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International Journal of Modern Physics B

AN ALGEBRAIC APPROACH TO THE INTERACTION OF A THREE-LEVEL ATOM INTERACTING WITH QUANTIZED RADIATION FIELD

Xu‐Bo Zou, Jing‐Bo Xu, Xiao-Chun Gao, Jian Bo Fu

Published Jun 10, 2000• 0 citations
Publisher

Abstract

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.

DOI
10.1142/s0217979200001370

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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.
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.
With the help of an SU(3) dynamical algebraic structure, we find an exact solution of the Milburn equation for the system of a three-level atom in the Ξ configuration interacting with one quantized field mode with arbitrary detuning. The exact solution is then used to discuss the influence of the intrinsic decoherence on the nonclassical properties of the system, such as collapes and revivals of the atomic populations, oscillations of the photon number distribution, and squeezing of the radiation field.
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.
In this article, by using the perturbation theory, we analytically solve the eigenvalue problem for the Hamiltonian describing the interaction of a Λ -type three-level atom with a single-mode radiation field without the rotating wave approximation (RWA). For this purpose, the atom–field interaction Hamiltonian, which contains the counter-rotating terms (CRTs), is transformed to an analytically solvable Hamiltonian by applying three successive unitary transformations. According to our calculations, the contribution of CRTs within the transformed Hamiltonian is in fact replaced by transforming the 'constant detuning' with the 'intensity-dependent detuning' in the first order, and the 'constant atom–field coupling' with the intensity-dependent coupling in the second order of the perturbation parameters. Then, by solving the eigenvalue problem for the transformed Hamiltonian, the eigenvector of the considered atom–field Hamiltonian is obtained analytically. Finally, after achieving the state vector of the atom–field system at an arbitrary time, a few nonclassical properties of the system state are investigated numerically. Meanwhile, we compare our results with the presence of RWA, from which the role of CRTs will be established.