Ergotropy-Based Quantum Thermodynamics

TL;DR

The paper develops an ergotropy-based quantum thermodynamics framework that directly links heat to entropy through changes in the passive state, yielding a heat notion invariant under passive transformations. It defines the infinitesimal heat and work via passive-state changes and ergotropy variation, derives a consistent nonequilibrium temperature, and decomposes entropy production into passive and non-passive components, recovering a generalized Clausius form for thermal maps. Applied to a single qubit, the approach demonstrates a positive, well-defined out-of-equilibrium temperature and a tight relation between heat and entropy across generalized amplitude-damping and phase-damping dynamics, including non-Markovian scenarios. The framework provides a robust non-Markovianity witness based on heat and temperature and suggests avenues for quantum thermal machines and energy-extraction resource theories in open quantum systems.

Abstract

We introduce an ergotropy-based formulation of quantum thermodynamics, which provides a strong connection between average heat and von Neumann entropy. By adopting this formulation, we can reinterpret the infinitesimal average heat in terms of the infinitesimal change of the passive state associated with the density operator behind the quantum dynamics. Such as entropy, this leads to a heat concept that is invariant under passive state transformations. As an application, the average heat can be used as a general non-Markovianity measure for unital maps. Moreover, a positive-semidefinite temperature naturally emerges in an out-of-equilibrium ergotropy-based scenario. Concerning the infinitesimal average work, it arises as the infinitesimal variation of ergotropy, as well as an extra passive work contribution in the case of a time-dependent Hamiltonian. As illustrations, we consider the thermodynamics of a single-qubit open system in the cases of generalized amplitude-damping and phase-damping channels.

Figures (3)

• Figure 1: (Color online) Dimensionless temperatures $k_BT_e/\omega_0$, $k_BT/\omega_0$, $k_B\mathbbm{T}/\omega_0$, and $k_B\mathcal{T}/\omega_0$ as functions of the dimensionless time $\omega_0t$ for a qubit under a Markovian GAD process with $\vec{r}_{-}(0)=(0.45,\,0.00,\,-0.80)$. Inset: Same functions for the initial state $\vec{r}_{+}(0)=(0.45,\,0.00,\,0.80)$. We have used $\gamma_0=1$.
• Figure 2: (Color online) Dimensionless heats $Q/\omega_0$, $Q_{op}/\omega_0$, $\mathbbm{Q}/\omega_0$, $\mathcal{Q}/\omega_0$, and entropy variation $\Delta S/k_B$ as functions of the dimensionless time $\omega t$ for a qubit under a Markovian PD process with $\vec{r}(0)=(0.5,\,0.7,\, 0.0)$ and $\gamma=\omega$.
• Figure 3: (Color online) Dimensionless heat-based non-Markovianity quantifiers $N_Q/\omega_0$, $N_{\mathbbm{Q}}/\omega_0$, and $N_{\mathcal{Q}}/\omega_0$ as functions of the ohmicity parameter $s$. Inset: Dimensionless ergotropy-based temperature $T$ as a function of $\omega_c t$ for $|\vec{r}(0)|=0.8$.