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Journal de physique

The dynamics of a four-level three-mode system. Operator solution

M. Kozierowski, A.S. Shumovsky

Published Jan 1, 1987• 3 citations
Publisher

Abstract

In the dipole and rotating wave approximation the operator solutions for the level populations and photon numbers are found. The numerical results for the time-evolution of the level populations for the case of initially coherent pumping modes 1 and 3 (mode 2 initially in vacuum) are reported and compared with those for a three-level atom in the ladder and lambda configurations.

DOI
10.1051/jphys:019870048090149300

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In the dipole and rotating wave approximation the operator solutions for the level populations and photon numbers are found. The numerical results for the time-evolution of the level populations for the case of initially coherent pumping modes 1 and 3 (mode 2 initially in vacuum) are reported and compared with those for a three-level atom in the ladder and lambda configurations.
The Schr\"odinger equation in the rotating-wave approximation has been solved nonperturbatively to study the temporal behavior of the occupation probabilities of different levels of a four-level atom interacting with four arbitrarily intense laser fields, such that the frequency of one of them equals the sum of the other three. The three-photon-excitation population dynamics, which is already complex, is strongly modified by one-photon excitation and detunings. Under certain conditions the three-photon excitation traps the population in the state initially populated; the population starts flowing into other states when one-photon excitation is also operative.
Recalling that the rotating wave approximation (RWA) is only valid in the weak coupling regimes, the purpose of this paper is to study the Hamiltonian dynamics describing the full quantum mechanical approach of the interaction between various configurations of three-level atoms Lambda, V and ladder distinctly with a two-mode radiation field, while the RWA is not considered; the counter-rotating terms (CRTs) are taken into account. Generally, the presence of CRTs in the Hamiltonian prevents one to achieve an analytical solution. Moreover, as we will show in the present work, using the perturbation theory, analytical solvable Hamiltonians can be successfully obtained. According to our calculations, the contribution of CRTs within the ordinary Hamiltonian is equivalent to the replacement of the constant detuning with a specific intensity dependent detuning in the first order, and the constant atom field coupling with a particular intensity dependent (f deformed) coupling in the second order of the associated perturbation parameter. Moreover, noticing that according to the initial expression of the Hamiltonian, each mode of the field interacts only with a specific pair of the allowed transitions of each type of the three-level atom, it is surprisingly observed that via applying the mentioned approach, the obtained intensity dependent coupling functions depend on both modes of the field, f(na,nb). In this way, it is seen that the CRTs are removed with the price of arriving at some intermixed intensity dependent atom field coupling functions of the two modes of the field. In this way, the obtained final Hamiltonians are analytically solvable. At last, by determining the time evolution of the atom field wave function, we study the effects of CRTs on a few nonclassical properties of the state of the system, including the atomic population inversion and photon statistics.
By using the full quantum theory, we investigate the time evolution of the atomic dipole squeezing parameters of a Λ-type three-level atom interacting with a single-mode coherent optical field, and study the influence of the initial coherent-field intensity, the initial atomic coherence, the initial populations and energy splitting of the two lower atomic levels on the atomic dipole squeezing. The influence of a classical external driving field coupling to the atom on the atomic dipole squeezing is also explored at the end of the paper.