Quantum thermodynamics of ergotopy for a relativistic battery as a witness to Unruh-Hawking thermality in curved (A)dS spacetimes

Abstract: We propose a relativistic quantum battery model consisting of an accelerated Unruh-DeWitt detector coupled to a massless scalar field in a de Sitter and anti-de Sitter spacetimes. The maximal amount of quantum extractable work, defined as the ergotropy, is used to probe Unruh-Hawking thermality induced by acceleration and spacetime curvature. Using the open quantum system approach, we study the dynamics of the ergotropy with respect to the Kubo-Martin-Schwinger temperature, spacetime boundary conditions, and dimensionality. It has been found that the asymptotic value of quantum work extraction is determined solely by the acceleration and curvature. The steady behavior in the two spacetimes can be unified to witness the global thermality which is independent of boundary conditions and dimensionality. From a local perspective, we investigate how the ergotropy evolves through different pathways as the battery gradually reaches the same thermal equilibrium state characterized by a certain KMS temperature. In dS spacetime, the evolution at large acceleration exhibits pronounced oscillations and differs from the fast thermal relaxation observed at low acceleration. Varying the boundary condition in AdS spacetime can improve the energy storage of the moving battery. When the dimension of AdS spacetime is increased, vacuum fluctuations can modestly amplify the ergotropy in the initial stage and facilitate the rapid thermalization. From the perspective of energy transfer, the relativistic quantum battery helps explore the thermal vacuum in curved spacetimes.

• The paper demonstrates that a driven, accelerated Unruh-DeWitt battery provides an operational probe of Unruh-Hawking thermality, with steady-state ergotropy determined universally by the KMS temperature.
• Transient ergotropy distinguishes spacetime curvature and AdS boundary conditions, showing stronger oscillations at low acceleration and faster thermal relaxation as acceleration or dimension increases.
• The results identify key limits of the proposal: it relies on coherent driving and weak-coupling Markovian dynamics, while non-Markovian effects, realistic implementation, and nonstationary trajectories remain open questions.

Overview and motivation

This paper constructs a relativistic quantum battery from an accelerated Unruh-DeWitt (UDW) detector coupled to a massless scalar field in de Sitter (dS) and anti-de Sitter (AdS) spacetimes, and uses the ergotropy—the maximal extractable work under cyclic unitary operations—as an operational witness of Unruh-Hawking thermality. The motivation is that while the KMS condition certifies thermality for the global Rindler horizon, it does not fix the shape of the response spectrum in curved spacetime, and existing probes (coherence, geometric phase, Fisher information, uncertainty relations) have mostly been applied in flat spacetime. The battery framework recasts vacuum structure detection as an energy-transfer problem: the fluctuating curved-spacetime vacuum acts as the environment of an open quantum system, and its influence is encoded directly in the Wightman function along the detector trajectory.

Model and formalism

The battery is a two-level system with Hamiltonian $H_0 = \omega_0 \sigma^+\sigma^-$, charged by a resonant classical drive with effective Rabi coupling $\Omega$ over a finite switching window. The ergotropy for a two-level state with Bloch vector $\mathbf{r}(\tau)$ reduces to $\xi(\tau) = \frac{1}{2}[r(\tau) + r_3(\tau)]$ in scaled units. The detector follows a uniformly accelerated trajectory in $\mathrm{(A)dS}_4$, treated perturbatively ($\lambda \ll \Omega$) via the Davies weak-coupling master equation with Born-Markov and rotating-wave approximations, yielding a Kossakowski-Lindblad equation whose coefficients $A$, $B$, $C$ are built from the response function $\mathcal{G}(\Omega)$—the Fourier transform of the Wightman function—and its Hilbert transform.

The two spacetimes enter through distinct response functions:

Spacetime	Temperature	Response features
$\Omega$0	$\Omega$1	Pure thermal (Bose-Einstein) spectrum; reduces to Gibbons-Hawking temperature $\Omega$2 at $\Omega$3
$\Omega$4	$\Omega$5	Thermal only for supercritical trajectories $\Omega$6; boundary-condition-dependent correction term controlled by $\Omega$7 (Dirichlet, transparent, Neumann)

The KMS condition fixes the asymptotic Bloch vector ratio $\Omega$8, so the steady state depends only on temperature—a structural fact that drives the paper's central claim about global thermality.

