Superradiant parametric conversion of spin waves

TL;DR

This paper introduces a scheme to generate spatially-structured superradiant states using atomic spin waves, enabling the conversion of long-lived spin excitations into photon pairs for advanced quantum optics applications.

Contribution

It presents a novel method to prepare superradiant states in atomic ensembles via spin-wave light storage, facilitating nonlinear optics with spin waves.

Findings

Demonstration of spatially-structured superradiant states

Conversion of atomic spin waves into photon pairs

Potential for nonlinear spin wave optics

Abstract

Atomic-ensemble spin waves carrying single-photon Fock states exhibit nonclassical many-body correlations in-between atoms. The same correlations are inherently associated with single-photon superradiance, forming the basis of a plethora of quantum light-matter interfaces. We devise a scheme allowing the preparation of spatially-structured superradiant states in the atomic two-photon cascade using spin-wave light storage. We thus show that long-lived atomic ground-state spin waves can be converted to photon pairs opening the way towards nonlinear optics of spin waves via multi-wave mixing processes.

Figures

Equations

$$\mathcal{S}(\mathbf{r})\equiv \sqrt{n(\mathbf{r})}{\rho }_{cg}(\mathbf{r})=\sqrt{n(\mathbf{r})}\beta e^{i(\mathbf{K}+{\mathbf{k}}_{P1}+{\mathbf{k}}_{P2})\mathbf{r}},$$

$${\psi }_k({\mathbf{k}}_{s\perp },{\mathbf{k}}_{i\perp })\propto \tilde{n}(\Delta {\mathbf{k}}_{\perp })e^{-\Delta k_z^2{\sigma }_z^2},$$

$$\begin{gathered} \Delta k_z\equiv K_z+\sqrt{k_{P1}^2-k_{P1\perp }^2}+\sqrt{k_{P2}^2-k_{P2\perp }^2} \\ -\sqrt{k_s^2-k_{s\perp }^2}-\sqrt{k_i^2-k_{i\perp }^2} \end{gathered}$$

$${\psi }_t(\tau )=\mathcal{N}{e}^{-\tau /2{\tau }_0}\Theta (\tau ),\tau =t_s-t_i.$$

$$R={\left(g_{si}^{(2)}\right)}^2/\left(g_{ss}^{(2)}{g}_{ii}^{(2)}\right)=986\pm 153\gg 1.$$

$${\bar{n}}_{\mathrm{coh}}({\mathbf{k}}_{i\perp })\propto \int e^{-2\Delta K_z({\mathbf{k}}_{s\perp },{\mathbf{k}}_{i\perp }{)}^2{\sigma }_z^2}{\mathrm{d}}^2{\mathbf{k}}_{s\perp }$$

$${\chi }_s=0,$$

$${\chi }_i=\frac{n}{{ϵ}_0\hslash }\frac{|d_{gd}{|}^2}{i{\Gamma }_d+2\Delta },$$

$${\chi }_s^{NL}=\frac{n}{{ϵ}_0\hslash }\frac{d_{dc}{d}_{dg}}{i{\Gamma }_d-2\Delta },$$

$${\chi }_i^{NL}=\frac{n}{{ϵ}_0\hslash }\frac{d_{gd}{d}_{cd}}{i{\Gamma }_d+2\Delta },$$

$${\mathcal{H}}_{\mathrm{eff}}={ϵ}_0{\int }_V{\mathrm{d}}^3\mathbf{r}{\chi }_i^{NL}{\mathcal{S}}^{+}{E}_s^{-}{E}_i^{-}+h.c.$$

$$E_j^{-}=\sum _{{\mathbf{k}}_j}{E}_j^{\ast }{a}_{{\mathbf{k}}_j}^{†}{e}^{-i({\mathbf{k}}_j\cdot \mathbf{r}-{\omega }_{j}t)}$$

$$|\Psi ⟩=|0⟩-\frac{i}{\hslash }\int \mathrm{d}t{\mathcal{H}}_{\mathrm{eff}}|0⟩=|0⟩+\sum _{{\mathbf{k}}_s}\sum _{{\mathbf{k}}_i}F({\mathbf{k}}_s,{\mathbf{k}}_i)a_{{\mathbf{k}}_s}^{†}{a}_{{\mathbf{k}}_i}^{†}|0⟩$$

