Abstract
Parallelized quantum information processing requires tailored quantum memories to simultaneously handle multiple photons. The spatial degree of freedom is a promising candidate to facilitate such photonic multiplexing. Using a singlephoton resolving camera, we demonstrate a wavevector multiplexed quantum memory based on a cold atomic ensemble. Observation of nonclassical correlations between Raman scattered photons is confirmed by an average value of the secondorder correlation function \(g_{{\mathrm{S,AS}}}^{{\mathrm{(2)}}} = 72 \pm 5\) in 665 separated modes simultaneously. The proposed protocol utilizing the multimode memory along with the camera will facilitate generation of multiphoton states, which are a necessity in quantumenhanced sensing technologies and as an input to photonic quantum circuits.
Introduction
Multiplexing in optical fibers or in freespace is essential in modern techniques for highthroughput transmission^{1,2}. Similarly, as quantum technologies mature, the necessity of multiplexing in photonbased quantum communication becomes clear^{3} and much effort is devoted to various schemes exploiting spatial^{4,5,6,7,8,9,10,11,12}, temporal^{13,14,15,16,17,18}, or spectral^{19,20} degrees of freedom. Utilization of many modes can finally allow efficient application of the Duan–Lukin–Cirac–Zoller (DLCZ) protocol^{19,21,22,23,24} and offer nearly deterministic generation of multiphoton states^{25,26,27} later applicable in quantumenhanced sensing technologies^{28,29} as well as optical quantum computation^{30}.
Regardless of substantial efforts, the task of achieving large number of modes remains a challenging endeavor especially in hybrid atomphoton systems. In purely photonic, memoryless systems, hundreds of modes have been obtained within the spatial domain of spontaneous parametric downconversion^{31,32,33} or by means of frequencytime entanglement^{18,34,35,36,37,38}. However, most applications such as the DLCZ protocol^{21}, enhanced photon generation^{26,27,39}, or even linear optical quantum computing^{30} require or greatly benefit from a multimode quantum memory. For instance, the largest number of temporal modes used for photon storage in cavitybased quantum memory^{35} is 30. Likewise, ensembles of dopants in crystals have been used to store externally generated photons^{40} in up to 100 modes or otherwise generate them^{23,24} in 12 modes.
Another mainstream trend is to build a multiplexed quantum repeater by splitting a trapped atomic ensemble into many cells^{5,26}. The idea was recently realized in two dimensions achieving 225 modes^{41}. These schemes however suffer from the limitation given by difficulty in trapping large ensembles as well as hinder heralded simultaneous excitation of all modes. In consequence, they are rendered useful only for the DLCZ quantum repeater^{21}, but neither for quantum imaging^{42,43,44} nor enhancing rate of the photonic state generation^{18,19,26,27}.
The purpose of this paper is a demonstration of massive improvement in the number of modes processed by the quantum memory. The experimental realization is accomplished through multiplexing of angular emission modes of a single quantum memory^{27} and by employing a spatially resolved singlephoton detection. Our experimental setup generates photons in 665 pairwisecoupled modes, exploring the regime of multimode capacity with simultaneous extremely low noiselevel achieved with stringent, spatially multimode yet simple and robust filtering. We use a singlephoton resolving camera to measure both correlations and autocorrelation unambiguously proving quantum character of light. Note that throughout our results, we do not perform accidental or noise background subtraction—in contrast to any previous experiments with singlephoton sensitive cameras^{12,27,31,32}. We achieve the quantum memory lifetime of >50 μs, which combined with the multimode capacity invites realtime feedback processing of stored excitations^{45} and paves the way toward promptly achieving fast generation of single and multiphoton states^{26,27}.
Results
Multiphoton generation
Here we propose the potential application of our scheme as a platform for multiphoton state generation. Figure 1 pictures a protocol utilizing the multipixel capability of the singlephoton resolving camera to enhance generation of multiphoton states. The essential advantage over recently introduced quantum memory arrays^{41} is simultaneous excitation and access to many modes. The protocol is being managed by a classical memory storing the wavevectors of registered photons and the whichmode information. This information is finally used to route the photons retrieved in the readout process from the quantum memory. A photonic switch is used to direct photons to a quantum circuit or to conditionally generate arbitrary states through multiphoton interference. Furthermore, since a small number of photons is generated per each frame, one can adapt in realtime the number of trials to create exactly the desired number of excitations in the quantum memory. By keeping the mean photon number per shot small, we virtually eliminate the malicious pairs in a single mode. This gives us an advantage over a simpler scheme^{27} in which a single excitation shot is used. In that scheme, the mean number of photons may be controlled but the multimode thermal statistics severely limits the fidelity of generation of nphoton state. Extensions of our new proposal are numerous, including usage of spinwave echos to conditionally manipulate the atomic excitations^{46}. The experimental results presented below constitute the most essential step toward realization of the proposed protocol.
