Proposal for a motional-state Bell inequality test with ultracold atoms

We propose and theoretically simulate an experiment for demonstrating a motional-state Bell inequality violation for pairs of momentum-entangled atoms produced in Bose-Einstein condensate collisions. The proposal is based on realizing an atom-optics analog of the Rarity-Tapster optical scheme: it uses laser-induced Bragg pulses to implement two-particle interferometry on the underlying Bell state for two pairs of atomic scattering modes with equal but opposite momenta. The collision dynamics and the sequence of Bragg pulses are simulated using the stochastic Bogoliubov approach in the positive-$P$ representation. We predict values of the Clauser-Horne-Shimony-Holt (CHSH) parameter up to $S\simeq 2.5$ for experimentally realistic parameter regimes, showing a strong violation of the CSHS-Bell inequality bounded classically by $S\le 2$.

Figure 1

Schematic diagram of the collision geometry and the proposed adaptation of the Rarity-Tapster scheme. (a) The two condensates in position space, counterpropagating along the $z$ axis with mean momenta $\pm {\mathbf{k}}_0$, are shown in the left upper corner; the same condensates in momentum space (or after a time-of-flight expansion) have a pancake shape and are shown on the north and south poles of the spherical halo of scattered atoms. The counterpropagating (along $y$) Bragg lasers are tuned to couple and transfer the population between two pairs of momentum modes, such as the pair ($\mathbf{p},\mathbf{q}$) and ($-\mathbf{q},-\mathbf{p}$), indicated on the equatorial plane of the scattering halo. A similar quartet of modes (not shown for clarity), coupled by the same Bragg lasers, can be identified on any other plane obtained by rotating the equatorial plane by an angle $\theta$ around the $y$ axis; together, all these quartets of modes form two opposing rings shown in red. (b) The Rarity-Tapster scheme for implementing the $\pi$ and $\pi /2$ Bragg pulses on pairs of momentum modes emanating from the source (S) and the arrangement of two independent relative phase setting ${\varphi }_L$ and ${\varphi }_R$ (respectively, between $\mathbf{p}$ and $\mathbf{q}$, and between $-\mathbf{p}$ and $-\mathbf{q}$) imposed in the left and the right arms of the setup. After being mixed by the final $\pi /2$ pulse, the output modes are detected by four atom detectors $D_i$ ($i=1,2,3,4$) and different coincidence counts $C_{ij}$ are measured for calculating the CHSH-Bell parameter $S$.

Figure 2

Illustration of typical results for the collisional halo in momentum space from the stochastic Bogoliubov approach in the positive-$P$ representation. The simulations were performed using the XMDS2 software package, see Ref. [38]. Shown here are three orthogonal slices (cuts through the origin) of the 3D momentum distribution $n\left(\mathbf{k}\right)$ at the end of the collision; the saturated (white) regions of the color map correspond to the high-density colliding condensates. The central figure is a discretized scatter plot of the 3D data (shown only for illustrational purposes and comparison with Fig. 1), in which the dots (pixels) represent random samples of the average, but still fluctuating within the sampling error, density distribution binned into pixels whose color coding scales with the atom number in the bin (only four color grades were used for clarity). For quantitative details of the same data on the equatorial plane, see Fig. 3.

Figure 3

Momentum distribution $n\left(\mathbf{k}\right)$ of scattered atoms on the equatorial plane of the halo and the correlation coefficient $E$. Momentum distribution is shown (a) after the collision, at $t_1=65 \mu \mathrm{s}$; (b) after the $\pi$ pulse chosen here to be a Gaussian, centered at $t_2=79 \mu \mathrm{s}$ and having a duration (rms width) of ${\tau }_{\pi }=3.5 \mu \mathrm{s}$; and (c) after the final $\pi /2$ pulse, centered at $t_3=139\mu \mathrm{s}$ and having a duration of ${\tau }_{\pi /2}=3.5\mu \mathrm{s}$. The momentum axes $k_{x,y}$ are normalized to the collision momentum $k_0\equiv \left|{\mathbf{k}}_{\mathbf{0}}\right|$ (in wave-number units), which in our simulations is $k_0=4.7\times {10}^6{\mathrm{m}}^{-1}$. The plotted results are for an initial BEC containing a total average number of $N=1.9\times {10}^4$ atoms of metastable helium (${}^4\mathrm{H}{\mathrm{e}}^{*}$) prepared in a harmonic trap of frequencies $({\omega }_x,{\omega }_y,{\omega }_z)/2\pi =(64,1150,1150)$ Hz and colliding with the scattering length of $a=5.3$ nm; all these parameters are very close to those realized in recent experiments [10, 11, 12]. The optimal timing of the final Bragg pulse differs slightly for condensates with different $N$; in particular, $t_3$ ranged from $135.5$ to $139\mu \mathrm{s}$ for the data in Fig. 4 (see Appendix pp2). The data are averaged over $\sim 30000$ stochastic trajectories on a spatial lattice of $722\times 192\times 168$ points. Panel (d) shows the correlation coefficient $E({\varphi }_L,{\varphi }_R)$ as a function of $\varphi \equiv {\varphi }_L-{\varphi }_R$, for the same detection bin sizes as in Fig. 4, blue circles. The data points are from numerical simulations (error bars of two standard deviations, representing sampling errors from 360 stochastic runs, are within the marker size), including averaging over $\sim 370$ quartets of distinct detection volumes on the two opposing rings of the scattering halo shown in Fig. 1, while the solid line is from the Gaussian-fit model, Eq. (7). A maximum amplitude of $E_0>1/\sqrt{2}$ (outside the shaded region) corresponds to a correlation strength that can lead to a Bell inequality violation, given the underlying sinusoidal behavior.

