Simulations and theory of power spectral density functions for time-dependent and anharmonic Langevin oscillators

Simulations and theory are presented for the power spectral density functions (PSDs) of particles in time-dependent and anharmonic potentials, including the effects of a thermal environment leading to damping and fluctuating forces. We investigate three one-dimensional perturbations to the harmonic oscillator of which two are time-dependent changes in the natural frequency of the oscillator, while the other is a time-independent extension of the quadratic potential to include a quartic term. We investigate the effect of these perturbations on two PSDs of the motion that are used in experiments on trapped nano-oscillators. We also derive and numerically test the PSDs for the motion of a spherical nanoparticle in a Paul trap. We found that the simple harmonic Langevin oscillator's PSDs are good approximations for the x and y coordinates' PSDs for small values of the parameter q of the Mathieu equation, but the difference can be more than a factor of two as "q" increases. We also numerically showed that the presence of a permanent electric dipole on the nanosphere can significantly affect the PSDs in the x and y coordinates.