Interfacing GHz-bandwidth heralded single photons with a warm vapour Raman memory

Photonics is a promising platform for quantum technologies [1]. However, photon sources [2] and two-photon gates [3] currently only operate probabilistically. Large-scale photonic processing will therefore be prohibitively inefficient without a multiplexing strategy to actively select successful events. For this reason, spatial [4, 5], spectral [6] and temporal [7, 8] multiplexing are being investigated. For temporal multiplexing, quantum memories, which are capable to store and retrieve single photons on-demand, are an efficient solution. In particular, on-demand recall of the stored light can be used to temporally synchronize different outputs of heralded probabilistic events. Such temporal multiplexing enables scalable photonic architectures using repeat-until-success strategies [8–10]. A key figure of merit for this task is the time-bandwidth-product $B=\tau \delta$, where τ is the memory lifetime and δ is the memory acceptance bandwidth. B corresponds to the maximum number of possible processor steps that can occur within the memory lifetime. Dramatic enhancements in the scaling of photonic networks become possible with $B\sim 1000$ [8]. The very long storage times [11–13] and large time-bandwidth products [14, 15] obtainable by optical storage in atoms make these systems prime candidates for temporal synchronization devices. Notably, besides atomic systems, broadband signal storage has also been achieved in solid-state systems. Examples for these are off-resonant Raman storage [8] in diamond [16, 17] and in cryogenic rare-earth ion doped crystals [18], using atomic-frequency-comb (AFC) echoes with preprogrammed storage times. The recently demonstrated spin-wave storage in AFC [19–22] could also offer a route to temporal multiplexing.

The first step towards memory-based synchronization of single-photon generation is the storage of single photons. This has been demonstrated by spin-wave storage in cold atomic ensembles [23–26], single atoms [27, 28], and by pre-programmed AFC echoes in cryogenic rare-earth ion doped crystals [18, 29–31]. Storage [32, 33] and retrieval [34] of non-classical light in the room-temperature regime has been achieved with narrow bandwidths on the order of a few MHz. However, a quantum memory suitable for large-scale temporal multiplexing is still elusive. Key desiderata are broad acceptance bandwidths, on-demand storage and retrieval, as well as room-temperature operation with low noise. The ability to store multiple temporal modes in a single device [35, 36] is also appealing, if multiple memories could not be operated in parallel [37]. Notably, when we mention temporal multiplexing in this work we refer to temporal synchronization using a single-mode memory [8], not time-mode-multiplexing within a multimode memory; see supplementary material (stacks.iop.org/njp/17/043006/mmedia) for a more detailed discussion.

In this paper we report the storage of GHz-bandwidth heralded single photons, generated by spontaneous parametric down-conversion (SPDC), in a warm caesium (Cs) vapour Raman memory, for which $B\gtrsim 1000$ [14, 38]. On-demand memory operation, which is crucial for synchronization tasks, is implemented by triggering the memory on the detection of a herald photon. The performance of our system is characterized by measuring the autocorrelation of the fields transmitted through, and retrieved from the memory. We present a simple theoretical model which describes the system dynamics and identify four-wave mixing (FWM) as the sole significant noise source. Suppressing this noise process will make it possible to use the presented technology to construct large-scale photonic networks.

Our memory is based on transient Raman absorption in a Cs vapour cell at $70\;{}^\circ {\rm C}$ [14, 38, 40–42] (figure 1 and appendix A). The memory's large acceptance bandwidth of $\delta =1.2$ GHz, set by the control pulse bandwidth, allows it to be directly interfaced with a traveling-wave SPDC source without the need to enhance the spectral brightness by an optical cavity [43]. Our source is based on type-II SPDC in a periodically-poled potassium-titanyl-phosphate waveguide (figure 1 (a)). The single-pass waveguide is pumped by a pulse of approximately 260 ps duration at a wavelength of 426 nm. SPDC produces photons in pairs, thus we herald the presence of a signal photon to be stored in the memory by detecting its partner idler photon (${{{\rm D}}_{{\rm T}}}$ in figure 1(a)). Optimization of the SPDC spatial mode (figures 1 (b)–(d)) and signal path transmission yields a heralding efficiency of ${{\eta }_{{\rm herald}}}=0.22$, which is the probability that a single photon is sent into the memory given a herald detection event (see supplementary material available at stacks.iop.org/njp/17/043006/mmedia). As shown in figure 1, we operate the memory in feed-forward configuration (see supplementary material), whereby herald photon triggers the control field that operates the memory. Control pulses are prepared by pulse picking with a Pockels cell. Spectrally filtering the idler projects the signal photon's marginal spectrum into Raman resonance with the control [44, 45]. The heralded signal photons have a full-width-half-maximum bandwidth of ∼1.8 GHz. Conditioning on herald detection ensures that storage and retrieval are only attempted when a spectrally matched photon is sent into the memory. To benchmark the storage of heralded single photons we also store coherent states with average photon numbers similar to ${{\eta }_{{\rm herald}}}$. These are generated by phase modulation of a weak control beam pick-off [14, 38, 41]. Behind the Cs cell, the control field is separated from the signal by polarization and spectral filtering. The signal is then split 50:50 on a beam splitter and directed to a pair of photon-counting detectors (${{{\rm D}}_{{\rm H}}}$, ${{{\rm D}}_{{\rm V}}}$ in figure 1 (a)). See supplementary material available at stacks.iop.org/njp/17/043006/mmedia for details.

