Energy distribution of the quantum harmonic oscillator under random time-dependent perturbations

This paper investigates the evolution of a quantum particle in a harmonic o scillator driven by time-dependent forces. The perturbations are small, but they act long enough so that we can solve the problem in the asymptotic fr amework corresponding to a perturbation amplitude that tends to zero and a perturbation duration that tends to infinity. We describe the effective evo lution equation of the state vector, which reads as a stochastic partial di fferential equation. We exhibit a closed-form equation for the transition p robabilities, which can be interpreted in terms of a jump process. Using st andard probability tools, we are then able to compute explicitly the probab ilities for observing the different energy eigenstates and give the exact s tatistical distribution of the energy of the particle. [S1063-651X(99)04810 -2].