When one sign is not enough: 2+1 circular motion Unruh effect at low energies

Abstract: We address the circular motion Unruh effect in 2+1 spacetime dimensions, as probed by a pointlike Unruh-DeWitt detector coupled to a massless scalar field. The effective temperature due to circular acceleration, operationally defined in terms of the detector's excitation and de-excitation probabilities, is known to be much smaller than the linear acceleration Unruh temperature when the detector's energy gap is small and the interaction lasts for a long time. It was shown by Parry et al. [Class. Quant. Grav. 42, 245012 (2025), arXiv:2508.19987] that a temperature of the order of the linear acceleration Unruh temperature can nevertheless be recovered in a simultaneous long-time-small-gap double limit, using suitable classes of detector-field couplings described by asymptotically scaled switching families (ASSFs). The successful constructions presented there required the coupling to change sign. Here we prove, within the ASSF framework and under certain technical boundedness and localisation conditions, that sign changes in the detector-field coupling are in fact necessary for obtaining a nonvanishing limiting effective temperature. Our analysis is motivated by current work towards an experimental verification of the circular motion Unruh effect in analogue spacetime experiments.

• The paper demonstrates that incorporating sign changes in the switching function is essential to achieve an effective Unruh temperature in low-energy regimes.
• Using asymptotically scaled switching families, the study rigorously establishes the small frequency suppression condition in the detector's response function.
• The results offer actionable guidance for analog gravity experiments, emphasizing the need for non-monotonic, sign-alternating protocols to simulate horizon thermality.

Effective Temperature Recovery and Sign Constraints in the $2+1$ Circular Motion Unruh Effect

Introduction and Context

This work examines the operational Unruh effect for detectors undergoing uniform circular motion in $2+1$-dimensional Minkowski spacetime, interacting with a massless scalar field. The chief focus is the parameter regime relevant for experimental analog gravity setups, where direct observation of the linear acceleration Unruh effect is infeasible due to the unattainably high accelerations needed for observable temperatures. Consequently, analog experiments, such as those based on Bose-Einstein condensates and superfluid Helium thin films, frequently employ circular acceleration and simulate quantum field dynamics in reduced dimensionality.

A key operational indicator of the Unruh effect is the effective temperature, inferable from the transition probabilities of a two-level Unruh-DeWitt (UDW) detector. However, an anomaly arises in the $2+1$-dimensional circular case: for conventional, monotonic detector-field switchings, the effective temperature in the long interaction/low gap regime is significantly reduced compared to the canonical Unruh temperature. This suppression arises from the pathologies in the small-gap limit, related to the weak decay of the Wightman function along circular worldlines in $2+1$D.

The paper builds on previous results showing that a suitable class of non-monotonic, asymptotically scaled switching families (ASSF) can recover an effective temperature of the correct parametric order. The main claim advanced and rigorously demonstrated here is that, under broad technical conditions, sign changes in the coupling function are not only sufficient but also necessary to avoid vanishing temperature in the scaling limit.

Detector Response, Switching Families, and the Low-Energy Limit

The system studied is an UDW detector traversing a stationary circular orbit, interacting linearly with a massless scalar field prepared in the Minkowski vacuum. The response function, to first-order in the coupling, is computed using the standard two-point Wightman function pulled back to the detector's worldline. This function exhibits a discontinuity at zero energy gap—a direct consequence of the invariant content of the Wightman correlation in reduced dimensions.

To formalize long-time and low-gap asymptotics, the authors introduce switching families $\chi_\lambda$, indexed by a parameter $\lambda$ encoding the duration of nontrivial coupling. The framework of ‘asymptotically scaled switching families’ (ASSFs) is employed to handle the double-scaling limit rigorously, controlling both $\lambda \to \infty$ (long interaction) and $E \to 0$ (small detector energy splitting), possibly in a $\lambda=\lambda(E)$ correlation.

The effective temperature is characterized via the detailed-balance relation in the Markovian regime. Under fixed, monotonic switchings, the effective temperature collapses to zero at small gaps, as previously observed. The authors formalize the regime of interest by requiring that the ‘scaled response function’ admit a nontrivial limit when pulled along suitable ASSFs.

