Fokker-Planck equation for a superradiant laser

Abstract

Recently (L2) a theoretical model for a supe, rradiant laser has been treated, which is fully described by the dynamics of ~m)mic operators only. We derivc in this paper a Fokker-Planck (FP) equation for a classical distribution function which is closely related to the atomic-density apcrator. Our equation is particularly suitable to discuss the stationary behaviour of the, superradiant laser, and generalizes the treatment given in ref. (a) because it includes also ptlase-destroying effecls wbicb arc neglected in the heatbath-atom interaction model assumed therein. Ore' slarting t)oint is the superradiance master equation (SME) obtained in ref. (1). That equation is too hard to be solved exactly because it contains simultaneously single-atom and collective operators. We, have been able to derive the associated FP equation by using a general procedure (h~cribed in ref. (% A significant check on the validity of our al)proximations is that from our FP equation we derive the same rate equations obtained from the SME in ref. (1). We deal with a, N-two-level-atom system which is formally ident, ical with a Nst)in-~-particle systcm (5). Therefore, the generic /~-th atom cart be described in terms of the spin-ttip operators lt',)~ and R~- and the inversion operator/~3,- The :V-atom system 2l N can be described by means of the collective (6) operators R::-. ~R~ and R s=Z R3/, which still have angular-momentum commutation relations, u~ ,,-~ The SME from which we start is (~)