1. Introduction

On-demand retrieval of pre-stored photons can be accomplished using quantum memories [1]. This is an indispensable technique for long distance quantum communication networks [2, 3] and for the enhanced generation of multi-photon states [4]. Such states find applications in a linear quantum computing scheme [5] or quantum simulators using linear optics [6, 7].

Implementations of quantum memories in room-temperature setups are among the most auspicious in terms of possible future applications due to their robustness. Until now room-temperature quantum memories have been realized in solid state systems [8, 9] and in warm atomic ensembles [1] such as gradient echo memory [10, 11], Raman memory [12, 13] or EIT memory [14–17].

In this paper we focus on Raman-type atomic memory implemented in warm rubidium-87 vapors [18–20]. In such memory photons are stored in atomic collective excitations called spin-waves. They are interfaced to photons via off-resonant Raman transitions. Storage times up to 30 ms in a single spin-wave mode was demonstrated [21]. Multimode storage, both temporal [10] and spatial [11, 19], is also feasible. With the use of multiple transverse spin-wave modes images can be stored and retrieved [22, 23]. The storage time of the multimode memory is limited by diffusional decoherence and thus can be prolonged by increasing the beam size [20, 23] at the expense of laser intensity.

Here we demonstrate experimentally the theoretical concept of “Hamiltonian design” proposed in [24]. We implement the Hamiltonian design at the readout from Raman atomic memory. Ideal readout from Raman atomic memory relies on a pure anti-Stokes scattering process which maps the spin-wave state onto photons state. In real systems additional Stokes scattering at the readout is virtually inevitable and it is a source of extra noise [13, 25]. By modifying the interaction Hamiltonian we are able to control the relative contributions of anti-Stokes and Stokes scattering processes. In the particular setting of our experiment this enables parametric amplification of the readout, albeit with extra noise.

In this paper we use simple theoretical model describing the coaction between anti-Stokes and Stokes scattering in readout. Previous approaches to describe theoretically the readout of spin-wave excitations in Raman scattering were focused only on pure anti-Stokes process. Theoretical descriptions of readout from Raman-type memory have already been given for a number of cases, including spatio-temporal evolution with losses [26], temporal eigen-modes [27], optimized retrieval [28], and spatial modes [29]. However, in those papers emphasis was put on the pure anti-Stokes scattering. In the present paper we use the extended description of the readout process [30], taking into account the four-wave mixing that includes the Stokes scattering.

In the experiment we detect scattered light with spatial resolution and temporal gating. This enables directly relating the experimental results to the theoretical predictions for temporal evolution of scattered light. Stokes and anti-Stokes light contributions can be distinguished through intensity correlation measurements and quantified via careful post-processing.

We provide evidence for a possibility of engineering a wide range of atom-light interfaces which can be described theoretically as simultaneous readout and parametric amplification. This may enable enriching purely optical quantum metrology [31] or communication [32] schemes with quantum storage capabilities.

This paper is organized as follows: In Sec. 2 we introduce a single-mode, theoretical model of temporal evolution of Stokes and anti-Stokes in readout, Sec. 3 describe experimental details, in Sec. 4 we present the phenomenological model utilized for data analysis and the results; finally Sec. 5 concludes the paper.

2. Theory

We begin by synthesizing a simple, plane-wave theoretical model of the readout by four-wave mixing from a Raman memory to qualitatively support our experimental results. The spin-wave excitation in the atomic medium can be created by different means. Light can be absorbed in a two-photon Raman transition to store previously prepared state of light [13, 25]. In our experiment we induced spontaneous Stokes scattering using write pulse with a wave vector k_W to populate the spin-waves, i.e. collective atomic excitations from the levels |0〉 to |1〉 as in Fig. 1(a). Both approaches results in the spin-wave excitation, however the latter which we use does not require special efforts to match the photons from the external source to the memory bandwidth. Ideally, the number of created spin-wave excitations n_b with a certain wave vector K_b equals the number of scattered Stokes photons with wave vector k_WS = k_W − K_b. We were able to estimate those numbers in each single iteration of the experiment.

