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Journal of Optics B Quantum and Semiclassical Optics

Nonlinear dynamics of the two-photon Dicke model

A. B. Klimov, J. Negro, R Farias, S. M. Chumakov

Published Sep 2, 1999• 23 citations
Publisher

Abstract

We derive an effective two-photon Hamiltonian for a collection of two-level atoms interacting with a quantized field in the regime of two-photon resonance. This Hamiltonian naturally includes the dynamical Stark shift terms. Considering, in detail, the case of an initially strong quantum field, we show that the effective Hamiltonian acquires nonlinear terms in the su (2) algebra generators. For this case we find the evolution operator and prove the wavefunction factorization for special initial states. We show that collective effects modify the dynamics of the atomic inversion leading to a modulation of the collapse-revival structure.

DOI
10.1088/1464-4266/1/5/312

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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.

•Jan 1, 1992•0 citations
We derive an effective two-photon Hamiltonian for a collection of two-level atoms interacting with a quantized field in the regime of two-photon resonance. This Hamiltonian naturally includes the dynamical Stark shift terms. Considering, in detail, the case of an initially strong quantum field, we show that the effective Hamiltonian acquires nonlinear terms in the su (2) algebra generators. For this case we find the evolution operator and prove the wavefunction factorization for special initial states. We show that collective effects modify the dynamics of the atomic inversion leading to a modulation of the collapse-revival structure.
We study the resonant interaction of A initially unexcited two-level atoms with single-mode Fock, coherent and thermal cavity fields of arbitrary intensity. Two mechanisms lead to a spread of the eigenfrequencies: one is due to the photon-number distribution in the initial state and the other reflects the cooperativity of the system. Both affect the evolution of the photon number and photon statistics. We present approximate expressions for the eigenvalues and eigenvectors of the system. Its regular dynamical regions are found and the difference in shape of the collapses and revivals connected with the above mechanisms is discussed.
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 state evolution in the two-photon atom-field interaction with large initial fields is studied. It is shown that under certain conditions, for the initial coherent field, at one-quarter and three-quarters of the revival time the atom is converged into unique pure states, and at the revival time both the atom and field return to their initial states. The possibility for the field macroscopic superposition states to be generated is discussed. The effects of the cavity damping, photon statistics, and dynamic Stark shifts are investigated. Our results give some further insight into the validity of the effective two-photon Hamiltonian which neglects the Stark shifts.
Using the adiabatic elimination we have managed to deal with the problem of the interaction between a single three-level atom and |NNN| two-level non-interacting atoms. The adiabatic elimination is used to obtain the effective Hamiltonian that describes a nonlinear interaction in the form of a Stark-like shift between a two-level atom and su|(2)N(2)N(2)N| two-level atoms. The time evolution operator is calculated in the presence of a Stark-like shift and the detuning parameters. The time-dependent wave function is determined through the evolution operator and consequently the density matrix is obtained. We adopt an initial wave function for the |NNN| two-level atoms that takes into account the structural degeneracies occurring in the system and thus generalize the atomic coherent state. The phenomenon of the collapses and revivals of the atomic population inversion for different values of the Stark shift parameters is considered. The degree of entanglement is also investigated and discussed, where the linear entropy is used. Finally we examine the variance squeezing for the system for different values of the involved parameters. It is shown that different values of the Stark shift parameters lead to different observations of the squeezing in the quadratures.
We investigate the effect of the Stark shifts on the atomic-inversion dynamics when a multilevel atom interacts with a two-mode quantized radiation field. We utilize Hamiltonians obtained by adiabatically eliminating all but two of the atomic levels and including terms that give rise to the Stark shifts for a Raman-type process as well as for equal- and unequal-frequency two-photon absorption. We find that the Stark effect is relatively unimportant for Raman scattering but should not be ignored in equal- or unequal-frequency absorption. In equal-frequency absorption, the atomic dynamics can be made exactly periodic through an appropriate choice of detuning.
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.