Beyond the Carnot limit: work extraction via an entropy battery

TL;DR

The paper addresses whether work can be extracted from a hot reservoir more efficiently than Carnot by transferring entropy across multiple conserved quantities. It introduces the entropy battery, a device comprising energy and spin reservoirs coupled by a unitary Raman transition, and analyzes how entropy exchange can yield energetic work beyond the classical Carnot limit at maximum power, without relying on induced coherence. By treating spin as an independent thermodynamic degree of freedom with its own temperature, heat capacity, and statistics (including Bose–Einstein behavior for bosons), the authors derive expressions for spin labor, spin therm, and the total work, showing that increasing spin-state degeneracy boosts energetic efficiency beyond Carnot, while accounting for the necessary coherence cost via spin labor. The study further develops a maximum-entropy framework to compare distinguishable, bosonic, and fermionic statistics for entropy storage, arguing that interactions are essential for entropy transfer and that bosons best realize the required entropy capacity. Collectively, these results establish a foundation for entropy-based quantum devices—potentially enabling high-density energy storage and more efficient heat management in quantum technologies—while highlighting that generalized thermodynamic limits remain bounded when all resource costs are included.

Abstract

Heat is a physical manifestation of entropy, where the removal of entropy from a thermal energy reservoir permits the conversion of heat into work. This entropy transfer is facilitated by the cold thermal energy reservoir in typical heat engines. Recent developments in quantum heat engines that operate between thermal energy and spin angular momentum reservoirs show that it is possible to transfer entropy out of energy and into a different conserved quantity. The implications of this type of entropy transfer have not been fully explored, especially on the work extractable using an ensemble with multiple conserved quantities. Using the aforementioned heat engines, we show that such an ensemble will transform heat into work beyond the Carnot efficiency limit while operating at maximum power. This result is obtained without induced quantum coherence, a technique commonly used in the field of quantum heat engines to achieve the same outcome. Without loss of generality, we also show that thermal spin reservoirs behave as thermodynamic baths with well-defined temperatures, heat capacities, and fluctuation-dissipation relations. Finally, our analysis of entropy capacity suggests that particle indistinguishability is necessary for inter-particle interactions, and for entropy to transfer between canonical ensembles. These results establish a foundation for entropy-based quantum devices that extract work from a hot thermal energy reservoir more efficiently than possible with a cold thermal energy reservoir. These devices also act as high energy-density batteries and efficient heat storage systems. Our results will have implications for quantum heat engines, spinor condenstates, spintronics, and quantum batteries.

Figures (8)

• Figure 1: Overview of (a) a conventional Carnot heat engine between hot and cold baths, and (b) the entropy battery. In (b), the equivalent conserved quantity 'heat' baths with general conserved quantity $\braket{A_m}$ are represented by black cubes aligned vertically. Energy couples to spin angular momentum within the entropy battery via a unitary 'heat engine', realized experimentally by coherent Raman transitions writeQHEoperating2018andre2025QSHEmanakil2025GSHE. Spin labor $\mathcal{L}$ within the entropy battery enables the conversion of heat energy into work. There aren't any equivalent unitary operations for the other conserved quantities that we are aware of. Entropy from the environmental reservoir is transferred into the battery’s internal reservoirs (energy, spin, etc.), enhancing work extraction until their entropy capacities are saturated or equilibrium is reached. As these other conserved quantity 'heat' baths reside within the battery’s particles, the entropy battery achieves a higher per particle work extraction density than possible with conventional heat engines.
• Figure 2: Schematic of the internal structure of the entropy battery. The device consists of two thermal, energy- and spin-coupled canonical ensembles: the working fluid and the reservoir, containing $N_\text{wf}$ and $N_\text{res}$ particles respectively. Each ensemble has two mutually independent conserved quantities: the external energy modes of the quantum harmonic oscillators, labeled in the number basis $\ket{n}$, and internal spin states, labeled using the arrows $\ket{\uparrow}$. The particle distributions across these states are illustrated with circles and horizontal lines, where circle size represents the population at the respective temperatures. Since $N_\text{res} \gg N_\text{wf}$, the two ensembles equilibrate rapidly in both spin and energy, so that $T_\text{res} \approx T_\text{wf}$ and $\tau_\text{res} \approx \tau_\text{wf}$. The reservoir also couples thermally to an external hot environment as in Fig.\ref{['fig: Extensive entropy storage']}, represented here by a canonical coupling in the bottom left. Particles in the reservoir have spin $S$, which sets the entropy capacity of the battery according to eqs.\ref{['eq:Distinguishable entropy capacity']}–\ref{['eq:Fermionic entropy capacity']}. In this schematic, incoherent properties (entropy) are represented with a spotted pattern, while coherent properties (information) are indicated by solid colors, illustrating the flow of entropy and information.
• Figure 3: Efficiency of energetic work extraction $\eta_\text{energy}$ for varying number of spin states per particle calculated using the entropy and heat equations \ref{['eq: Boson analytic entropy']} and \ref{['eq:Bosonic heat']}. In these simulations we used $d_\text{env}=d_{\text{batt},E}=400$ without loss in accuracy. Also shown are 4 sets of initial temperatures for the ensembles energy and spin baths, with $\tau_\text{batt}=\tau_C=\tau_s$, while the environments initial temperature was set to $\tau_H=0.6$ for all cases. These unitless temperatures were chosen as it would correspond to a temperature range between room temperature, 300 K, and 600 K with a Debye cut off frequency of $21$ THz, which is a typical maximum frequency for metals rogers2005einstein. These initial temperatures also correspond to the spin reservoir having a polarization fidelity of roughly 99.97%. 300-600 K is a typical temperature range for coal and nuclear power plants, and concentrated solar farms. As the number of spin states increases, the energy efficiency approaches 100%, with all initial conditions reaching approximately this limit at 5 states. This is well above the Carnot efficiency limit $\eta_\text{Carnot}=1-\tau_C/\tau_H$ which is shown by the red bars when $d=0$, with values between 16%-50% for the varying initial ensemble temperatures.
• Figure 4: An example entropy distribution over spin temperature for the different particle statistics, distinguishable, bosonic and fermionic with 7 states. In (a) and (b) we show example distributions with $N=4$ and $N=6$ particles respectively, illustrating how distinguishable and bosonic maximum entropies increase with $N$ without bound, while the fermionic maximum entropy decreases as saturation is approached $N=d$ where $\mathbb{S}_\text{Fermi}=0$ for all temperature. Maximum entropy, or the systems entropy capacity is given by the entropy at infinite temperature $\tau=\infty$, which we have labeled for the different ensemble statistics.
• Figure 5: Mapping of spin temperature $\tau$ to polarization, eq.\ref{['eq: S=1/2 tau to polarisation']}, for different ensembles of distinguishable particles with per particle spins $S$. This plot shows up to $S=200$, but trends towards a steepening slope around $\gamma=0$, where in the limit of $S\rightarrow\infty$ one would expect a step function centred at $\gamma=0$