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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Evolution operator calculation and transition probability study of molecules interacting with an electromagnetic field

José Luis Paz, J. Récamier

Published Aug 14, 2001• 0 citations
Publisher

Abstract

Transition probabilities and mean property values of a molecular systems interacting with a low-intensity electromagnetic field are studied by algebraic methods developed in a the present contribution. Lie algebra and a harmonic-oscillator description is used as a first characterization of the system. The SU(2) algebra is used later to characterize the same harmonic system in the dipolar classical approximation for the radiation-matter interaction.

DOI
10.1117/12.437247

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Transition probabilities and mean property values of a molecular systems interacting with a low-intensity electromagnetic field are studied by algebraic methods developed in a the present contribution. Lie algebra and a harmonic-oscillator description is used as a first characterization of the system. The SU(2) algebra is used later to characterize the same harmonic system in the dipolar classical approximation for the radiation-matter interaction.
Transition probabilities between quantum states of a molecular system interacting with a low intensity time dependent field were obtained. The molecular system is modeled as a quartic anharmonic oscillator, whereas the interaction with the electric field is described by the electric-dipole approximation. We apply Lie algebraic techniques and make use of iterative Bogoliubov transformations to derive an approximate expression for the evolution operator. The methodological procedure proposed by Yuen allows us to evaluate the final time evolution operator in a closed for, using the optical properties of the two- photon coherent state of the radiation field.
The harmonic oscillator is a useful model for the interpretation of many physical phenomena. It is used as a first approximation to the vibration of molecules and crystal lattices and for the quantization of electromagnetic fields, among other applications. One can solve the time-independent Schrodinger equation for the harmonic oscillator in many different ways; operator methods give rise to recurrence relations and generating functions for the stationary states. The coordinate representation renders the coordinate operator diagonal. In this particular representation the eigenvectors of the harmonic oscillator are functions of the coordinate that are commonly expressed in terms of Hermite polynomials. The chapter shows that calculation with the eigenfunctions of the harmonic oscillator is remarkably simple in the realm of the Lie algebras. The application of the bosonic algebra in theoretical physics and chemistry is so vast that it is impossible even to give a reasonable summary. An important application of the bosonic algebra is the modeling of simple quantum-mechanical systems.
We consider the description of the quantum dynamics of a two-level system interacting with a single Glauber coherent mode of the radiation field. An exact solution (within the rotating wave and electric dipole approximations) is provided for both the time-independent and time-dependent Green’s functions for the coupled matter–radiation system. The atom (molecular) transition probabilities are shown to be Poisson averages of the corresponding semiclassical problem when the latter is likewise treated in the rotating wave approximation. At low radiation intensities, the quantum and semiclassical results differ, but at high intensities, they become the same when the average photon number n0 in the field satisfies the condition n01/2≫1. A similar behavior is displayed for the expectation value of the electric field of the coherent mode in the presence of the atom. The spectrum of the radiation, resonantly scattered by the atom, is shown to coincide with the results of Mollow.
The approximation of a short-time evolution operator for a molecular system with the Hamiltonian is constructed in terms of the evolution operator of an unperturbed system, a perturbative potential V and commutators of and V . A description of a powerful method for such a construction based on the Lie-algebra technique is given. It is demonstrated that efficient representation for the evolution operator can be obtained using approximations like where . This relation is accurate up to the third order in time. The simplest approximations of this type are listed.