Wavevector multiplexing
For the quantum memory, we use an engineered atomic ensemble of cold rubidium87 atoms at T = 22 ± 2 μK generated within a magnetooptical trap (MOT) and cooled using polarizationgradient cooling (PGC) scheme, as depicted in Fig. 2. With the 1 cmlong cigarshape ensemble of diameter w = 0.6 ± 0.1 mm (taken as 1/e ^{2} diameter of the atomic column density) containing N = 10^{8} atoms, we achieve optical depth OD = 40, which limits the memory readout efficiency^{17,47}. Quantum memory operates once atoms are released from MOT with the magnetic field gradients switched off. We prepare 70% of atoms in the F = 1, m _{ F } = 1 state and the rest of atoms in the F = 1, m _{ F } = 0 state through optical pumping. Atomphoton interface is achieved with two lasers: write, which is reddetuned from 5^{2} S _{1/2}, F = 1 → 5^{2} P _{3/2}, F = 2 transition and read laser tuned to 5^{2} S _{1/2}, F = 2 → 5^{2} P _{1/2}, F = 2 transition.
To generate the multimode, multiphoton state, we illuminate the ensemble with a write pulse containing 10^{7} photons with wavevector k _{w} tilted at an angle of 2° to the ensemble axis in the xz plane. We take the axis defined by counterpropagating read and write beams as the z axis of the frame of reference. Stokes (S) photons scattered in the Raman process are registered on the IsCMOS camera located in the farfield with respect to the atomic ensemble. A scattered photon with a transverse wavevector k _{S} is accompanied by a collective atomic excitation (spin wave) with a spatial phase dependance: \(N^{  1/2}\mathop {\sum}\nolimits_{j = 1}^N \,{\mathrm{exp}}\left( {i{\bf{K}} \cdot {\bf{r}}_j} \right)\left {g_1 \ldots h_j \ldots g_N} \right\rangle\), where K = k _{S} − k _{w} is the spinwave wavevector, r _{ j } is the position of jth atom, \(\left {g_j} \right\rangle\) \(\left( {\left {h_j} \right\rangle } \right)\) corresponds to the 5^{2} S _{1/2}, F = 1 (F = 2) state, and the summation is carried over all N atoms in the ensemble. To learn about the spin wave, we convert it to an antiStokes (AS) photon through resonant Raman scattering (readout process) with a read pulse with wavevector k _{r} containing 10^{8} photons. Wavevector of the AS photon is determined by the stored atomic excitation k _{AS} = K + k _{r}. We estimate the readout efficiency as χ _{R} = 35 ± 2% (taken as the ratio of coincidence rate to S photons rate and accounted for losses). AS photons are registered on a separate region of the same IsCMOS sensor.
Spatially insensitive filtering is essential for the memory to take advantage of its inherent multimode capability. Commonly used frequency filtering cavities^{5,11,13} transmit only one spatial mode. To overcome this issue, we use two separate optically pumped hot rubidium vapor cells with buffer gas and paraffin coating. The cells are pumped by strong lasers during the cooling and trapping period of the MOT. Additional interference filters are used to separate stray pump laser light from single photons (see Methods for details).
Finally, photons originating from the atomic quantum memory are imaged onto the IsCMOS sensor through a nearly diffractionlimited imaging setup. The sensor is located in the Fourier plane of the atomic ensemble. Positions of photons registered on the camera are calibrated as transverse emission angles, directly proportional to transverse wavevector components. The IsCMOS camera has the quantum efficiency of 20% and the combined average transmission of the imaging and filtering system is 40% (see Methods for details). The net efficiencies in S and AS arms are equal η _{S} ≈ η _{AS} ≈ 8%.
