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arXiv:2308.14246 · The European Physical Journal C

Using nanokelvin quantum thermometry to detect timelike Unruh effect in a Bose–Einstein condensate

Zehua Tian, Jiliang Jing

Published Nov 11, 2023• 6 citations
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Abstract

Abstract It is found that the Unruh effect can not only arise out of the entanglement between two sets of modes spanning the left and right Rindler wedges, but also between modes spanning the future and past light cones. Furthermore, an inertial Unruh–DeWitt detector along a spacetime trajectory in one of these cones may exhibit the same thermal response to the vacuum as that of an accelerated detector confined in the Rindler wedge. This feature thus could be an alternative candidate to verify the “Unruh effect”, termed as the timelike Unruh effect correspondingly. In this paper we propose to detect the timelike Unruh effect by using an impurity immersed in a Bose–Einstein condensate (BEC). The impurity acts as the detector which interacts with the density fluctuations in the condensate, working as an effective quantum field. Following the paradigm of the emerging field ofquantum thermometry, we combine quantum parameter estimation theory with the theory of open quantum systems to realize a nondemolition Unruh temperature measurement in the nanokelvin ( $$\textrm{nK}$$ nK ) regime. Our results demonstrate that the timelike Unruh effect can be probed using a stationary two-level impurity with time-dependent energy gap immersed in a BEC within current technologies.

DOI
10.1140/epjc/s10052-023-12191-6

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Abstract It is found that the Unruh effect can not only arise out of the entanglement between two sets of modes spanning the left and right Rindler wedges, but also between modes spanning the future and past light cones. Furthermore, an inertial Unruh–DeWitt detector along a spacetime trajectory in one of these cones may exhibit the same thermal response to the vacuum as that of an accelerated detector confined in the Rindler wedge. This feature thus could be an alternative candidate to verify the “Unruh effect”, termed as the timelike Unruh effect correspondingly. In this paper we propose to detect the timelike Unruh effect by using an impurity immersed in a Bose–Einstein condensate (BEC). The impurity acts as the detector which interacts with the density fluctuations in the condensate, working as an effective quantum field. Following the paradigm of the emerging field ofquantum thermometry, we combine quantum parameter estimation theory with the theory of open quantum systems to realize a nondemolition Unruh temperature measurement in the nanokelvin ( nK\textrm{nK}nK nK ) regime. Our results demonstrate that the timelike Unruh effect can be probed using a stationary two-level impurity with time-dependent energy gap immersed in a BEC within current technologies.
The Unruh effect states an accelerated particle detector registers a thermal response when moving through the Minkowski vacuum, and its thermal feature is believed to be inseparable from Lorentz symmetry: Without the latter, the former disappears. Here we propose to observe the analog circular Unruh effect using an impurity atom in a quasi-two-dimensional Bose-Einstein condensate (BEC) with dominant dipole-dipole interactions between atoms or molecules in the ultracold gas. Quantum fluctuations in the condensate possess a Bogoliubov spectrum ${\ensuremath{\omega}}_{\mathbf{k}}={c}_{0}|\mathbf{k}|f(\ensuremath{\hbar}{c}_{0}|\mathbf{k}|/{M}_{*})$, working as an analog Lorentz-violating quantum field with the Lorentz-breaking scale M∗{M}_{*}M∗​, and the impurity acts as an effective Unruh-DeWitt detector thereof. When the detector travels close to the sound speed, observation of the Unruh effect in our quantum fluid platform becomes experimentally feasible. In particular, the deviation of the Bogoliubov spectrum from the Lorentz-invariant case is highly engineerable through the relative strength of the dipolar and contact interactions, and thus a viable laboratory tool is furnished to experimentally investigate whether the thermal characteristic of the Unruh effect is robust to the breaking of Lorentz symmetry.
The Unruh effect can not only arise out of the entanglement between modes of left and right Rindler wedges, but also between modes of future and past light cones. We explore the geometric phase resulting from this timelike entanglement between the future and past, showing that it can be captured in a simple Λ system. This provides an alternative paradigm to the Unruh-deWitt detector. The Unruh effect has not been experimentally verified because the accelerations needed to excite a response from Unruh-deWitt detectors are prohibitively large. We demonstrate that a stationary but time-dependent Λ-system detects the timelike Unruh effect with current technology.
The Unruh effect predicts that an accelerating observer perceives the Minkowski vacuum as a thermal bath, yet direct detection remains experimentally inaccessible. Its timelike counterpart, arising from the entanglement of massless fields between the future and past light cones, offers a more feasible route but requires a detector whose transition frequency follows a specific conformal-time scaling. We propose and analyze a practical implementation of such a detector using superconducting fluxonium circuits, which naturally provide two quasi-degenerate ground states and a tunable excited state, forming an effective ΛΛΛ-system. By modulating the excited-state transition frequency in Minkowski time, the detector accumulates a geometric phase associated with the timelike Unruh effect. Open-system simulations predict ∼10%\sim 10\%∼10% shift in the ground-state population within 530530530 ns, representing a three-order-of-magnitude sensitivity enhancement over two-level Unruh-DeWitt detectors. These results establish a realistic quantum-circuit platform for experimentally probing the timelike Unruh effect and, more broadly, for testing fundamental nature of quantum fields using engineered quantum systems.
We introduce a novel minimally disturbing method for sub-nK thermometry in a Bose-Einstein condensate (BEC). Our technique is based on the Bose polaron model; namely, an impurity embedded in the BEC acts as the thermometer. We propose to detect temperature fluctuations from measurements of the position and momentum of the impurity. Crucially, these cause minimal backaction on the BEC and hence, realize a nondemolition temperature measurement. Following the paradigm of the emerging field of quantum thermometry, we combine tools from quantum parameter estimation and the theory of open quantum systems to solve the problem in full generality. We thus avoid any simplification, such as demanding thermalization of the impurity atoms, or imposing weak dissipative interactions with the BEC. Our method is illustrated with realistic experimental parameters common in many labs, thus showing that it can compete with state-of-the-art destructive techniques, even when the estimates are built from the outcomes of accessible (suboptimal) quadrature measurements.
The Unruh effect can not only arise out of the entanglement between modes of left and right Rindler wedges, but also between modes of future and past light cones. We explore the geometric phase resulting from this timelike entanglement between the future and past, showing that it can be captured in a simple Λ\LambdaΛ-system. This provides an alternative paradigm to the Unruh-deWitt detector. The Unruh effect has not been experimentally verified because the accelerations needed to excite a response from Unruh-deWitt detectors are prohibitively large. We demonstrate that a stationary but time-dependent Λ\LambdaΛ-system detects the timelike Unruh effect with current technology.
While the Unruh effect has traditionally been studied under the assumption of uniform acceleration, a simplification motivated by experimental considerations, it is not necessarily true for all non-inertial motions. We propose a novel approach for the indirect detection of the Unruh effect without relying on the former restriction. Previous studies have shown that probing the decoherence of an Unruh-DeWitt detector can significantly reduce the acceleration required for observing the effect by several orders of magnitude compared to earlier proposals. Building on this idea, we develop a theoretical framework describing a non-inertial observer equipped with a detector undergoing non-uniform, time-dependent acceleration. We show that, in a non-uniformly accelerated Rindler spacetime, the particle distribution perceived in the Minkowski vacuum acquires a time-dependent modification of the standard Unruh spectrum. Furthermore, we demonstrate that the inclusion of quantum entanglement leads to a deformation of the Minkowski vacuum into squeezed states.