ABSTRACT

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.