Multisetting protocol for Bell correlated states detection with spin-$f$ systems

TL;DR

The work introduces a multisetting, two-body Bell test tailored to spin-$f$ Bose-Einstein condensates, leveraging parallel spin-nematic squeezing across $f$ SU(2) subspaces. A data-driven, permutationally invariant Bell inequality is derived, yielding an additive, block-structured correlator whose violation improves as the number of measurement settings grows, with analytical targets and numerical validation. For $f=1$ the scheme achieves a notable 12% enhancement in Bell violation when using three settings, and simulations for $f=2,3$ show increased setting counts are attainable through tuning interaction parameters, enabling richer multipartite correlations. The protocol offers a scalable route to robust Bell correlations in high-spin ultracold systems and has potential extensions to other platforms and multiparameter tasks, supported by available numerical tools.

Abstract

We propose a multisetting protocol for the detection of two-body Bell correlations, and apply it to spin-nematic squeezed states realized in $f$ pairs of SU(2) subsystems within spin-$f$ atomic Bose-Einstein condensates. Experimental data for $f=1$, alongside with numerical simulations using the truncated Wigner method for $f=1,\,2,\,3$, demonstrate the effectiveness of the proposed protocol. Our findings extend the reach of multisetting Bell tests in ultracold atomic system, paving the way for extended quantum information processing in high-spin ensemble platforms.

Figures (12)

• Figure 1: The generalized Bloch spheres in SU(2) subsystems are spanned by the triplet of operators $\{ \hat{J}^{(\sigma)}_{x,\mu}, \hat{J}^{(\sigma)}_{y,\mu}, \hat{J}^{(\sigma)}_{z,\mu} \}$ which are defined in Eq.(\ref{['eq:spin operatorsx']})-(\ref{['eq:spin operatorsz']}). There are $f$ ($\mu = 1,\, 2,\cdots, f$) pairs of subsystems, each corresponding to a symmetric ($\sigma = s$) and an antisymmetric ($\sigma = a$) component. On each sphere, ‌constant-energy contours‌ under the mean-field approximation are plotted as black curves, with arrows indicating the ‌direction of time evolution. The Husimi function is represented by color, illustrating the squeezed states generated simultaneously in each subsystem.
• Figure 2: The Bell correlator in the given subsystem $\mu$, normalized by the total number of atoms, $\mathcal{L}_\text{opt}(k_\mu)/N$, versus the number of measurement settings $k_\mu$ when angles $\vec{\theta}$ are optimal given by Eqs. (\ref{['eqM:thetasigma']}) and (\ref{['eqM:thetasigma2']}) (marked by black point) and evenly distributed in the region $[\gamma,\pi-\gamma]$ for $\gamma\approx0.22\pi$ (blue points). The orange solid line represents the function with the fitted parameters listed in the legend‌.
• Figure 3: The evolution of the Bell correlator with three measurement settings $L^{(3)}_{{\rm opt}}/E_{\rm max}$ given by Eq. (\ref{['eqM:thetavarphiN']}). The blue points mark the results obtained using the experimental data in Ref. cao2023jointestimationtwophasespin, where squeezing parameters $\xi^2_{1, s}$ and $\xi^2_{1, a}$ were measured simultaneously for $N=26500$ and $q=|c_1|$. Numerical simulations using the TWM are shown with (solid line) and without (dashed line) additional noise from microwave measurement pulses, respectively. The shaded region indicates enhancement with respect to two measurement settings.
• Figure 4: The Bell correlator (\ref{['eqM:BelloverEmaxgeneral']}) given by the evolution using the noiseless TWM for $N=26400$, $q=c_1$ and $f=2$ (a) and $f=3$ (b). The blue curves correspond to the interaction coefficients of Rb and Cr atoms with their natural scattering lengths KAWAGUCHI2012253Widera_2006Chomaz_2023. The red curves correspond to the adjusted scattering length required to achieve equal initial squeezing rates across all subsystems. The weights of each block $w_\mu$ are listed in legend.
• Figure 5: The entangled state $\hat{\rho}$ is shared by $N$ parties. For each party, one can choose between $k$ local measurement settings. For each party and setting chosen, there are two measurement results possible $\pm 1/2$. In the case of spinor BEC, each party corresponds to an individual spin-$f$ boson‌ that shares ‌a squeezed state‌ within its respective subsystem $\mu$. The measurement settings $(\alpha, \mu)$ are defined in the respective subsystems (\ref{['eq:s01']}), each leading to the two measurement outcomes.