Full text

I Introduction

Spatially extended atomic ensembles are a particularly versatile medium allowing generation and control of light with widely varying properties. Fulfillment of the phase-matching (PM) condition facilitates efficient generation, storage and retrieval of single photons. Superradiance is inherently linked to the phase-matched emission and, in particular, spin waves (SW) that store information about light in the atomic coherence are superradiant Dicke states $N^{-1/2}\sum_{j}e^{i\mathbf{K}\mathbf{r}_{j}}|g_{1}\ldots h_{j}\ldots g_{N}\rangle$ Dicke (1954). In practice, the $\Lambda$ scheme of atomic levels, which forms the basis of the Duan-Lukin-Cirac-Zoller quantum entanglement distribution protocol (Duan et al., 2001), is well-known for its capabilities to generate photon pairs (Kuzmich et al., 2003; van der Wal et al., 2003) with both non-trivial temporal (Zhao et al., 2016) and spatially-multimode structure (Parniak et al., 2017; Pu et al., 2017). There, light is interfaced with a coherence between two metastable ground-state sublevels.

An alternative ladder or diamond schemes allow generation of two-color photon pairs, and attract much attention (Gulati et al., 2015; Guo et al., 2017; Srivathsan et al., 2013; Zhang et al., 2014) also for single-photon storage (Kaczmarek et al., 2018; Finkelstein et al., 2018) and in Rydberg-blockaded media (Lukin et al., 2001; Gaëtan et al., 2009). In those cases the associated SWs lie between a ground state and an excited atomic state and are intermediate steps in the generation of a photon pair. More complex manipulations of those SWs, such as temporal (Mazelanik et al., 2019; Reim et al., 2012; Saglamyurek et al., 2018; Campbell et al., 2014; Yang et al., 2018) and spatial (Parniak et al., 2019) mutli-SW beamsplitters demonstrated for ground-state SWs, remain elusive. Such control would allow SW-based engineering of photon pair emission, possibly also extensible to deterministic quantum nonlinear optics based on Rydberg atoms (Li et al., 2016).

The atomic-ensemble-based schemes hold some advantages over spontaneous parametric down-conversion processes (SPDC) in nonlinear crystals and optical parametric oscillators (Kwiat et al., 1995; Neergaard-Nielsen et al., 2007; Takeno et al., 2007), that also rely on engineered PM, as the photons generated in atomic ensembles are inherently narrowband and atom-resonant, making them suitable for quantum metrology (Qiu et al., 2016) and repeater-based communication (Duan et al., 2001; Sangouard et al., 2011; Niizeki et al., 2018). With purely atomic photon-pair source, potentially difficult engineering of cavity-based SPDC can be avoided (Wolfgramm et al., 2008, 2011).

Here we show that the properties of a phase-matched cascaded atomic decay can be engineered via proper preparation of the atomic SW state. In particular, we treat the SW prepared in the Raman process as one of the fields participating in the parametric conversion, similarly as a pump field in a spontaneous parametric down-conversion (SPDC) in nonlinear crystals (Kwiat et al., 1995). Hitherto schemes necessarily required that the atom starts and ends the wave-mixing processes in the same ground state, as dictated by the principles of energy and momentum conservation. This requirement can be leveraged if the ensemble is first prepared in a superposition state, or in other words some coherence is present. Our scheme allows for complex engineering of two-photon emission, with atom-resonant photons without the need of cavities.

We generate a strong atomic coherence (SW) between $|g\rangle$ and $|h\rangle$ via Raman interaction in the $\Lambda$ scheme, as depicted in Fig. 1(a). The two pump fields (P1 and P2), as in Fig. 1(b), then transfer the coherence $|g\rangle\langle h|$ via a two-photon process to the $|g\rangle\langle c|$ involving the highest excited state. The standard coherent two-photon decay follows by the end of which the atom returns to its ground state $|g\rangle$. The entire process can be viewed as a two-step spontaneous six-wave mixing (6WM) with a time delay, previously demonstrated for classical beams Parniak et al. (2016a), which overall conserves both momentum and energy.