Figure 4

CHSH-Bell parameter $S$ as a function of the correlation strength $h$ (see text); the value of $h$ can be controlled by varying the total average atom number $N$ in the initial BEC. For the data points shown here, $N$ was varied between $1.9\times {10}^4$ (largest $h$) and $7.4\times {10}^4$ (smallest $h$). The two sets of data correspond to two different detection bin sizes: $(\mathrm{\Delta }{k}_x,\mathrm{\Delta }{k}_y,\mathrm{\Delta }{k}_z)=(0.052,0.53,0.47)\mu {\mathrm{m}}^{-1}$ circles (blue) and $(0.12,1.24,1.10)\mu {\mathrm{m}}^{-1}$ squares (red). The vertical error bars on data points indicate the stochastic sampling errors [40]; the horizontal error bars are the sampling errors on the value of $h$. The results are compared to the analytic predictions (solid lines) of Eq. (8); uncertainty (shaded regions) is due to the uncertainty in determining ${\sigma }_d$. The inset shows the explicit dependence of $S$ on $\mathrm{\Delta }{k}_x$ (in units of $2{\sigma }_x=0.068\mu {\mathrm{m}}^{-1}$), for fixed $(\mathrm{\Delta }{k}_y,\mathrm{\Delta }{k}_z)=(0.77{\sigma }_y,0.89{\sigma }_z)=(0.53,0.47)\mu {\mathrm{m}}^{-1}$ and $N=1.9\times {10}^4$ ($h\simeq 27$). For a typical time-of-flight expansion time of $t_{exp}\sim 300$ ms, which maps the atomic momentum distribution into position space density distribution, and which is when the atoms are experimentally detected, these detection bin sizes convert to position space distances of $(\mathrm{\Delta }x,\mathrm{\Delta }y,\mathrm{\Delta }z)\simeq (0.32,2.5,2.2)$ mm (where we have taken ${\lambda }_x=1$ for definitiveness), which are several times larger than the three orthogonal resolutions of multichannel plate detectors used in ${}^4\mathrm{H}{\mathrm{e}}^{*}$ experiments [12, 41].

Figure 5

(a) Correlation amplitude $E_0$ predicted by the Gaussian-fit model [Eq. (B11)] as a function of the integration bin size $\mathrm{\Delta }{k}_y$ and the free propagation time $\mathrm{\Delta }{t}_{\mathrm{free}}$. Calculations were performed for an initial condensate of $N=1.9\times {10}^4$ atoms and other parameters as per the main text with $h$ and ${\sigma }_d$ extracted from the stochastic numerical results. The central ridge corresponds to Eq. (B13) where the phase drift term $\phi (\mathbf{k},{\mathbf{k}}^{\prime })$ is eliminated. (b) Amplitude of the correlation function $E_0$ as a function of free propagation time $\mathrm{\Delta }{t}_{\mathrm{free}}$ for an integration volume $(\mathrm{\Delta }{k}_x,\mathrm{\Delta }{k}_y,\mathrm{\Delta }{k}_z)=(0.052,0.53,0.47)\mu {\mathrm{m}}^{-1}$ and simulation parameters are as per (a). The predictions of the Gaussian-fit analytic model Eq. (B10) (gray shaded region) are compared to the numerical results from stochastic simulations (black circles). The error bars on data points indicate the stochastic sampling error of two standard deviations obtained from $\sim 800$ trajectories, while for the analytic prediction the uncertainty in $E_0$ (shaded region) is due to the uncertainty in the values $h$ and ${\sigma }_d$ extracted from the numerical simulations.

Figure 6

Optimal free propagation time $\mathrm{\Delta }{t}_{\mathrm{free}}$ for a range of initial BEC atom number. Numerical results (black circles) are compared to the prediction of Eq. (B12) from the perturbative model (dashed line). The range of $N$ in the initial BECs corresponds to those in the main text, while the integration volume is the same as Fig. 5.