 

Fig. 1 Atomic levels and phase matching in Λ-scheme Raman scattering induced by classical laser field ℰ_W and ℰ_R. (a) In the spontaneous write-in process Stokes photons ( ${\widehat{a}}_{\text{WS}}^{†}$ - mode) and spin-wave excitation (b̂^† - mode) are created pairwise. (b) Four-wave mixing in readout consists in simultaneous anti-Stokes and Stokes scattering into modes ${\widehat{a}}_{\text{RA}}^{†}$ and ${\widehat{a}}_{\text{RS}}^{†}$. (c) Phase matching condition or momentum conservation dictates the wave vectors of single photons coupled to a spin-wave excitation with a certain wave vector K_b. Write beam with k_W wave vector is scattered as Stokes photon with k_WS wave vector, while read beam with k_R wave vector either scatters Stokes or couples anti-Stokes of the respective wave vectors k_RS, k_RA.

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Here we focused on the retrieval stage at which the spin-wave excitations are converted to photons in four-wave mixing induced by the read laser pulse depicted in Fig. 1(b). The read laser is assumed to be plane-wave with a wave vector k_R. Spin-wave with a certain wave vector K_b are coupled to anti-Stokes and Stokes fields with wave vectors k_RA = k_R + K_b and k_RS = k_R − K_b respectively. In the experiment those weak light fields illuminated distinct pixels of the camera which was located in the far field. They were also shifted with respect to the initial Stokes photons with the wave vector k_WS = k_W − K_b due to different direction of the write beam k_W. We summarize the phase matching condition relating the wave vectors in Fig. 1(c).

To describe the interaction at the readout stage we use bosonic operators of the weak field light modes â_RA and â_RS. The spin-wave can be described with the Holstein-Primakhoff approximation by a bosonic annihilation operator b̂ which removes one excitation from spin-wave with the wave vector K_b [33]. The Hamiltonian describing both the Stokes and anti-Stokes scattering at the readout stage can be obtained by eliminating the excited level adiabatically [30, 34]:

The above Hamiltonian leads to coupled Maxwell-Bloch equations, which can be integrated as e.g. in [30, 33]. This yields input-output relations linking the operators of the incoming light fields and the initial spin-wave state to the outgoing fields and the final spin-wave state.

We used the input-output relations to compute the quantities observed in the experiments, that is the mean number of scattered photons as a function of time in anti-Stokes $〈{\widehat{a}}_{\text{RA}}^{†}(t){\widehat{a}}_{\text{RA}}(t)〉$ and Stokes fields $〈{\widehat{a}}_{\text{RS}}^{†}(t){\widehat{a}}_{\text{RS}}(t)〉$ respectively. With the assumption that the incoming light fields are in a vacuum state and the mean number of initial spin-wave excitations equals n_b = 〈b̂^†(0)b̂(0)〉, the result is:

The underbraced terms G_RA(t) and G_RS(t) represent the readout gains, that is the number of scattered photons per each input spin-wave excitation. In turn, the terms S_RA(t) and S_RS(t) represent the spontaneous scattering into anti-Stokes and Stokes quantum fields, which is independent of the initial state of quantum fields. Just as expected, they appear only in the presence of Stokes scattering ξ ≠ 0. Note that both coupling coefficients χ and ξ appear in both formulas (2) and (3), confirming that anti-Stokes and Stokes scattering cannot be treated separately and should be viewed together as a full four-wave mixing process.

Note that the above results were derived neglecting all decoherence. In the experiment, the most important sources of decoherence are diffusion of atoms and spontaneous emission from the excited state in random direction. Both can be neglected provided the optical depth is large and the duration of the interaction short.

Let us proceed to examining the evolution of scattered anti-Stokes and Stokes light in a few typical cases.

