Quantum Battery Could Reveal Hidden Heat of Accelerating Observers in Curved Spacetimes

Physicists have long sought ways to detect the strange thermal effects that arise when quantum mechanics meets curved spacetime. Now, a team of theorists has proposed an unusual witness for this hidden heat: a relativistic quantum battery. In a study published in The European Physical Journal C, Xiang Hao of Suzhou University of Science and Technology and colleagues, together with Yin-Zhong Wu, modeled a tiny two-level quantum system—an accelerated Unruh–DeWitt detector—coupled to a massless scalar field in de Sitter and anti-de Sitter spacetimes. Their central quantity is the ergotropy, the maximum amount of work that can be extracted from the battery by cyclic unitary operations. Remarkably, they found that the long-time value of this extractable work depends only on the observer’s acceleration and the curvature of spacetime, making it a clean, operational probe of what is known as Unruh–Hawking thermality.

The physics behind the proposal traces back to two landmark discoveries. In 1976, William Unruh showed that an observer accelerating through the vacuum of flat spacetime perceives that vacuum as a warm thermal bath, with a temperature proportional to the acceleration. A year earlier, Stephen Hawking had demonstrated that black holes radiate thermally. Shortly afterward, Gibbons and Hawking found that observers in an expanding de Sitter universe—our best mathematical description of a dark-energy-dominated cosmos—also perceive a thermal bath at a temperature set by the cosmological constant. These effects share a common signature: the correlation functions of the quantum field obey the Kubo–Martin–Schwinger (KMS) condition, the mathematical fingerprint of thermal equilibrium. Detecting them directly, however, is extraordinarily difficult because the temperatures involved are vanishingly small for any achievable acceleration.

The new work turns the problem around. Instead of measuring particles directly, the researchers treat the accelerated detector as a quantum battery that can be charged by an external classical driving field while simultaneously interacting with the fluctuating vacuum. A quantum battery stores energy in quantum states, and the ergotropy quantifies how much of that stored energy is genuinely extractable as useful work. For a two-level battery with transition frequency, the ergotropy is determined by the Bloch vector describing the battery’s quantum state: specifically, it equals half of the sum of the vector’s magnitude and its third component. As the battery evolves, vacuum fluctuations in the surrounding spacetime cause dissipation and decoherence, gradually reshaping the state and therefore the extractable work.

Using the open quantum system approach, the team derived a Kossakowski–Lindblad master equation in the weak-coupling limit, valid under the Born–Markov approximation. The coefficients of this equation are fixed by the response function of the detector—the Fourier transform of the field’s Wightman function along the accelerated trajectory. In de Sitter spacetime, this response takes a purely thermal form with a temperature that combines the Unruh contribution from acceleration and the Gibbons–Hawking contribution from curvature: the effective temperature is proportional to the square root of the acceleration squared plus the curvature scale squared. In anti-de Sitter spacetime, by contrast, thermality appears only when the acceleration exceeds the curvature scale—a supercritical condition—because the negative curvature prevents the formation of the relevant horizon below that threshold.

The most striking result concerns the asymptotic behavior. After a long charging time, the battery thermalizes with the field, and its ergotropy settles at a steady value equal to one half of the hyperbolic tangent of the transition frequency divided by twice the effective temperature. Because this expression depends only on the KMS temperature, it is completely independent of the spacetime dimension and, in anti-de Sitter, of the boundary conditions imposed on the quantum field. The hotter the effective bath—whether from higher acceleration or stronger curvature—the smaller the steady ergotropy. Crucially, the same formula unifies the two spacetimes: once the acceleration greatly exceeds the curvature scale, de Sitter and anti-de Sitter batteries converge to identical steady behavior, providing a single witness for Unruh–Hawking thermality across both geometries.

The transient dynamics, however, tell a richer story. In de Sitter spacetime, the ergotropy exhibits pronounced oscillations during its evolution at low accelerations, ringing before settling to its steady value, while at high accelerations it relaxes rapidly and smoothly to equilibrium. In anti-de Sitter spacetime, the picture changes dramatically with the boundary conditions applied to the scalar field at the spacetime boundary. The team examined three cases—Dirichlet, transparent, and Neumann—and found that the oscillatory behavior of the ergotropy is strongest under Dirichlet conditions, weaker for transparent boundaries, and weakest for Neumann boundaries. Notably, choosing the right boundary condition can improve the energy storage of the moving battery, and the boundary-induced corrections become less important as the acceleration grows.

Dimensionality adds another twist. Extending the calculation to six-dimensional anti-de Sitter spacetime, the researchers showed that vacuum fluctuations can modestly amplify the ergotropy during the initial charging stage, because the detector’s response rate increases with the number of spacetime dimensions. A short-time expansion of the ergotropy reveals that it grows quadratically at early times, with a coefficient proportional to the response function, which is larger in six dimensions than in four. Higher dimensions also accelerate thermalization, stabilizing the oscillations more quickly. Yet the asymptotic ergotropy remains the same regardless of dimension, again dictated solely by acceleration and curvature through the KMS condition. The team also notes that in odd-dimensional anti-de Sitter spacetimes, a curious inversion of quantum statistics occurs—bosonic fields behave as if fermionic—but this anomaly does not destroy the underlying thermal nature witnessed by the battery.

The sensitivity of the probe differs between the two geometries in an instructive way. Comparing the derivatives of the steady ergotropy with respect to acceleration, the researchers found that the anti-de Sitter battery responds more sharply to changes in acceleration than its de Sitter counterpart, and this heightened sensitivity persists regardless of boundary conditions. At very short charging times, by contrast, the acceleration has almost no effect, because the thermal response of the vacuum is still too weak and the extractable work is dominated by the external driving field. This separation of timescales suggests a practical strategy: short-time measurements characterize the charging protocol and the geometry-dependent corrections, while long-time measurements isolate the universal thermal fingerprint encoded in the KMS condition.

Beyond its conceptual appeal, the framework connects several active research frontiers: quantum thermodynamics, relativistic quantum information, and the physics of open quantum systems in curved spacetime. Previous studies had used uncertainty relations, quantum coherence, geometric phases, and Fisher information as probes of the Unruh effect in flat spacetime, and recent work had explored quantum batteries near black holes. The present study extends this program to the two maximally symmetric curved spacetimes of constant positive and negative curvature, offering an operational interpretation of thermality in terms of energy transfer. While an actual laboratory demonstration remains distant—the Unruh temperatures involved are minuscule—the model provides theorists with a concrete, work-based observable that distills the essence of horizon thermodynamics. In the researchers’ view, the relativistic quantum battery is not merely an energy storage device but a functional probe of vacuum fluctuations, opening a new window onto the thermal structure of spacetime itself.

Subject of Research: Quantum thermodynamics of relativistic quantum batteries as probes of Unruh–Hawking thermality in curved spacetimes

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

Article References: Hao, X., Gan, T.-F., Wu, C.-T., Ren, T.-X., Zhang, W.-W., & Wu, Y.-Z. (2026). Quantum thermodynamics of ergotopy for a relativistic battery as a witness to Unruh–Hawking thermality in curved (A)dS spacetimes. The European Physical Journal C, 86(10), Article 1137. https://doi.org/10.1140/epjc/s10052-026-16400-w

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16400-w

Keywords: quantum battery, ergotropy, Unruh effect, Hawking radiation, de Sitter spacetime, anti-de Sitter spacetime, Unruh–DeWitt detector, KMS condition, quantum thermodynamics, curved spacetime, open quantum systems, relativistic quantum information