Universal efficiency boost in prethermal quantum heat engines at negative temperature

TL;DR

This work unveils a universal, temperature-dependent efficiency boost for prethermal quantum heat engines operating in many-body systems with conserved quantities. By deriving adiabatic flow equations for generalized Gibbs ensembles and applying hydrodynamic projections, the authors prove that infinitesimal cycles favor prethermalization at negative temperatures but favor thermalization at positive temperatures. They corroborate these findings with finite-cycle analyses in integrable Ising and XXZ spin chains using Thermodynamic Bethe Ansatz and Generalized Hydrodynamics. The results are shown to hold beyond integrability and are argued to be observable with current quantum simulators, offering a practical path to exploiting prethermal phases for enhanced engine performance. Overall, the paper provides a rigorous, broadly applicable framework linking conservation laws, prethermal dynamics, and thermodynamic efficiency in quantum heat engines.

Abstract

Heat engines near the adiabatic limit typically assume a working medium at thermal equilibrium. However, quantum many-body systems often showcase conservation laws that hinder thermalization, leading to prethermalization in exotic stationary phases. This work explores whether prethermalization enhances or reduces engine efficiency. We investigate Otto cycles in quantum systems with varying numbers of conserved quantities. We find that additional conservation laws reduce efficiency at positive temperatures, but enhance it in regimes of negative temperatures. Our findings stem from general thermodynamic inequalities for infinitesimal cycles, and we provide evidence for integrable models undergoing finite cycles using the theoretical framework of Generalized Hydrodynamics. The relevance of our results for quantum simulators is also discussed.

Figures (6)

• Figure 1: Thermal vs prethermal Otto cycles. The thermodynamic ensemble (thermal plane) describing the adiabatic strokes of a thermalizing working medium is described by a few conserved charges, including the energy, and the control parameter ${\chi}$. Prethermal working matter is characterized by a larger number of charges, exploring prethermal phases.
• Figure 2: Thermal vs Prethermal infinitesimal Otto cycles in integrable models. We show the relative efficiency of thermal-prethermal infinitesimal Otto cycles in the Ising \ref{['eq_Ising']} and antiferromagnetic $J=-1$ XXZ \ref{['eq_XXZ']} chains with average magnetization $\langle \sigma^z\rangle=0.45$. We use as tunable parameters the magnetization $h$ and anisotropy $\Delta$, respectively. To plot efficiencies independent of the cycle size, we focus on skewed cycles where the absorbed heat is much larger than the work, $\mathcal{W} \ll \mathcal{Q}_\text{abs}$, or equivalently $|\delta\chi|\ll|\delta\beta|$. In this regime, the efficiency scales as $\eta\sim \delta \chi$ and $\eta^{\text{pth}}/\eta^{\text{th}} - 1\sim \delta \chi/\delta \beta$, and the data is rescaled to factor out the explicit dependence on the size of the infinitesimal stroke. Explicit formulas are obtained from Eq. \ref{['eq_Den']} and reported in SI suppmat. Notice that the regions of large relative efficiency are related to regions of vanishing thermal efficiency.
• Figure 3: Finite Otto cycles in integrable models. On the horizontal and vertical axes, we show the control parameter and the energy difference from the ground state (normalized with respect to the system size), respectively. In each panel, both cycles are represented and the values of the efficiencies are reported. Panels (a,b): Ising chain with cold(hot) reservoirs at temperatures $\beta^{-1}_\text{C}$($\beta^{-1}_\text{H}$) respectively. Specifically, in (a) we consider negative temperatures $(\beta^{-1}_\text{C},\beta^{-1}_\text{H})=(-0.70,-0.69)$ and in (b) positive temperature $(\beta^{-1}_\text{C},\beta^{-1}_\text{H})=(0.30,0.48)$. Panels (c,d): analog cases for the XXZ chain \ref{['eq_XXZ']} with $J=-1$ and fixed magnetization $\langle\sigma^z \rangle$. The temperatures of the baths in (c) and (d) are $(\beta^{-1}_\text{C},\beta^{-1}_\text{H})=(-0.175,-0.150)$ and $(\beta^{-1}_\text{C},\beta^{-1}_\text{H})=(0.5,2.0)$. These examples show that the general conclusions for infinitesimal cycles remain valid also for finite operations. A quantitative measure of the GGE's departure from canonical equilibrium and further cases are reported in SI suppmat.
• Figure S1: Comparison between the momentum distributions in the Ising model at the end of finite strokes and relative distance of GGEs from equlibrium. Panel (a-d): Ising model's momentum distributions $\rho(\lambda)$ at the end of the strokes in the cycles in Fig. 3 of the main text. In each panel, three momentum distributions are plotted: $\rho_\text{GGE}$ is the one unitary evolved, equal to the initial thermal state due to the adiabatic theorem. The final state along the thermal stroke is $\rho_\text{th}$, while $\overline{\rho_\text{th}}$ is thermal with the same energy as the prethermal one. The latter is useful both for quantifying non-ergodicity and comparing the two cycles. Panels a) and b) refer to strokes in the negative temperature cycle, as one can see since the more energetic modes are excited, while panels c) and d) refer to the positive temperature cycle. In this case, despite the thermal momentum distributions are not very different, an appreciable energy difference appears (see Fig. 3. Panel (e,f): evolution along the stroke of the GGE's distance from equilibrium, measured as the relative distance in the $L_2$ norm between $\rho_\text{GGE}$ and $\overline{\rho}_\text{th}$. For the explicit formula, see Eq. \ref{['eq_GGEdist']}. The orange and the black lines refer to the two different strokes of the prethermal cycle. The maximum relative distance goes from the $8\%$ of the strokes at negative temperature, up to $50\%$ in the first stroke at positive temperature, meaning the system is considerably driven out of equilibrium.
• Figure S2: Thermal vs Prethermal infinitesimal Otto cycle in the XXZ chain. Relative efficiency of thermal-prethermal infinitesimal skewed Otto cycles for the easy-axis antiferromagnetic $J=-1$ XXZ chain from Eq. \ref{['eq_releffXXZ']}. The plot is at a small fixed average magnetization $\langle\sigma^z\rangle=0.05$ to enhance the string formation at small positive temperatures. As for negative temperatures, regions of large enhancement correspond to vanishing thermal efficiency.