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Physical Review A

Effective Raman theory for a three-level atom in the Λ configuration

Ying Wu

Published Aug 1, 1996• 190 citations
Publisher

Abstract

It is shown that a three-level atom in the \ensuremath{\Lambda} configuration with arbitrary detunings can be exactly reduced to a two-level system with an effective Raman coupling, which depends nonlinearly on the intensity of the two radiation fields. This is done by exactly evaluating the unitary transformation introduced by Alexanian and Bose [Phys. Rev. A 52, 2218 (1995)] for a three-level atom coupled to two modes of the radiation field. We obtain an exact transformed Hamiltonian in which one of the three levels is decoupled for all values of the detunings. In particular, our result is then valid for any ratios of the coupling constants to detunings, even for zero detuning, in contrast to earlier work which requires that these ratios be small. We find the the eigenvalues of the exact transformed Hamiltonian and study its population dynamics. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.1586

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It is shown that a three-level atom in the \ensuremath{\Lambda} configuration with arbitrary detunings can be exactly reduced to a two-level system with an effective Raman coupling, which depends nonlinearly on the intensity of the two radiation fields. This is done by exactly evaluating the unitary transformation introduced by Alexanian and Bose [Phys. Rev. A 52, 2218 (1995)] for a three-level atom coupled to two modes of the radiation field. We obtain an exact transformed Hamiltonian in which one of the three levels is decoupled for all values of the detunings. In particular, our result is then valid for any ratios of the coupling constants to detunings, even for zero detuning, in contrast to earlier work which requires that these ratios be small. We find the the eigenvalues of the exact transformed Hamiltonian and study its population dynamics. \textcopyright{} 1996 The American Physical Society.
It is shown that the system of a three-level atom coupled with two modes of quantized cavity fields in the V-configuration with arbitrary detunings can be exactly reduced to a two-level system with a nonsingular effective coupling which depends nonlinearly on the intensity of the two cavity fields. By performing a unitary transformation, we obtain an exact transformed Hamiltonian in which one of the three levels is decoupled for all values of the detunings including the zero-detuning. We also present the analytical expressions of energy eigenvdues and eigenvectors, evolution operator, time-varying atomic inversion operator and photon number operators of the two modes for the effective two-level model.
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.
The system of a three-level atom coupled to two quantized field modes with arbitrary detuning and density-dependent multiphoton interaction is studied by a dynamical algebraic method. We find an alternative su(3) dynamical algebraic structure within the system. Based on this su(3) structure, we diagonalize the Hamiltonian by making use of unitary transformations and obtain the eigenvalues, eigenstates, and time evolution operator for the system. The effective two-level Hamiltonian is also derived by unitary transformation. Finally the system governed by the Milburn equation is discussed and the exact solution of the Milburn equation presented.
The system of a three-level atom in the Ξ configuration coupled to two quantized field modes with arbitrary detuning and density-dependent multiphoton interaction is studied by dynamical algebraic method. With the help of an su(3) algebraic structure, we diagonalize the Hamiltonian by making use of unitary transformations and obtain the eigenvalues, eigenstates and time evolution operator for the system. Based on this su(3) structure, we also show that the system of a three-level atom in the Ξ configuration can be exactly transformed to an effective two-level Hamiltonian by an unitary transformation. Finally, we show that there exist an su (N) algebraic structure in the system of a N-level atom interacting with N-1 field modes.