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

The power spectral densities (PSDs) of trapped nanoparticles is conventionally compared to those of a simple harmonic Langevin oscillator (SHLO) assuming that the particle is oscillating in the harmonic regime. We show numerically that for a particle in a Paul trap that is oscillating in the stable region of the Mathieu equation, its PSDs can deviate significantly from those of the SHLO. We derive analytic expressions for the PSDs and show that they are in agreement with the numerical PSDs obtained by simulating the motion for a sufficiently long period of time in comparison to the damping time. We also derive analytic expressions for the PSDs of multiple cases of perturbation to a SHLO and show that they agree with the numerically obtained PSDs even when the resulting PSD strongly deviates from that for a SHLO. We consider a linearly drifting or an oscillating frequency that could be a result of experimental configuration. We also consider the case of a particle oscillating in a slightly anharmonic regime. Our results show that a modified version of the PSD which was utilized in cite{Barker} changes less than the conventional PSD under the above mentioned perturbations to the SHLO.

References

[1] A Pontin, NP Bullier, M Toro?s, and PF Barker. Ultranarrow-linewidth levitated nano-oscillator for testing dissipative wave-function collapse. Physical Review Research, 2(2):023349, 2020.