Data analysis
The spatial degree of freedom provides an advantage over singlemode experiments^{8,9}. If one considers each mode as a separate realization of the experiment, we are able to collect statistics at a rate of 3 × 10^{5} effective experiments per s. This rate is very similar to what is obtained in singlemode experiments, however the multimode scheme offers much more versatility as increasing the memory storage time to many μs decreases the rate very insignificantly, contrasted with a dramatic drop of the rate in the singlemode experiments. For example, with 30 μs storage time, our effective experimental rate remains at 300 kHz, as it is anyway limited by the readout speed of the sCMOS camera. For the corresponding singlemode experiment, the absolute maximum stands at 33 kHz. With faster camera acquisition rate, the advantage of the multimode scenario would become overwhelming.
Here, to obtain proper statistics we have collected 10^{7} camera frames. For a pair of small conjugate squareshaped region of interests (ROIs) with side length κ = 160 mm^{−1} and a net S photon detection probability of 4 × 10^{−2}, we register very few accidental coincidences, i.e., 90% of coincidences come from conjugate modes. This figure of merit changes with a mean photon number and thus the number of observed modes, as due to limited detection efficiencies, we will sometimes register a pair of photons from two different pairwisecoupled modes. For two conjugate ROIs with a side lengths κ = 340 mm^{−1} (i.e., 43 mrad) corresponding to a nearly full field of view composed of hundreds of modes, we have registered a total number of 1.6 × 10^{5} coincidences of which 4.4 × 10^{4} came from conjugate mode pairs.
Collection of photon counts with a multipixel detector requires new experimental and data analysis tools^{48}. To verify the anticorrelation between momenta of S and AS photons both in x and y coordinates, we count the coincidences for each pair of camera pixels corresponding to wavevector coordinates (k _{ x,S}, k _{ y,S}) and (k _{ x,AS}, k _{ y,AS}). Figure 3a portrays the number of coincidences for a large field of view as a function of (k _{ y,S}, k _{ y,AS}) momenta summed over the x coordinates. Notably, thanks to a very high signaltonoise ratio, we do not subtract the accidental and noise background in contrast to hitherto schemes^{12,27,31,32}. We observe a clear anticorrelated behavior that we model by the quantum amplitude of the generated Stokes–antiStokes photon pair in y dimension, given by:
where σ _{ y } is a correlation length in the y dimension and \({\cal N}\) is a normalization constant. In turn, the number of coincidences is proportional to \(\left {{\mathrm{\Psi }}_y\left( {k_{y,{\mathrm{S}}},k_{y,{\mathrm{AS}}}} \right)} \right^2\). An identical expression describes photons behavior in x dimension—see inset in Fig. 3a. For the Gaussianshaped atomic ensemble, the size of the emission mode should be related to ensemble transverse dimension w = 0.6 ± 0.1 mm, corresponding to wavevector spread of 2/w = 3.3 ± 0.5 mm^{−1} in the far field for light at the wavelength of S photons. To precisely determine the mode widths σ _{ x,y }, in Fig. 3b, we plot the coincidences in terms of sum of wavevector variables. Gaussian fit yields values of σ _{ x } = 4.45 ± 0.02 mm^{−1} and σ _{ y } = 4.76 ± 0.02 mm^{−1} for x and y dimension, respectively. Consequently, we can consider the generated entangled state^{10,12} to be nearly symmetrical in terms of x and y spatial dimensions. This wavevector spread is very close to the limit 2/w given by the diffraction at the atomic ensemble and confirms the quality of the imaging system for conjugate modes.
Capacity estimation
From the fundamental point of view, multimode states of light can be considered either as continuousvariable systems^{49} or highly dimensional entangled states^{50} offering large dimensionality of available Hilbert space and in turn providing high informational capacity. Estimation of the informational capacity of continuousvariable entangled states of light has attracted some attention of its own due to broad applications of such states^{12,32}. Various measures of this capacity have been discussed, e.g., on the informationtheoretical basis^{51}. Here we estimate the number of independent mode pairs observed in S and AS arms using the Schmidt mode decomposition^{52,53}. For a singledimensional photon pair amplitude given by Eq. (1) and cropped to a finite region, we find a decomposition into the Schmidt modes as:
where λ _{ j } are singular values corresponding to contributions of each mode, while u _{ j } and \(v_j^*\) are orthogonal sets of eigenfunctions. Effective number of independent mode pairs is given by \(M = 1{\mathrm{/}}\mathop {\sum}\nolimits_{j = 0}^\infty \lambda _j^4\), which may be also described in terms of modes given in an orbitalangularmomentum^{7,8,50} or another orthogonal basis. We find the effective number of modes M _{ x,y } is well approximated by an inverse relation M _{ x,y } = 0.565 κ/2σ _{ x,y } (see Methods for justification) and obtain M _{ x } = 26.7 ± 0.1 and M _{ y } = 24.9 ± 0.1. Finally, for the total number of modes M, which is the product of the number of modes in each spatial dimension, we get M = M _{ x } M _{ y } = 665 ± 4.
