulse is stored we combine the pulses in t − x space. As in previous cases, the K_{z} = 0 component of the processed spin wave becomes the first port of the Mach–Zehnder interferometer. To sample the second port (which is distributed into successive diffraction orders \(c_{n \ne 0}^{{\mathrm{sq}}}\)) we use sawtooth grating in the zdirection \(\varphi _{{\mathrm{acS}}}^{{\mathrm{saw}}}\left( {k_{{\mathrm{saw}}}z} \right)\) with \(k_{{\mathrm{saw}}} = {\bf{k}}_{{\mathrm{sq}}} \cdot \hat e_z = 5\,{\mathrm{rad/mm}}\) (pattern B in Fig. 5b) to transfer the 1st order to readable K_{z} = 0 space, and perform the readout. The simulation reveals that in this protocol the two output ports turn out to not be in perfectly opposite phases. The same behavior is observed in the experiment, as demonstrated in Fig. 5d. We attribute this effect to imperfect retrieval of the first port which in turn partially leaks to the second readout operation. We envisage that further simulations will facilitate a more elaborate scheme that could yield two output ports that are perfectly in opposite phases, as in experiments described in Sec. “Programmable beamsplitting of stored optical pulses” and “Transverse space interference and manipulation”. In this case, the theoretical maximal efficiency for perfect readout is 100% for the first and 40% for the second port. However, as already mentioned we only sample the second port using sawtooth grating. We envisage that using more complex grating the second port could also be retrieved with 100% efficiency in the ideal case.
Discussion
We have demonstrated a reprogrammable device that processes atomic spin waves through interference. Starting with the application of the acStark shift control facilitating a temporally multimode coldatomic memory for light, we have extended the concept of acStark control to enable interference of coherent spinwave states stored in the memory. In particular, the processing is performed simultaneously in two dimensions of the wavevector space. With this, we simultaneously exploit temporal and spatial multiplexing. We show how to perform spinwave interference between light pulses stored both at different times, as well as sent to the memory at different angles. By switching only a pair of patterns we achieve a substantial degree of reprogrammability and control (see Fig. 6), which paves the way towards creating complex unitary quantum networks through spinwave interference.
The demonstrated SMIP lends itself to many critical schemes in quantum and classical telecommunication, including the quantum memoryenabled superadditive communication^{31,32,33,34} or implementation of a receiver operating with an error rate below the standard quantum limit^{35}, as well as quantum metrology through collective measurements on many optical pulses.^{36,37} The ability of programming interference of stored states provides a robust tool for probing fundamental properties of quantum systems. Recently, a tunable beamsplitter transformation has been used to demonstrate Hong–Ou–Mandel interference between two microwave quantum memories.^{38} The techniques presented here pave the way towards programmable complex interference experiments which can be used to reveal fundamental properties of a given quantum system.
The acStark control owes its versatility to the possibly a very high speed of switching and operation, as compared with the magnetic field gradients. This feature makes it applicable to recently developed shortlived quantum memories that operate in the ladder atomic scheme in warm atomic vapors and achieve very low noise levels.^{39,40} The high speed of the acStark control also facilitates realtime feedback processing that could lead to the realization of an even broader class of operations, including enhanced singlephoton generation through multiplexing.^{6,41,42,43} Here, such a scheme could also include engineering of photonic spatial and temporal mode. This could be taken even further with techniques used in stationarylight experiments, where the amplitude of the stored spinwave is nondestructively reshaped using a multilaser field.^{44,45}
The universality of the current approach is limited by access only to pure phase modulation of spin waves, that inherently produces many diffraction orders, rather than a specific desired pattern. An interesting approach to achieve universality of performed operations, rather than just control of phase in the wavevectordomain, would be to implement a threestep protocol. It has been demonstrated that for optical frequencybin modes a pair of electrooptics modulator and a pulse shaper^{46} are sufficient to perform an arbitrary unitary operation. In our case to obtain arbitrary inputoutput relation for many stored pulses we would first store the pulses in the memory and apply the phase modulation as presented in the current work. Next, the pulses would be simultaneously readout and stored again in the memory with an electrooptic modulator changing the phase of each timebin. Final modulation in the spinwave domain would yield an arbitrary operation, achieving universality in terms of timebin inputoutput operation. In turn, the linearoptical scheme of quantum computation becomes feasible.^{47} The number of modes that can be efficiently processed would be limited by the optical depth. In general, a set of concatenated memories, as proposed in,^{22} could facilitate universal operations within our acStark modulation protocol.
