Operator algebra and Floquet states

We begin by considering the time-dependent Schrddinger equation 

This represents a harmonic oscillator of zero-point energy wq interacting with a one-mode laser of angular frequency m. From the Floquet theory of differential equations, it is well known that the solutions of equation (44) are 

In order to find these solutions by operator algebra we use the well-known boson operators to obtain [31] 

By using Louisell s technique [2], which introduces the concept of normal order, equation (48) becomes a differential equation involving only complex numbers for the function f/ (a, a, t), where a is a complex number and U a, a, t) is obtained from the normal order form of a+, t) through 

Substitution of equation (51) into equation (50) gives a set of four coupled linear differential equations, whose solutions can be obtained by standard procedures. We note that once we have obtained a, t) we recover 

---