This unorthodox view leads to numerous implications, as the SW may be manipulated in various ways, including usage of magnetic (Sparkes et al., 2013; Campbell et al., 2012; Higginbottom et al., 2015), electric (Hétet et al., 2008) and optical fields (Parniak et al., 2019; Mazelanik et al., 2019; Lipka et al., 2019; Ham, 2017; Sparkes et al., 2010). Reshaping the SW structure will subsequently result in a modified atomic state during emission of the idler photon. Such a modification can serve to change the properties of superradiance facilitating better understanding of this collective process, and harnessing it in bulk atomic ensembles. From the quantum information perspective, both polarization, temporal and spatial properties of the generated biphoton wave-function can be modified, leading to complex multi-degree-of-freedom entangled states.

The ladder (Lee et al., 2017, 2016; Jeong et al., 2017) and diamond type (Parniak and Wasilewski, 2015; Becerra et al., 2008; Jen, 2017a) systems have previously been successfully used to generate photons with particularly good characteristics. The key advantages of these schemes applied to Rubidium are the possibility to generate one of the photons at telecom wavelength (Chanelière et al., 2006; Radnaev et al., 2010; Ding et al., 2015) and relaxed requirements for filtering, even in room-temperature ensembles (Zhang et al., 2014; Willis et al., 2011; Park et al., 2017), as compared with the $\Lambda$ scheme. Previous preliminary studies have suggested that excited-to-excited state transition could be used for readout of a ground-state atomic coherence Parniak et al. (2016a); Ding et al. (2012).

A striking feature of the two-photon decay is the superradiant enhancement of emission (Dicke, 1954; Srivathsan et al., 2013; Lee et al., 2017), observed both in ultracold and warm atomic ensembles at high optical depths. Superradiance, while previously observed in many systems, here is shown to be controlled within the spatial domain. Our system thus contributes to recent fundamental studies of superradiant emission in cold atoms (Kuraptsev et al., 2017; Roof et al., 2016; Ortiz-Gutiérrez et al., 2018). Among many intricacies, this behavior proves the role of atomic coherence in the process.

II Experiment

We start the experiment by employing two coherent optical fields (Write and Seed, both 795 nm) to generate macroscopic ground-state atomic coherence $\rho_{hg}$ between states $|g\rangle$ and $|h\rangle$ in a cold ${}^{87}\mathrm{Rb}$ atomic ensemble (see Fig. 1(a), and Appendix A for details). This coherence, commonly called a spin wave has a nontrivial spatial dependence $\rho_{hg}(\mathbf{r})\propto\exp(i\mathbf{K}\mathbf{r})$, where $\mathbf{K}=\mathbf{k}_{W}-\mathbf{k}_{Sd}$ is a SW wavevector and $\mathbf{k}_{W}$, $\mathbf{k}_{Sd}$ are wavevectors of the Write and Seed field respectively.

In next step we use a pair of strong laser fields (Pump 1 and Pump 2) to transform substantial part of excitations of state $|h\rangle$ into the state $|c\rangle$, creating an excited-state SW: $\rho_{cg}(\mathbf{r})\propto\rho_{hg}\exp(i(\mathbf{k}_{P1}+\mathbf{k}_{P2})\mathbf{r})$, where $\mathbf{k}_{P1}$, $\mathbf{k}_{P2}$ are the wavevectors of the Pump 1 and 2 respectively. For clarity, assuming large diameters of Pump 1&2 beams, we define the excited-state SW as:

[TABLE]

where the complex SW amplitude $\beta$ incorporates all the phase shifts between pump fields, and by $n(\mathbf{r})$ we denote the local atom density. The presence of $\mathcal{S}$ enables spontaneously induced coherent two-photon transition to the original ground state level $|g\rangle$ closing the 6WM scheme.

The transition happens in two steps: first the spontaneously emitted signal photon transfers one collective excitation of $|c\rangle$ to the intermediate state $|d\rangle$, conditionally creating a Fock-state SW between levels $|d\rangle$ and $|g\rangle$, from which the superradiantly enhanced emission of the idler photon occurs. Thus, the idler photon is emitted always after the signal photon.

