Time-frequency Wigner evolution of the quantum Langevin equation

Abstract

We obtain an explicit equation for the time-frequency Wigner spectrum of momentum governed by the Langevin equation when the random driving term is quantum noise. The equation is solved exactly and includes both the transient and the stationary part. The time-dependent Wigner spectrum generalizes the result of Wang and Uhlenbeck wherein they showed that for the white noise driving force the power spectrum in the stationary state regime is Lorenzian. We show that our solution reduces to the classical solution when the parameters of the quantum noise are such that the white noise limit is approached and when the long time limit is taken.