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Waiting around for Unruh

Leo J. A. Parry, Diego Vidal-Cruzprieto, Christopher J. Fewster, Jorma Louko

TL;DR

This work analyzes the circular Unruh effect for a massless scalar field in 2+1 dimensions using an Unruh–DeWitt detector and a rigorous long-time limit via Asymptotically Scaled Switching Families (ASSF). It shows that with sign-changing switching, a positive small-gap temperature can be recovered in a simultaneous long-time and small-gap limit, provided the switching satisfies a Small Frequency Suppression (SFS) condition and the duration scales as an inverse power of the gap; the finite temperature is set by the motion through a velocity-dependent factor, $T_0 = a/(2\pi I(v))$. Without sign changes, the small-gap temperature vanishes in this double limit for both adiabatic and plateau scalings, highlighting the essential role of switching profiling in observing the circular-motion Unruh effect in analogue experiments. The results thus broaden the experimental prospects for observing circular Unruh physics while delivering a rigorous framework for long-time asymptotics and switching design. They also connect to broader contexts like entanglement harvesting, where sign-changing switching profiles can naturally arise.

Abstract

How long does a uniformly rotating observer need to interact with a quantum field in order to register an approximately thermal response due to the circular motion Unruh effect? We address this question for a massless scalar field in 2+1 dimensions, defining the effective temperature via the ratio of excitation and de-excitation rates of an Unruh-DeWitt detector in the long interaction time limit. In this system, the effective temperature is known to be significantly smaller than the linear motion Unruh effect prediction when the detector's energy gap is small: the effective temperature tends to zero in the small gap limit, linearly in the gap. We show that a positive small gap temperature at long interaction times can be regained via a controlled long-time-small-gap double limit, provided the detector's coupling to the field is allowed to change sign. The resulting small gap temperature depends on the parameters of the circular motion but not on the details of the detector's switching. The results broaden the energy range for pursuing an experimental verification of the circular motion Unruh effect in analogue spacetime experiments. As a mathematical tool, we provide a new implementation of the long interaction time limit that controls in a precise way the asymptotics of both the switching function and its Fourier transform.

Waiting around for Unruh

TL;DR

This work analyzes the circular Unruh effect for a massless scalar field in 2+1 dimensions using an Unruh–DeWitt detector and a rigorous long-time limit via Asymptotically Scaled Switching Families (ASSF). It shows that with sign-changing switching, a positive small-gap temperature can be recovered in a simultaneous long-time and small-gap limit, provided the switching satisfies a Small Frequency Suppression (SFS) condition and the duration scales as an inverse power of the gap; the finite temperature is set by the motion through a velocity-dependent factor, . Without sign changes, the small-gap temperature vanishes in this double limit for both adiabatic and plateau scalings, highlighting the essential role of switching profiling in observing the circular-motion Unruh effect in analogue experiments. The results thus broaden the experimental prospects for observing circular Unruh physics while delivering a rigorous framework for long-time asymptotics and switching design. They also connect to broader contexts like entanglement harvesting, where sign-changing switching profiles can naturally arise.

Abstract

How long does a uniformly rotating observer need to interact with a quantum field in order to register an approximately thermal response due to the circular motion Unruh effect? We address this question for a massless scalar field in 2+1 dimensions, defining the effective temperature via the ratio of excitation and de-excitation rates of an Unruh-DeWitt detector in the long interaction time limit. In this system, the effective temperature is known to be significantly smaller than the linear motion Unruh effect prediction when the detector's energy gap is small: the effective temperature tends to zero in the small gap limit, linearly in the gap. We show that a positive small gap temperature at long interaction times can be regained via a controlled long-time-small-gap double limit, provided the detector's coupling to the field is allowed to change sign. The resulting small gap temperature depends on the parameters of the circular motion but not on the details of the detector's switching. The results broaden the energy range for pursuing an experimental verification of the circular motion Unruh effect in analogue spacetime experiments. As a mathematical tool, we provide a new implementation of the long interaction time limit that controls in a precise way the asymptotics of both the switching function and its Fourier transform.

Paper Structure

This paper contains 46 sections, 3 theorems, 127 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Field-detector model
  4. Stationary response and the long time limit
  5. Effective temperature
  6. Asymptotically scaled switching families
  7. 2+1 circular motion
  8. Response
  9. Successive long time and small gap limits
  10. $\lambda\to\infty$ followed by $E\to0$
  11. $E\to0$ followed by $\lambda\to\infty$
  12. Summary
  13. Obstruction to a simultaneous small gap and long time limit
  14. Interlude: adiabatic and plateau scaled switching functions
  15. Adiabatic scaling
  16. ...and 31 more sections

Key Result

Proposition 3.1

Let $\chi_\lambda\in C^1(\mathbb{R})$$(\lambda>0)$ be a family of absolutely integrable switching functions with the property that $\lambda^{-1}\widehat{\chi}_\lambda(u/\lambda)$ converges almost everywhere in $u$ as $\lambda\to\infty$. Suppose that there exists $\eta\in L^2(\mathbb{R},\mathrm{d} u/ for almost all $u\in\mathbb{R}$. Furthermore, if $V:\mathbb{R}^2\to \mathbb{C}$ is any bounded meas

Figures (4)

  • Figure 1: Plots of $\chi(\tau)$\ref{['eq: chi example 1']} for $q=1$, $q=7.5$ and $q=20$. Each curve crosses the horizontal axis at least once at positive $\tau$ and at least once at negative $\tau$, as is required by the property $\int_{-\infty}^{\infty} \chi(\tau) \, d\tau =0$ and the evenness of $\chi$.
  • Figure 2: Plots of $\widehat{\chi}(\omega)$\ref{['eq: chihat example 1']} for $q=1$, $q=7.5$ and $q=20$. Each curve displays a characteristic "double-peak" structure around the global minimum at $\omega=0$ where $\widehat{\chi}(0)=0$.
  • Figure 3: Plots of the compact support switching function $\chi$\ref{['eq: example 2 adiabatic chi']} for selected values of $m$ and $n$. The support $-n \le S\tau \le n$ and and the parity ${(-1)}^{n-m}$ of are clear in the plots.
  • Figure 4: Plots of $\mathrm{i}^{m-n}\widehat{\chi}$\ref{['eq: chi hat example 2']} that correspond to two of the switching functions plotted in Figure \ref{['fig: chi example 2']}. A dominant double peak within $-\pi < \omega/S < \pi$ is clear in the plots, with power-law suppressed peaks at larger $|\omega/S|$.

Theorems & Definitions (7)

  • Proposition 3.1
  • Definition 3.2
  • proof
  • Proposition 7.1
  • proof
  • Proposition C.1
  • proof