Zoom In Zoom Out Reset image size

The memory efficiency and noise are determined by measuring the arrival time histograms of signal photons transmitted through and retrieved from the memory. As shown in figure 2, the performance is similar for both signal types, with a total memory efficiency ${{\eta }_{{\rm tot}}}$ of 29% for coherent states and 21% for heralded single photons, limited by the control pulse energy, mode-matching between signal and control, and optical pumping (see supplementary material (stacks.iop.org/njp/17/043006/mmedia)). In addition to the signal, there is noise present in the read-in and the read-out time bins. It is visible as a pulse emitted from the memory when only the control field is applied. These noise pulses contain on average ${{\epsilon }_{{\rm in}}}=(6\pm 2)\times {{10}^{-2}}$ photons per read-in and ${{\epsilon }_{{\rm out}}}=(15\pm 5)\times {{10}^{-2}}$ photons per read-out control cycle. The noise originates from FWM, which is a two-step process: first, the control generates spurious spin-wave excitations by spontaneous anti-Stokes scattering (figure 1(f), appendix B.1). Second, subsequent retrieval of these excitations by the control leads to the emission of Stokes photons into the signal mode, which cannot be separated from the signal. The FWM noise is higher in the read-out bin, where it contains contributions from both the read and the write control pulses. The Raman memory's large bandwidth allows for memory operation with consecutive control pulses from the Ti:Sa pulse train, separated by 12.5 ns. In this regime decoherence of the spin-wave over the storage time is negligible, as is the effect of the optical pumping beam, which is, for convenience, on continuously. We have also explored longer storage times, where the retrieved signal vanishes because of the dephasing of the spin-wave. The noise then reduces to the level observed in the read-in time bin. This is characteristic of FWM, where spurious spin-wave excitations, generated during read-in, decohere at the same rate as the stored signal. As discussed in the supplementary material (stacks.iop.org/njp/17/043006/mmedia), the unconditional noise floor does not contain significant contributions from spontaneous Raman scattering emitted into the Stokes channel. The signal-to-noise ratio (SNR), i.e. the ratio of the number of retrieved signal photons to the number of noise photons, is ${\rm SNR}={{\eta }_{{\rm tot}}}{{\eta }_{{\rm herald}}}/{{\epsilon }_{{\rm out}}}=0.3\pm 0.1$. It should be noted that, with FWM present, the SNR, measured with the signal present and removed, may not be a faithful measure of memory performance due to a small amount of FWM gain during the memory operation. We have analysed this in the supplementary material, using a worse case model (stacks.iop.org/njp/17/043006/mmedia) and found only a small perturbation to the SNR presented here. Notably, the sub-unity heralding efficiency ${{\eta }_{{\rm herald}}}$ sets tight tolerances on the unconditional noise floor of the memory. For future implementations it is thus vital to understand the effects of noise on the properties of stored single photons.

