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arXiv:2409.02873 · Physical Review A

Bounds on the detection of Bell correlations with entangled ultracold atoms in optical lattices under occupation defects

Tanausú Hernández Yanes, Y. Bamaara, Alice Sinatra, Emilia Witkowska

Published Feb 11, 2025• 6 citations
arXivHTMLPDF

Abstract

Bell nonlocality stems from quantum correlations effectively identified using inequalities. Spin chains, simulated with ultracold atoms in optical lattices, Rydberg atoms in tweezer arrays, trapped ions, or molecules allow single-spin control and measurement. Therefore, they are suitable for studying fundamental aspects of these correlations and nonlocality. Occupation defects, such as vacancies or multiple atoms occupying a single site due to imperfect system preparation, limit the detection of Bell correlations. We study their effects with the help of a simplified toy model parametrized by the probability p of having a single occupation for a given site. Within this model, and for entangled systems obtained by one-axis twisting evolution from an initial factorized state, we derive two Bell inequalities, one based on many-site correlations and the other on two-site correlations, and identify the smallest probability p that allows the Bell inequalities violation to be detected. We then consider two physical realizations using entangled ultracold atoms in optical lattices where the parameter p is related to a nonunitary filling factor and nonzero temperature. We test the predictions of the toy model against exact numerical results.

DOI
10.1103/physreva.111.023312

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Bell nonlocality stems from quantum correlations effectively identified using inequalities. Spin chains, simulated with ultracold atoms in optical lattices, Rydberg atoms in tweezer arrays, trapped ions, or molecules allow single-spin control and measurement. Therefore, they are suitable for studying fundamental aspects of these correlations and nonlocality. Occupation defects, such as vacancies or multiple atoms occupying a single site due to imperfect system preparation, limit the detection of Bell correlations. We study their effects with the help of a simplified toy model parametrized by the probability p of having a single occupation for a given site. Within this model, and for entangled systems obtained by one-axis twisting evolution from an initial factorized state, we derive two Bell inequalities, one based on many-site correlations and the other on two-site correlations, and identify the smallest probability p that allows the Bell inequalities violation to be detected. We then consider two physical realizations using entangled ultracold atoms in optical lattices where the parameter p is related to a nonunitary filling factor and nonzero temperature. We test the predictions of the toy model against exact numerical results.
We develop criteria to detect three classes of nonlocality that have been shown by Wiseman et al. [Phys. Rev. Lett. 98, 140402 (2007)] to be nonequivalent: entanglement, EPR steering, and the failure of local hidden-variable theories. We use the approach of Cavalcanti et al. [Phys. Rev. Lett. 99, 210405 (2007)] for continuous variables to develop the nonlocality criteria for arbitrary spin observables defined on a discrete Hilbert space. The criteria thus apply to multisite qudits, i.e., systems of fixed dimension ddd, and take the form of inequalities. We find that the spin moment inequalities that test local hidden variables (Bell inequalities) can be violated for arbitrary ddd by optimized highly correlated nonmaximally entangled states provided the number of sites NNN is high enough. On the other hand, the spin inequalities for entanglement are violated and thus detect entanglement for such states, for arbitrary ddd and NNN, and with a violation that increases with NNN. We show that one of the moment entanglement inequalities can detect the entanglement of an arbitrary generalized multipartite Greenberger-Horne-Zeilinger state. Because they involve the natural observables for atomic systems, the relevant spin-operator correlations should be readily observable in trapped ultracold atomic gases and ion traps.
Nonlocal entanglement between pair-correlated particles is a highly counter-intuitive aspect of quantum mechanics, where measurement on one particle can instantly affect the other, regardless of distance. While the rigorous Bell’s inequality framework has enabled the demonstration of such entanglement in photons and atomic internal states, no experiment has yet involved motional states of massive particles. Here we report the experimental observation of Bell correlations in motional states of momentum-entangled ultracold helium atoms. Momentum-entangled pairs are first generated via s-wave collisions. Using a Rarity-Tapster interferometer and a Bell-test framework, we observe atom-atom correlations required for violation of a Bell inequality. This result shows the potential of ultracold atoms for fundamental tests of quantum mechanics and opens new avenues to studying gravitational effects in quantum states. Momentum-entangled atom pairs are used to demonstrate quantum non-locality, where changing one atom in an entangled pair instantly alters the state of the other atom. This result paves the way to study interactions between quantum states and gravity.
Intensive studies of entanglement properties have proven essential for our understanding of quantum many-body systems. In contrast, much less is known about the role of quantum nonlocality in these systems because the available multipartite Bell inequalities involve correlations among many particles, which are difficult to access experimentally. We constructed multipartite Bell inequalities that involve only two-body correlations and show how they reveal the nonlocality in many-body systems relevant for nuclear and atomic physics. Our inequalities are violated by any number of parties and can be tested by measuring total spin components, opening the way to the experimental detection of many-body nonlocality, for instance with atomic ensembles.
We derive experimentally measurable lower bounds for the two-site entanglement of the spin-degrees of freedom of many-body systems with local particle-number fluctuations. Our method aims at enabling the spatially resolved detection of spin-entanglement in Hubbard systems using high-resolution imaging in optical lattices. A possible application is the observation of entanglement generation and spreading during spin impurity dynamics, for which we provide numerical simulations. More generally, the scheme can simplify the entanglement detection in ion chains, Rydberg atoms, or similar atomic systems.
Summary form only given. Ultra-cold atomic gases trapped in optical lattices offer an unprecedented playground for studying the quantum phases of many-body systems. In particular, quantum states of ultra-cold lattice gases with spin degrees of freedom may be used to simulate quantum magnetism and to investigate physics relevant for our understanding of high-Tcsuperconductivity. While enormous progress has been made towards engineering such systems, achieving the regime of high-Tcsuperconductivity remains experimentally extremely challenging because of the low temperatures required [1].Here, we describe a new technique for the preparation of quantum spin-correlations in a lattice gas of ultracold atoms using atom-light interaction of the kind routinely employed in quantum spin polarization spectroscopy (QPS) [2], a promising technique for detecting quantum phases in lattice gases via quantum non-demolition (QND) measurement. Motivated by recent experimental work demonstrating the generation of spin-squeezing and entanglement in atomic ensembles via QND measurement [3], and by the recent extension of these ideas to unpolarized ensembles [5], we propose an alternative approach to preparing quantum spin-correlations, demonstrating that a simple modification of the experimental scheme of Ref. [4] allows for the on-demand preparation of spatial spin-correlations in a quantum lattice gas. Our method is based on entropic cooling via QND measurement and feedback, and allows the creation and detection of quantum spin-correlations, as well as a certain degree of multipartite entanglement, which we verify by deriving a novel generalization of the entanglement witness decribed in Ref. [5]. The proposed technique works with an unpolarized ensemble of non-interacting spins such as may be obtained by loading ultracold atoms into a deep optical lattice. We illustrate the procedure with examples drawn from the bilinear-biquadratic Hamiltonian, which can be modeled by a 1D chain of spin-1 atoms, showing that it is possible to prepare exponentially- and algebraically-decaying correlations, as well as spatial correlation signatures of more exotic quantum phases such as quantum criticalities.