For ξ ≃ 0 there is virtually no Stokes scattering while the anti-Stokes scattering appears instantaneously and decays exponentially as depicted in Fig. 2(a). There is virtually no spontaneous noise S_RA ≃ 0 and the time integrated gain reaches unity Ḡ_RA = ∫ G_RAdt = 1. In the ideally case of ξ = 0 this leads to the readout Hamiltonian of form ${\widehat{H}}_{\text{R}}~{\widehat{a}}_{\text{RA}}^{†}\widehat{b}+H.c.$ which represent purely anti-Stokes scattering process.

 

Fig. 2 (a) to (c) - temporal evolution of gains G_i and spontaneous noises S_i building up the light fields. Data plotted for different values of coupling coefficients χ and ξ correspond to detuning parameters Δ_R= 0.3, 0.6 and 0.8. (d) Integrated readout gains Ḡ_i and spontaneous noises S̄_i versus detuning Δ_R.

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In real systems Stokes interaction is typically unavoidable. The case where χ > ξ > 0 is depicted in Fig. 2(b). Here the readout appears in both the anti-Stokes and Stokes fields with comparable intensity and decays over time. Time-integrated gain may reach over unity Ḡ_i = ∫ G_idt > 1, yet with spontaneous noise that slowly builds up S_i > 0, for i = RA, RS. This is the setting utilized in [37].

A situation where Stokes interaction dominates, ξ > χ, is depicted in Fig. 2(c). In this case readout goes predominantly to Stokes field with exponentially increasing gain. Yet, the noise intensity virtually equals the gain G_i ≃ S_i for both fields. In the limit of ξ ≫ χ we can put χ = 0 which leads to the readout Hamiltonian ${\widehat{H}}_{\text{R}}~{\widehat{a}}_{\text{RS}}^{†}{\widehat{b}}^{†}+H.c.$ consists only the contribution of Stokes scattering process.

It is instructive to integrate gains G_RA(t) and G_RS(t), and noises S_RA(t) and S_RS(t) over the time of interaction and inspect them as functions of coupling coefficients. Coupling coefficients χ and ξ are inversely proportional to the detuning of the read laser from the excited level Δ_R [33, 35]. One coupling coefficient can be increased at the expense of the other by tuning the frequency of the read laser when it is in between the |0〉 ↔ |e〉 and |1〉 ↔ |e〉 resonances as in Fig. 1(b).

In Fig. 2 (d) we plot the time integrated gains Ḡ_i = ∫ G_idt and noises S̄_i = ∫ S_idt as a function of the detuning Δ_R from |0〉 ↔ |e〉 transition in the units of ground state splitting. We assumed χ ∝ 1/Δ_R and ξ ∝ 1/(1 − Δ_R). Different characters of anti-Stokes and Stokes interaction causes asymmetry of the plot in Fig. 2 (d). The integrated readout gain in the anti-Stokes domination regime χ ≫ ξ remains equal to unity Ḡ_RA = 1 and the noise is suppressed S̄_RA ≪ 1. These are the conditions for perfect unamplified readout.

On the contrary, in the Stokes domination regime χ ≪ ξ the integrated gain Ḡ_RS varies almost exponentially with the Stokes coupling ξ entailing elevated noise S̄_RS. In this domain four-wave mixing enhances the anti-Stokes emission and Ḡ_RA rises although the relative contribution Ḡ_RA/Ḡ_RS diminishes. Note that this is the regime we use to populate the spin-waves by inducing spontaneous Stokes scattering with write pulse.

Finally, let us consider a single realization of the four-wave mixing process. Assume that the number of initial spin-wave excitations n_b has been measured by counting the Stokes photons n_WS scattered during prior write-in, ideally n_WS = n_b. In this particular iteration the number of photons in the anti-Stokes and the Stokes field equals

3. Experiment

We implemented Raman-type atomic memory in rubidium-87 vapors. We applied spontaneous write-in process into memory similarly as in previous works [18,38]. The spontaneous write-in process had been proven to efficiently create the spin-wave excitations, thus we could focus on the retrieval characteristics. Here we extended the setup and measurements schemes applied and explained in detail in our previous papers [19, 20].