Zoom In Zoom Out Reset image size

We characterize these effects by measuring the ${{g}^{(2)}}$ autocorrelation of all field combinations shown in figure 2 for both time bins. From the detector count statistics, the normalized autocorrelation [44], conditional on herald detection, is given by ${{g}^{(2)}}={{p}_{{\rm her},{\rm H},{\rm V}}}/\left( {{p}_{{\rm her},{\rm H}}}\cdot {{p}_{{\rm her},{\rm V}}} \right)$ (see supplementary material available at stacks.iop.org/njp/17/043006/mmedia). Here, ${{p}_{{\rm her},\;{\rm H}/{\rm V}}}$ is the probability that the heralding detector (${{{\rm D}}_{{\rm T}}}$) and one of the signal detectors (${{{\rm D}}_{{\rm H}}}$ or ${{{\rm D}}_{{\rm V}}}$) detect a photon in coincidence. ${{p}_{{\rm her},{\rm H},{\rm V}}}$ is the probability for a coincidence event between all three detectors (${{{\rm D}}_{{\rm T}}}$, ${{{\rm D}}_{{\rm H}}}$ and ${{{\rm D}}_{{\rm V}}}$). Blocking the control, we measure ${{g}^{(2)}}=1.01\pm 0.01$ for coherent-state inputs (average over all input photon numbers $N_{{\rm in}}^{{\rm COH}}$) and ${{g}^{(2)}}=0.016\pm 0.004$ for heralded single photons, which are close to the ideal values of 1 and 0. When the input signal is blocked, we measure ${{g}^{(2)}}=1.62\pm 0.04$ and ${{g}^{(2)}}=1.70\pm 0.02$ for the noise in the input and output time bins, respectively. With normal memory operation, i.e. signal and control applied, the photon statistics are modified by the accompanying FWM process. The ${{g}^{(2)}}$ of the transmitted and retrieved signals increase, yielding the results displayed in figure 3. In the input time bin, transmitted coherent-state signals converge towards the ideal ${{g}^{(2)}}=1$ for larger input photon numbers of ${{N}_{{\rm in}}}\sim 2.5$, as the amount of signal increases compared to the fixed amount of noise. Heralded single photons show ${{g}^{(2)}}=0.92\pm 0.02$, slightly below the classicality boundary. Looking at the memory read-out (figure 3(b)), we find that coherent states with an input photon number of ${{N}_{{\rm in}}}=0.23\approx {{\eta }_{{\rm herald}}}$ have ${{g}^{(2)}}=1.69\pm 0.02$ and are thus indistinguishable from the noise. Importantly, heralded single photons however show ${{g}^{(2)}}=1.59\pm 0.03$. This is a drop in ${{g}^{(2)}}$ by more than three standard deviations compared to coherent states and noise (see supplementary material available at stacks.iop.org/njp/17/043006/mmedia). The lower ${{g}^{(2)}}$ for heralded single photons reveals the influence of the non-classical SPDC input photon statistics in the memory read-out.

Zoom In Zoom Out Reset image size

To investigate the observed photon statistics we compare two models: first a model based on incoherent addition of signal and noise [46]. It treats both as independent fields with different ${{g}^{(2)}}$ values, whose mixing ratio is determined by the SNR (see appendix B.4). Second, we develop a model, which takes into account the full coherent off-resonant interaction between the incident light fields and the spin-wave excitation in the Cs ground states. The model assumes noise only through FWM and, unlike the incoherent addition, does not contain any free parameters. It uses the important realization that, while Raman storage and FWM are two different processes, they are coupled through the shared signal field. Their contributions to the signal mode are thus not independent. Signal and noise are generated by the same Hamiltonian from which the photon statistics in the Stokes output mode is predicted. To aid model comparison we also investigate the photon statistics without spin-wave storage by turning off the optical pumping for atomic state preparation, while still sending signal and control pulses into the Cs cell (figure 3(c)). While the incoherent model completely underestimates the ${{g}^{(2)}}$ and fails to describe the data (see appendix B.4), the coherent model agrees well with the measurements in all three cases, as the solid lines in figure 3 show. The bar graphs in figures 3(d)–(f) illustrate its excellent prediction of the observed ${{g}^{(2)}}$ difference between single photons and coherent states. Having the same good agreement for both signal types also displays how well the SPDC source can be mode matched to the Raman memory. Importantly, this result means that the FWM process has a much greater influence on the photon statistics than naively expected from the SNR. To benchmark memory performance, one consequently needs to investigate the photon statistics and the influence of noise thereon in addition to the SNR. However, since the ${{g}^{(2)}}$ increase originates solely from FWM, eliminating it would result in the noise free case [40] with full photon statistics conservation (see appendix B.1).