Nonclassical photonnumber correlations
The presented spatial correlations at a singlephoton level require further analysis to confirm actual generation of multiphoton quantum states of light. Quantumness of the correlations (and hence the memory) may be assessed by looking at the secondorder correlation function:^{54}
where p _{S} and p _{AS} are the probabilities of registering a Stokes and an antiStokes photon in their respective regions, while p _{S,AS} is the Stokes–antiStokes coincidence probability. Since the singlemode statistics of Stokes and antiStokes light are thermal^{55}, the maximum value of local \(g_{{\mathrm{S,S}}}^{(2)}\) and \(g_{{\mathrm{AS,AS}}}^{(2)}\) autocorrelation functions is 2. Consequently, a value of \(g_{{\mathrm{S,AS}}}^{(2)} >2\) yields violation of the CauchySchwarz inequality:^{54}
and thus proves nonclassical character of the generated state of light.
To perform the measurements, we utilize the photonnumber resolving capability of the IsCMOS detector^{48}. We verify nonclassical photonnumber correlations in many modes by selecting a set of ROIs in both S and AS arms and calculating \(g_{{\mathrm{S,AS}}}^{(2)}\) for all accessible combinations. Results presented in Fig. 4a, b clearly confirm the multimode capacity discussed in previous sections. For the experimental data presented in Fig. 4b, we obtain \(g_{{\mathrm{S,AS}}}^{(2)} = 72 \pm 5\) at the diagonal compared with \(g_{{\mathrm{S,AS}}}^{(2)} = 1.0 \pm 0.4\) for a set of uncorrelated regions, where the errors correspond to one standard deviation. Next, we select a single pair of squareshaped conjugate ROIs in S and AS arms. Figure 4c presents the measured \(g_{{\mathrm{S,AS}}}^{(2)}\) at t = 0 storage time for varying size of ROI with a constant photon flux per pixel. With the decreasing size of ROI, the S photon detection probability p _{S} decreases and we observe \(g_{{\mathrm{S,AS}}}^{(2)}\) crosscorrelation well above the classical limit of 2, which perfectly matches our theoretical predictions (see Methods for detailed theory of \(g_{{\mathrm{S,AS}}}^{(2)}\) measured with a multipixel detector). We compare this result with a maximum value achievable without noise in the AS arm as well as maximum theoretical value for twomode squeezed vacuum state for the given p _{S}, achievable only if coherent spatial filtering (using e.g., singlemode fibers or cavities) is applied.
Even though we expect the photon statistics in S and AS arms to exhibit maximum values of autocorrelation functions of 2, to implicitly demonstrate violation of Cauchy–Schwartz inequality (4), we have performed additional measurements of \(g_{{\mathrm{S,S}}}^{(2)}\) and \(g_{{\mathrm{AS,AS}}}^{(2)}\) using a slightly modified experimental setup (see Methods for details). Due to inherently low number of S–S and AS–AS coincidences, we have increased the mean photon number in the S arm to 1.2 obtaining an average value of R = 4.0 ± 0.2, significantly violating inequality (4) and proving both \(g_{{\mathrm{S,S}}}^{(2)},g_{{\mathrm{AS,AS}}}^{(2)} \le 2\) (see Supplementary Fig. 2 for spatially resolved maps).
