1/f^{3/2} Spectral Density at the Phonon Bottleneck

TL;DR

This paper demonstrates that a $1/f^{3/2}$ spectral density naturally arises in the response of two-level systems coupled to a heat bath via phonons, explaining a common anomalous relaxation phenomenon in condensed matter physics.

Contribution

It provides an analytical model showing how $1/f^{3/2}$ noise emerges from the phonon bottleneck effect in a non-equilibrium steady state of phonons.

Findings

Derives the frequency dependence of the response analytically.

Shows $1/f^{3/2}$ behavior over a diverging frequency range.

Connects the model to experimental observations of anomalous relaxation.

Abstract

The common observation of anomalous `$1/f^\alpha$' relaxation with $\alpha<2$ constitutes one of the enduring mysteries of condensed matter physics. Here it is shown that a $1/f^{\alpha}$ spectral density, with $\alpha = 3/2$, can arise in the response of an ensemble of two--level systems coupled to a heat bath by means of a system of Bosonic quasiparticles. The model considered is the classic model of Faughnan and Strandberg of the phonon bottleneck, and the anomalous response is associated with an approximate non-equilibrium steady state of the phonons maintained by slow spin relaxation. The frequency dependence of the response to an applied field is calculated analytically, revealing the emergence, in the limit of a strong bottleneck, of $\alpha=3/2$ behaviour over a diverging range of frequencies. The application of this result to experimental systems is discussed and comparisons are drawn with other systems that exhibit anomalous relaxation.