Analytic solution for quasiprobability distributions of the Jaynes-Cummings model with cavity damping

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Abstract
An analytic expression for the quasiprobability distributions of Cahill and Glauber [Phys. Rev. 177, 1882 (1969)] of the Jaynes-Cummings model is derived in the rotating-wave approximation for a vanishing thermal photon number. First the equations of motion for the density operator are transformed to a system of partial differential equations for the quasiprobability distributions. By a suitable expansion of these distributions into a Fourier series and into Laguerre functions a system of ordinary tridiagonally coupled differential equations for the expansion coefficients is obtained. By an appropriate choice of a scaling parameter and by a proper elimination procedure it is shown that the coefficients are only coupled to coefficients with the same index and to coefficients with the next upper index. Because of this coupling the Laplace transform can be given analytically. By using a partial fraction decomposition the eigenvalues and eigenvectors can be obtained in closed forms, thus leading to a simple analytic solution for the long-time behavior of the quasiprobability distributions. Finally it is shown that the method can also be applied if additional atomic damping is present.