Luckily, FWM is not intrinsic to the memory interaction. The correlation between Stokes and anti-Stokes emission in FWM allows one to reduce Stokes noise by suppressing the anti-Stokes scattering. This enables noise reduction, while leaving the Raman memory unaffected. In principle, FWM can be suppressed by terminating the control field coupling to the initial atomic ground state via polarization selection rules. Yet, this approach is ineffective in alkali vapours far from resonance [47]. As FWM has phase-matching constraints similar to SPDC [48], an alternative could be to introduce dispersion between the Stokes and anti-Stokes frequencies, or to use a storage medium with larger Stokes shifts [17]. For alkali vapours, the approach of choice is to reduce the density of states at the anti-Stokes frequency by placing the memory inside a low-finesse cavity or photonic-bandgap structure [42]. As shown in figure 3(g), a FWM suppression by a factor of ∼2.5, readily achievable with the aforementioned techniques, would preserve the nonclassical signature of retrieved heralded single photons (see also appendix B.3). The resulting control over FWM will enable operation of our system as a noise-free memory for scalable photonics in the near future.

In conclusion, we have, for the first time, interfaced sub-ns heralded single-photons with a warm atomic vapour memory in a practical configuration suitable for temporal multiplexing; a key goal for photonics quantum information science. Using this new device we have investigated the storage and retrieval of broadband heralded single photons in the memory. We have demonstrated that FWM is the only, ultimately important, noise source in EIT- and Raman-type memories. Its influence on the photon statistics is more severe than naively expected, but possible to mitigate.

We thank A Lvovsky, E Poem and C Kupchak for helpful discussions on FWM in Raman systems. We acknowledge Nathan Langford for early contributions to the design of the source, Michał Karpiński for assistance with the source optimization. We thank Justin Spring for assistance with the FPGA. This work was supported by the UK Engineering and Physical Sciences Research Council (EPSRC; EP/J000051/1 and Programme Grant EP/K034480/1), the Quantum Interfaces, Sensors, and Communication based on Entanglement Integrating Project (EU IP Q-ESSENCE; 248095), the Air Force Office of Scientific Research: European Office of Aerospace Research & Development (AFOSR EOARD; FA8655-09-1-3020), EU IP SIQS (600645), the Royal Society (to J.N.), EU ITN FASTQUAST (to P.S.M.), and EU Marie-Curie Fellowships (PIIF-GA- 2011-300820 to X.-M.J.; PIEF-GA-2012-331859 to W.S.K.; PIIF-GA-2013-629229 to D.J.S.). I.A.W. acknowledges an ERC Advanced Grant (MOQUACINO).

As shown in figures 1 (a) and (e) of the main text, storage of a signal field, which is blue-detuned by $\Delta =15.2$ GHz from the Cs ${{D}_{2}}$-line, is triggered by a co-propagating, orthogonally polarized control pulse tuned into two-photon resonance with the ground-state hyperfine transition (${{\Delta }_{{\rm s}}}=9.2\;{\rm GHz}$ hyperfine ground state splitting). Signal storage transfers Cs atoms, initially prepared in the ${{6}^{2}}{{{\rm S}}_{\frac{1}{2}}}F=4$ hyperfine ground state ($|1\rangle$) by optical pumping with a counter-propagating resonant diode laser, to the ${{6}^{2}}{{{\rm S}}_{\frac{1}{2}}}F=3$ hyperfine ground state ($|3\rangle$), exciting a spin-wave coherence between both states [38–40]. Re-application of the control field drives the reverse process, producing a read-out signal and returning the atoms to $|1\rangle$.

B.1. Coherent ${{g}^{(2)}}$ model

Figure 3 in the main text shows our measurements of the ${{g}^{(2)}}$ field autocorrelations alongside the predictions of a theoretical model that includes the effects of FWM seeded by spontaneous anti-Stokes scattering. The agreement is extremely good, which suggests that FWM is the only significant source of noise in the memory. Here we briefly outline the theoretical model, which is based on [14, 17], and is in fact identical to [49]. We consider one-dimensional propagation along the z-axis, normalized so that z runs from 0 to 1, of Stokes (signal) and anti-Stokes fields S, A through an ensemble of three-level Λ-type atoms (see figures 1 (e), (f)) in the presence of a control pulse with time-dependent Rabi frequency $\Omega (\tau )$, where $\tau =t-z/c$ is the local time, at time t, in a frame propagating with the pulse at velocity c. After adiabatic elimination of the excited state $|2\rangle$ [50, 51], the Maxwell–Bloch equations describing the Raman interaction of the fields with the ground state coherence are given by

