Nonequilibrium thermodynamics of quantum coherence beyond linear response

TL;DR

Nonequilibrium quantum thermodynamics of coherence is addressed by developing a dynamic-Bayesian-network framework that preserves quantum coherence and yields a quantum Crooks-type fluctuation relation and a quantum maximum-work theorem for both closed and open dynamics. The formalism introduces a coherence contribution $\mathcal{C}$ and a relative-entropy term $\mathcal{D}$, along with a fluctuation relation $\langle e^{\beta(w-\Delta F)-\Delta c-\Delta d}\rangle=1$ and a maximum-work bound $\beta W_{\max} = -\beta \Delta F - \Delta \mathcal{C} - \Delta \mathcal{D} = -\beta \Delta \mathcal{F}$, highlighting how initial coherence can drive work while coherence generated during evolution and decoherence can hinder it. The approach is illustrated on a driven qubit, revealing a nonequilibrium regime where maximum work extraction is enhanced by coherence for fast processes beyond linear response, yet strong thermalization suppresses it. These results provide concrete criteria and timescales for coherence-assisted energy conversion and offer guidance for reservoir-engineering strategies to realize quantum-enhanced nanomachines.

Abstract

Quantum thermodynamics allows for the interconversion of quantum coherence and mechanical work. Quantum coherence is thus a potential physical resource for quantum machines. However, formulating a general nonequilibrium thermodynamics of quantum coherence has turned out to be challenging. In particular, precise conditions under which coherence is beneficial to or, on the contrary, detrimental for work extraction from a system have remained elusive. We here develop a generic dynamic-Bayesian-network approach to the far-from-equilibrium thermodynamics of coherence. We concretely derive generalized fluctuation relations and a maximum-work theorem that fully account for quantum coherence at all times, for both closed and open dynamics. We obtain criteria for successful coherence-to-work conversion, and identify a nonequilibrium regime where maximum work extraction is increased by quantum coherence for fast processes beyond linear response.

Figures (3)

• Figure 1: Quantum coherence for a periodically driven qubit. Change of relative entropy of coherence, $\Delta \mathcal{C}$, for a two-level system in a rotating magnetic field as a function of driving frequency $\omega$ and driving amplitude $g$ (with $\beta=0.5$): $\Delta \mathcal{C}=0$ along the orange lines (in particular, in the adiabatic limit $\omega\rightarrow 0$). For small driving amplitude $g$, $\Delta \mathcal{C}<0$ close to resonance $\omega\simeq\omega_0$, enabling coherence-to-work conversion.
• Figure 2: Coherence-to-work conversion for a periodically driven qubit. a) Without initial coherence ($a=0$), work is consumed, $\beta W>0$, as quantum coherence is created, $\Delta \mathcal{C}>0$, during unitary evolution. b) With initial coherence ($a=0.3$), work is efficiently extracted from quantum coherence, $\beta W<0$ and $\Delta \mathcal{C}<0$. Maximum work production occurs at half the Rabi time, $\tau_W=\Omega/\pi$, where $\Omega$ is the Rabi frequency. This time lies beyond the linear response regime; insets show deviations from the quantum fluctuation-dissipation relation for work. c) For nonunitary dynamics ($\gamma \neq0$), coherence-to-work is hampered by decoherence, which reduces $\Delta \mathcal{C}$, and by nonequilibrium entropy production, which leads $\beta W$ to deviate from the variation, $\Delta \mathcal{C} + \mathcal{D}_t$, of coherence and athermality. Parameters are $\omega = 1$, $g = 0.005$, $\beta = 0.5$ and $\delta = \omega_0-\omega=- 0.005$.
• Figure 3: Influence of decoherence on work extraction. Coherence-to-work conversion, $\beta W<0$, is only effective when the work extraction time $\tau_W$ is much smaller than the decoherence timescale $\tau_D$, $\tau_W\ll \tau_D$. No work can be extracted for strong thermalization, $\tau_D\ll\tau_W$ (vertical lines indicate the decoherence time $\tau_D$). Same parameters as Fig. 2.