The whole quantum amplitude $\psi(t_{s},t_{i},\mathbf{k}_{s\perp},\mathbf{k}_{i\perp})$ of the generated biphoton state exhibits two features: the spatial and temporal correlation. Here, however, we study these features separately and thus we will define two proper objects: the time-averaged spatial wave function $\psi_{k}(\mathbf{k}_{s\perp},\mathbf{k}_{i\perp})=\langle\psi(t_{s},t_{i},\mathbf{k}_{s\perp},\mathbf{k}_{i\perp})\rangle_{t}$, and the temporal wave function $\psi_{t}(t_{s},t_{i})$ for fixed pair of $\left\{\mathbf{k}_{s\perp},\mathbf{k}_{i\perp}\right\}$.

From quantum perturbation theory (see Appendix B for the derivation using Hamiltonian treatment) for weak signal and idler fields, assuming that the atom density $n(\mathbf{r})$ is described by the Gaussian distribution with longitudinal width equal $\sigma_{z}$ we can write the spatial part as:

[TABLE]

where by $\tilde{n}$ we denote the Fourier transform of the atomic density, the transverse wavevector mismatch is defined as follows: $\Delta\mathbf{k}_{\perp}=\mathbf{K}_{\perp}+\mathbf{k}_{P1\perp}+\mathbf{k}_{P2\perp}-\mathbf{k}_{s\perp}-\mathbf{k}_{i\perp}$, and by

[TABLE]

we denote the longitudinal wavevector mismatch.

The temporal part can be approximated with the following form (Srivathsan et al., 2014):

[TABLE]

Importantly, the time constant $\tau_{0}$ depends on particular geometrical configuration, i.e. the PM, exposing the non-trivial connection of the temporal and spatial properties of the generated state.

The experimental setup details are provided in Fig. 1(c). The signal, idler and all of the control beams (including Pumps as well as Write and Seed) are combined in the far field (FF) of the ensemble and propagate along the $z$-axis. Since the signal and idler fields closing the 6WM process have orthogonal polarizations (respectively: $\sigma^{+}$and $\sigma^{-}$), we separate them using quarter-wave plate and Wollaston prism. Then after passing the interference filters (IF), depending on situation, they are either detected on the Intensified-sCMOS camera (Lipka et al., 2018) situated in the far-field of the ensemble or coupled to single-mode fibers connected to single photon counting modules (SPCM). To filter out uncorrelated 795 nm photons coming from $5P_{1/2},F=2\rightarrow 5S_{1/2},F=2$ decay path, we employ an additional filtering consisting of glass cell containing optically pumped to $5S_{1/2},F=2$ state rubidium-87 vapor and buffer gas (nitrogen, 10 Torr). For alignment purposes we send an additional beam indicated as incoming idler in the Fig. 1(c) to seed the signal-idler (SI) generation process and observe the classically generated signal beam. For the temporal correlations measurements we first observe the generated signal beam and maximize its amplitude in the sense of satisfying the PM condition $\Delta k_{z}=0$. Then we couple this signal beam to the single-mode mode fiber of the signal SPCM. The particular experimental configuration i.e. control beam angles and propagation direction was chosen to provide broad PM spatial spectrum, while the particular choice of atomic states and polarizations was motivated by minimization of loss and noise. Essentially, by selecting $F=3$ manifold as the first intermediate state $|b\rangle$ we avoid performing read-out of the SW with the Pump 1 field. Still, the Pump 1 field re-scatters excitation to other magnetic states of the $F=2$ ground-state manifold causing effective decoherence of the SW. In the Fig. 1(d) we present the time sequence of the experiment including all the steps described above.

III Temporal correlations

We first present the measurement of temporal correlations in a particular, pre-selected wavevector configuration. Fig. 2(a) depicts a histogram of SI correlation as a function of mutual delay, where we fit the second-order cross correlation as $g_{si}^{(2)}(\tau)=1+\alpha|\psi_{t}(\tau)|^{2}\ast h(\tau)$. By introducing convolution with $h(\tau)=(\Theta(\tau)-\Theta(\tau-\delta t))/\delta t$ we account for finite time resolution $\delta t=3.85\ \mathrm{ns}$ of the time tagging module. We observe very strong correlation $g_{si}^{(2)}(0)=35.3\pm 0.8$, with simultaneously measured auto-correlations $g_{ii}^{(2)}(0)=1.11\pm 0.06$ and $g_{ss}^{(2)}(0)=1.13\pm 0.16$. These two measurements constitute a strong violation of the Cauchy-Schwarz inequality (Paul, 1982) with:

