Dissipation-induced transition of a simple harmonic oscillator

We investigate the dissipation-induced transition probabilities between any two eigenstates of a simple harmonic oscillator. Using the method developed by Yu and Sun [Phys. Rev. A 49, 592 (1994)], the general analytical expressions for the transition probabilities are obtained. The special cases: transition probabilities from the ground state to the first few excited states are then discussed in detail. Different from the previous studies in the literature where only the effect of damping was considered, it is found that the Brownian motion makes the transitions between states of different parity possible. The limitations of the applicability of our results are also discussed.