Storage capabilities
Cold atomic ensemble prepared in MOT typically offers μs up to ms coherence times, limited mainly by atomic motion, atom losses, and stray magnetic fields. We characterize the memory storage time by analyzing the \(g_{{\mathrm{S,AS}}}^{(2)}\) correlation function when the read laser is applied after a variable storage time t following the write pulse. Figure 4d presents the average \(g_{{\mathrm{S,AS}}}^{(2)}\) calculated for 1000 pairs of correlated squareshaped ROIs with side length κ = 21 mm^{−1} and p _{S} = 1.9 × 10^{−3} per entire ROI, each comprising approximately five modes. Data sets in Fig. 4d correspond to two different angles at which the photons were scattered, hence spin waves with different wavevectors—higher scattering angles (and thus spin waves with larger wavenumbers) correspond to shorter decay times. We observe a quantumbeating oscillation on a double exponential decay of correlations due to the presence of two types of spin waves arising as a result of imperfect optical pumping (see Methods for details). Due to the axial magnetic field of 36 mG, the two types of spin waves accumulate different phases over the storage time that leads to their constructive or destructive interference at the readout stage. We observe this interference effect as an oscillation with a Larmor period of T = 2π/ω = 19.5 μs. For all spin waves, we measure lifetimes larger than 50 μs. In particular for the clocktransition spin wave (between F = 1, m _{ F } = 1 and F = 2, m _{ F } = −1 states) with small K _{ x } = 100 mm^{−1}, we obtain the lifetime of over 100 μs. The main source of decoherence is the random atomic motion governed by the Maxwell–Boltzmann velocity distribution^{56}. The sharp drop in \(g_{{\mathrm{S,AS}}}^{(2)}\) in the very beginning (two initial experimental point) is attributed to increase of noise fluorescence as a result of an influx of unpumped thermal atoms into the interaction region. This noise might be eliminated by optical pumping of thermal atoms or by using a twostage MOT with differential pumping. See Supplementary Fig. 3 for the measured temporal evolution of noise fluorescence.
Discussion
We have demonstrated a quantum memoryenabled source of spatially structured nonclassical light based on a principle of wavevector multiplexing. Simultaneous operation on many collective atomic excitations allows us to generate a multimode quantum state of light. The memory preserves nonclassical correlations up to 50 μs and exhibits excellent noise properties, in contrast to the hitherto used warmatomic vapor schemes^{12,27}. Simultaneous detection using a stateoftheart singlephoton resolving camera is an ideal scheme to implement the enhanced photon generation protocols^{26,27,45}. Additionally, a twodimensional detector is both necessary and wellsuited to the access high quantum information capacity of multimode states of light, which is unachievable with singlemode fibers^{31}. Furthermore, simultaneous detection of the entire transverse field of view is essential in fundamental tests such as demonstration of the Einstein–Podolsky–Rosen paradox^{57} without the Bell sampling loophole^{32}.
Our results clearly demonstrate the ability of multimode quantum memory to emit a single photon with high probability. In particular, we measured S photon detection probability of 0.21 and simultaneously extremely low probability of registering a photon per mode equal 3.8 × 10^{−4} that drastically minimizes the probability of generating a photon pair in a single mode and proves memory efficacy in enhanced generation of photons. Excellent quality of single photons has been verified through measurements of \(g_{{\mathrm{S,AS}}}^{(2)}\) crosscorrelation function. Our quantum memory also exhibits an excellent timebandwidth product of >500, which is an important figure of merit in terms of probability of retrieving all the photons stored in the memory (Fig. 1), as well as prospective integration with timebin multiplexing^{26}. We envisage that hundreds of μs memory lifetime, contrasted with noisefree yet low storagetime solutions^{58}, and 100 ns operation time are excellent parameters when it comes to integration with fast electronic or photonic circuits for realtime feedback^{25,59}. With these technical difficulties overcame, we expect that the proposed enhanced multiphoton generation protocol would be readily realizable. Integration of existing schemes^{17} with readout efficiency χ _{R} of nearly 0.9 and the probability to generate n photons equal (χ _{R})^{n}, will make our protocol highly competitive. Keeping a low probability of generating a photon–atomic excitation pair per mode p ≈ 0.01, our setup can emit n = pM ≈ 6.6 photons on average and thus could efficiently generate even sixphoton states in the memory. Consequently, the number of modes M places a fundamental yet possibly distant limit on generating multiphoton states.