We used a 10 cm glass cell containing pure ⁸⁷Rb isotope with krypton as a buffer gas under pressure of 1 torr. The cell was heated by bifilar windings to 90°C equivalent to the optical density of 135. The cell was magnetically shielded to avoid decoherence produced by stray magnetic field and the main source of decoherence was due to diffusion [20].

We used three external cavity diode lasers: pump, write and read laser, operating at D1 line of ⁸⁷Rb (795 nm). The frequencies of all laser beams and the scattered light are sketched against the ⁸⁷Rb level scheme in Fig. 3. The pump laser was resonant to F=2 →F’=2 transition, while the write and read laser were detuned to the red from transition F=1→F=1′. The detuning of the write laser was Δ_W = 1.77 GHz and it was kept constant throughout all measurements while the detuning of the read laser Δ_R varied. The pump and write lasers were frequency locked using the DAVLL setup [39]. The exact values of the write and read lasers detunings were set repeatedly and measured precisely using Doppler-free saturated absorption spectroscopy inside an auxiliary rubidium cell and their reference beat-note signal was measured on a fast photodiode. Inside the cell write, read and pump lasers had the power of 6.8 mW, 4.5 mW and 75 mW respectively. During the experimental sequence the lasers power fluctuated at a level of 5% and the frequencies were changing by no more than 50MHz.

 

Fig. 3 Configuration schemes of ⁸⁷Rb hyperfine energy levels at D1 line (795nm) using write-in and readout respectively. Laser beams and scattered light detunings in both processes are also depicted.

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The simplified schematic of the setup is shown in Fig. 4(a). All of the three horizontally polarized beams overlapped inside the cell where their diameters were 6 mm for the pump beam and 4 mm for both write and read beams. The write and read beams were tilted at the angle θ = 2 mrad. Acousto-optic modulators were used to shape rectangular pulses with rising time of 1 μs from laser beams. The pulse sequence applied in the experiment is depicted in Fig. 4(b). The sequence was initiated by optical pumping of rubidium atoms into the ground state of ⁸⁷Rb S^1/2 F=1. The 700 μs long rectangular pulse of 75 mW power yielded a pumping efficiency of 98%. Then we applied a 10 μs long rectangular write pulse to create spin-wave excitations between F=1 and F=2 levels together with the Stokes scattering. The rectangular read pulse started right after the end of the write pulse and its duration was set to 40 μs. The read pulse generated both anti-Stokes and Stokes scattering. The total time duration of write and read pulse corresponds to the average atomic displacement of c.a. 0.5 mm due to diffusion [20] which was small enough to keep the atoms in the range of the pump beam size.

 

Fig. 4 (a) Experimental setup: write and read laser beams propagate forward to the intensified camera and the pump beam propagates backward. ⁸⁷Rb - atomic memory cell, ⁸⁵Rb - absorption filter, blue arrows correspond to scattered light. (b) Pulse sequence with long intensifier gate covers the whole write and a part of the read pulse (both rectangular). Exponential shapes in front of write and read pulses are typical time-resolved intensities of the scattered light observed on the intensified camera, WS and RS stand for Stokes while RA for anti-Stokes scattering respectively. (c) Scheme of the measurement of the temporal evolution of the readout light using short gate duration τ = 250 ns.

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We separated the horizontally polarized laser beams from the vertically polarized scattered fields by polarization and spectral filtering, which yielded the total attenuation factor for laser beams that exceeded 10⁹. For spectral filtering we applied a ⁸⁵Rb absorption filter at 130°C placed in magnetic field which increase the absorption by broadening and shifting the ⁸⁵Rb spectral lines. The attenuation of laser beams inside the 30 cm length absorption filter was at least 10⁴. We also measured the transmission of the filter at frequencies corresponding to the scattered light generated at write-in and readout stages. They were found to be 12% for Stokes scattering in write-in and 76% for all applied frequencies of anti-Stokes and Stokes in readout.