Asymptotic ergotropy: a dimension-independent witness

In the long-time limit the scaled ergotropy approaches $\Omega$9. Three results follow. First, the steady ergotropy decreases monotonically with acceleration: stronger acceleration produces hotter thermal noise and more decoherence, degrading extractable work; lower temperature yields greater asymptotic storage. Second, because $\mathbf{r}(\tau)$0 depends on temperature alone through the KMS condition, it is independent of boundary conditions and of spacetime dimension—the authors present this as a unified witness of global thermality across both spacetimes. Third, at fixed acceleration the AdS ergotropy exceeds the dS value and decays more slowly, reflecting the different curvature dependence of the two temperatures. Notably, in AdS the threshold structure survives: nonzero asymptotic ergotropy requires $\mathbf{r}(\tau)$1, so the battery operationally reproduces the known absence of Unruh response on subcritical AdS trajectories. A caveat stated by the authors: without the external driving field ($\mathbf{r}(\tau)$2), the asymptotic ergotropy vanishes identically since the thermal equilibrium state satisfies $\mathbf{r}(\tau)$3; the probe therefore relies on coherent charging coexisting with vacuum-induced dissipation.

Transient dynamics: pathways to the same equilibrium

The time-dependent ergotropy,

$\mathbf{r}(\tau)$4

with $\mathbf{r}(\tau)$5, exhibits qualitatively different transient behavior in the two spacetimes even though all trajectories terminate at the same KMS-determined fixed point. In $\mathbf{r}(\tau)$6, low-acceleration evolution shows pronounced oscillations before relaxation, whereas large acceleration suppresses oscillations and produces fast thermal relaxation. In $\mathbf{r}(\tau)$7, the boundary conditions modulate the oscillation amplitude, ordered as Dirichlet ($\mathbf{r}(\tau)$8) strongest, transparent intermediate, Neumann weakest; Dirichlet conditions also enhance energy storage at low accelerations. These boundary-condition corrections diminish at larger accelerations, consistent with the asymptotic universality. At very short charging times the ergotropy is nearly insensitive to acceleration, since it is dominated by the external coherent field rather than vacuum fluctuations—an honest limitation on how early the probe can discriminate spacetimes.

Dimensionality effects in AdS

Extending to $\mathbf{r}(\tau)$9, the authors use the Jennings Wightman function along supercritical trajectories. For odd dimensions the second (curvature-induced) term of the response is analytically awkward, but the KMS-type relation $\xi(\tau) = \frac{1}{2}[r(\tau) + r_3(\tau)]$0 still certifies thermality, albeit with fermionic character—an instance of the statistics inversion known in odd-dimensional settings. An explicit analytic response function is derived for $\xi(\tau) = \frac{1}{2}[r(\tau) + r_3(\tau)]$1 with Dirichlet boundary conditions. Numerical comparison of $\xi(\tau) = \frac{1}{2}[r(\tau) + r_3(\tau)]$2 and $\xi(\tau) = \frac{1}{2}[r(\tau) + r_3(\tau)]$3 shows that higher dimension modestly amplifies the initial-stage ergotropy (the response rate grows with dimension) and accelerates the stabilization of oscillations, i.e., faster thermalization. Crucially, the asymptotic ergotropy remains identical to the four-dimensional expression for any dimension, reinforcing the interpretation that only the KMS temperature governs the steady state. The authors argue that the odd-dimensional statistics inversion does not destroy the thermal nature witnessed by the battery.

Limitations and open questions

The analysis rests on several assumptions that bound the strength of the conclusions. The weak-coupling, Born-Markov treatment neglects the Lamb shift and non-Markovian memory effects, which may matter for short-time transients where the discriminative power between spacetimes is claimed. The battery is pointlike, so no smearing or spatial structure is considered, and the scalar field is massless and conformally coupled. The claim that ergotropy witnesses "global" thermality is established only for stationary uniformly accelerated trajectories in maximally symmetric constant-curvature backgrounds; whether the same steady-state universality holds for non-stationary trajectories or non-conformal vacua remains open. The dimensionality study is numerical beyond $\xi(\tau) = \frac{1}{2}[r(\tau) + r_3(\tau)]$4 and restricted to Dirichlet conditions for the $\xi(\tau) = \frac{1}{2}[r(\tau) + r_3(\tau)]$5 analytic result, leaving the combined effect of odd dimensions and alternative boundary conditions unexplored. Finally, the proposal is theoretical: no experimental protocol or realistic noise model for implementing such a battery is provided, so the operational interpretation of "witness" remains at the level of principle.

Conclusion

The paper demonstrates that the ergotropy of an accelerated UDW battery provides a thermodynamic handle on Unruh-Hawking thermality in curved spacetimes. Its main result is a clean separation of scales: the asymptotic ergotropy is a universal, dimension- and boundary-independent function of the KMS temperature set by acceleration and curvature, while the transient dynamics encode local information about curvature sign, boundary conditions, and dimensionality. Within its idealized assumptions, the work offers a consistent operational framework in which quantum work extraction doubles as a diagnostic of vacuum structure in (A)dS backgrounds.