Proposal for a Bell Test with Entangled Atoms of Different Mass

TL;DR

The paper proposes a Bell-test that uses momentum-entangled atom pairs of different masses (3He* and 4He*) generated by interspecies collisions, with independent Bragg-based momentum control for each species to realize a CHSH experiment in external degrees of freedom. The entanglement is modeled as a mass-momentum state expressible through a two-mode squeezing framework, and a post-selected subspace enables a complete CHSH test. Numerical simulations predict a significant Bell-inequality violation under realistic conditions, with an ideal $\mathcal{S}$ around $2.46$ and a degraded but still super-classical $\mathcal{S}$ near $2.12$ when considering experimental imperfections. Beyond nonlocality, the setup offers a platform to probe gravity-quantum interfaces and weak equivalence principle tests with entangled masses, potentially constraining collapse models and related theories, thereby opening a new frontier in quantum tests with massive, mass-different systems.

Abstract

We propose a Bell test experiment using momentum-entangled atom pairs of different masses, specifically metastable helium isotopes 3He* and 4He*, though the method extends to other atom species. Entanglement is generated via collisions, after which the quantum states are manipulated using two independent atom interferometers, enabling precise phase control over each species. Numerical simulations predict a significant violation of Bell's inequality under realistic conditions. This proposal opens a new paradigm to study the intersection of quantum mechanics and gravity.

Figures (3)

• Figure 1: Schematic of our proposed Bell test. (a) Initially, we prepare a $\mathop{\mathrm{{}^3\mathrm{He}^\ast}}\nolimits$ DFG (${\mathcal{A}}$ - shown in red) and a $\mathop{\mathrm{{}^4\mathrm{He}^\ast}}\nolimits$ BEC (${\mathcal{B}}$ - shown in blue) overlapping in a trapping potential (not shown). The potential is switched off, and we apply laser beams with wavevectors $k_1^{\mathcal{B}}$ and $k_2^{\mathcal{B}}$ to induce a Bragg process, transferring $\mathop{\mathrm{{}^4\mathrm{He}^\ast}}\nolimits$ (${\mathcal{B}}$) into an equal superposition of two momentum states $\pm \hbar ({\bm k}_2^{\mathcal{B}}-{\bm k}_1^{\mathcal{B}})=: \pm {\bm p}_k$. These momentum components travel through the at-rest ${\mathcal{A}}$ atoms and cause individual pairs of ${\mathcal{A}}$ and ${\mathcal{B}}$ atoms to undergo $s$-wave collisions. (b) These collisions form two scattering halos in velocity space (faded red and blue spheres) for each of the two initial momenta $\pm {\bm p}_k$. By energy-momentum conservation, each ${\mathcal{A}}$ atom at a particular point on the ${\mathcal{A}}$ scattering halo will have a corresponding ${\mathcal{B}}$ atom on the diametrically opposite point on the corresponding ${\mathcal{B}}$ halo. Two example pairs $\ket{{\mathcal{A}}\nwarrow}\ket{{\mathcal{B}}\nearrow}$ from the $+{\bm p}_k$ halos and $\ket{{\mathcal{A}}\swarrow}\ket{{\mathcal{B}}\searrow}$ from the $-{\bm p}_k$ halos are indicated. (c) The addition of a second pair of Bragg beams resonant with ${\mathcal{A}}$ atoms that overlap with the ${\mathcal{B}}$ beams $\hat{{\bm k}}_{1,2}^{\mathcal{A}}=\hat{{\bm k}}_{1,2}^{\mathcal{B}}$, allows us to couple the upper and lower halo atoms of the same species $\ket{{\mathcal{A}}\nwarrow} \leftrightarrow \ket{{\mathcal{A}}\swarrow}$ and $\ket{{\mathcal{B}} \nearrow}\leftrightarrow \ket{{\mathcal{B}}\searrow}$ independently, which enables the mixing between states for a Bell test. (d) Corresponding momentum space distribution of the collision generated states $\ket{\psi}_\mathrm{upper}$ and $\ket{\psi}_\mathrm{lower}$ as described in Eqn \ref{['eq:psi-halo-upper']}. See main text for details.
• Figure 2: Numerical simulations of the momentum density distribution of (a)$\mathop{\mathrm{{}^3\mathrm{He}^\ast}}\nolimits$ (${\mathcal{A}}$) and (b)$\mathop{\mathrm{{}^4\mathrm{He}^\ast}}\nolimits$ (${\mathcal{B}}$) immediately after the initial Bragg diffraction and collision show the scattering halos generated from $s$-wave collisions between atoms of both species. (c) The back-to-back correlations present in the dual-species double halo, calculated by slicing the $2D\otimes 2D$ total momentum wavefunction to $1D\times1D$, i.e., $|\braket{p^{{\mathcal{A}}}_{z},p^{{\mathcal{B}}}_{z}}{\Phi(p^{{\mathcal{A}}}_{x},p^{{\mathcal{A}}}_{z},p^{{\mathcal{B}}}_{x},p^{{\mathcal{B}}}_{z})}|^2$. By integrating around the selected momentum states, we can calculate each joint detection probability between different momentum states of interest, shown in the overlaid text. These probabilities are used to calculate the Bell correlation (see Eqn. \ref{['eq:e-corr-def']}) of this initial state to be $E=0.94$.
• Figure 3: (a) The Bell correlator $E(\theta_{\mathcal{A}},\theta_{\mathcal{B}})$ for different transfer proportions of $\theta_{\mathcal{A}}$ and $\theta_{\mathcal{B}}$. The simulations show excellent agreement with the quantum prediction of $E=+\cos(\theta_{\mathcal{A}}+\theta_{\mathcal{B}})$. The four insets show the correlator for the four different phase settings used to calculate the CHSH parameter with a maximal violation. For the four values highlighted, the CHSH value exceeds the classical limit of $2$.