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Chinese Journal of Physics

Treatment of a Damped Two-level Atom in an Electromagnetic Wave by the Path Integral Formalism

Mekki Aouachria, L. Chétouani

Published Oct 1, 2002• 4 citations
OpenAlex

Abstract

The applications of the path integral formalism have widely increased since a large class of potentials have been resolved [1]. However, it is quite rare to find an explicit calculation for interactions that include spin [2]. As is commonly known, spin is a fundamental physical notion for the comprehension of atomic propertics. Recently, the behavior of a damped two-level atom with an electromagnetic wave of circular polarization has been studied using the Schrodinger equation approach [3]. The same problem has been considered without damping [4]. The purpose of this article is to deal with this same problem using the path integral formalism. The two-level atom with a mass m and an angular frequency ! has a dipole moment D. The dynamics of the atom in interaction with a wave is described by the following Hamiltonian:

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The applications of the path integral formalism have widely increased since a large class of potentials have been resolved [1]. However, it is quite rare to find an explicit calculation for interactions that include spin [2]. As is commonly known, spin is a fundamental physical notion for the comprehension of atomic propertics. Recently, the behavior of a damped two-level atom with an electromagnetic wave of circular polarization has been studied using the Schrodinger equation approach [3]. The same problem has been considered without damping [4]. The purpose of this article is to deal with this same problem using the path integral formalism. The two-level atom with a mass m and an angular frequency ! has a dipole moment D. The dynamics of the atom in interaction with a wave is described by the following Hamiltonian:
In this paper, we study the behavior of a two-level atom in an electromagnetic wave of circular polarization. A general solution to the Schr\"odinger equation is obtained, and the action of the electromagnetic wave on the atom is analyzed in detail. It is shown that the levels of the atom would be split, with the energy difference between the split levels the same for the two states of the atom. The atom would acquire an average momentum, with its direction the same as the propagating direction of the electromagnetic wave, showing the expelling action of the electromagnetic wave on the atom; or with its direction opposite to the propagating direction of the electromagnetic wave, showing the trapping action of the electromagnetic wave on the atom. The properties of the action are dependent on the initial state of the atom. The average expelling and trapping forces exerted on the atom by the electromagnetic wave are determined. Two cases of ``resonance'' and ``strict detuning'' are discussed particularly. Under the resonance case, the transition probability reaches its maximum for a given time. In the strict detuning case, the atom is not influenced by the electromagnetic wave.
Abstract We introduced the formalism of path integrals to describe the dynamics of a system consisting of two particles. One is located in a periodic potential well (first subsystem), while the other particle is outside the well and subjected to a circular electromagnetic field (second subsystem). Thanks to this formalism, we are able to describe the dynamics of our system by a propagator which is the product of two subsystems of propagators. The propagator of the first subsystem is put in the standard form exp[Action] as a result of the coherent spin states. This form of propagator is possible by replacing the spin with a unit vector aligned along the polar and azimuthal directions. The second subsystem propagator come with the use of the Feynman-Vernon functional on the subsystem. Thanks to its two propagators, we have succeeded in explicitly and analytically evaluating the dynamics of our dissipative quantum system by a single propagator. Using this fundamental result, we evaluated the special case where the second subsystem bathed in a super-ohmic environment.
In this chapter a quantum mechanical treatment of atomic response to the fields of an electromagnetic wave is presented. Absorption and emission of energy by atoms involves transition between their quantized energy levels driven by an interaction Hamiltonian. Schrodinger equation is used to derive the equations of motion for the density matrix of the atom, which is better suited to include the effects of mechanisms other than the interaction Hamiltonian – collisions, level decays, level shifts, etc. that also affect the state of the atom. Using the density matrix equation of motion and focusing on two atomic levels, induced dipole moment for a single two-level atom and polarization and susceptibility for a collection of two-level atoms, including the effects of level decays, and collisions, are calculated. Equations of motion for level populations are also derived. These expressions naturally predict stimulated emission as well as absorption. Expressed in terms of spontaneous emission rate they confirm the validity of equations derived in Chapter 5 using the dipole oscillator model of atomic response. The limit in which quantum mechanical equations for atomic response reduce to the classical dipole oscillator equations is also discussed.