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Physical Review A

Quantum theory of optical multistability in a two-photon three-level Λ-configuration medium

Zhi‐Cheng Wang, Wenan Guo, Songyi Zheng

Published Dec 1, 1992• 3 citations
Publisher

Abstract

An effective Hamiltonian for the two-photon two-level model is derived form the microscopic Hamiltonian of a three-level atom in a \ensuremath{\Lambda} configuration, interacting through two-photon transitions with a light field. The effective Hamiltonian includes a term describing the optical Stark shift. With this effective Hamiltonian, we study the steady-state equation and the squeezing spectrum for a cavity-field mode interacting with an ensemble of three-level ``\ensuremath{\Lambda}-configuration'' atoms by a two-photon transition. It is shown that significant changes in the transmission characteristics and the squeezing spectrum for the output field result from the inclusion of the Stark shift.

DOI
10.1103/physreva.46.7235

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Journal of Physics B Atomic Molecular and Optical Physics•Mar 10, 2008•24 citations
An effective Hamiltonian for the two-photon two-level model is derived form the microscopic Hamiltonian of a three-level atom in a \ensuremath{\Lambda} configuration, interacting through two-photon transitions with a light field. The effective Hamiltonian includes a term describing the optical Stark shift. With this effective Hamiltonian, we study the steady-state equation and the squeezing spectrum for a cavity-field mode interacting with an ensemble of three-level ``\ensuremath{\Lambda}-configuration'' atoms by a two-photon transition. It is shown that significant changes in the transmission characteristics and the squeezing spectrum for the output field result from the inclusion of the Stark shift.
The analysis presented in part I of this paper is applied to the two-photon interaction of a cavity field mode with N three-level atoms in a Lambda configuration instead. Reduction of the three-level two-photon model to an effective two-level two-photon model leads to a given effective interaction Hamiltonian, which incorporates the optical Stark effect. Using the technique developed by Collett, Cardinear, Collett and Walls, the authors calculate the squeezing spectra for the output field for different Stark shift in the good cavity limit. For the same driving field strength, the inclusion of Stark shift changes the transmission strength.
The squeezing spectrum for a cavity field mode interacting with an ensemble of three-level ``\ensuremath{\Lambda}-configuration'' atoms by an effective two-photon transition is calculated. The advantage of the three-level \ensuremath{\Lambda} system as a squeezing medium, that is, optical nonlinearity without atomic saturation, has recently been pointed out by Reid, Walls, and Dalton. We predict perfect squeezing at the turning points for dispersive optical bistability and good squeezing for a range of other cases. Three-level ladder atoms interacting by an effective two-photon transition are also shown to give perfect squeezing in the dispersive limit.
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.
The Hamiltonian is derived for the interaction of a Λ-type three-level atom with a two-mode quantum cavity field from the general interaction Hamiltonian between a multi-level atom and a multi-mode radiation field, and it is reduced to an effective two-mode Raman-coupled model under a large detuning condition. We propose a modified effective Hamiltonian for the two-mode Raman-coupled model, find the time-dependent state vectors, and present the validity conditions for the involved interaction Hamiltonians. It is shown that in the study of the two-mode Raman-coupled model it is not enough to retain only the usually used effective Hamiltonian, one must also take into account the ac Stark shift of the atomic levels (at least one of the levels). Finally, we study the atomic dynamics in the interaction of a Λ-type three-level atom with a two-mode quantum cavity field and in the two-mode Raman-coupled model. We find that the number of collapse-revivals, the collapse time and the revival time show new characteristics.
Starting from the microscopic Hamiltonian of the three-level model and making use of the slowly varying amplitude approximation and the two-photon rotating wave approximation, we derive an effective Hamiltonian for the two-level two photon model under conditions where the two-photon process dominates the other radiartive processes, so that an exact relation between the three-level model and the two-level two-photon model is established. The effective Hamiltonian includes a term come from the optical Stark effect which was neglected in the conventional treatment. The coefficient of this term may have the same order of magnitude as the effective coupling constant. Based on this effective Hamiltonian, the standard method of Haken's laser theory is applied to study the threshold, the mean photon number, the self-induced frequency shift, the photon statistics and the fluctuation in steady states. Results are compared with previous ones which neglect the Strk effect.
We study the entanglement properties of two three-level Rydberg atoms passing through a single-mode cavity. The interaction of an atom with the cavity field allows the atom to make a transition from the uppermost (lowermost) to the lowermost (uppermost) level by emission (absorption) of two photons via the middle level. We employ an effective Hamiltonian that describes the system with a Stark-shifted two-photon atomic transition. We compute the entanglement of formation of the joint two-atom state as a function of Rabi angle gt . It is shown that the Stark shift can be used to enhance the magnitude of atomic entanglement over that obtained in the resonant condition for certain parameter values. We find that though the two-atom entanglement generally diminishes with the increase of the two-photon detuning and the Stark shift, it is possible to sustain the entanglement over a range of interaction times by making the detuning and the Stark shift compensate each other. Similar characteristics are also obtained for a thermal state cavity field.