Measuring the equation of state of trapped ultracold bosonic systems in an optical lattice with in-situ density imaging

TL;DR

This paper demonstrates how in-situ density imaging techniques can accurately measure thermodynamic properties of trapped ultracold bosonic systems in optical lattices, enabling extraction of temperature and chemical potential with high precision.

Contribution

It introduces methods to determine thermodynamic parameters from single-shot images and fluctuation analysis, improving measurement accuracy for Bose-Hubbard systems.

Findings

Single-shot imaging can determine chemical potential and temperature in large normal states.

Fluctuation-dissipation theorem allows temperature extraction in narrow normal states.

Reducing sample variance is possible with short density-density correlation lengths.

Abstract

We analyze quantitatively how imaging techniques with single-site resolution allow to measure thermodynamical properties that cannot be inferred from time-of-light images for the trapped Bose-Hubbard model. If the normal state extends over a sufficiently large range, the chemical potential and the temperature can be extracted from a single shot, provided the sample is in thermodynamic equilibrium. When the normal state is too narrow, temperature is low but can still be extracted using the fluctuation-dissipation theorem over the entire trap range as long as the local density approximation remains valid, as was recently suggested by Qi Zhou and Tin-Lun Ho [arXiv:0908.3015]. However, for typical present-day experiments, the number of samples needed is of the order of 1000 in order to get the temperature at least $10 \%$ accurate, but it is possible to reduce the variance by 2 orders of magnitude if the density-density correlation length is short, which is the case for the Bose-Hubbard model. Our results provide further evidence that cold gases in an optical lattices can be viewed as quantum analog computers.