[TABLE]

We measure coincidences during the P1+P2 pulse transferring the ground-state SW to the exited state $|c\rangle$, that last approx. 300 ns, as well as during a subsequent decay. We observed that if the excitation pulse is applied continuously, decoherence of the SW $\rho_{hg}$ by pump P1 is strong and the number of coincidences decreases. Quite similarly, even during the decay of the $|c\rangle$ state we observed that the initial period yields highest signal-to-noise ratio effectively represented by $g_{si}^{(2)}$. We attribute this effect to a thermal decay (Zhao et al., 2009) of $\mathcal{S}$. In Fig. 2(b) we present additional data showing the temporal evolution of signal-idler correlation during the two-photon coherence transfer (creation of $\mathcal{S}$) and the subsequent decay. Indeed, even with negligible $K$ (length of ground-state SW wavevector), the longitudinal component of $\mathcal{S}$ is $K_{\mathcal{S},z}=2\pi(\frac{1}{\lambda_{P1}}+\frac{1}{\lambda_{P2}})\approx 1.6\times 10^{4}\ \mathrm{rad/mm}$, yielding motional characteristic Gaussian decay time of $\tau_{T}^{-1}=|\mathbf{K}_{\mathcal{S}}|\sqrt{k_{B}T/m_{\mathrm{Rb}}}\approx 0.7/\mu\mathrm{s}$.

We measure the correlation time $\tau_{0}=9.8\pm 0.2$ ns which suggest a strong role of superradiance in the process, as without superradiant emission enhancement we would expect $\tau_{0}=27.7$ ns, which is a natural lifetime of state $|d\rangle$. Furthermore, thermal decoherence of SW $|h\rangle\langle d|$ and any magnetic decoherence are much slower. With this we experimentally prove that superradiance is inherently linked to PM emission (Jen, 2015), and can occur in any wave mixing process. Here, it is demonstrated in 6WM for the first time to our knowledge.

IV Wavevector-domain correlations

In the second experiment we analyze SI correlations in the wavevector space. Using the I-sCMOS camera we collected 5M frames for each of five different values of the SW wavevector $\mathbf{K}$. Here, the $g_{si}^{(2)}(\mathbf{k}_{\perp}=\mathbf{k}_{P1\perp}+\mathbf{k}_{P2\perp}-\mathbf{k}_{s\perp}-\mathbf{k}_{i\perp})$ is calculated in the wavevector-sum coordinates with shifted variables by the sum of transverse components of pumps wavevectors (see Fig. 3 for the full wavevector configuration). In this configuration, we expect that the correlation peak will appear in the spot $\mathbf{k}_{\perp}=\mathbf{K}_{\perp}$ determined by the initial SW. This is indeed what we observed in Figs. 4(a-e). Note that the correlations are averaged over the entire duration of the pulse, which we denote by $\langle g_{si}^{(2)}\rangle_{\tau}$. The averaging, with the pair-generation rate yielding 0.5 signal photon detected per frame on average, leads to significantly lower values for the cross-correlation than in Fig. 2(a).

The PM condition renders the SI emission cones and thus, we expect that correlated signal and idler pairs will appear in ring-shaped regions on the camera. To verify that we filter SI events by choosing only those for which $|\Delta\mathbf{k}_{\perp}|\leq\delta k$, where by introducing $\delta k=15\ \mathrm{rad\,mm^{-1}}$ we account for a finite spread of the correlation peak. In insets (a,b) of Fig. 3 we plot signal and idler accumulated events after the filtering procedure. We see that the correlated photons mostly come from the ring-shaped regions.