$$\begin{eqnarray*}&&[{{\partial }_{z}}+{\rm i}\kappa ]S={\rm i}CB,\end{eqnarray*}$$

$$\begin{eqnarray*}&&{{\partial }_{z}}{{A}^{\dagger }}=-{\rm i}C^{\prime} B,\end{eqnarray*}$$

$$\begin{eqnarray}&&[{{\partial }_{\epsilon }}+{\rm i}s]B={\rm i}w[CS+C^{\prime} {{A}^{\dagger }}],\end{eqnarray} \tag{ B.1 }$$

where $B={{\sum }_{j:{\rm atoms}\,{\rm at}\,z}}|{{3}_{j}}\rangle \langle {{1}_{j}}|{{{\rm e}}^{{\rm i}{{k}_{B}}z}}$ is the annihilation operator for spin wave excitations with wavevector ${{k}_{B}}={{k}_{A}}-{{k}_{S}}$, with ${{k}_{S,A}}$ the Stokes, anti-Stokes wavevectors. The effective time $\epsilon =\epsilon (\tau )=\alpha \int _{-\infty }^{\tau }|\Omega (\tau ^{\prime} ){{|}^{2}}\;{\rm d}\tau ^{\prime}$, with α such that $\epsilon (\infty )=1$, parameterizes the adiabatic following of the control pulse [40]. With this coordinate transformation, the dynamic Stark shift is $s={{(\alpha \Delta )}^{-1}}+{{(\alpha \Delta ^{\prime} )}^{-1}}$, where $\Delta ^{\prime} =\Delta +{{\Delta }_{{\rm s}}}$ is the anti-Stokes detuning (figure 1 (e)). The dimensionless coupling constant C is given by ${{C}^{2}}=d\gamma /\alpha {{\Delta }^{2}}$, where d is the resonant optical depth and γ the homogeneous linewidth of the excited state [38, 51]. C' is identical to C, except that Δ is replaced by $\Delta ^{\prime}$. Both are inversely proportional to the respective detuning of the channel they couple to (Stokes or anti-Stokes). $C^{\prime} =0$ corresponds to a vanishing anti-Stokes coupling which represents the noiseless case [40]. The population inversion is $w={{p}_{1}}-{{p}_{3}}$, where ${{p}_{1}}=\langle |1\rangle \langle 1|\rangle$ and ${{p}_{3}}=\langle |3\rangle \langle 3|\rangle$ are the initial occupation probabilities of the ground states $|1\rangle$, $|3\rangle$, respectively. Finally, we have defined the FWM phase mismatch $\kappa =2{{k}_{\Omega }}-{{k}_{S}}-{{k}_{A}}$, where ${{k}_{\Omega }}=L{{\omega }_{{\rm c}}}/c+d\gamma ({{p}_{3}}/\Delta +{{p}_{1}}/\Delta ^{\prime} )$ is the wavevector of the control with frequency ${{\omega }_{{\rm c}}}.$ We also have ${{k}_{S}}=L{{\omega }_{S}}/c+d\gamma [{{p}_{1}}/\Delta +{{p}_{3}}/(\Delta -{{\Delta }_{{\rm s}}})]$, and ${{k}_{A}}=L{{\omega }_{A}}/c+d\gamma [{{p}_{3}}/\Delta ^{\prime} +{{p}_{1}}/(\Delta +2{{\Delta }_{{\rm s}}})]$. Here the length L of the cell appears, to account for the normalization of the z-coordinate. These expressions are simply derived by considering the refractive index for each field, due to off-resonant interaction with the atomic transitions. Note that on Raman resonance we have ${{\omega }_{S}}={{\omega }_{{\rm c}}}-{{\Delta }_{{\rm s}}}$ and ${{\omega }_{A}}={{\omega }_{{\rm c}}}+{{\Delta }_{{\rm s}}}$. Our experiments were done with a 7.5 cm Cs cell held at 70 °C, giving a resonant optical depth $d\approx 1800$. We focus 10 nJ control pulses with duration ${{T}_{{\rm c}}}=360$ ps into the cell with a waist ∼300 μm, giving a peak Rabi frequency ${{\Omega }_{{\rm max} }}=4.2$ GHz, so that $\alpha =0.31$ ns. We operate blue-detuned from resonance with the D₂ line, whereby the detuning is $\Delta =15.2$ GHz and the Stokes shift in Cs is ${{\Delta }_{{\rm s}}}=9.2$ GHz. Finally, the homogeneous linewidth is $\gamma =16$ MHz, so that we have C = 0.82. When the memory is operated normally, the atoms are optically pumped into the ground state $|1\rangle$, and we set ${{p}_{3}}=0$ (w = 1). For the case where the optical pumping beam is blocked, the atoms thermalize and we set ${{p}_{3}}=0.5$ (w = 0).

