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

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.