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We study a Bose-Einstein condensate (BEC) in a double-well potential subjectto an unsharp continuous measurement of the atom number in one of the twowells. We investigate the back action of the measurement on the quantumdynamics and the viability to monitor the ensuing time evolution. For vanishinginter-atomic interactions, mainly the expectation values of the measured localobservable can be inferred from the measurement record. Conversely, in thepresence of moderate inter-atomic interactions, the entire many-body state--modified by the measurement-- is monitored with unit fidelity and, at thesame time, the measurement effects a transition from quantum to mean-field(classical) behavior of the BEC. We show that this perfect state estimation ispossible because the inter-atomic interactions enhance the information gainedvia the measurement.

Unsharp continuous measurement of a Bose-Einstein condensate: full quantum state estimation and the transition to classicality