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 quantum particle in a harmonic oscillator driven by a time-dependent electric field. 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 framework corresponding to a field amplitude which tends to zero and a field duration which tends to infinity. We describe the effective evolution equation of the state vector, which reads as a stochastic partial differential equation. We explicitly describe the transition probabilities, which are characterized by a polynomial decay of the probabilities corresponding to the low-energy eigenstates, and give the exact statistical distribution of the energy of the particle.