Furthermore, note that here we did not use the acStark shift during write and read operations of the optical memory, and thus the twophoton absorption line is not broadened. Thus, the Gradient Echo Memory advantage of avoiding reabsorption of stored light is not yet exploited. Combined with larger optical densities this could significantly improve the efficiency of the presented memory.^{14}
Finally, by bringing the presented techniques to spin waves that involve a Rydberg state,^{48,49,50,51,52} the attainable range of operations between storage modes could be enriched with nonlinear interactions in order to realize efficient and deterministic quantum gates for photonic states. This could be particularly advantageous in engineering complex correlations within the spatial domain of a Rydberg atomic ensemble.^{53}
Methods
Lightatom coupling simulations
To correctly predict efficiencies during storage and retrieval as well as nontrivial shapes of spin waves created in the atomic ensemble. We choose to describe the system within the threelevel model described by an interaction picture Hamiltonian within the rotating wave approximation. It is subsequently reduced using the typical adiabatic elimination approach by setting the time derivatives of all excitedstate coherences and populations to zero. The coupled equations are then most conveniently expressed in terms of Rabi frequency of the signal field \({\mathrm{\Omega }} = {\cal {E}}d_{{\rm{eg}}}{\mathrm{/}}\bf{\hbar}\), where d_{eg} is the dipole moment of the relevant transition, and the coupling field \({\mathrm{\Omega }}_C = {\cal {E}}_Cd_{{\rm{eh}}}{\mathrm{/}}\bf{\hbar}\). With the coupling coefficient equal \(g = \omega \left {d_{eg}} \right^2N{\mathrm{/}}2\mathbf{\hbar} c\epsilon _0\) and in the frame comoving with the pulses (t → t − z/c) the equations take the following form (see refs. ^{54,55,56}):
where we have also introduced Γ = 2π × 6 MHz as the excitedstate e〉 decay rate and γ ≈ 2π × 10 kHz as the intrinsic spinwave decoherence rate, dominated by motional dephasing. The onephoton Δ = 2π × 20 MHz and twophoton δ detunings are defined as in Fig. 1b. In the first equation, the two terms in the nominator correspond to the twophoton and onephoton processes, respectively. In the second equation, the first term corresponds to the Raman interaction, while the second term is the free evolution under the acStark shift Hamiltonian due to the coupling light, which includes both the additional phase acquired as well as the power broadening. Even though the acStark modulation is applied only during dark periods of the memory, for completeness we include its influence as an additional term in Eq. (11B) given by iΔ_{acS}S. Note that in all cases the atom number density N is implicitly zdependent, and so is the coupling constant g. In the simulation we model this dependence as a Gaussian function in the z dimension with a width of 5 mm. Finally, we also add a small imaginary component to the acStark shift Δ_{acS} → (1 + isgn(Δ_{acS})γ_{acS})Δ_{acS} with γ_{acS} ~ 0.1, which effectively simulates dephasing due to inhomogeneous acStark light intensity.
We determine the coupling constant g by observing singlephoton offresonant absorption. This allows us to experimentally determine its peak value as g_{0} ≈ 200 cm^{−1} μs^{−1}, which corresponds to the optical depth OD ≈ 11. For the coupling field, we take short pulses with smooth slopes (modeling ~100 ns experimental rise times) and peak Ω_{C} ≈ 2π × 9 MHz = 1.5Γ. Typical signal field intensities correspond to peak Ω ≈ 2π × 50 kHz. The evolution is simulated using the XMDS package^{57} on a twodimensional z–t grid, or threedimensional x–z–t grid for the results in Section F. For this case, we also include a diffraction term in Eq. (11A), although the diffraction effects prove to be negligible for the planewave modes we work with.
