Analytic solution for inversion and intensity of the Jaynes-Cummings model with cavity damping

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Abstract
An analytic expression for the inversion and intensity of the Jaynes-Cummings model with cavity damping is derived in the rotating-wave approximation for vanishing thermal photon numbers. Using the s-parametrized quasiprobability distributions of Cahill and Glauber [Phys. Rev. 177, 1882 (1969)], the equation of motion for the density operator is transformed into c-number equations for the quasiprobability functions. By a suitable expansion into a Fourier series and into Laguerre functions, we obtain ordinary tridiagonal coupled differential equations for the expansion coefficients. By an appropriate choice of a scaling parameter and by a proper elimination procedure, it is shown that the coefficients that determine the inversion and the mean intensity 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. Furthermore, it is shown that the eigenvalues and eigenvectors can also be calculated, thus leading to an analytic solution for the inversion of the Jaynes-Cummings model.