In our system we generated and retrieved spatially multimode light as described in detail in [19, 33]. This light was detected in the far field by a gated image intensifier coupled to sCMOS camera [40]. We used the intensified camera in two different operational schemes. In the first scheme we applied a long gate pulse which covered both the write and the read pulse as in Fig. 4(b). We set the gain of the intensifier to a low value. Then the response of the camera system was proportional to the intensity of scattered light. We calibrated the excess noise contributed by the image intensifier and made sure that it was insignificant as compared to the shot-to-shot intensity fluctuations of scattered light. In the second scheme we used a short, delayed gate of τ = 250 ns duration as depicted in Fig. 4(c). Here we set the image intensifier to high gain and the camera system was sensitive to single photons [40]. We also utilized an intensified camera as a photon number resolving detector [41] and we counted all photons with a quantum efficiency of 20% in the region of readout scattering. Representative images: single shots and averaged intensities obtained typically in those regimes are depicted in Fig. 5. Note that in Fig. 5(a) the number and localization of speckles are random so they average to the smooth intensity profile as shown in Fig. 5(b). Both measurement regimes are phase insensitive although spatially resolved, homodyne-type detection was also reported [37] and then it could be used to directly measure phase noise and squeezing properties of the generated light.

 

Fig. 5 Representative images of the retrieved field, write-in and read-out in upper and lower parts respectively. (a) Intensity map in a single shot obtained using a long gate in the linear regime of camera operation and (b) the average over 10⁴ frames. (c) Photon positions in a single shot [40] obtained using a short gate positioned in the readout stage and (d) the total number of photons per sCMOS pixel summed over 2000 frames.

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4. Results

4.1. Temporal evolution of readout intensity

At first we measured the temporal evolution of the mean number of photons emitted during readout stage. For this purpose we counted the scattered photons using an intensified camera with a gate duration of τ = 250 ns. We collected all photons in the specified circular camera region around the point corresponding to the center of the read beam i.e. wave vector k_R. Thus, we measured the anti-Stokes and the Stokes scattering together, both their spontaneous and stimulated parts. In Fig. 6 we present the results of measurement for mean number of photons versus gate delay with exponential fits. Each data point was obtained by averaging 2000 frames in the area of angular diameter 0.5 mrad and the time separation between the data points was 1 μs. The sequence was repeated for different detunings Δ_R of the read beam from F=2→F’=2 resonance.

 

Fig. 6 Average number of photon counts detected at a large solid angle around read beam per a gate duration of 250 ns for different readout laser detunings Δ_R from ⁸⁷Rb F=2 → F’=2 resonance. The curves represent exponential fits to the first and last data points. The size of errorbars is comparable with marker size.

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The results plotted in Fig. 6 vividly depict a transition from decay to growth of the readout scattered light intensity. For a small detuning Δ_R = 1.67 GHz the read beam is close to the F=2→F’=2 resonance and we expected the anti-Stokes coupling to dominate. Indeed, we observed an exponential decay as predictied. At the opposite extreme Δ_R = 4.19 GHz, in turn, the read beam is much closer to the F=1→F’=1 transition and here it was the Stokes field that we expected to dominate. Again, the total number of scattered photons increases as expected, compare Fig. 2(c). Nonetheless, in the experiment the growth starts after the first 4 μs of the evolution during which the number of scattered photons slightly decreases. In the intermediate cases we observed initial decay followed by growth. These cases might be associated with the case presented in Fig. 2(b), where stimulated field contributions decay while the spontaneous contributions rise. Moreover, in the observed evolution the spin-wave decoherence certainly plays an important role, which, however, is not accounted for by our simplified model.

In Fig. 7 we plot decay and growth coefficients as functions of detuning Δ_R of the read beam from F=2→F’=2 resonance. The coefficients were calculated by fitting exponential decay or growth to the first or the last 10 μs, as depicted in Fig. 6. Fitting entire 40 μs with a single exponential fails because the significant influence of decoherence and multimode character of scattering on this time scale is not capured by our simple thoeretical model. The dependence of the growth and decay coefficients on the detuning Δ_R agree qualitatively with predictions of the theoretical model as they rise along with Δ_R. However, there is a discontinuity between the growth and decay coefficients at Δ_R = 3.8 GHz that may be attributed to spin-wave decoherence influence.

