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Europhysics Letters (EPL)

A New Stable Static Solution of the Jaynes-Cummings Model with Cavity Damping

Martin Munz, Adam Kujawski

Published Sep 15, 1990• 2 citations
Publisher

Abstract

The semi-classical Jaynes-Cummings model including damping inside the cavity without the rotating-wave approximation is analysed. It is shown that, depending on the values for the coupling constant and the detuning parameter, a new stable static solution of the equations of motion can exist, where the electric field does not vanish and the atoms are not in their ground state. By numerical simulations of the model the analytical results are confirmed.

DOI
10.1209/0295-5075/13/2/002

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The semi-classical Jaynes-Cummings model including damping inside the cavity without the rotating-wave approximation is analysed. It is shown that, depending on the values for the coupling constant and the detuning parameter, a new stable static solution of the equations of motion can exist, where the electric field does not vanish and the atoms are not in their ground state. By numerical simulations of the model the analytical results are confirmed.
In the semiclassical version of the Jaynes-Cummings1 model without the rotating-wave approximation, an exact periodic solution has been obtained and studied for some specific values of the coupling constant and the detuning. It is shown that when the atomic frequency is equal to three times the cavity frequency and the coupling constant is arbitrary, the oscillation amplitudes are maximal. The Fourier spectrum of the inversion is discrete. As is known (see, for example, Ref. 2) chaos has been found for the semiclassical Jaynes-Cummings model.
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.
An alternative approach to the Jaynes-Cummings model (JCM) with dissipation at a finite enviromental temperature is presented in terms of a new master equation under Born-Markovian approximations. An analytic solution of the dissipation JCM is obtained. A variety of physical quantities of interest are calculated analytically. Dynamical properties of the atom and the field are investigated in some detail. It is shown that both cavity damping and environmental temperature strongly affect nonclassical effects in the JCM, such as collapse and revivals of the atomic inversion, oscillations of the photon-number distribution, quadrature squeezing of the field and sub-Poissonian photon statistics.
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.