Bistability in a Quantum Nonlinear Oscillator Excited by a Stochastic Force

We present an approximate analytical method of analysis of stationary states of nonlinear quantum systems with noise. As an example, we consider a quantum nonlinear oscillator excited by a fluctuating force and obtain a range of parameters with more than one stationary solution. The existence of such a range is a necessary condition for bistability. We neglect fluctuations in the amplitude of oscillations but do not neglect fluctuations in its phase. Then, the oscillator noise power spectrum depends on the oscillator mean energy n, which leads to a nonlinear integral equation for n. We can find an analytical solution of this equation. We derive the oscillator stationary states for various spectra of fluctuations of the exciting force. Linear stability analysis of stationary states was carried out. This approach is a generalization of our previous analysis of thresholdless lasers.