Unruh and analogue Unruh temperatures for circular motion in $3+1$ and $2+1$ dimensions

Unruh and analogue Unruh temperatures for circular motion in $3 + 1$ and $2 + 1$ dimensions

Steffen Biermann, Sebastian Erne, Cisco Gooding, Jorma Louko, Jörg Schmiedmayer, William G. Unruh, and Silke Weinfurtner

Phys. Rev. D 102, 085006 – Published 13 October 2020

Abstract

The Unruh effect states that a uniformly linearly accelerated observer with proper acceleration $a$ experiences Minkowski vacuum as a thermal state in the temperature $T_{lin} = a / (2 π)$ , operationally measurable via the detailed balance condition between excitation and deexcitation probabilities. An observer in uniform circular motion experiences a similar Unruh-type temperature $T_{circ}$ , operationally measurable via the detailed balance condition, but $T_{circ}$ depends not just on the proper acceleration but also on the orbital radius and on the excitation energy. We establish analytic results for $T_{circ}$ for a massless scalar field in $3 + 1$ and $2 + 1$ spacetime dimensions in several asymptotic regions of the parameter space, and we give numerical results in the interpolating regions. In the ultrarelativistic limit, we verify that in $3 + 1$ dimensions $T_{circ}$ is of the order of $T_{lin}$ uniformly in the energy, as previously found by Unruh, but in $2 + 1$ dimensions, $T_{circ}$ is significantly lower at low energies. We translate these results to an analogue spacetime nonrelativistic field theory in which the circular acceleration effects may become experimentally testable in the near future. We establish in particular that the circular motion analogue Unruh temperature grows arbitrarily large in the near-sonic limit, encouragingly for the experimental prospects, but the growth is weaker in effective spacetime dimension $2 + 1$ than in $3 + 1$ .

Received 29 July 2020
Accepted 21 September 2020

DOI:https://doi.org/10.1103/PhysRevD.102.085006

Physics Subject Headings (PhySH)

Quantum fields in curved spacetime Unruh effect

Particles & FieldsGravitation, Cosmology & Astrophysics

Authors & Affiliations

Steffen Biermann^1,*, Sebastian Erne ^1,2,3,†, Cisco Gooding^1,‡, Jorma Louko ^1,§, Jörg Schmiedmayer ^2,∥, William G. Unruh^4,5,¶, and Silke Weinfurtner^1,6,**

¹School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, United Kingdom
²Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, Stadionallee 2, A-1020 Vienna, Austria
³Wolfgang Pauli Institut, c/o Fak. Mathematik, Universität Wien, Nordbergstrasse 15, 1090 Vienna, Austria
⁴Department of Physics and Astronomy, The University of British Columbia, Vancouver V6T 1Z1, Canada
⁵Hagler Institute for Advanced Study and Institute for Quantum Science and Engineering, Texas A&M University, College Station, Texas 77843-4242, USA
⁶Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham NG7 2RD, United Kingdom

^*steffen.biermann@nottingham.ac.uk
^†erne@atomchip.org
^‡cisco.gooding@nottingham.ac.uk
^§jorma.louko@nottingham.ac.uk
^∥schmiedmayer@atomchip.org
^¶unruh@physics.ubc.ca
^**silke.weinfurtner@nottingham.ac.uk

Interferometric Unruh Detectors for Bose-Einstein Condensates

Cisco Gooding, Steffen Biermann, Sebastian Erne, Jorma Louko, William G. Unruh, Jörg Schmiedmayer, and Silke Weinfurtner

Phys. Rev. Lett. 125, 213603 (2020)

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Vol. 102, Iss. 8 — 15 October 2020

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Images

Figure 1
Relativistic spacetime $T_{rat} ≔ T_{circ} / T_{lin}$ as a function of $v$ and $E_{red} ≔ E / a$ , for $0.1 \leq v \leq 0.95$ and $0.1 \leq E_{red} \leq 3$ . The plotting range was chosen for numerical stability, avoiding small and large values of $v$ and small and large values of $E_{red}$ . Left in $3 + 1$ dimensions, evaluated from (2.6) with (3.2); right in $2 + 1$ dimensions, evaluated from (2.6) with (4.4). In the limit $E_{red} \to 0$ , outside the plotted range, the $3 + 1$ graph tends to a nonzero value, as seen from (3.9), while the $2 + 1$ graph has a significant drop, tending to zero linearly in $E_{red}$ , as seen from (4.10). The continuations of the graphs to the ultrarelativistic limit $v \to 1$ , outside the plotted range, are shown in Fig. 4.
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Figure 2
Analogue spacetime $T_{rlab} ≔ {\hat{T}}_{circ} / {\hat{T}}_{lin}$ as a function of $v$ and $E_{rlab} ≔ \hat{E} / \hat{a}$ , for $0.1 \leq v \leq 0.95$ and $0.1 \leq E_{rlab} \leq 3$ . The plotting range was again chosen for numerical stability, avoiding small and large values of $v$ and small and large values of $E_{rlab}$ . Left in $3 + 1$ dimensions; right in $2 + 1$ dimensions. The data are as in Fig. 1, and $T_{rlab} = T_{rat}$ , but $E_{rlab} = γ E_{red}$ . In the limit $E_{red} \to 0$ , outside the plotted range, it is again the case that the $3 + 1$ graph tends to a nonzero value while the $2 + 1$ graph has a significant drop, tending to zero linearly in $E_{red}$ . In the near-sonic limit $v \to 1$ , outside the plotted range, the $3 + 1$ graph tends to the constant value $π / (2 \sqrt{3}) \approx 0.9$ , as seen from (5.5a), but the $2 + 1$ graph drops to zero proportionally to $- 1 / \ln (1 - v^{2})$ , as seen from (5.5b); within the plotted range, this drop shows as incipient for $0.9 ≲ v \leq 0.95$ .
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Figure 3
Relativistic spacetime $T_{rat} ≔ T_{circ} / T_{lin}$ as a function of $v$ in the limits of large and small $| E |$ , showing the continuation of the Fig. 1 plots to these limits. The dashed (blue) curve shows the large $| E |$ limit, in both $3 + 1$ and $2 + 1$ dimensions, evaluated from (2.6) with (3.7). The solid (brown) curve shows the small $| E |$ limit in $3 + 1$ dimensions, evaluated from (3.9). In $2 + 1$ dimensions, the small $| E |$ limit vanishes, as seen from the analytic formula (4.10).
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Figure 4
Relativistic spacetime $T_{rat} ≔ T_{circ} / T_{lin}$ as a function of $E_{red} ≔ E / a$ in the ultrarelativistic limit, $v \to 1$ , showing the continuation of the Fig. 1 plots to this limit. The dashed (blue) curve is for $3 + 1$ dimensions, evaluated from (3.12), interpolating between $π / \sqrt{3} \approx 1.8$ as $E_{red} \to \infty$ and $π / (2 \sqrt{3}) \approx 0.9$ as $E_{red} \to 0$ , as previously found in [42]. The solid (red) curve is for $2 + 1$ dimensions, evaluated from (4.18), interpolating between $π / \sqrt{3} \approx 1.8$ as $E_{red} \to \infty$ and 0 as $E_{red} \to 0$ , showing the falloff proportional to $1 / \ln (1 / E_{red})$ (4.19a) as $E_{red} \to 0$ .
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