Atomic number fluctuations in a falling cold atom cloud

We evaluate the effective number of atoms in experiments where a probe laser beam with a Gaussian transverse profile passes through an atomic medium consisting of a cold atom cloud released from a magneto-optical trap. Considering the case where the initial distribution is a Gaussian function of position and of velocity, we give a quantitative description of the time variation of the effective number while the cloud is exploding and falling. We discuss the two cases where the effective number is defined from the linear and nonlinear phase shifts, respectively. We also evaluate the fluctuations of the effective atomic number by calculating their correlation functions and the associated noise spectra. Finally, we estimate the effect of these fluctuations on experiments where the probe beam passes through a cavity containing the atomic cloud.