Quantum-Mechanical Harmonic Oscillator under a Random Quadratic Perturbation

The behavior of a quantum-mechanical harmonic oscillator under a random perturbation of the form $f\left(t\right)\mathrm{}{q}^2$ is discussed. Equations are obtained governing the time evolution of the Wigner function and the weight function for the $P$ representation. The latter function satisfies a fourth-order differential equation, in contrast to the simpler second-order equation obeyed by the Wigner function.