Phase Diffusion and Lamb-Shift-Like Spectrum Shift in Classical Oscillators

TL;DR

This paper solves the nonlinear Langevin equation for classical oscillators, revealing how nonlinearity affects phase diffusion and causes a frequency shift similar to the Lamb shift in quantum electrodynamics.

Contribution

It introduces a perturbation method from quantum mechanics to solve the nonlinear Langevin equation, uncovering effects of nonlinearity on oscillator linewidth and frequency.

Findings

Slower phase diffusion leading to linewidth narrowing.

Discovery of a frequency shift analogous to the Lamb shift.

Enhanced understanding of nonlinear effects in classical oscillators.

Abstract

The phase diffusion in a self-sustained oscillator, which produces oscillator's spectral linewidth, is inherently governed by a nonlinear Langevin equation. Over past 40 years, the equation has been treated with linear approximation, rendering the nonlinearity's effects unknown. Here we solve the nonlinear Langevin equation using the perturbation method borrowed from quantum mechanics, and reveal the physics of the nonlinearity: slower phase diffusion (linewidth narrowing) and a surprising oscillation frequency shift that formally corresponds to the Lamb shift in quantum electrodynamics.