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arXiv:0904.0205

New Structures in the Theory of the Laser Model

Fabio Bagarello, Geoffrey L. Sewell

Published Apr 1, 2009• 0 citations
arXivHTMLPDF

Abstract

In a recent article, Alli and Sewell formulated a new version of the Dicke-Hepp-Lieb laser model in terms of quantum dynamical semigroups, and thereby extended the macroscopic picture of the model. In the present article, we complement that picture with a corresponding microscopic one, which carries the following new results. (a) The local microscopic dynamics of the model is piloted by the classical, macroscopic field, generated by the collective action of its components; (b) the global state of the system carries no correlations between its constituent atoms after transient effects have died out; and (c) in the latter situation, the state of the system at any time $t$ maximises its entropy density, subject to the constraints imposed by the instantaneous values of its macroscopic variables.

DOI
10.48550/arxiv.0904.0205

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Consider a two-level atom of energy states |1〉 and |2〉 driven by a laser field. The atom can be represented as a spin − 1 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429092053/5e12ab0e-ba42-4caa-9bdf-6325e9117d3e/content/ineqn14_1.tif"/> particle and the laser field can be treated as a classical field. The Hamiltonian of the system is given by ( 14.1 ) H ^ = 1 2 ℏ ω 0 σ ^ Z − 1 2 i ℏ Ω ( σ ^ + e − i ω L t − σ ^ − e i ω L t ), https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429092053/5e12ab0e-ba42-4caa-9bdf-6325e9117d3e/content/eqn14_1.tif"/>

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In a recent article, Alli and Sewell formulated a new version of the Dicke-Hepp-Lieb laser model in terms of quantum dynamical semigroups, and thereby extended the macroscopic picture of the model. In the present article, we complement that picture with a corresponding microscopic one, which carries the following new results. (a) The local microscopic dynamics of the model is piloted by the classical, macroscopic field, generated by the collective action of its components; (b) the global state of the system carries no correlations between its constituent atoms after transient effects have died out; and (c) in the latter situation, the state of the system at any time ttt maximises its entropy density, subject to the constraints imposed by the instantaneous values of its macroscopic variables.
In a recent article, Alli and Sewell [J. Math. Phys. 36, 5598 (1995)] formulated a new version of the Dicke–Hepp–Lieb laser model in terms of quantum dynamical semigroups, and thereby extended the macroscopic picture of the model. In the present article, we complement that picture with a corresponding microscopic one, which carries the following new results. (a) The local microscopic dynamics of the model is piloted by the classical, macroscopic field, generated by the collective action of its components; (b) the global state of the system carries no correlations between its constituent atoms after transient effects have died out; and (c) in the latter situation, the state of the system at any time t maximizes its entropy density, subject to the constraints imposed by the instantaneous values of its macroscopic variables.
A basic formulation of the laser system is given in the form most convenient to study the time-dependent cooperative ( symmetry breaking ) behavior of the system in the far-from-equilibrium states from the microscopic and fully quantum-statistical-mechanical point of view. The quantum mechanical master equation, derived from a microscopic model Hamiltonian, is mapped onto a c-number function space with the use of the Boson and spin coherent states representations. The distribution function (mapped from the density operator) is expanded in terms of the Hermite polynomials and the spherical harmonics to obtain differential equations for the expansion coefficients with which time evolution of the laser system can be determined completely.
I. Introduction.Statistical mechanics is the science of explaining, predicting and understanding the gross, macroscopic attributes of matter (which may be taken to mean mechanical systems with essentially an infinite number of degrees of freedom) in terms of the elementary dynamical laws governing its atomic constituents.The problems that arise are sufficiently complex and intriguing, but at the same time sufficiently well posed, that the subject is nowadays as much a part of mathematics as of physics.The fields of information theory and ergodic theory had their genesis in statistical mechanical modes of thought and are now well established in the mathematics literature; there will be more to come.Ludwig Boltzmann, who died in 1906, was one of the principal founders of statistical mechanics, and his monument in Vienna contains the following eloquent testimonial to his scientific creativity:(1) S = k log W.Surely, this hypothesis of Boltzmann [1] is one of the most important and daring in statistical mechanics, for it relates S, the macroscopic entropy of a system, to W, the number of microscopic states of the system which have the same, given macroscopic properties.The number k is a universal constant, called Boltzmann's constant, and, for our purposes, we can consider it to be 1.In these lectures we shall explore some of the abstract properties of entropy, after first giving a precise formulation of it, and will include some recent results (with M. B. Ruskai) which extend formerly known facts about the strong subadditivity of entropy from the domain of classical mechanics to the quantum-mechanical domain.The presentation here will be sketchy and the reader is referred to the original papers [3], [4], [5] for more details.An expanded version of an invited address delivered before the M.I.T.
Consider a two-level atom of energy states |1〉 and |2〉 driven by a laser field. The atom can be represented as a spin − 1 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429092053/5e12ab0e-ba42-4caa-9bdf-6325e9117d3e/content/ineqn14_1.tif"/> particle and the laser field can be treated as a classical field. The Hamiltonian of the system is given by ( 14.1 ) H ^ = 1 2 ℏ ω 0 σ ^ Z − 1 2 i ℏ Ω ( σ ^ + e − i ω L t − σ ^ − e i ω L t ), https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429092053/5e12ab0e-ba42-4caa-9bdf-6325e9117d3e/content/eqn14_1.tif"/>