Temporal fluctuations in the bosonic Josephson junction as a probe for phase space tomography

We study the long-time dynamics of the reduced one-particle Bloch vector S of a two-mode Bose-Hubbard model in the Josephson interaction regime, as a function of the relative phase and occupation imbalance of an arbitrary coherent preparation. We find that the variance of the long-time fluctuations of S can be factorized as a product of the inverse participation number 1/M that depends only on the preparation, and a semiclassical function C(E) that reflects the phase space characteristics of the pertinent observable. Temporal fluctuations can thus be used as a sensitive probe for phase space tomography of quantum many-body states.