Finally, we show that the PM patterns are most easily recovered by looking at the absolute intensity of registered idler light channel to phase-matched regions by superradiance. The intensity in perfectly correlated regions is increased approximately two-fold (see Figs. 4(f-j)). This can be understood as an interplay between coherent and incoherent emission. The signal photon is emitted in a random direction, leading to a creation of a random SW $\rho_{dg}$. In most cases this SW either does not lead to phase-matched emission (see Fig. 3) or decoheres rapidly due to atomic motion. A photon associated with such excitation is thus emitted in a random direction, in particular also in the PM cone. This cone is quite narrow, and there, the PM leads to expected coherent emission of idler photon. Thus, starting with a mean SW number $\bar{n}$ per pixel, we arrive at still roughly $\eta\bar{n}$ incoherent idler photons per pixel and additional $\bar{n}_{\mathrm{coh}}=\eta\bar{n}$ coherent emission into the phase-matched region. This number, and thus the ratio $(\bar{n}+\bar{n}_{\mathrm{coh}})/\bar{n}$ may get decreased due to incoherent excitation of SW field $\rho_{hg}$, incoherent transfer to $\mathcal{S}$, as well as its motional decoherence. The actual value which we obtain in the experiment is $(\bar{n}+\bar{n}_{\mathrm{coh}})/\bar{n}\approx 1.4$.

We compare the expected shape of phase-matched regions described by:

[TABLE]

with the observed shapes of regions of increased idler counts rate. Without fitting (since all parameters are determined from correlation measurement), we observe very good agreement of the shape between experiment and theoretical prediction.

In Figure 5 we compare circular averages of the normalized idler counts for all cases from Fig. 4. Each point represents an average number of photon counts within a ring at radius $\left|\,\mathbf{k}_{\perp}^{i}-\mathbf{k}_{\perp c}^{i}\,\right|$ with unit width, where by $\mathbf{k}_{\perp c}^{i}$ we denote the position of the center of each phase-matching pattern from Fig. 4(f-j)). The actual values of $\mathbf{k}_{\perp c}^{i}$ are derived from Eq. 6. With this detailed view we show explicitly that correlation phase-matching patters, ranging from Gaussian-like shapes to tight rings can be engineered by controlling the spin-wave wavevector $\mathbf{K}$. Additionally, one can imagine that even more complex patterns could be obtained by preparing the initial spin wave in a superposition in $k$-space or by reshaping the spin-wave spatial structure using for example ac-Stark shift (Parniak et al., 2019; Mazelanik et al., 2019; Lipka et al., 2019).

V Conclusions and perspectives

In conclusion we have demonstrated SW based control of superradiance in the six-wave mixing process. Our experiments paves the way towards new schemes involving specifically prepared atomic states that lead to non-standard emission, such as for example sub-radiance (Jen, 2017b) as well as interference in the superradiant emission. Interesting perspectives arise due to the increased range of possibilities in the wavevector space, compared with low-dimensional interfaces. With those features, applications such as optical quantum amplifiers (Svidzinsky et al., 2013) or superradiant clocks (Norcia et al., 2016) can be extensively studied and optimized in atom-ensemble systems where SW control is feasible. Nowadays, extensive studies of superradiance in atomic arrays or one-dimensional interfaces (Solano et al., 2017; Asenjo-Garcia et al., 2017; Hood et al., 2016; Goban et al., 2015; Sørensen et al., 2016), where atoms are strongly coupled to each other, distinctively reveal associated many-body quantum effects, such as cooperative resonances and cooperative Lamb shift (Shahmoon et al., 2017; Meir et al., 2014). Remarkably however, even in optically-dense extended free-space atomic ensembles (Srivathsan et al., 2013) as seen in this work, superradiance plays a crucial role.

Spatial engineering of the demonstrated superradiance extends beyond fundamental aspects. In particular, control of the enhancement of the emission rate in a particular direction helps extend the bandwidth of a photon and potentially match it to a particular receiver. On the other hand, canceling the phase matching allows selective retrieval of stored atomic coherence, and thus temporal-multimodality.

In the more practical context, our experiment shows that photons resonant to excited-to-excited state transitions couple to the ground-state coherence. While preliminary demonstrations for classical light have been made Parniak et al. (2016a); Ding et al. (2012), here we have demonstrated quantum correlations in this scheme. This opens the path to embedding the two functions of a frequency-converted quantum repeater node Radnaev et al. (2010) in a single atomic ensemble. Practically it is beneficial to use a telecom transition (for rubidium for example 1529 nm and 1475 nm transitions leading to $4\mathrm{D}_{3/2}$ manifold).

