First-passage time statistics for stochastic parametric oscillators with time-dependent frequency

Noise-driven parametric oscillators describe a wide range of physical systems. One example of current interest (as a possible hardware platform for quantum computing) concerns the motion of a single ion in a harmonic oscillator potential produced by suitably arranged electrodes. In many of these systems, additive noise is also present and this inevitably leads to (oftentimes undesirable) heating of the oscillator. Here we study the possibility to alter/control these heating effects by suitable modulation of the oscillator frequency on time scales that are long compared with the period of the oscillator. This separation of time scales allows us to derive an effective Fokker-Planck equation for the slow heating dynamics from which we are able to obtain analytic expressions for first-passage time distributions (FPTDs). Interestingly, when the time-dependent frequency varies as an Ornstein-Uhlenbeck (OU) process with a sufficiently large time constant of order the mean first passage time, we find that the FPTD develops heavy long-time tails. This implies that long lived trial members survive even longer in the presence of a noisy frequency than for the case where the frequency is constant. Theoretical predictions are comparable to results of numerical simulations of the noise-driven oscillator dynamical equations.