 

Fig. 7 Exponential decay and growth coefficients for scattered light found by fitting data as in Fig. 6 versus read laser detuning Δ_R. For triangle points the size of errorbars are smaller than the marker size. Inset: ⁸⁷Rb D1 line absorption spectrum with read and write laser frequencies marked.

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4.2. Spatial resolving of anti-Stokes and Stokes scattering

The analysis of intensity correlations between light scattered in various directions provides valuable insight into the four-wave mixing at the readout. To set the stage we shall first recall the relations between scattered photons directions. The scattering generated during the write pulse is emitted around the write beam and falls upon a circular region of angular diameter 0.5 mrad in the upper part of the camera. Similarly, the readout light illuminates the lower part of the camera of angular diameter 0.5 mrad placed in the far field. Both regions are contained in a small central part of the camera sensor. The angle between the write and read beams equal to θ = 2 mrad inside the ⁸⁷Rb cell is precisely the distance between the centers of both circular regions on the camera. Due to phase matching illustrated in the Fig. 1(c) the scattered fields coupled to the same spin-wave with a wave vector K_b propagate in different directions. They are detected in the far field on three different pixels of the camera.

For a K_b oriented rightward a pixel located left of the center in the write region gathers Stokes-coupled light of an intensity proportional to the number of photons n_WS. Another pair of pixels, right and left of the center in the read region, gathers anti-Stokes and Stokes coupled light of intensities proportional to n_RA and n_RS respectively. Note that the latter pair of pixels also gathers light coupled to the spin-wave with an opposite wave vector −K_b, and their role is reversed. That is, the left and the right pixels gather anti-Stokes and Stokes scattering respectively. These contributions add extra noise when correlated with Stokes scattering n_WS generated in the write-in process.

In order to perform correlation measurements the camera intensifier was operated in linear regime and long gate pulses were used. We detected the light scattered during both the write-in and the readout processes. In each iteration of the experiment we registered entire, randomly varying camera images. The signal I_i at the i–th pixel of the camera was proportional to the intensity of the incoming light. We calculated the correlation coefficient between intensities of light registered by any two pixels, i–th and j–th, of the camera:

In Fig. 8(a) we present the maps of correlation coefficients C_ij as in Eq. (5) between one pixel marked by a cross referred as i = WS and all other pixels j in the image. The pixel i = WS corresponds to the emission of Stokes field coupled to the spin-wave with the wave vector |K_b| = 45.8 cm⁻¹. This value was set, so as to spatially resolve anti-Stokes and Stokes correlation regions. Each correlation map was calculated basing on 10⁵ single-shot images. We limited our calculations to the circles marked in the maps. They correspond to the regions where the scattering was registered, centered around the directions of the write and read beams. In the upper red circles we registered the Stokes scattering emitted in the write-in process whereas in the bottom green circles light from readout was observed.

 

Fig. 8 (a) Spatial correlation maps for different read laser detunings from ⁸⁷Rb F=2 → F’=2 resonance. Red and green circles mark the respective areas where photons scattered at the write and read stages fell. (b) Correlation coefficients at the peaks corresponding to anti-Stokes C_WS,RA and Stokes C_WS,RS versus detuning Δ_R with (quadratic) trend.

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In each map in Fig. 8(a) three spots appear. The maxima of the spots marked by the crosses on pixels i = WS, RA, RS correspond to the emission of scattered fields coupled to the same spin-wave with the wave vector K_b and their positions agree with the phase matching conditions illustrated in Fig. 1(c).

