Spin- and Momentum-Correlated Atom Pairs Mediated by Photon Exchange and Seeded by Vacuum Fluctuations

Engineering pairs of massive particles that are simultaneously correlated in their external and internal degrees of freedom is a major challenge, yet essential for advancing fundamental tests of physics and quantum technologies. In this Letter, we experimentally demonstrate a mechanism for generating pairs of atoms in well-defined spin and momentum modes. This mechanism couples atoms from a degenerate Bose gas via a superradiant photon-exchange process in an optical cavity, producing pairs via a single channel or two discernible channels. The scheme is independent of collisional interactions, fast, and tunable. We observe a collectively enhanced production of pairs and probe interspin correlations in momentum space. We characterize the emergent pair statistics and find that the observed dynamics is consistent with being primarily seeded by vacuum fluctuations in the corresponding atomic modes. Together with our observations of coherent many-body oscillations involving well-defined momentum modes, our results offer promising prospects for quantum-enhanced interferometry and quantum simulation experiments using entangled matter waves.

Figure 1

(a) Microscopic pair-production mechanism. Upper: two $m=0$ atoms in the BEC (green) are converted into a pair of $m=+1$ (blue) and $m=-1$ (red) via two Raman processes involving absorption and emission of a ${\sigma }_{+}$ drive photon and a shared virtual cavity $\pi$ photon; arrows on the spheres highlight the acquired spin. With the magnetic quantization axis along $z$, the $y$-polarized running-wave drive is a superposition of ${\sigma }_{\pm }$ photons. Thus, pairs can be generated via two channels with rates ${\chi }_{\pm }$, as shown in momentum space (lower): the Raman processes impart recoil momentum $+k$ along the drive ($z$) on the first atom ($m=\pm 1$ for ${\chi }_{\pm }$ channel) and opposite momentum $-k$ on the second, generating correlated pairs both in spin and in momentum. Because of the standing-wave cavity mode, the pairs acquire also momentum symmetrically along $\pm x$ (illustrated here as half spheres). (b) Energy-level diagram of the four-photon pairing mechanism, composed of two superradiant Raman processes, each involving a ${\sigma }_{\pm }$-polarized drive and a $\pi$-polarized cavity photon, as indicated by straight and curly arrows, respectively. The two intermediate modes are split by twice the linear Zeeman shift ${\omega }_z$ and correspond to two discernible channels with coupling rates ${\chi }_{\pm }$, depending on the detunings ${\delta }_{\pm }$ and the cavity loss rate $\kappa$. The pair energy offset ${\omega }_0$ is set by the kinetic and internal energy of the pair constituents. (c) Spin-momentum pair creation as atomic parametric amplifier, with two pump atoms (green) being converted into a pair of signal (blue) and idler (red) atoms via two nonlinear channels ${\chi }_{\pm }$. (d) Measured mean number of pairs $⟨N_{\mathrm{p}}⟩$ after $t=65\mathrm{\mu }\mathrm{s}$, showing a superlinear growth with the initial atom number $N$. Unless specified otherwise, the error bars indicate the standard error. The solid line shows our numerical simulations; see Supplemental Material, also for experimental parameters [43].

Figure 2

Pair statistics seeded by vacuum fluctuations. (a),(b) Exemplary spin-resolved momentum distributions for the (a) single-channel and (b) two-channel configuration. The orange and yellow boxes indicate the modes $|\pm k{⟩}_{\pm 1}$ and $|\pm k{⟩}_{\mp 1}$, respectively. (c),(d) Pair statistics, generated through the ${\chi }_{+}$ (orange) and ${\chi }_{-}$ (yellow histograms) process, for (c) the single-channel and (d) two-channel configurations. The solid lines correspond to fitted Bose-Einstein distributions with experimentally determined mean $⟨N_{\mathrm{p}}⟩$ (purple bin) convolved with our detection noise [43]; the dashed line in (c) is consistent with a distribution with zero mean pairs and Gaussian detection noise of $\sim 200$ pairs. The arrows indicate the standard deviation of the resulting distributions, demonstrating $⟨N_{\mathrm{p}}⟩\approx \sigma (N_{\mathrm{p}})$. (e) Simulated pair statistics convolved with detection noise [43] for a single channel and different classical seeds $N_{\mathrm{p}}^{\mathrm{s}}$ in the pair modes. The solid line corresponds to the fitted data of (c) and is compatible with the pair distribution being seeded by vacuum fluctuations ($N_{\mathrm{p}}^{\mathrm{s}}=0$).

Figure 3

Correlated formation of atomic pairs. (a),(b) Momentum-space interspin correlation maps ${\mathcal{C}}^{+1,-1}(k_{+1}^z,k_{-1}^z)$ for the (a) single-channel and (b) two-channel configuration, demonstrating the correlated nature of the produced pairs. We attribute the side patterns close to the correlation peaks to residual density-dependent imaging artifacts. (c) Anticorrelations ${\mathcal{C}}^{+1,-1}(\pm k,\pm k)$ for realizations with $N_{\mathrm{p}}>N_{\mathrm{p}}^{\min }$ for a single channel (light blue) and two channels (dark blue), with the solid lines showing the results from our numerical simulations. The anticorrelations are obtained by averaging a region around ${\mathcal{C}}^{+1,-1}(\pm k,\pm k)$ on the order of the mode sizes [43] as schematically shown for the first data points (squares) corresponding to (a) and (b) by the gray squares. The two-channel anticorrelations increase with $N_{\mathrm{p}}^{\min }$ due to pump-mode depletion. The inset displays a representative correlation map for $N_{\mathrm{p}}^{\min }=7\times {10}^3$. The error bars indicate the standard deviation of the averaged region.

Figure 4

Coherent many-body oscillations and tunable dissipation. (a) Time evolution of mode occupations in the single-channel configuration [${\delta }_{+}/(2\kappa )=-9.08(4)$], exhibiting oscillatory dynamics. Photon loss results in a progressive imbalance between the $|+k{⟩}_{+1}$ and $|-k{⟩}_{-1}$ populations. Inset: representative momentum-space distribution at $t=138\mathrm{\mu }\mathrm{s}$. (b) Mean number of pairs $⟨N_{\mathrm{p}}⟩$ (upper) and imbalance $⟨N_{\mathrm{imb}}⟩$ (lower, gray) and number of photons lost from the cavity $⟨N_{\mathrm{ph}}⟩$ (lower, orange) [56] for $t=80\mathrm{\mu }\mathrm{s}$ and ${\chi }_{+,\mathrm{fix}}=-2\pi \times 0.50(2)\mathrm{Hz}$ as a function of the detuning ${\delta }_{+}/(2\kappa )$, which controls the coherent and dissipative processes. The solid curves show numerical simulations [43], with the dashed line indicating the regime where the adiabatic cavity field elimination becomes invalid, i.e., $|{\delta }_{+}|\lesssim 2\kappa$. We attribute both the deviation from theory at large quench times as well as the excess photon numbers ($⟨N_{\mathrm{ph}}⟩>⟨N_{\mathrm{imb}}⟩$) to superradiant decay to higher-order momentum modes in $m=+1$, which are outside the field of view ($\sim 2.2k$) along $x$ [56]. Inset: illustration of superradiant scattering.