7 - Laser Induced Non-linearMultiphoton Processes

Summary

Introduction

The electromagnetic field−matter interactions are studied using both perturbative as well as non-perturbative methods. Both these methodologies have their advantages. Perturbative methods are suitable and are very efficient in electromagnetic field−matter interaction in the regime where the interaction term is negligibly small as compared to the unperturbed Hamiltonian of the system. Prturbative techniques fail when the interaction term (perturbation) becomes comparable to the Hamiltonian of the system[1] like in the case of laser−matter interaction. Further, Cohen- Tannoudji et al.[2] have shown that perturbative methods are valid only when the energy difference between initial and final states is very large as compared to the corresponding transition matrix element. In the present day scenario, very high intensity lasers are available, made possible by the advances in short pulse laser technology. We have the prerogative to study high intensity laser interactions with matter, bulk and nanostructures as well as atoms and molecules. In this high intensity regime of laser, the interactions are best studied using non-perturbative methods. Floquet theory is one of the most accurate non-perturbative methods. It relates the solution of the Schrödinger equation involving a periodic Hamiltonian to the solution of another equation with a time-independent Hamiltonian represented by an infinite dimensional matrix (called the Floquet matrix). It also has the advantage of reducing the problem into a time-independent one. This approach has earlier been applied in various atomic systems[3−6], and has the distinct advantage of handling both discrete and continuum states in a single treatment.

The Floquet theorem, as proposed by Floquet[7] describes the solution of a second order partial differential equation, with one independent variable time ‘t’, and containing a time-periodic coefficients of periodicity ‘T’ as a combination of a periodic function (r, t), with the same periodicity T, and another appropriate function. The Floquet method then was used extensively by Poincare in the 1890s[8]. Later, Autler and Townes[9] used it for obtaining the wave functions of a two level system in terms of infinite continued functions. However, it was only in the 1960s, when the Floquet method found a way into quantum mechanical approaches. Shirley[10] contributed significantly to the further development of the technique by reformulating the time-dependent problem of interaction of a two level quantum system with an oscillating classical field into an equivalent time-independent problem involving an infinite-dimensional matrix, the Floquet matrix.