The Small Frequency Suppression (SFS) Criterion

A central technical contribution is the identification and analysis of the small frequency suppression (SFS) condition—a necessary and sufficient quantitative constraint ensuring that the odd-in-energy contributions to the response function, associated with the discontinuity of the Wightman function, are suppressed compared to the even components in the joint long-time/low-gap limit. Explicitly, SFS requires that

$$\frac{1}{|E|\lambda(E)} \int_{-|E|}^{|E|} |\widehat{\chi}_{\lambda(E)}(\omega)|^2 d\omega \to 0$$

as $2+1$0.

In cases where the SFS criterion is met, the limiting effective temperature is

$2+1$1

with $2+1$2 denoting the instantaneous proper acceleration and $2+1$3 a velocity-dependent factor (see eqs. (5.12–5.13)). The explicit computation reveals that for typical physical values, $2+1$4 remains order one, so the effective temperature is parametrically in line with the conventional Unruh temperature.

Necessary and Sufficient Conditions: The Role of Sign Changes

The core result of the paper is a rigorous necessity proof: within the class of ASSFs observing mild boundedness and localization properties—including all physically realistic, temporally localized switching profiles—a sign change in the coupling function is required to meet the SFS condition and thus to realize the desired effective temperature. The proof exploits the observation that, for nonnegative (or nonpositive) switching functions, the Fourier transform remains macroscopically nonzero in a window around zero frequency, and thus the relevant small-frequency integrals cannot be suppressed.

Explicitly, Theorem 5.1 shows that, for any scaling $2+1$5 with $2+1$6, and for any positive, localized switching profile, SFS fails. Conversely, switchings satisfying an average-zero constraint (vanishing Fourier component at zero), which necessarily entail sign changes, can be constructed (including compactly supported or Gaussian profiles as in prior work), and examples are detailed. Thus, the analytic and operational capacity to recover Unruh-like behavior in circular motion at low energy hinges precisely on the sign structure of the physical switching/coupling protocol.

Practical Implications and Experimental Protocols

The necessity of sign-changing switching functions constrains experimental designs aiming to realize Unruh-type effects through quantum detector-field interactions. For instance, simple adiabatic or plateau switchings (always-on or single-bump) are theoretically excluded as means for temperature recovery in the circular, $2+1$7-dimensional regime. However, the constructions are not unphysical: protocols involving coherent superpositions of positive and negative coupling amplitudes—readily implemented via entanglement-harvesting scenarios with multi-pulse control—naturally realize the required non-uniform sign structure.

This extends directly to analog spacetime systems where detector-field couplings are engineered—e.g., through tailored laser pulses—opening a direct set of prescriptions for nontrivial detector responses in the low-gap, long-time regime.

Theoretical Implications and Future Directions

The demonstrated necessity of indefinite sign for the coupling underscores a fundamental and previously unappreciated feature of the Unruh effect in reduced dimensions and noninertial trajectories: operational thermality is not guaranteed by acceleration alone but depends intricately on the details of how the system is probed and on the ultraviolet structure of the switching protocol. This injects a new degree of freedom into proposals for both theoretical modeling and experimental verification of horizon thermality, especially in analog quantum simulation platforms.

It remains to fully characterize how minimal or how ‘weak’ the sign change must be, and whether further relaxation of localization assumptions might permit alternative routes to thermalization. The authors conjecture that the necessity is general, and ongoing work may further refine the constraints and applicability in both relativistic and condensed matter settings. Extensions to higher dimensions, other field content, or nonstationary couplings represent fruitful avenues.

Conclusion

This paper provides a comprehensive analysis of operational thermality for UDW detectors under circular acceleration in $2+1$8-dimensional spacetime. By formulating the SFS criterion and proving the necessity of sign-changing switching functions for temperature recovery, it both resolves a technical puzzle regarding the vanishing of effective temperature in conventional switchings and prescribes a concrete requirement for experimental realization of Unruh-like effects in analog gravity contexts. The results supply both theoretical clarity and actionable prescriptions for future analog gravity experiments and quantum simulation protocols targeting horizon thermality.