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Journal of Physics B Atomic Molecular and Optical Physics

Modelling light-induced kinetic effects in gases with eigenvalues of collision operators

S Kryszewski, Gerard Nienhuis

Published Nov 14, 1989• 10 citations
Publisher

Abstract

The authors consider a system of two-level atoms coupled to a radiation field and immersed in a buffer gas. The corresponding Bloch-like equations are augmented by collision operators stemming from the linear Boltzmann equation. Collision operators describe velocity-changing collisions that give rise to various kinetic effects of light. They study the general properties of collision operators, the main one consisting of the fact that such operators are Hermitian in a suitably defined space of velocity distribution functions They introduce a set of functions that serve as approximations to the eigenfunctions. The eigenvalues are connected with transport coefficients (they give an example of such a relation), but they keep eigenvalues as free parameters of the approach. In this way the authors can describe the whole class of collision operators, the Keilson-Storer model being a particular example. Employing the introduced eigenfunctions they derive and formally solve the equations of motion for the expansion coefficients of the elements of the density matrix of active atoms.

DOI
10.1088/0953-4075/22/21/009

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The authors consider a system of two-level atoms coupled to a radiation field and immersed in a buffer gas. The corresponding Bloch-like equations are augmented by collision operators stemming from the linear Boltzmann equation. Collision operators describe velocity-changing collisions that give rise to various kinetic effects of light. They study the general properties of collision operators, the main one consisting of the fact that such operators are Hermitian in a suitably defined space of velocity distribution functions They introduce a set of functions that serve as approximations to the eigenfunctions. The eigenvalues are connected with transport coefficients (they give an example of such a relation), but they keep eigenvalues as free parameters of the approach. In this way the authors can describe the whole class of collision operators, the Keilson-Storer model being a particular example. Employing the introduced eigenfunctions they derive and formally solve the equations of motion for the expansion coefficients of the elements of the density matrix of active atoms.
The authors consider a system of optically active two-state atoms immersed in a buffer gas and irradiated by a light beam. The corresponding Bloch equations are augmented by Keilson-Storer (KS) collision kernels (1952) which account for velocity-changing collisions. Expansion of the density operator elements in the eigenfunctions of KS kernels allows one to obtain exact solutions for the Laplace transforms of the density operator elements which may be expressed as rational approximants with any prescribed degree of accuracy. As an example they apply this method to discussion of the light-induced drift effect. In particular they give analytic expressions for the drift velocity, which are illustrated by numerical results. Finally, they briefly discuss possible generalisations and applications of the presented method.
The present paper includes the influence of collisions in the high-intensity laser theory by Stenholm and Lamb. The history of an atom is divided into field- and collision-dominated intervals. The collisions are treated as instantaneous independent events (impact approximation). The equation of motion for the density matrix of an ensemble of atoms undergoing collisions is derived. The collisions manifest themselves as an integral operator in the equation of motion. As a model, collisions which do not change the populations of the atomic levels are treated. Physical conditions on the kernel matrix of the collision integral are used to construct a simple nontrivial model. To simplify the treatment, the phase and velocity changes are assumed to take place in different collisions. This model is used to solve the problem of a high-intensity laser with collisions. Only the lowest nonvanishing terms in the Fourier expansion of the density matrix elements are used. Typical numerical results are evaluated and shown in the figures. Connections with other papers and generalizations are discussed.
The density operator of the arbitrary physical system must be positive definite. Employing the general master equation technique which preserves this property, we derive equations of motion for the density operator of an active atom which interacts collisionally with the reservoir of perturber atoms. The obtained general relations applied to the two-level atom yield Bloch-Boltzmann equations (BBE) which, as it is the case with master equation approach, are linear in the matrix elements of the active-atom density operator. The obtained BBE guarantee that positivity is preserved, which needs not to be the case with the results known from literature. We argue that our results are the correct ones and as such should be used in practical applications. Moreover, the structure and the terms which appear in our set of BBE seem to allow simpler and more straightforward physical interpretation.
The arguments of Chapter 2, as well as of those of subsequent chapters, have as their foundation the formalism based on the state vector of a system of which the evolution is described by the Schrödinger equation. In fact, such an approach is badly suited to the case in which the coupling between an atom and its environment (for example through collisions with other atoms or spontaneous emission into formerly empty modes of the electromagnetic field) cannot be neglected. If the correlations induced by these interactions between the atom and its environment do not concern us and we are only interested in the evolution of the atom, the formalism of the density matrix must be employed. This provides a description at all times of the state of the atom, although a state vector for the atom alone cannot be defined. In this formalism the effect of the environment on the atom is accounted for by the introduction of suitable relaxation terms (Section 2C.1) in the equation of evolution of the density matrix. An important application of the density matrix is to the case of a two-level atomic system for which the relaxation terms lead to its deexcitation to a level of lower energy. We shall show that in this case the density matrix can be represented by a vector, known as the Bloch vector , which will allow us to give simple geometrical pictures of the evolution of the system.
The quantum kinetic equation describing slow atomic motion in laser light is derived by an operational method which provides mathematical expressions with a transparent physical structure. The authors prove in a general way that the coefficients appearing in this equation, which is of a Fokker-Planck type, are simply related to the mean value and to the correlation functions of the Heisenberg radiative force of the semiclassical approach, where the atomic position is treated classically. They derive in particular a new theoretical expression for the damping force responsible for radiative cooling and they interpret it in terms of linear response theory. The authors also obtain a new crossed r-p derivative term, which does not appear in semiclassical treatments, but which they find to be very small in most situations. Finally, all the theoretical expressions derived are valid for any J g to J e transitions and are not restricted to two-level atoms.