Finally, it would be particularly interesting to replace one of the strong driving fields in the 6WM process by a quantum field. Essentially, generation of a single SW excitation can be heralded by detecting a photon scattered in the Raman process. Such configuration directly leads to generation of a correlated three-photon state. Note that the process could also be reversed - first, a single SW would be generated by heralding detection of a photon pair. The resulting state would be a three-partite entangled state of a photon pair and a collective atomic excitation. We envisage that both configurations can be achieved by increasing the two-photon excitation efficiency

• currently limited by decoherence caused by the sole Pump 1 field. Additionally, it would be also interesting to couple signal field to the cavity enhancing the emission to the phase-matched region, preferably in the co-linear SI configuration (Fig. 4(a)). In this way, the brightness could even surpass the photon down-conversion probability in three-photon SPDC experiments (Ding et al., 2015; Hübel et al., 2010). Such a tripartite entanglement generation scheme lends itself to two-dimensional quantum repeaters, as proposed by in Ref. (Wallnöfer et al., 2016). Our scheme would hold a direct advantage over other three-photon entanglement generation schemes in the quantum repeater context, as one of the photons is directly created as a stored spin wave, akin to the manner of the original DLCZ scheme. Furthermore, the SW could also be prepared in a deterministic way using Rydberg blockade (Wei et al., 2011; Li et al., 2016), and non-linear Rydberg interactions can play an intricate role in superradiance

Acknowledgements.

We thank K. Banaszek for the generous support and J. H. Müller for insightful discussion. This work has been supported by the Polish Ministry of Science “Diamentowy Grant” Projects No. DI2013 011943 and DI2016 014846, National Science Centre (Poland) grants No. 2016/21/B/ST2/02559, No. 2017/25/N/ST2/01163, and No. 2017/25/N/ST2/00713. The research is supported by the Foundation for Polish Science, cofinanced by the European Union under the European Regional Development Fund, as a part of the “Quantum Optical Technologies” project carried out within the International Research Agendas programme. M.P. was supported by the Foundation for Polish Science within the START programme scholarship.

Appendix A Experimental setup details

We use an elongated ($\sigma_{x}\times\sigma_{y}\times\sigma_{z}=0.3\times 0.3\times 2.5\,\mathrm{mm}^{3}$) ensemble of cold ($T\approx 22\ \mu\mathrm{K}$) rubidium-87 atoms in the magnetooptical trap (MOT), described in details in (Parniak et al., 2017; Dąbrowski et al., 2018). Before the actual 6WM experiment the atoms are optically pumped to the $|g\rangle$ state ($5S_{1/2},F=1,m_{F}=1$ , see Fig.1(a)). To generate the macroscopic ground-state coherence $\rho_{gh}(\mathbf{r})$, where $|h\rangle$ = $5S_{1/2},F=2,m_{F}=-1$ we use two phase-coherent optical fields (Write and Seed, both 795 nm). The $\rho_{hg}(\mathbf{r})$ generation process can be understood as a stimulated (seeded with Seed light) Raman scattering in the $\Lambda$ system driven by the strong Write light. Here, the $\Lambda$ system is composed of levels $|g\rangle$, $|h\rangle$ and $|a\rangle$ ($5P_{1/2},F=2,m_{F}=0$). In such process, both the Seed light and atomic coherence are amplified at the expense of the Write field.

The Seed field is derived from the Write laser using frequency-shifting setup consisting of electro-optic modulator (EOM) and filtering Fabry-Perot cavity (see Supplementary Information of Ref. (Parniak et al., 2019)). Remaining lasers (excluding Pump 2) are frequency-stabilized to either cooling-and-trapping laser (780 nm) or repumping laser (795 nm) using the optical frequency locked loop described in (Lipka et al., 2017). The Pump 2 laser at 776 nm is stabilized to the Pump 1 laser using modified version of the two-photon lock (see Fig. 6) presented in Ref. (Parniak et al., 2016b). The modification employs technique called modulation transfer spectroscopy, to keep the Pumps 1&2 in two-photon resonance between levels $|g\rangle$ and $|c\rangle$.

