  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.
