Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Steven T. Bramwell

TL;DR

This paper identifies a concrete mechanism for anomalous $1/f^{\alpha}$ relaxation within the phonon bottleneck framework, showing that a strong bottleneck ($\sigma\gg1$) in the canonical two-level spin–phonon model yields a robust $1/\omega^{3/2}$ spectral density. Using a limiting reduction of the FS equations and linear-response theory, the authors derive an analytic expression for the complex susceptibility $\chi(\omega)$ and connect it to a time-domain relaxation $F(t)$ expressed with the Lambert $W$-function. They demonstrate three distinct frequency regimes, with the intermediate $\alpha=3/2$ window emerging over a wide range for large $\sigma$, and validate the theory against experimental data from Standley and Wright and Roinel et al. that show good agreement with the predicted $1/\omega^{3/2}$ behavior. The result provides a minimal, physically realistic route to non-Debye relaxation in quantum systems and suggests observable implications for ac-susceptibility measurements and noise in two-level systems, with potential relevance to qubits and nanoscale magnets.

Abstract

The common observation of anomalous `$1/f^α$' relaxation with $α<2$ constitutes one of the enduring mysteries of condensed matter physics. Here it is shown that a $1/f^α$ spectral density, with $α= 3/2$, 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 $α=3/2$ 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.

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

TL;DR

This paper identifies a concrete mechanism for anomalous relaxation within the phonon bottleneck framework, showing that a strong bottleneck () in the canonical two-level spin–phonon model yields a robust spectral density. Using a limiting reduction of the FS equations and linear-response theory, the authors derive an analytic expression for the complex susceptibility and connect it to a time-domain relaxation expressed with the Lambert -function. They demonstrate three distinct frequency regimes, with the intermediate window emerging over a wide range for large , and validate the theory against experimental data from Standley and Wright and Roinel et al. that show good agreement with the predicted behavior. The result provides a minimal, physically realistic route to non-Debye relaxation in quantum systems and suggests observable implications for ac-susceptibility measurements and noise in two-level systems, with potential relevance to qubits and nanoscale magnets.

Abstract

The common observation of anomalous `' relaxation with constitutes one of the enduring mysteries of condensed matter physics. Here it is shown that a spectral density, with , 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 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.

Paper Structure

This paper contains 5 sections, 23 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Review of the phonon bottleneck
  3. Analytic form of the relaxation
  4. Application to real systems
  5. Discussion and Conclusions

Figures (4)

  • Figure 1: Phonon bottleneck relaxation according to the model of ref. FS with parameters $a=100, \sigma = 500$ (parameters defined in Section \ref{['Review']}). Fast Fourier transform (FFT) of the numerically evaluated relaxation function $v(|t|)$ (blue dots) compared with a line (orange) of slope $3/2$ on a log-log scale, to illustrate an extensive range of $\sim 1/\omega^{3/2}$ relaxation.
  • Figure 2: Frequency dependence of the derived complex susceptibility for $\sigma= 500$. Upper plot: real part (orange dotted line, Eq. \ref{['chip']}) and imaginary part (full blue line, Eq. \ref{['chipp']}) of the the analytic complex susceptibilty, showing three distinct frequency regimes. Lower plot: the corresponding function $2|\chi"|/\chi_T\omega$ in the region of the high frequency 'crossover', comparing the 'exact' Eq. \ref{['chipp']} (blue line), the intermediate-frequency approximation Eq. \ref{['asymp']} (orange dashed line) and the high frequency approximation $2/\omega^2$ (gray dashed line).
  • Figure 3: Phonon bottleneck relaxation in the limiting case with bottleneck parameter $\sigma = 50$. Fast Fourier transform (FFT) of the exact relaxation function $v(|t|)$ (blue dots, Eq. \ref{['Lambert']}) compared with the analytic $2|\chi"|/\chi_T\omega$ (magenta line, Eq. \ref{['chipp']}), showing distinct regimes of $\sim1/\omega^0, \sim 1/\omega^{3/2}$ and $2/\omega^2$ (Eq. \ref{['asymp']}) behaviour. In the analytic curve, the value of $\sigma$ is rescaled by a factor of $2$ to exactly match $v(\omega = 0)$. With this rescaling, and no fitting, the two curves are almost coincident at all frequencies, illustrating a rare instance where a '$1/f^{\alpha}$' spectral density (with $\alpha = 3/2$) can be calculated analytically.
  • Figure 4: Replotted experimental data StandleyRoinel on a reduced timescale $t_0$ defined in the text. The lower figure is an enlargement of the upper one for short times. The black points are the data of Ref. Standley, and the orange points are the data of Ref. Roinel. The blue lines are the exact solution, Eq. \ref{['Lambert']} for (top to bottom) $\sigma = 1.57$Standley, $\sigma = 17$Roinel and $\sigma = 500$, a large value for comparison. The figure illustrates how closely experimental systems approach the exact limiting form, Eq. \ref{['Lambert']}.