Bounds on detection of Bell correlations with entangled ultra-cold atoms in optical lattices under occupation defects
Bell non-locality stems from quantum correlations effectively identified using inequalities. Spin chains, simulated with ultra-cold 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 non-locality. 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 parameterised 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 factorised 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 ultra-cold atoms in optical lattices where the parameter $p$ is related to a non-unitary filling factor and non-zero temperature. We test the predictions of the toy model against exact numerical results.