Physical Review A

Quasiprobability distributions for the Jaynes-Cummings model with cavity damping

J. Eiselt, H. Risken

Published Jan 1, 1991• 185 citations

Abstract

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.

DOI: 10.1103/physreva.43.346

Related papers

Quasiprobability distributions for the cavity-damped Jaynes-Cummings model with an additional Kerr medium

M. J. Werner, H. Risken

A cavity-damped Jaynes-Cummings model with a Kerr-like medium filling the cavity is investigated in the rotating-wave approximation. We introduce six operators with respect to the light field whose equations of motion are transformed to six coupled partial differential equations using the s-parametrized quasiprobability distributions of Cahill and Glauber [Phys. Rev. 177, 1882 (1969)]. Equations of motion for expansion coefficients of the distribution functions are solved by a Runge-Kutta procedure for vector tridiagonal relations. Starting with an initial coherent state for the cavity field and the atom in its upper state, we find that revivals of the atomic inversion are more pronounced for a given damping constant compared to the case of no Kerr medium. Also, quadrature squeezing is less affected by weak cavity damping and thermal noise compared to the standard Jaynes-Cummings model. The effect of damping on interesting non-Gaussian structures is also discussed.

Physical Review AOct 1, 199174 citations

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

B. Daeubler, H. Risken, L. Schoendorff

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.

Physical Review AAug 1, 199236 citations

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

B. Daeubler, H. Risken, L. Schoendorff

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.

Physical Review ANov 1, 199314 citations

Asymptotic approximations for theQfunction in the Jaynes-Cummings model

C. A. Miller, J. Hilsenbeck, H. Risken

The Q function in the undamped Jaynes-Cummings model for a coherent initial state in the rotating-wave approximation is investigated. By applying the method of steepest decent and saddle-point integration in the limit of high photon numbers, closed analytic expressions for the Q function at different times are given, from which analytic expressions for the Rabi frequency, revival times, envelope shapes of the inversion, beat frequencies, etc. are derived. As Eiselt and one of us have shown [J. Eiselt and H. Risken, Phys. Rev. A 43, 346 (1991)], the Q function first splits into two peaks which counterrotate in the complex plane and then collide on the opposite side of the plane. This behavior of the Q function leads to an interpretation of the collapses and revivals of the Rabi oscillations and a better understanding of expressions for the Rabi frequency, revival times, envelope shapes of the inversion, beat frequencies, etc. The results are generalized to N-photon processes; in particular, the cases N=2 and 4 are treated in some detail.

Physical Review AOct 1, 199234 citations

Origin of splits inQfunctions for the Jaynes-Cummings model

Kuniaki Matsuo

A splitting in the Q distribution functions for the Jaynes-Cummings model is explained, based on the interpretations of the density matrix. It has been shown [J. Eiselt and H. Risken, Phys. Rev. A 43, 346 (1991)] that the Q function splits into two peaks, which counterrotate in phase space and that the collision on the opposite side of the space and the behavior of the Q function are closely connected to the collapse of the Rabi oscillations. This paper reveals the origin of the splitting of the Q distribution functions in terms of the interpretation of the density-matrix formalism. The effects of the initial atomic state on the Q distribution functions and on the quadrature fields are emphasized. It will be shown that the nonrotating Q function is responsible both for the collapse and revival of the Rabi oscillations and for the high-frequency components of the quadrature fields. In this analysis it is assumed that the initial light field of the model is a coherent state and that an atom is prepared in a coherent superposition of the excited and ground states. Three-dimensional figures of the Q distribution functions are presented as a function of the interaction time.

Physical Review AJul 1, 199411 citations

Exact solution of the Jaynes-Cummings model with cavity damping

A. J. van Wonderen

Operating in Laplace language and making use of a representation based on photon-number states, we find the exact solution for the density operator 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 equation for the density operator can be replaced by two algebraic recursion relations for vectors of dimension 2 and 4. These vectors are built up from suitably chosen matrix elements of the density operator. By performing an iterative procedure, the exact solution for each matrix element is found in the form of an infinite series. We demonstrate that all series are convergent and discuss how they can be truncated when carrying out numerical work. With the help of techniques from function theory, it is proved that our solutions respect the following conditions on the density operator: conservation of trace, Hermiticity, convergence to the initial state 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 and find that their analytic structure becomes much simpler. Finally, it is shown that the exact atomic density matrix converges to the state of maximum von Neumann entropy if the time, the square of the initial electromagnetic energy density, and the inverse of the cavity-damping parameter tend to infinity equally fast. The initial condition for the atom can be chosen freely, whereas the field may start from either a coherent or a photon-number state.

Physical Review AOct 1, 199725 citations