As we discussed in conclusion of Sec. 2, the correlation between the write scattered light and two other fields scattered at the readout stage is mediated by the spin-wave, which caries random number of excitations n_b in each iteration of the experiment. This number is proportional to the signal on the write pixel I_WS and contributes to the signals I_RA and I_RS. The signals on pixels far away are correlated to the excitations of other spin-waves, which vary independently. We verified that correlation maps displayed characteristic three-spot pattern for arbitrarily chosen reference points except when two spots in the readout merge. This guarantees that, indeed, we resolve the Stokes and anti-Stokes scattering in the readout. Correlation measurements are a practical alternative to spectral filtering in other experiments with quantum memories [37, 42].

Now we can proceed to examine how the anti-Stokes and Stokes scattering contributions change with the read laser detuning Δ_R. In Fig. 8(a) we depicted three correlation maps for different read laser detunings Δ_R. Transition from the anti-Stokes to Stokes scattering domination can be achieved by adjusting of the detuning, here from Δ_R = 1.17 GHz to Δ_R = 3.28 GHz. To quantify the effect, we calculated the correlation between the Stokes scattering in write-in and the anti-Stokes scattering — C_WS,RA or the Stokes scattering in readout — C_WS,RS. The values of C_WS,RA, C_WS,RS were taken as the marked maxima of the respective anti-Stokes/Stokes regions for various detunings. Figure 8(b) summarizes the results and illustrates the transition.

4.3. Effective gains of anti-Stokes and Stokes readout

The measured correlation values together with mean intensities and their variances can be used to extract information about time integrated readout gains for the anti-Stokes Ḡ_RA and Stokes Ḡ_RS separately. This is accomplished by tracing the origins of signal observed at respective pixels.

The signal I_WS registered at pixel i = WS was calibrated to the number of photons. It consists of the part proportional to the number of photons n_WS generated during the write-in to the spin-wave of the wave vector K_b and the camera system read noise f_WS:

The signal in the readout at pixel i = RA can be decomposed into the following contributions:

We can write an analogous formula for the components of the signal registered at the pixel i = RS:

In the actual experiment the number of created spin-wave excitations n_b is smaller than the number of scattered photons n_WS due to decoherence at the write-in stage n_b = η_Wn_WS. That factor η_W is write-in efficiency. In turn, the decoherence at the readout stage leads to a limited efficiency η_R during this process. We include those efficiencies in the formula Eq. (4), which yields the phenomenological formulas for the number of retrieved photons:

Equations (7)–(10) and simple observations that corr(n_WS, η_Wη_RḠ_RAn_WS)= 1 and corr(n_WS, Š_i)= corr(n_WS, n′_i)= corr(n_WS, f_i)= 0 allow us to calculate the effective gains from the measured quantities:

In Fig. 9 we present the results of the effective gains of the anti-Stokes scattering η_Wη_RḠ_RA and the Stokes scattering η_Wη_RḠ_RS for different read laser detunings Δ_R. The efficiency due to decoherence in the write-in η_W is fixed. The efficiency of the readout η_R is mostly determined by the decoherece during this process, thus it also weakly depends on the detuning Δ_R. Thus we expect that the experimentally determined effective gains vary mostly due to the change of the time integrated gains Ḡ_i and can be compared with the theoretical predictions.

 

Fig. 9 The effective gains of the anti-Stokes scattering η_Wη_RḠ_RA and the Stokes scattering η_Wη_RḠ_RS. For the smallest detuning Δ_R absorption in rubidium cell starts to contribute.

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Our results show that in our configuration the Stokes scattering during readout is an important contribution to the scattered light. The highest observed efficiency of the anti-Stokes retrieval was η_Wη_RḠ_RA =22% for Δ_R = 1.37 GHz whereas the efficiency of the Stokes scattering η_Wη_RḠ_RS grew with Δ_R. Note that the observed dependence of the anti-Stokes and Stokes contributions on Δ_R agrees qualitatively with our simple theoretical model for the integrated gains Ḡ_i presented in Fig. 2(d). Note that for large detunings the anti-Stokes scattering would decay if there was no Stokes contribution. We suspect that the efficiencies values η_Wη_RḠ_RA > 10% for the anti-Stokes for Δ_R > 2 GHz originated from the enhancement due to the full four-wave mixing process. The switch from domination of anti-Stokes to Stokes scattering occurs at Δ_R ≃ 1.5 GHz.

