Quantum system driven by periodic external field

A quantum Floquet Theory for a periodically driven system is studied. For this purpose the periodic external fields are changed to have an increasing amplitude with exp(t), where is an infinitestimal positive number to be taken to be zero at the end. By using the expansions in terms of inverse powers of the driving frequency, periodic factors of the time evolution operator are factorized successively. Each step corresponds to a periodically driven system with different strength of external field. This approach produces a time-independent effective Hamiltonian. The effectiveness of the method is examined by applying it to simple models; 1) a forced harmonic oscillator, 2) a particle in the double-well potential, and 3) a hydrogen atom in an electric potential.