The number of available modes is limited by the imaging system. In a cold atomic ensemble generated within a released MOT, we expect that the final limit for the number of modes will be set by the lifetime of longwavevector spin waves as well as the phasematching at the retrieval stage. To keep the lifetime within tensofmicroseconds regime the maximum scattering angles should be smaller than 6 degrees while the phasematching happens to place a similar limitation^{4,60}. We thus predict that the number of readily available modes may reach thousands under realistic experimental conditions. However, with novel spinwave manipulation techniques^{61} or by placing the atoms in an optical lattice^{62} at least another orderofmagnitude improvement could be achieved, allowing our setup to serve as a universal platform for quantum state preparation.
While we have focused on the application of our quantum memory as a light source in multiplexed communication and computation protocols^{26,27}, our scheme is also perfectly matched to expedite quantum communication in freespace^{1} or with multimode or multicore fibers^{2}, quantum imaging, and image processing at the singlephoton level, as well as quantumenhanced metrology^{28,29}. Spatial photonnumber quantum correlations are readily applicable in quantum imaging techniques and the memory capability could help quantum ghost imaging or subshot noise imaging along the way to practical applications^{43,44}. Furthermore, the quantumbeat signal between two spinwave excitations demonstrates the ability of our quantum memory to store a superposition of a few spin waves in many modes and paves the way toward manipulations within and between the Zeeman sublevels as well as with the spatial degree of freedom.
Methods
Experimental sequence
Our experimental sequence depicted in Fig. 2c starts with trapping the atoms in MOT for 1.4 ms in an octagonal doubleside AR coated glass chamber (Precision Glassblowing) with cooling laser reddetuned from 5^{2} S _{1/2}, F = 2 → 5^{2} P _{3/2}, F = 3 transition by 16 MHz followed by 300 μs phase of PGC (cooling laser detuning increased to 35 MHz by tuning the doublepass acoustooptic modulator) that brings the temperature from roughly 100 μK to 22 ± 2 μK. Fast MOSFETbased (metaloxide semiconductor field effect transistor) coil switch turns off the coil current in <5 μs from 125 A down to zero, and thus the MOT is turned off during memory operation. Small stray magnetic fields due to eddy currents take another 200 μs to decay but are compensated to nearly zero in the position of the atomic ensemble by a shorted compensation coils, which we verified by taking the freeinduction decay measurements^{63}. Compensation coils maintain a constant axial magnetic field of 36 mG during both the optical pumping and the memory operation stages. The cooling and trapping period is followed by 40 μs stage of optical pumping carried out by three lasers to ensure emptying of the memory state \(\left h \right\rangle\) and maximizing of Zeeman population in the F = 1, m _{ F } = 1 state. Two lasers empty the F = 2 manifold: a strong pulse of the read laser (10 mW, depletion stage) as well as another laser (hyperfine pump—HF) tuned to 5^{2} S _{1/2}, F = 2 → 5^{2} P _{1/2}, F′ = 2 transition, incident from four directions (with various polarizations) with a total power of 10 mW. Another laser (Zeeman pump) with a power of 7 mW is resonant to 5^{2} S _{1/2}, F = 1 → 5^{2} P _{3/2}, F′ = 1 transition and transfers the population from m _{ F } = −1 and m _{ F } = 0 to m _{ F } = 1 state of the F = 1 manifold. Dark period of 1 μs follows the optical pumping to ensure all light from lasers and from the atomic ensemble is extinguished. Next, a 100 ns write pulse is applied (leftcircular polarization, reddetuned by 20 MHz from 5^{2} S _{1/2}, F = 1 → 5^{2} P _{3/2}, F′ = 2 transition). Due to small detunings from respective energy levels, the influence of deleterious processes of readout (writein) with the write (read) laser is negligible. After a variable memory storage time, a 200 ns read pulse (rightcircular polarization, resonant with 5^{2} S _{1/2}, F = 2 → 5^{2} P _{1/2}, F = 2 transition) is applied. All lasers are locked to either cooler or repumper laser through a beatnote offset lock^{64}.
