Phase destruction in quantum non-demolition measurement of the photon number

The decay of the off-diagonal elements of the density matrix for the field that accompanies the quantum non-demolition measurement, recently proposed by Brune et al (Phys. Rev. Lett., vol.65, p.976, 1990) of the number of photons present in a microwave cavity, is studied. Averaging over the time of flight of the atoms, a simple estimate of the number of atoms needed for a precise measurement, is found. It agrees well with the result obtained by Brune et al by numerically simulating the measurement, and, moreover, allows a determination, in a systematic manner, of the optimum values for the relevant physical parameters.