Bell Correlations in Spin-Squeezed States of 500 000 Atoms

Bell correlations, indicating nonlocality in composite quantum systems, were until recently only seen in small systems. Here, we demonstrate Bell correlations in squeezed states of $5\times 10^5$ ${}^{87}\mathrm{Rb}$ atoms. The correlations are inferred using collective measurements as witnesses and are statistically significant to 124 standard deviations. The states are both generated and characterized using optical-cavity aided measurements.

Figure 1

(a) Illustration of a squeezed spin state. The example is a Wigner distribution of a 10 dB squeezed state with 30 atoms, polarized along the $x$ axis. Squeezing is along the $z$ direction, while antisqueezing is along the $y$ direction. Also shown is the axis $\mathbf{n}$ used to calculate the Bell witness in Eq. (2). (b) The sequence used for squeezing. The initial state preparation consists of a composite $\pi /2$ pulse and a presqueezing procedure that squeezes the state in $S_{\mathbf{z}}$ such that the initial uncertainty is smaller than the cavity linewidth. The two QND measurements then follow, before a final fluorescence measurement that measures the atom number. (c) Histogram of the differences in $S_z$ between the first and second measurements for 18.5(3) dB squeezed states of $6.5\times 10^5$ atoms.

Figure 2

Rabi oscillations of squeezed states of $6.5\times 10^5$ atoms. (Upper panel) ${\mathcal{J}}_{1,\mathbf{n}}$ as a function of the microwave pulse time. The fit is sinusoidal and is used to extract the angle for the witness function in Fig. 3. The fit shows a contrast of 94.9(1)%. (Lower panel) Residuals from subtracting the sine fit from the data points. The increased noise at the ${\mathcal{J}}_{1,\mathbf{n}}\approx \pm 1$ points is due to antisqueezing. Fluorescence detection noise dominates at short pulse times, while microwave amplitude noise takes over at longer pulse times. Pulse times below $5\mu \mathrm{s}$ were not achievable due to control system limitations.

Figure 3

The data points show the Bell correlation witness $⟨W⟩$ as a function of $\theta$. The $\theta$ values are extracted from the fit in Fig. 2. The error bars show the combined statistical error from the measured ${\mathcal{J}}_{1,\mathbf{n}}$ and the total error in the estimated ${\mathcal{J}}_{2,\mathbf{z}}$ value. Points below the dashed red line show a violation of the inequality in Eq. (2). The highest violation is from the point shown in red (and also in the inset) which is 56 standard deviations from the boundary. The solid blue line is calculated from the contrast of the fit to the Rabi fringe and the squeezing level. For a maximally squeezed state with 100% coherence, the minimum of the witness function would approach $-0.25$.

Figure 4

Entanglement depth and Bell correlation boundaries. The red line shows the Bell violation boundary according to Eq. (3). The blue lines show the boundary for $k=2^n$ entanglement depth for $n=1,\dots ,9$ (labeled below each line). The area below the black line contains entangled states according to the Wineland criterion for entanglement [30]. The data points, taken with $5\times 10^5$ atoms and approximately 450 measurements each, have measurement strengths going from higher on the left to lower on the right. The error bars represent 68% confidence intervals. The open-square data point shows the most statistically significant demonstration of Bell correlations (the inset is an enlarged version of this data point). The open-diamond data point shows the result from a data set of 3286 runs with unconditional squeezing.