EXACT SOLUTION OF THE JAYNES-CUMMINGS MODEL WITH CAVITY DAMPING

Operating in Laplace language and making use of a representation based on photon-number states, we find the exact solution for the density o perator that belongs to the Jaynes-Cummings model with cavity damping. The detuning parameter is set equal to zero and the optical resonator does not contain any thermal photons. It is shown that the master equ ation for the density operator can be replaced by two algebraic recurs ion relations for vectors of dimension 2 and 4. These vectors are buil t up from suitably chosen matrix elements of the: density operator. By performing: an iterative procedure, the exact solution for each matri x element is found in the form of an infinite series, We demonstrate t hat all series are convergent and discuss how they can be truncated wh en carrying out numerical work. With the help of techniques from funct ion theory, it is proved that our solutions respect the following cond itions on the density operator: conservation of trace, Hermiticity. co nvergence to the initial stale for small times, and convergence to the ground state for large times. We compute some matrix elements of the density operator for the case of weak damping rind find that their ana lytic structure becomes much simpler, Finally, it is shown that the ex act atomic density matrix converges to the state of maximum von Neuman n entropy if the rime, the square of the initial electromagnetic energ y density, and the inverse of the cavity-damping parameter tend to inf inity equally fast. The initial condition for the atom can be chosen f reely, whereas the field may start from either a coherent or a photon- number state.