In this modulation transfer spectroscopy technique (Negnevitsky and Turner, 2013; de Escobar et al., 2015), as opposed to the simpler frequency modulation spectroscopy, we do not modulate the probe light. Instead, we modulate the coupling (776 nm) light using a phase-only EOM equipped with a custom resonant circuit, and probe the system with an unmodulated beam, here at 780 nm (Pump 1 laser). The modulation is thus only transferred to the probe beam, and detectable in the demodulated signal if the two beams interact in the vapor cell. The probe beam is then measured and its RF (18.5 MHz) signal demodulated (mixed) with the pump modulation. Finally, the error signal is obtained after low-pass filter. A significant advantage of this scheme is the stability of the error signal offset, as out of resonance we will simply not have any signal. Furthermore, we become nearly immune to the residual amplitude modulation of the EOM, as the amplitude modulation is transduced weakly by the atoms.

The obtained error signal as we scan the 776 nm laser is presented in Fig. 6(a). All four levels of the highest excited state $|c\rangle$ (Fig. 6(b)) manifold are visible. By changing the single photon detuning within the Doppler-broadened line corresponding to the $F_{h}=2$ manifold we observe relative changes of various components that arise due to selection rules.

The time domain correlation measurements were performed using Perkin-Elmer SPCMs ($\sim 50\%$ quantum efficiency) connected to a single-mode fibers (SMF). The SMF are projecting the output states on a Gaussian beam modes with waists of radius equal 0.15mm overlapping at the center of the atomic cloud. The FPGA-based time-tagger which we use provides 3.85 ns time resolution. The spatially resolved detection is performed using home-built I-sCMOS camera situated in the far field of the ensemble. The far-field imaging setup was designed using ray-tracing software (ZEMAX) to maximize the field of view and resolution in $k$-space (see (Dąbrowski et al., 2018) for more details). Detailed description of the I-sCMOS camera itself can be found in Ref. (Lipka et al., 2018).

Appendix B Spinwave downconversion

Treating the atom-light interaction induced by the weak quantum signal and idler fields as a small perturbation to the free atom density operator evolution in the presence of $\mathcal{S}$, we can obtain expressions for the coherences relevant in the signal idler generation process: $\rho_{cd}$ and $\rho_{dg}$. Next, we obtain the slowly varying amplitudes of polarization operators: $\mathcal{P}_{s}=nd_{dc}\rho_{cd}$, $\mathcal{P}_{i}=nd_{gd}\rho_{dg}$. Then, treating the excited-state spin wave as one of the interacting fields, we use the relationship: $\mathcal{P}=\epsilon_{0}\chi E+\epsilon_{0}\chi^{NL}\mathcal{S}E+...$ to define the linear $\chi$ and nonlinear $\chi^{NL}$ susceptibilities. Under the assumption of small number of excitations $|\rho_{hg}|\ll 1$ and long lifetime of the excited state $|c\rangle$: $\Gamma_{c}\ll\Gamma_{d}$, we get:

[TABLE]

where $\Delta=\omega_{gd}-\omega_{i}$. As we focus on the signal-idler generation process only the nonlinear part is of our interest. Especially, in the interaction picture the effective Hamiltonian for this process is given by (Wen et al., 2007a):

[TABLE]

where $\mathcal{S}^{+}=\mathcal{S}e^{-i\omega_{cg}t}$, and $E_{j}^{-}$ stands for the quantized fields operators with negative frequency (Wen et al., 2007a, b) defined as follows:

[TABLE]

where $E_{j}=i\sqrt{\hbar\omega_{j}/2\epsilon_{0}n_{j}^{2}V_{q}}$ with the quantization volume $V_{q}$ and index of refraction $n_{j}$. The first-order perturbation theory gives then the state vector:

[TABLE]

where $F$ is the well known from the SPDC experiments two-photon spectral function (Rubin et al., 1994), from which we can then obtain the approximate formulas for temporal (fixed $\{\mathbf{k}_{s},\mathbf{k}_{i}\}$) and wavevector-domain (time-averaged) wave functions $\psi_{k}(\mathbf{k}_{s\perp},\mathbf{k}_{i\perp})$ and $\psi_{t}(t_{s},t_{i})$ respectively.