Pattern preparation and imaging
The acStark laser is frequencystabilized using an offset beatnote lock.^{58} It is then spatially filtered using a singlemode fiber and amplified using a tapered amplifier (Toptica, BoosTA) to 1.5 W. The output beam is then reshaped using a cylindrical lens to better fit the shape of the elongated atomic ensemble. Simultaneously, an acoustooptical modulator (AOM) situated in the far field of the spatiallight modulator (SLM) is used to control the position of the beam at the SLM and carve out ~2 μs long pulses. With this setup, depicted in Fig. 6, we may select which region of the SLM is illuminated by changing the frequency of the AOM, which is done in real time using a direct digital synthesizer (DDS). On the SLM matrix, we display two patterns in two disparate regions. The SLM surface is then imaged onto a Dshape mirror which sends each pattern on a different path. The two paths are joined on a polarizing beamsplitter (PBS) before the vacuum chamber and now the two patterns overlap. Note that in the current configuration we loose half of the power at the final PBS. An additional mirror placed in the far field of the SLM is used for fine adjustment of grating position in the vertical direction with the help of a piezoelectric transducer (PZT), which is used to scan the grating phase ζ as in Sec. “Transverse space interference and manipulation”.
By flipping the flip mirror patterns can also be observed on a camera situated at the same image plane as the atomic ensemble. The camera provides feedback to the computer program that controls the SLM, which is used to actually generate the desired pattern, with a particular focus on intensity homogeneity. The program operates by first mapping the SLM coordinates onto the camera pixels using a National Instruments Vision module, and then iteratively adjusting the SLM display to achieve an intensity distribution closest to the target one. The optimization procedure begins with a white rectangle displayed on the SLM. Next, at each iteration, the error distribution i.e., the difference between pattern observed on camera an the target is calculated. According to that error, the SLM pattern is changed proportionally.
As demonstrated, at this point our setup already allows rapid switching between two patterns that could be used in shortlived quantum memories.^{39,40} We envisage that the setup may be extended to feature more patterns that could be rapidly reprogrammed, by for example using a twodimensional AOM to scan the beam through the atomic ensemble, or by using a set of AOMs to transfer many multiplexed images displayed with an SLM. This would unavoidably increase the complexity of the setup. In consequence, fast rapid spatiallight modulators and deformable mirrors that can already be reprogrammed during ~100 μs could serve as a highly viable solutions.^{59} On the other hand, simpler patterns could also be rapidly generated by sweeping the frequencies in a setup based on acoustooptic deflectors.^{60,61}
Code availability
Computer code used to process data in this study is available from the corresponding authors upon reasonable request.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
We thank K. Banaszek for generous support. This work has been funded by the National Science Centre, Poland (NCN) (Grants No. 2015/19/N/ST2/01671, 2016/21/B/ST2/02559, 2017/25/N/ST2/01163 and 2017/25/N/ST2/00713) and by the Polish MNiSW “Diamentowy Grant” (Projects No. DI2013 011943 and DI2016 014846). The research is supported by the "Quantum Optical Technologies'' project 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.
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M.M., M.P., and W.W. planned the experiment. M.M. and M.P. performed the experiment, analyzed the data, and wrote the manuscript with input from other authors. M.P. performed the simulation. M.M., M.P., A.L., and M.L. built the experimental setup. W.W. supervised the project. M.M. and M.P. contributed equally to this work.
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Mazelanik, M., Parniak, M., Leszczyński, A. et al. Coherent spinwave processor of stored optical pulses. npj Quantum Inf 5, 22 (2019). https://doi.org/10.1038/s4153401901360
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