The system of coupled partial differential operator equations (B.1) is linear, and the solutions can therefore be written as a linear mapping from initial input to final output fields via the system's Green's functions G_jk. For example, the signal field transmitted through the memory during the storage interaction is

$$\begin{eqnarray}\begin{array}{rcl} {{S}_{{\rm trans}}}(\epsilon ) & = & \int _{0}^{1}{{G}_{SS}}(\epsilon ,\epsilon ^{\prime} ){{S}_{{\rm in}}}(\epsilon ^{\prime} ){\rm d}\epsilon ^{\prime} +\int _{0}^{1}{{G}_{AS}}(\epsilon ,\epsilon ^{\prime} )A_{{\rm vac}}^{\dagger }(\epsilon ^{\prime} ){\rm d}\epsilon ^{\prime} \\ {} & {} & +\int _{0}^{1}{{G}_{BS}}(\epsilon ,z){{B}_{{\rm therm}}}(z){\rm d}z, \\ \end{array}\end{eqnarray} \tag{ B.2 }$$

where 'vac' and 'therm' denote initial vacuum and thermal states for the anti-Stokes and spin-wave fields. The signal field retrieved from the memory after a storage time T is given by a similar expression,

$$\begin{eqnarray*}&&{{S}_{{\rm ret}}}(\epsilon )=\int _{0}^{1}{{G}_{SS}}(\epsilon ,\epsilon ^{\prime} ){{S}_{{\rm vac}}}(\epsilon ^{\prime} ){\rm d}\epsilon ^{\prime} +\int _{0}^{1}{{G}_{AS}}(\epsilon ,\epsilon ^{\prime} )A_{{\rm vac}}^{\dagger }(\epsilon ^{\prime} ){\rm d}\epsilon ^{\prime} +\int _{0}^{1}{{G}_{BS}}(\epsilon ,z){{B}_{T}}(z){\rm d}z,\end{eqnarray*}$$

where to a good approximation in our experiment we may neglect decoherence and set ${{B}_{T}}={{B}_{{\rm out}}}$, since T = 12.5 ns, while the memory lifetime is $\gt 1$ μs [14]. Here, the spin-wave at the end of the storage interaction is given by

$$\begin{eqnarray*}&&{{B}_{{\rm out}}}(z)=\int _{0}^{1}{{G}_{SB}}(z,\epsilon ){{S}_{{\rm in}}}(\epsilon ){\rm d}\epsilon +\int _{0}^{1}{{G}_{AB}}(z,\epsilon )A_{{\rm vac}}^{\dagger }(\epsilon ){\rm d}\epsilon +\int _{0}^{1}{{G}_{BB}}(z,z^{\prime} ){{B}_{{\rm therm}}}(z^{\prime} ){\rm d}z^{\prime} .\end{eqnarray*}$$

The ${{g}^{(2)}}$ autocorrelation for a short-pulsed time-dependent field measured by slow detectors is given by

$$\begin{eqnarray}&&g_{x}^{(2)}=\frac{\int \int {\rm d}t\;{\rm d}t^{\prime} \langle S_{x}^{\dagger }(t)S_{x}^{\dagger }(t^{\prime} ){{S}_{x}}(t^{\prime} ){{S}_{x}}(t)\rangle }{{{\left[ \int {\rm d}t^{\prime\prime} \langle S_{x}^{\dagger }(t^{\prime\prime} ){{S}_{x}}(t^{\prime\prime} )\rangle \right]}^{2}}}=\frac{\int \int {\rm d}\epsilon \;{\rm d}\epsilon ^{\prime} \langle S_{x}^{\dagger }(\epsilon )S_{x}^{\dagger }(\epsilon ^{\prime} ){{S}_{x}}(\epsilon ^{\prime} ){{S}_{x}}(\epsilon )\rangle }{{{\left[ \int {\rm d}\epsilon ^{\prime\prime} \langle S_{x}^{\dagger }(\epsilon ^{\prime\prime} ){{S}_{x}}(\epsilon ^{\prime\prime} )\rangle \right]}^{2}}},\end{eqnarray} \tag{ B.3 }$$

