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

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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>p</a:mi></a:math> 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 <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>p</b:mi></b:math> 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 <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>p</c:mi></c:math> is related to a nonunitary filling factor and nonzero temperature. We test the predictions of the toy model against exact numerical results.