Heating Through Phonon Excitation Implied by Collapse Models

Abstract

We calculate the rate of heating through phonon excitation implied by the noise postulated in mass-proportional-coupled collapse models, for a general noise power spectrum. For white noise with reduction rate $λ$, the phonon heating rate reduces to the standard formula, but for non-white noise with power spectrum $λ(ω)$, the rate $λ$ is replaced by $λ_{\rm eff}=\frac{2}{3 π^{3/2}} \int d^3w e^{-\vec w^2} \vec w^2 λ(ω_L(\vec w/r_c))$, with $ω_L(\vec q)$ the longitudinal acoustic phonon frequency as a function of wave number $\vec q$, and with $r_C$ the noise correlation length. Hence if the noise power spectrum is cut off below $ω_L(|\vec q| \sim r_c^{-1})$, the heating rate is sharply reduced.