We propose a measurement scheme that allows determination of even-moments of a Bose-Einstein condensate (BEC) atom number, in a ring cavity, by continuous photodetection of an off-resonant quantized optical field. A fast cavity photocounting process limits the heating of atomic samples with a relatively small number of atoms, being convenient for BECs on a microchip scale applications. The measurement back-action introduces a counting-conditioned phase damping, suppressing the condensate typical collapse and revival dynamics.

We propose a scheme to measure the cross-correlations and mutual coherence of optical and matter fields. It relies on the combination of a matter-wave detector operating by photoionization of the atoms and a traditional absorption photodetector. We show that the double-detection signal is sensitive to cross-correlation functions of light and matter waves

Atomic collisions are included in an interacting system of optical fields and trapped atoms allowing field amplification. We study the effects of collisions on the system stability. Also a study of the degree of entanglement between atomic and optical fields is made. We found that, for an atomic field initially in a vacuum state and optical field in a coherent state, the degree of entanglement does not depend on the optical field intensity or phase. We show that in conditions of exponential instability the system presents at long times two distinct stationary degree of entanglement with collisions affecting only one of them.