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

We analyze quantitatively how imaging techniques with single-site resolution allow to measure thermodynamical properties that cannot be inferred from time-of-flight 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 Q. Zhou and T.-L. Ho [arXiv:0908.3015]. However, for typical present-day experiments, the number of samples needed is on 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 optical lattices can be viewed as quantum analog computers.