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Heisenberg Equations for Non-Degenerate Two-Photon Jaynes-Cummings Model with Dynamic Stark-Shifts

Kuniaki Matsuo, Hiroshi Iwasawa

Published Dec 1, 2004• 0 citations

Abstract

One of the simplest models for the interactions between an atom and a quantized electromagnetic field is known as the Jaynes-Cummings model GCM) [1]. In this model, the atom is assumed to have two levels, and the electromagnetic field is a single mode. For the past several years, we have investigated the model for the singlephoton, the two-photon, and the multi-photon cases. First, we studied the characteristic functions and their time behaviors for a simple JCM[Z] using the density matrix approach, and then, by using the Heisenberg equation method, we exmined the degenerate two-photon JCM[31, where the operator differential equations and their solutions are obtained. We further studied the cases of the non-degenerate two-photon JCM[4] and the mUlti-photon JCM[51. Moreover, the atomic spectrum for mUlti-photon JCM[61 and the field spectrum for multiphoton JCM[71 were explored, and these spectra were presented in the 3-dimensional pictures. We also solved the degenerate two-photon JCM with Stark-shifts [8] , and further we just published the results of the non-degeneratetwo-photon JCM with Stark-shifts[91. In this paper, we show a derivaition of the product operator alZ ( t ) = al ( t ) az ( t ). We explicitly use the idea of the constant motion operator. Since the derivations are quite complicated and it is not streightword, we showed them in detail here. The present paper is organized as follows. In section II, we show the effective non-degenerate Hamiltonians and the constant of motion operators. In section Ill, the second-order differential equation for a product operator alZ ( t ) = al ( t ) az ( t) of Mode 1 and Mode 2, will

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One of the simplest models for the interactions between an atom and a quantized electromagnetic field is known as the Jaynes-Cummings model GCM) [1]. In this model, the atom is assumed to have two levels, and the electromagnetic field is a single mode. For the past several years, we have investigated the model for the singlephoton, the two-photon, and the multi-photon cases. First, we studied the characteristic functions and their time behaviors for a simple JCM[Z] using the density matrix approach, and then, by using the Heisenberg equation method, we exmined the degenerate two-photon JCM[31, where the operator differential equations and their solutions are obtained. We further studied the cases of the non-degenerate two-photon JCM[4] and the mUlti-photon JCM[51. Moreover, the atomic spectrum for mUlti-photon JCM[61 and the field spectrum for multiphoton JCM[71 were explored, and these spectra were presented in the 3-dimensional pictures. We also solved the degenerate two-photon JCM with Stark-shifts [8] , and further we just published the results of the non-degeneratetwo-photon JCM with Stark-shifts[91. In this paper, we show a derivaition of the product operator alZ ( t ) = al ( t ) az ( t ). We explicitly use the idea of the constant motion operator. Since the derivations are quite complicated and it is not streightword, we showed them in detail here. The present paper is organized as follows. In section II, we show the effective non-degenerate Hamiltonians and the constant of motion operators. In section Ill, the second-order differential equation for a product operator alZ ( t ) = al ( t ) az ( t) of Mode 1 and Mode 2, will

The dynamics of the degenerate two-photon Jaynes-Cummings model in Heisenberg picture is investigated. By solving the Heisenberg equations, the exact solutions of the photon annihilation operator a( t), the atomic dipole moment operator σ-( t), and the population difference operator σ z ( t) are obtained. We demonstrate that the correlation functions for the photon annihilation and dipole moment operators can be directly evaluated from these exact solutions. The solutions of the time-evolution operators are shown to be quite useful in the calculation of two-time correlation functions. Furthermore, some results of the dynamical behavior of the average values for the photon number <n (t)>, the in-phase field component <X 1(t)>, and the population difference <σ z (t)> are demonstrated by computer simulations.

The driven two-photon Jaynes-Cummings model (JCM) exhibits a collapse and revival phenomenon in its mean photon number on a time scale much larger than the periodic revival time for two-state inversion in the usual two-photon JCM. The time scale of revivals for these oscillations is much larger than the corresponding time scale for the driven one-photon JCM and can be explained with the Hamiltonian of type ${H}_{\ensuremath{\eta}}=\ifmmode\pm\else\textpm\fi{}\sqrt{{(J}_{\ensuremath{\eta}}^{\ifmmode\dagger\else\textdagger\fi{}}+\ensuremath{\alpha}{P}_{\ensuremath{\eta}}{)}^{2}{(J}_{\ensuremath{\eta}}+\ensuremath{\alpha}{P}_{\ensuremath{\eta}}{)}^{2}}$ constructed explicitly for this purpose. The effect of Stark shifts is also studied and it is observed that the dynamics is strongly influenced by such Stark shifts both at early stages as well as at late stages of the evolution but in different ways, and thus Stark shifts play a significant role in determining the dynamical evolution of the system.

The Jaynes-Cummings model (JCM) of two-level atom interacting with the photon mode in ideal cavity plays an essential role in modern quantum optics. In previous papers 1,2 an exact form of density matrix of the JCM with fixed atom position and photons dissipation was found. Here, taking into account the classical motion of the atom through the cavity, it is considered a case of nonideal cavity with zero temperature. We have obtained an exact expression for density matrix and calculated photon spectra and spectra of the mean number of photons in a cavity and and time dependencies of some values relevant for the one-atom maser theory.