Imaging
The angles at which the photons are emitted from the atomic ensemble are imaged on the photocathode of the image intensifier with two separate (for S and AS) complex telescopes composed of six lenses, each having an effective focal length of f _{eff} = 50 mm. Total length of the single system is 2 m as optically pumped atomic filters need to fit along the photons path. The linear size of one pixel of the sCMOS camera corresponds to transverse wavevector size of 2.1 mm^{−1} or angle of 265 μrad. Lens apertures limit our field of view to a ROI with κ = 420 mm^{−1} or total solid angle of Ω = 2.76 msr, which later determines maximum number of observable modes. The imaging system was calibrated with custom Ronchi rulings. The IsCMOS device, composed of image intensifier (Hamamatsu V7090D) and an sCMOS sensor (Andor Zyla 5.5 MP), is gated (Photek GM3003 gating module) only during writein and readout of atomic excitations. The sequence is repeated at the rate of 500 Hz, which is limited by the frame rate of the sCMOS camera. The combined image intensifier gate duration is 400 ns for both writein and readout stages. The probability of registering a dark count for the combined S and AS fields of view is ~5 × 10^{−3}—much less than typical photon detection probability of 0.2–0.3.
Filtering
Two separate rubidium vapor cells are used to filter out stray write and read laser light from S and AS photons. The 10cmlong cells are paraffincoated and contain 99.4% isotopically pure ^{87}Rb as well as buffer gases (Precision Glassblowing, 1 torr Kr for both S and AS filters) that keep the pumped atoms in the interaction region. The cells are pumped with 50 mW of resonant laser light (with 5^{2} S _{1/2}, F = 2 → 5^{2} P _{1/2} and 5^{2} S _{1/2}, F = 1 → 5^{2} P _{3/2} for S and AS filters, respectively) in a doublepass configuration with collimated 1cmwide beams. The optical pumping is active at all times except when the image intensifier gate is open. Importantly, write and read lasers are filtered with FabryPérot cavities before illuminating the ensemble to eliminate amplified spontaneous emission from laser diodes that would not be filtered with the hot atomic cell. Figure 5 presents a characterization of the S filter (we measured the comparable characteristics also for the AS filter). Both filters are characterized by OD >70 for the laser light and approx. sixtyfive percent transmission for single photons generated inside the atomic quantum memory.
Autocorrelation measurement
Part of the experimental setup has been modified to allow measurement of autocorrelation functions \(g_{{\mathrm{S,S}}}^{(2)}\) and \(g_{{\mathrm{AS,AS}}}^{(2)}\). A high extinctionratio Wollaston prism was placed in front of the image intensifier and a pair of halfwave plates was used to rotate the polarization of S and AS photons. The Wollaston prism split the photons into two beams (both for S and AS arm) at the 50:50 ratio in the vertical direction, so four distinct regions were observed on the camera (S1, S2, AS1, and AS2). After compensating for the change in angle of incidence due to refraction at the Wollaston prism, we have analyzed the correlations between regions S1–S2 and AS1–AS2 to obtain estimates of autocorrelation functions. The results are presented in Supplementary Fig. 2.
Coincident counts
Let us consider a collection of M squeezed modes pairs. Assuming the probability p of generating S photon in a single mode and efficiencies in the S and AS arms equal η _{S} and η _{AS} χ _{R}, respectively, with χ _{R} being the retrieval efficiency, we obtain the probability of registering a coincidence from any two conjugate modes pMη _{S} η _{AS} χ _{R}. If we now consider a pair of squareshaped ROIs with the side length κ containing M modes for which we again assume the probability p per mode to generate a photon pair, the coincidence rate is reduced, as not all coincidences will fall into the ROI. This effect is more pronounced for the smaller size of ROI. In particular, if we consider that the S photon is detected inside its respective ROI, we seek the probability that its conjugate AS photon will be detected in conjugate ROI (i.e., with conjugate center). We may calculate this probability by considering photon pairs distributed in momentum space according to Eq. (1). By considering S photons in the given ROI, we calculate the conditional probability f(κ) of registering AS photon in the conjugate ROI in the AS arm, which gives us:
where squaring is due to the twodimensional character of the problem. Finally, to estimate the net coincidence probability, we additionally consider the total number of accidental coincidences, which is very well approximated by a product of probabilities in S and AS arms p _{S} p _{AS} ^{12,24,56,65,66}.
