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Journal of the Physical Society of Japan

Rigorous Solution to a Microscopic Laser Equation

Chikako Uchiyama, Fumiaki Shibata

Published Apr 15, 1993• 4 citations
Publisher

Abstract

A microscopic laser equation is solved rigorously. In order to transform the original equation of the density matrix into a more tractable partial differential equation, spin and boson coherent states are used. The resulting equation for the probability density is solved to give a continued fraction with the use of orthogonal polynomials. As a preliminary result, dynamics of certain physical quantities are calculated.

DOI
10.1143/jpsj.62.1089

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A microscopic laser equation is solved rigorously. In order to transform the original equation of the density matrix into a more tractable partial differential equation, spin and boson coherent states are used. The resulting equation for the probability density is solved to give a continued fraction with the use of orthogonal polynomials. As a preliminary result, dynamics of certain physical quantities are calculated.
Molecular coherent excitation calculations are performed using a simple model of quantum N + 1-levels systems. An exact solution of the corresponding system of differential equations is obtained without their integration. For this, the discrete Fourier transform is applied: the sought-for functions – probability amplitudes a[n](t) of a quantum system are represented with Fourier images F[n] (ω) ,i.e. spectra that are described by some corresponding system of discrete orthogonal polynomials. Fourier spectra are calculated using the polynomials constructed. We find the required a[n](t) by calculating the final sum from 0 to N. Based on a one-to-one correspondence: polynomial characteristics vs equations coefficients, we find all the characteristics of quantum systems, the dynamics of which are described by the obtained solution. The construction of various polynomial systems of a discrete variable makes it possible to obtain solutions for quantum systems with various characteristics, including systems with non-equidistant arrangement of energy levels, which are typical for real molecules.
A pragmatic mechanism to obtain exact analytical solutions of equations describing the coherent dynamics of quantum multilevel systems by laser radiation is implemented. The technique shows interesting features and significant connections between different mathematical structures for constructing exact solutions for a large family of quantum systems, and is based on discrete orthogonal polynomials which are used to express the Fourier spectra of the probability amplitudes of an excited quantum system.
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The paper presents and implements an algorithm for constructing exact analytical solutions of the corresponding differential equations. The solutions are obtained without integration of the equations but with the use of methods of a discrete mathematics. The discrete Fourier transform of the equations and discrete orthogonal polynomials in Fourier space are used. Specially constructed polynomials are used and the solutions of the coherent dynamics equations of some quantum systems excited by laser radiation are given.