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Journal of the Physical Society of Japan

Exact Solutions for Two-Photon Jaynes-Cummings Model with Dynamic Stark-Shifts

Hiroshi Iwasawa, Kuniaki Matsuo

Published Oct 1, 1999• 3 citations
Publisher

Abstract

The solutions for the degenerate two-photon Jaynes-Cummings model are investigated in the presence of the dynamic Stark-shifts. By using the Heisenberg equations of motion, the operator differential equations are established from the effective Hamiltonian. The exact solutions for these operators are derived and the effects of the Stark-shifts on the atomic and field dynamics are discussed. Especially, the behavior of the atomic population difference for the simplest case is demonstrated.

DOI
10.1143/jpsj.68.3439

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Modern Physics Letters B•May 20, 1995•4 citations
The solutions for the degenerate two-photon Jaynes-Cummings model are investigated in the presence of the dynamic Stark-shifts. By using the Heisenberg equations of motion, the operator differential equations are established from the effective Hamiltonian. The exact solutions for these operators are derived and the effects of the Stark-shifts on the atomic and field dynamics are discussed. Especially, the behavior of the atomic population difference for the simplest case is demonstrated.
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.
In this thesis, we investigate the effect of the dynamical Stark shift on the quantum electrodynamics of an atom undergoing two-photon transitions, interacting with a quantized radiation field in an ideal cavity. To begin with, we discuss the dynamics of the electromagnetic field and the atom in the presence of the Stark shift. We derive closed expressions for the second-order coherence function, quadrature variances, atomic population inversion and dipole squeezing under the approximation of a large initial photon number and coherent or squeezed field input. We show that for an atom initially in the ground state, the inclusion of Stark shift reduces the field squeezing as well as the antibunching. Both the population inversion and the dipole squeezing exhibit phase sensitivity in the presence of Stark shift and for a particular choice of the relative phase of the two atomic levels "trapping" occurs in a two-level atom and the dipole is modified in such a way that it is permanently squeezed. We then discuss the effect of the stark shift on the emission spectrum of an atom and the cavity field spectrum. The inclusion of Stark shift results in asymmetric vacuum field Rabi splitting. We also study the dynamics of the combined atom-field wave function in the presence of the Stark shift and show that both the electromagnetic field and the atom periodically evolve to pure states which are radically different from those in the absence of the Stark shift. Finally, we study the nondegenerate two-photon Jaynes-Cummings model (NTPJCM) in the presence of the Stark shift and show that the dynamics of the NTPJCM in the presence of Stark shift is qualitatively different from the case of the absence of the Stark shift and radically different from the case of degenerate two-photon JCM in the presence of Stark shift.
A generalized Jaynes–Cummings model including the Stark shifts is investigated where the transition is mediated by two different modes of photons. For two different types of correlated field states, the pair coherent states and two-mode SU(1, 1) coherent states, the effect of including the Stark shift on the dynamical behavior of atomic inversion, atomic squeezing parameters, second order coherence function, and photon number distribution is investigated. Our results indicate significant changes in the behavior of these quantities for large and small average photon number <n> in the presence and absence of Stark shift and depending on the type of correlated field involved.