Precision measurements of temperature and chemical potential of quantum gases

We investigate the sensitivity with which the temperature and the chemical potential characterizing quantum gases can be measured. We calculate the corresponding quantum Fisher information matrices for both fermionic and bosonic gases. For the latter, particular attention is devoted to the situation close to the Bose-Einstein condensation transition, which we examine not only for the standard scenario in three dimensions, but also for generalized condensation in lower dimensions, where the bosons condense in a subspace of Hilbert space instead of a unique ground state, as well as condensation at fixed volume or fixed pressure. We show that Bose-Einstein condensation can lead to sub-shot-noise sensitivity for the measurement of the chemical potential. We also examine the influence of interactions on the sensitivity in three different models and show that meanfield and contact interactions deteriorate the sensitivity but only slightly for experimentally accessible weak interactions.