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Chinese journal of atomic and molecular physics

The Heisenberg equation of motion of atom-field interaction systems in coherent state representation

Bian Qiu

Published Jan 1, 2000• 0 citations
OpenAlex

Abstract

The Heisenberg equation of motion of atom field interaction systems is formulated by the use of the representation of coherent state direct product. After introducting a newly defined operator amplitude fuction, the Heisenberg equation of motion for any operator is converted into partial differential equation of the type of separating variables. Formal solution of this differential equation is presented, and the time evolution operator is written down in the present representation.

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The Heisenberg equation of motion of atom field interaction systems is formulated by the use of the representation of coherent state direct product. After introducting a newly defined operator amplitude fuction, the Heisenberg equation of motion for any operator is converted into partial differential equation of the type of separating variables. Formal solution of this differential equation is presented, and the time evolution operator is written down in the present representation.
In the master equation method, we have already illustrated a powerful technique for the calculation of the dynamics of an atomic system interacting with the vacuum field. Another technique for calculating the dynamics of an atomic system coupled to the elec- tromagnetic (EM) field involves Heisenberg equations of motion for the system’s operators. A difference between the master equation and the Heisenberg equations is that the later involves dynamics of the operators, which allows to analyse the evolution of an atomic system in terms of the field and atomic operators. This creates some problems with handling the Heisenberg equations as, in general, operators do not commute and then in the course of solution of the equations we may face the problem of ordering of the operators. It is usually resolved by putting the operators in the normal order. We have gained some experience with the Heisenberg equations of motion in Chapter 6, Example 6.3, where we studied squeezing generation in the nonlinear degenerate parametric amplifier (DPA) process. Here, we illustrate the technique on the standard model of a two-level atom interacting with a multi-mode field. We then generalize the technique to some specific models such as Lorenz- Maxwell and Langevin equations, the derivation of which involves some approximations that can be applied only in some limited cases. We also present in detail the Floquet approach, which is usually applied to problems determined by differential equations with time- dependent coefficients.
Quantum statistical mechanics of spin systems is formulated by the use of the representation of atomic coherent state of spin, which is specified by a complex number. After introducing a newly defined distribution function of spin, the Bloch equation for the equilibrium density matrix, the von Neumann equation for the nonequilibrium density matrix and the Heisenberg equation of motion are converted into partial differential equations of the type of separating variables respectively. Formal solutions of them are presented. Any operator which is composed of spins can be converted uniquely into a differential operator. Thus we succeed in formulating equilibrium and nonequilibrium quantum statistical mechanics in the complex variable space, which differs from the previous one in the diagonal representation of density matrix. We further introduce the interaction representation for the perturbational expansion. The linear response formula is written down in the present representation.
The position operator in the Heisenberg representation obeys the usual classical equation of motion. The similarity between the Heisenberg equations of motion and the classical Hamiltonian equations of motion arises from the fact that the commutator of the operators for the physical quantities plays a similar role in quantum mechanics, to that of the Poisson bracket of the classical quantities A and B, in classical mechanics. Generally, the Heisenberg equations of motion are more difficult to solve than the corresponding classical equations because of the lack of commutation of quantum mechanical operators. The chapter explains the notion of a time-ordered product of operators, and then can be written much more compactly.
A Brief Survey of Analytical Dynamics, Discovery of Matrix Mechanics, Mathematical Preliminaries Postulates of Quantum Theory Equations of Motion, Hamiltonian Operator and the Commutation Relations Symmetries and Conservation Laws, Classical and Quantum Integrability Classical and Quantum Mechanical Degeneracies Bound State Energies for One-Dimensional Problems Exactly Solvable Potentials, Supersymmetry and Shape Invariance Supersymmetry and Superpotential Shape Invariant Potentials Two-Body Problem Determination of the Angular Momentum Eigenvalues Methods of Integration of the Heisenberg Equations of Motion Discrete-Time Formulation of the Heisenberg's Equations of Motion Quantum Tunneling Using Discrete-Time Formulation Integration of the Operator Differential Equations Perturbation Theory Other Methods of Approximation Heisenberg Correspondence Principle, Quantization of the Equations of Motion with Higher Derivatives Determinantal Method in Potential Scattering Time-Dependent Scattering Theory The Scattering Matrix Impact Parameter Representation of the Scattering Amplitude Transition Probability Quantum Diffraction, Motion of a Charged Particle in Electromagnetic Field The Aharanov-Bohm Effect Neutron Interferometry Quantum Many-Body Problem Hartree and Hartree-Fock Approximations Second Quantization Bogoliubov Transformation for a Many-Boson System Bogoliubov Transformation for Fermions Interacting Through Pairing Forces Quantum Theory of Free Electromagnetic Field Casimir Force Interaction of Radiation with Matter Heisenberg Equations for Interaction of an Atom with Radiation Bell's Inequalities
We discuss the Schr\"odinger equation with a time-dependent Hamiltonian that can be written as a linear combination of operators which span a finite-dimensional Lie algebra. The equations of motion for the operators in the Heisenberg representation are shown to be useful in calculating matrix elements and transition probabilities, as well as in obtaining the time-evolution operator. A general time-dependent one-dimensional bilinear Hamiltonian is considered as an illustrative example, and the product form for the time-evolution operator is shown to be global.