An analytical result about co-operative spontaneous emission

Abstract

In previous papers (a-s) the co-operative spontaneous decay of N two-level atoms was described by an equation for the density operator of the atomic system. That equation was solved analytically; the solution, however, is so complicated that cannot be used to obtain explicit expressions for the mean values of the various quantities. Hence, those expressions have been evaluated by machine calculations, for the cases of 100and 10000 atoms. The most striking result was that if the atoms are initially all excited, the variance of the atomic energy increases as N 2 as the system approaches the superradiant state, i.e. that state in which )7/2 atoms are on the ground level. A similar behaviour has been found for the radiated intensity too. This result is not in agreement with re{. (4.5), in which it is stated that the variance of the atomic energy, in the aame conditions referred to above, is proportional to N. In this letter we give an analytical proof of the result of re{. (3), using a method of calculation very similar to that developed by PR~:PARATA and one of us (~). In re{. (~-3) the following equation for the probability of having n+ atoms up and n_ atoms down (with n+ + n_ = N) was derived (7):