Quantum noise operators for theN-level system

Abstract

We consider aN-level system of one electron being subject to a heatbath or fluctuating forces as well as to classical or quantum mechanical driving forces. We assume that the equation of the density matrix averaged over the heatbath is already determined with all damping constants and transition probabilities. By this procedure the electron transition operatorsa _i ⁺ (t) a _k (t) loose, however, their property to form an algebraic ring with the relationa _i ⁺ a _k a _l ⁺ a _m =δ _kl a _i ⁺ a _m (1) (in the one-particle subspace).

We reintroduce now fluctuating forces (which have disappeared by the averaging) into the density matrix equation and require that (1) holds if it is averaged over the heatbath. We show, that this requirement determines uniquely all second order correlation functions of the fluctuating forces, which are assumed to be Markoffian. An explicit formula is given, by which these correlation functions can be calculated by a simple sum over averaged density matrix elements multiplied with the coefficients of the averaged density matrix equation. Usually, the fluctuating forces stem from many, statistically independent subsystems of the heatbath, so that they are to a very good approximation Gaussian. In this case our treatment fixes also all higher correlation functions. Equivalent results are found in a simultaneous paper byRisken andSchmid, who use a different method. If we specialize to a 2-level system the results agree with those ofSauermann.

As will be shown in a forthcoming paper these results lend themselves in a straight-forward way to determine the laser linewidth and amplitude fluctuations of atoms with several levels.