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Physical Review A

Analytic solution of the density-operator equation for the Raman-coupled model with cavity damping

L. Schoendorff, H. Risken

Published May 1, 1990• 28 citations
Publisher

Abstract

A Raman-coupled model, in which a single atom is coupled to a single cavity mode by Raman-type transitions, is treated. Taking damping of the cavity mode into account, we first transform the equation for the density operator into equations for quasiprobability densities. It is shown that these equations can be solved analytically. Various plots of the inversion and of quasiprobability distributions are presented and discussed.

DOI
10.1103/physreva.41.5147

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A Raman-coupled model, in which a single atom is coupled to a single cavity mode by Raman-type transitions, is treated. Taking damping of the cavity mode into account, we first transform the equation for the density operator into equations for quasiprobability densities. It is shown that these equations can be solved analytically. Various plots of the inversion and of quasiprobability distributions are presented and discussed.
The Jaynes-Cummings model with cavity damping is investigated in the rotated-wave approximation. First we introduce six appropriate combinations of the matrix elements of the density operator, which are still operators with respect to the light field. With the help of the s-parametrized quasiprobability distributions of Cahill and Glauber [Phys. Rev. 177, 1882 (1969)] the equations of motion for the density operator transform to six coupled partial-differential equations. By expanding the quasiprobability distributions into two suitable sets, we obtain six tridiagonally coupled differential equations for the expansion coefficients, which are solved by a Runge-Kutta method. Starting with an initial coherent state of the cavity field and the atom in its upper state, we find that the initially one-peak quasiprobability function splits into two peaked functions counterrotating in the complex plane and, depending on the damping constant, spiraling into the origin. Revivals of the inversion oscillation are found for those times, when the two peaks collide. The time dependence of the inversion and the intensity as well as some special distributions of interest are also discussed.
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.
Exact expressions are obtained for the density matrix in the operator form and for the quasiprobabilities for the system of an atom undergoing Raman transitions between two degenerate levels on interaction with a quantized field in a lossy cavity driven by an external field. The model is generalized to include the effects of atomic dephasing collisions. The spectral properties of the radiation are discussed, and the equation of motion of the density matrix is derived in the presence of collisions and an external drive. The equation is found to be of second order with respect to time in the absence of the driving field and also analytically solvable.
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.
We present the complete analytical solution of the cavity QED of a two-level atom and a field mode at zero temperature. It includes both dissipation of the field due to a finite Q value of the cavity and incoherent decay mechanisms for the atom. This analytical solution is provided by a powerful method for treating general master equations that appear in quantum optics. As distinct from the usual approaches we first deal with that part of the master equation which describes the dissipative coupling of the field and the atom to their reservoirs. Rather than using number-state or dressed-state bases we expand the density operator into the eigenstates of the nonunitary parts of the master equation which model the dissipative part of the dynamics. The set of these eigenstates is the damping basis. With the aid of this expansion we find the eigenvalues and eigenstates of the total Liouville operator. The evolution of an arbitrary initial state is then known. We employ these results to give an exact solution of the dynamics of the photon field in realistic experiments with one-atom masers at very low temperatures. It includes detuning, cavity leakage effects, spontaneous decay mechanisms for the atoms, a Fizeau-type velocity distribution for the atomic beam, and a statistical parameter for the probability of the excitation of incoming atoms, covering the limits of Poissonian pumping and of regular pumping. On the same grounds one can treat the one-atom laser, consisting of a single atom which stays in permanent interaction with the field mode and which is continuously pumped by external heat baths.