Nonlinear Power Spectral Densities for the Harmonic Oscillator

TL;DR

This paper introduces a general method to calculate nonlinear power spectral densities of the harmonic oscillator in both classical and quantum regimes, extending known results and verifying them through the fluctuation-dissipation theorem.

Contribution

A novel general procedure for computing kth-order PSDs of the harmonic oscillator in classical and quantum contexts, including damping effects and high-temperature limits.

Findings

Reproduces known linear PSD results for k=1

Extends to second-order PSD relevant for quantum measurement

Shows quantum PSDs match classical ones in high-temperature limit

Abstract

In this paper, we discuss a general procedure by which nonlinear power spectral densities (PSDs) of the harmonic oscillator can be calculated in both the quantum and classical regimes. We begin with an introduction of the damped and undamped classical harmonic oscillator, followed by an overview of the quantum mechanical description of this system. A brief review of both the classical and quantum autocorrelation functions (ACFs) and PSDs follow. We then introduce a general method by which the kth-order PSD for the harmonic oscillator can be calculated, where $k$ is any positive integer. This formulation is verified by first reproducing the known results for the $k = 1$ case of the linear PSD. It is then extended to calculate the second-order PSD, useful in the field of quantum measurement, corresponding to the $k = 2$ case of the generalized method. In this process, damping is included into each of the quantum linear and quadratic PSDs, producing realistic models for the PSDs found in experiment. These quantum PSDs are shown to obey the correspondence principle by matching with what was calculated for their classical counterparts in the high temperature, high-Q limit. Finally, we demonstrate that our results can be reproduced using the fluctuation-dissipation theorem, providing an independent check of our resultant PSDs.