where the label 'x' is either 'ret' or 'trans'. Since the initial field operators satisfy bosonic commutation relations (or in the case of $\langle [{{B}_{{\rm in}}}(z),B_{{\rm in}}^{\dagger }(z^{\prime} )]\rangle =w\delta (z-z^{\prime} )$, boson-like) and their expectation values on the initial state of the atomic-optical system are known, the expectation values of normally ordered products of the output field operators can be computed from products of the Green's functions [14, 49], enabling the calculation of the ${{g}^{(2)}}$. The Green's functions can be found analytically [49], although for convenience we use a previously-developed numerical code. Full details will be given elsewhere. However here we note that the ${{g}^{(2)}}$ predicted by this calculation depends on the statistics of the input field, and on its brightness, through taking expectation values involving the field operator S_in in equation (B.2).

B.2. Monte-Carlo error propagation

B.3. Effect of reduced anti-Stokes scattering

B.4. Incoherent ${{g}^{(2)}}$ model

We alternatively model the combined photon statistics $\left( g_{{\rm tot}}^{(2)} \right)$ as a superposition between FWM noise $\left( g_{{\rm noise}}^{(2)} \right)$ and the input signal $\left( g_{{\rm sig}}^{(2)} \right)$, whereby both fields are imagined to be combined on a beam-splitter. The expected value for $g_{{\rm tot}}^{(2)}$ is derived following the argumentation presented in [46], yielding

$$\begin{eqnarray}&&g_{{\rm tot}}^{(2)}=\frac{{{\left( {{N}_{{\rm sig}}} \right)}^{2}}\cdot g_{{\rm sig}}^{(2)}+2{{N}_{{\rm sig}}}\cdot {{N}_{{\rm noise}}}+{{\left( {{N}_{{\rm noise}}} \right)}^{2}}\cdot g_{{\rm noise}}^{(2)}}{{{\left( {{N}_{{\rm sig}}}+{{N}_{{\rm noise}}} \right)}^{2}}}.\end{eqnarray} \tag{ B.4 }$$

It depends on the number of signal photons, N_sig, and noise photons, N_noise, per pulse contributing to the mixture. If the memory is on, N_sig represents the transmitted fraction of the signal, ${{N}_{{\rm sig}}}=(1-{{\eta }_{{\rm read}-{\rm in}}})\cdot {{N}_{{\rm in}}}$, in the read-in time bin and the retrieved fraction of the signal, ${{N}_{{\rm sig}}}={{\eta }_{{\rm tot}}}\cdot {{N}_{{\rm in}}}$, in the read-out time bin. N_in is the respective input photon number, ${{\eta }_{{\rm read}-{\rm in}}}$ is the memory read-in efficiency and ${{\eta }_{{\rm tot}}}$ is the total memory efficiency (storage and retrieval). Figure B1 shows the results of equation (B.4) alongside the experimental data and the coherent ${{g}^{(2)}}$-model predictions. Clearly the incoherent model predictions significantly underestimate the measured ${{g}^{(2)}}$ data in all three cases.

Zoom In Zoom Out Reset image size

Notably, if the photon statistics for the input signal during memory storage was not conserved, it would not be justified to use the input signal's $g_{{\rm sig}}^{(2)}$ in equation (B.4) for calculating $g_{{\rm tot}}^{(2)}$ of the memory output. Instead, a modified $g_{{\rm sig}}^{(2)}$ would be required, which is not directly accessible experimentally. This potential flaw is circumvented by also measuring $g_{{\rm tot}}^{(2)}$ with the optical pumping switched off (figures 3(c) and B1(c)). Since there is no Raman storage, no modification of $g_{{\rm sig}}^{(2)}$ occurs. Moreover, N_sig corresponds directly to the input photon number ${{N}_{{\rm in}}}$, without additional memory efficiency factors. Yet, equation(B.4) also fails to describe this situation, despite the fact that all variables are known by direct measurement. Hence we can reject this model and its associated implication. Consequently, both light fields cannot be considered as individual entities; i.e. describing the system requires to take the full dynamics between the Cs atoms and the input light fields into account.