Bose-Einstein condensates relative phase measurements through atomic homodyne detection

The dynamics of a two-mode Bose-Einstein condensate trapped in a double-well potential results approximately in an effective Rabi oscillation regime of exchange of population between both wells for sufficiently strong overlap between the modes functions and sufficiently large condensates. Facing the effective Rabi oscillation as a temporal atomic beam splitter we propose a nondestructive measurement process allowing an atomic homodyne detection, thus yielding indirect relative phase information about one of the two-mode condensates. This proposal is achieved through a secondary optical homodyne detection, in which the field is allowed to dispersively interact with one of the condensate modes inside an optical cavity.