The net S–AS coincidence probability thus equals:
Secondorder correlation
We model the evolution of \(g_{{\mathrm{S,AS}}}^{(2)}\) correlation function following Zhao et al.^{56}, but including the effect of interference of different spin waves as well as the reduced number of coincidence counts due to incoherent spatial filtering. Finally, we end up with the following expression for the secondorder correlation function:
where ξ is a contribution of noise in the AS arm. The retrieval efficiency is modeled as an interference of two fields arising due to two atomic coherences by the following timedependent expression:
where α _{1} and α _{2} are contributions of spin waves between F = 1, m _{ F } = 1 ↔ F = 2, m _{ F } = −1 and F = 1, m _{ F } = 0 ↔F = 2, m _{ F } = −2 transitions, respectively. The fit yields α _{1} = 0.58 and α _{2} = 0.04, clearly confirming dominant role of the clocktransition spin wave. The relative phase between the two spin waves changes as one of them accumulates additional phase due to a Zeeman energy shift ħω = 2πħ × 51 kHz in the axial magnetic field of 36 mG. The lifetimes are bounded by wavevectordependent decoherence rate \({\mathrm{\Gamma }}_D = \left {\bf{K}} \rightv\), with \(v = \sqrt {\frac{{k_{\mathrm{B}}T}}{{m_{{\mathrm{Rb}}}}}} \approx 1.45\,{\mathrm{cm}}\,{\mathrm{s}}^{  {\mathrm{1}}}\).
Uncertainty estimation
For the data presented in Fig. 4a to obtain a single \(g_{{\mathrm{S,AS}}}^{(2)}\) map, we have selected 100 regions in a column (oriented in the ydirection) in S and AS arms and calculated value of \(g_{{\mathrm{S,AS}}}^{(2)}\) for each pair of regions (note these regions are partially overlapping). We collect these results for 25 different conjugate positions of columns in the xdirection (i.e., k _{ x,S} + k _{ x,AS} = 0) and estimate mean and standard deviation. A completely analogous procedure was applied to obtain Fig. 4b and corresponding errorbars. Maps of standard deviation are included as Supplementary Fig. 1. To calculate the value of \(g_{{\mathrm{S,AS}}}^{(2)}\) for Fig. 4c, we have additionally averaged over all conjugate regions (corresponding to averaging over the diagonal in Fig. 4b) and inferring the errorbars (one standard deviation). Similar procedure was used to calculate values and standard deviations in Fig. 4d, however we used far less regions to average as we required that regions correspond to appropriate spinwave wavevector K _{ x }. Furthermore, in this measurement less frames were collected for each point and thus we obtain relatively high uncertainty.
Eigenmode decomposition
To correctly determine the number of modes, we use a similar procedure as proposed by Law and Eberly^{53}. Focusing on one dimension, we generate a normalized biphoton amplitudes according to Eq. (1), with various widths σ, on a square twodimensional k _{ y,S} − k _{ y,AS} grid. We numerically find the eigenmode decomposition of the generated matrix and calculate the number of modes according to Eq. (2). Figure 6 presents example of results for κ = 420 mm^{−1} while the solid line corresponds to a fit of Aκ/2σ relation, which we verified numerically for various sets of parameters and obtained A = 0.565. Note that for a biphoton amplitude on a rectangular (nonsquare) grid numerical singular value decomposition might be used to give similar results.
Data availability
The data that support the findings of this study are available from M.P. upon reasonable request.
Additional information
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Acknowledgements
We acknowledge insightful discussions about the MOT with L. Pruvost, G. Campbell, S. Du, G. Roati, M. Zawada, and W. Gawlik, assistance of R. Chrapkiewicz with optical system design, careful proofreading of the manuscript by K. T. Kaczmarek, M. Jachura, R. Chrapkiewicz, R. Łapkiewicz, and M. Semczuk, and generous support of K. Banaszek. The project has been funded by National Science Center (Poland) grant nos. 2015/19/N/ST2/01671, 2016/21/B/ST2/02559, and Polish MNiSW “Diamentowy Grant” Project nos. DI2013 011943, DI2016 014846.
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Institute of Experimental Physics, Faculty of Physics, University of Warsaw, Pasteura 5, 02093, Warsaw, Poland
 Michał Parniak
 , Michał Dąbrowski
 , Mateusz Mazelanik
 , Adam Leszczyński
 , Michał Lipka
 & Wojciech Wasilewski
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Contributions
M.P., M.D., M.M. and A.L. performed the measurements and analyzed the data. M.P., M.D. and M.M. wrote the manuscript assisted by other authors. M.P., M.D., M.M., A.L., M.L. and W.W. contributed to building of the experimental setup. W.W. managed the project.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to Michał Parniak or Michał Dąbrowski.
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