We describe an apparatus designed to make non-demolition measurements on a
Bose-Einstein condensate (BEC) trapped in a double-well optical cavity. This
apparatus contains, as well as the bosonic gas and the trap, an optical cavity.
We show how the interaction between the light and the atoms, under appropriate
conditions, can allow for a weakly disturbing yet highly precise measurement of
the population imbalance between the two wells and its variance. We show that
the setting is well suited for the implementation of quantum-limited estimation
strategies for the inference of the key parameters defining the evolution of
the atomic system and based on measurements performed on the cavity field. This
would enable {\it de facto} Hamiltonian diagnosis via a highly controllable
quantum probe.Comment: 8 pages, 5 figures, RevTeX4; Accepted for publication in Phys. Rev.