In our setup we were able to manipulate the gate duration and thus check how the values of the effective gains depend on the time of interaction. We measured the effective gains for two different gate durations covering 10 μs write pulse and a part of 40 μs read pulse selected by a gate duration τ = 25 μs or τ = 50 μs, as we depicted in Fig. 4(b).

We measured the effective gains for two different detunings Δ_R and for the two values of the gate duration τ. We checked that the anti-Stokes effective gain η_Wη_RḠ_RA = 12% remained independent of the interaction time which indicates that light that has been correlated with the stored spin-waves is always retrieved in the beginning stage of the readout. On the other hand, the efficiency of the Stokes scattering varies, as we summarized in Table 1. As we can see, longer interaction time increases the effective gain η_Wη_RḠ_RS and in particular we can obtain more correlated light than we did at the write-in stage η_Wη_RḠ_RS > 100%. This corresponds to the case where the integrated gain Ḡ_RS significantly exceeded unity and overcompensated for the losses, albeit bringing a large amount of accompanying noise S̄_RS. Such a high gain of the obtained signal can find an application in various operations with quantum memories. For instance the dominant Stokes scattering process can be utilized for testing if the memory is empty. No photons retrieved from the memory at the readout may suggest that memory was initially empty with a certain probability. This probability is much higher for amplified readout with Stokes interaction dominant as opposed to conventional pure anti-Stokes readout process of limited efficiency.

Table 1. The effective gains of the Stokes scattering η_Wη_RḠ_RS for different gate durations.

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5. Conclusions

In conclusion, we have presented an experimental demonstration of manipulation of the Hamiltonian in the readout from Raman-type atomic memory. We measured the temporal evolution of the readout light and the spatial correlations between the Raman scattering in the write-in and the readout. Our measurements confirm the adjustability of coupling parameters corresponding to the anti-Stokes and Stokes scattering. The results match our simple theoretical model of a full four-wave mixing process. We resolved the anti-Stokes and the Stokes scattering contributions to the readout thanks to the phase matching in the atomic vapor which dictated directional correlations with the Stokes photons during write-in.

Our results provide a very simple framework for interpretation of extra noise in experiments on storing light in atomic vapor. When anti-Stokes scattering is used to map the spin-wave states onto the states of light, the accompanying Stokes scattering creates unwanted random photons and atomic excitations. Our results show that, though inevitable, this contribution can be estimated by our model and perhaps suppressed by adjusting the coupling light frequency to the other side of the atomic resonance. There is also an optimal duration for the anti-Stokes interaction. Beyond the optimum, the spontaneous noise contribution increases. It may be favorable to switch to noncollinear configuration where control and quantum fields enter the atomic medium at a small angle. Then the Stokes scattering photons will become directionally distinguishable from the anti-Stokes, which may lower the noise in some experiments.

The design of the Hamiltonian we demonstrated can be implemented in many types of quantum memories at little or no extra cost. The amplification of readout signal by Stokes scattering may be very useful in some applications especially if extra noise is not crucial. This is the case when we use detectors of small quantum efficiency or we are focused on other properties of retrieved light e.g. in retrieval of stored images [23, 43]. For instance the amplification in the readout can be utilized as a robust single-shot projective test to see whether the atomic memory is in the ground (empty) state. A complete absence of Stokes signal on an inefficient detector is a relatively rare occurrence if the extra Stokes gain overcompensates for the losses at the detection stage. Notably then, complete absence of signal ensures us that the memory was in the ground state.

Our results can be useful for suppressing unconditional noise floor in readout of Raman-type quantum memories at the single photon level [13, 25, 44]. We provide evidence for a possibility of engineering a wide range of atom-light interfaces which can be described theoretically as simultaneous readout and parametric amplification e.g. quantum non-demolition interaction while coupling coefficients are equal. The facility to continuously tune the Hamiltonian coefficients sets the scene for developing new quantum protocols in room-temperature atomic memories.