Quantum Otto engines at relativistic energies

TL;DR

The work addresses how relativistic quantum dynamics influence nanoscale heat engines by analyzing an endoreversible quantum Otto cycle with a working medium described by a Dirac/Klein-Gordon oscillator. The authors derive a relativistic canonical ensemble with a factorized partition function $Z = Z^+ Z^-$ and show that positive/negative energy sectors act as independent components due to Dirac sea stability. They find that relativistic effects increase work output but reduce efficiency, yielding an efficiency at maximum power identical to the Curzon-Ahlborn form $\eta_{CA} = 1 - \sqrt{T_c/T_h}$ for both regimes, with $\eta_{\mathrm{rel}} = 1 - \sqrt{\kappa}$ and $\eta_{\mathrm{nonrel}} = 1 - \kappa$ in respective limits. The results highlight the role of the light-cone restrictions and the linear-in-frequency nature of the Dirac oscillator in shaping thermodynamic performance, and point to experimental platforms such as trapped ions, Dirac materials, and microwave resonators for realizing relativistic quantum engines and exploring relativistic shortcuts to adiabaticity.

Abstract

Relativistic quantum systems exhibit unique features not present at lower energies, such as the existence of both particles and antiparticles, and restrictions placed on the system dynamics due to the light cone. In order to understand what impact these relativistic phenomena have on the performance of quantum thermal machines we analyze a quantum Otto engine with a working medium of a relativistic particle in an oscillator potential evolving under Dirac or Klein-Gordon dynamics. We examine both the low-temperature, non-relativistic and high-temperature, relativistic limits of the dynamics and find that the relativistic engine operates with higher work output, but an effectively reduced compression ratio, leading to significantly smaller efficiency than its non-relativistic counterpart. Using the framework of endoreversible thermodynamics we determine the efficiency at maximum power of the relativistic engine, and find it to be equivalent to the Curzon-Ahlborn efficiency.

Figures (8)

• Figure 1: Illustration of the combinatorics of distributing energy quanta across $k$ different relativistic oscillators. The top (blue partitions) and bottom (red partitions) rows represent the positive and negative energy solutions of magnitude $\epsilon_i$, respectively. Note that here we have accounted for the "stability of the Dirac sea", which requires that the positive and negative energy states cannot be simultaneously occupied.
• Figure 2: Equilibrium (a) internal energy, (b) free energy, (c) entropy, and (d) heat capacity for the one-dimensional Dirac oscillator using numerical methods (blue, dashed) and the continuum approximation (red, solid). Parameters are set such that $\lambda = 1$.
• Figure 3: Energy frequency diagram of the endoreversible quantum Otto cycle.
• Figure 4: Compression ratio as a function of the bath temperature ratio for $T_A = 0.05$ (brown, long dashed), $T_A = 0.25$ (blue, dot-dashed), and $T_A = 1$ (green, short dashed). The high temperature limit of $T_A/T_B = \sqrt{\kappa}$ (black, lower solid) and low temperature limit of $T_A/T_B = \kappa$ (red, upper solid) are given for comparison.
• Figure 5: Efficiency as a function of the compression ratio for an endoreversible quantum engine with a relativistic oscillator working medium in the relativistic limit (green, dashed) and non-relativistic limit (red, solid).