Quantum thermodynamics in nonequilibrium

TL;DR

This work addresses nonequilibrium quantum thermodynamics in the presence of quantum coherence by integrating the quantum resource theory of coherence with thermodynamics to derive a novel entropy-balance relation. The authors identify the thermodynamic entropy with the energy entropy ${\cal S}(t)$, distinct from the von Neumann entropy $S(t)$ away from equilibrium, and decompose entropy production into a heat-flux term $\Phi_Q$ and a coherence-induced term $\Phi_C$. They establish dynamic definitions of temperature $T(t)$ and free energy $F(t)$, showing the first and second laws hold far from equilibrium, and demonstrate that equilibrium thermodynamics emerges dynamically in the weak-coupling limit using an exactly solvable open quantum system (a single bosonic mode with a Fano-Anderson-type reservoir). The results provide a coherent, operational foundation for nonequilibrium quantum thermodynamics and clarify the thermodynamic role of quantum coherence, with potential applications to driven dissipative quantum devices.

Abstract

Understanding thermodynamics far from equilibrium at the quantum scale remains a fundamental challenge, particularly in the presence of quantum coherence. Here we develop a first-principles framework for nonequilibrium quantum thermodynamics by integrating quantum resource theory of coherence with thermodynamic laws. We derive a previously unexplored entropy balance relation that explicitly separates entropy flux due to heat exchange from entropy production arising from the loss of quantum coherence. This formulation identifies the appropriate thermodynamic entropy in nonequilibrium quantum processes as the energy entropy associated with energy measurements, demonstrating that the von Neumann entropy does not, in general, represent thermodynamic entropy away from equilibrium. Within this framework, dynamical temperature, free energy, work, and heat are consistently defined, and both the first and second laws are shown to hold far from equilibrium. Applying the theory to an exactly solvable open quantum system, we reveal how equilibrium thermodynamics emerges dynamically in the weak-coupling limit. Our results establish a unified and operational foundation for nonequilibrium quantum thermodynamics and clarify the fundamental thermodynamic role of quantum coherence.

Figures (3)

• Figure 1: Entropy balance at far from equilibrium: (a) thermodynamic entropy ${\cal S}(t)$, von Neumann entropy $S(t)$, and their derivatives $d{\cal S}(t)/dt$ and $dS(t)/dt$ are shown as a function of time (b) total entropy rate $\Sigma(t)$, entropy production due to coherence loss $\Phi_{C}(t)$, and entropy flux rate due to heat exchange $\Phi_{Q}(t)$ are shown as a function of time much before the system reaches equilibrium. The system is initially prepared in a coherent state $\vert \alpha_0 \rangle$, with coherent state amplitude $\alpha_0=1$. The initial temperature of the reservoir is fixed at $kT_0=20 \hbar \omega_0$. The cutoff frequency $\omega_c=10\omega_0$, and the system-reservoir coupling strength $\eta=0.1 \eta_c$
• Figure 2: Quantum thermodynamic quantities under nonequilibrium dynamics: Time evolution of (a) the internal energy $U(t)$, (b) the thermodynamic entropy ${\cal S}(t)$, (c) the dynamical temperature $T(t)$, and (d) the free energy $F(t)$, evaluated from Eqs. (\ref{['entroc']})–(\ref{['fengy2']}). The system is initially prepared in a coherent state $\vert \alpha_0 \rangle$, with $\alpha_0=1$. In each panel, the three curves correspond to different initial temperatures of the thermal reservoir: $kT_0=15 \hbar \omega_0$ (blue), $20 \hbar \omega_0$ (green), and $25 \hbar \omega_0$ (red). The system-reservoir coupling strength is fixed at $\eta=0.1 \eta_c$, with a reservoir cutoff frequency $\omega_c=10\omega_0$.
• Figure 3: Nonequilibrium dynamics of internal energy, work, and heat exchange: Time evolution of (a) the rate of change of internal energy $dU(t)/dt$, (b) the work rate $dW(t)/dt$, and (c) the heat flow rate $dQ(t)/dt$, evaluated from Eqs. (\ref{['rngyc']}), (\ref{['workcoher']}), and (\ref{['heatcoher']}), respectively. The system is initially prepared in a coherent state $\vert \alpha_0 \rangle$ with $\alpha_0=1$. We consider three different initial temperatures of the thermal reservoir: $kT_0=15 \hbar \omega_0$ (blue), $20 \hbar \omega_0$ (green), and $25 \hbar \omega_0$ (red). The work-rate dynamics exhibits negligible sensitivity to reservoir temperature $T_0$. The system-reservoir coupling strength is fixed at $\eta=0.1 \eta_c$, with a reservoir cutoff frequency $\omega_c=10\omega_0$.