Asymptotic behavior of the quantum harmonic oscillator driven by a random time-dependent electric field

This paper investigates the evolution of the state vector of a charged quan tum particle in a harmonic oscillator driven by a time-dependent electric f ield. The external field randomly oscillates and its amplitude is small but it acts long enough so that we can solve the problem in the asymptotic fra mework corresponding to a field amplitude which tends to zero and a field d uration which tends to infinity. We describe the effective evolution equati on of the state vector, which reads as a stochastic partial differential eq uation. We explicitly describe the transition probabilities, which are char acterized by a polynomial decay of the probabilities corresponding to the l ow-energy eigenstates, and give the exact statistical distribution of the e nergy of the particle.