Geometric Thermodynamics in Open Quantum Systems: Coherence, Curvature, and Work

Abstract

We formulate a geometric framework for quasistatic thermodynamics in open quantum systems by parameterizing the dynamics on a control manifold. In the quasistatic limit, the system follows a manifold of stationary states, and the work performed over a cycle is given by the flux of a curvature two-form, $W \sim \int Ω$, defined by the parametric response of the stationary state, establishing an open-system analog of classical thermodynamic area laws. For thermal stationary states, the curvature is isotropic and depends only on the instantaneous energy scale, yielding a population-driven geometry in which environmental parameters reshape how work is distributed across the control manifold. Beyond this limit, nonequilibrium stationary states can retain coherence in the energy representation; using a fixed-basis Lindblad model, we show that this coherence reshapes the curvature, making it anisotropic and sign-changing, so that work depends sensitively on the placement and orientation of the cycle. Quantum coherence therefore partitions the control manifold into regions of opposite curvature, producing geometric cancellation of work and allowing the net work over a cycle to be reduced or reversed despite dissipative dynamics. Thermodynamic work thus emerges as a curvature flux whose structure is set by thermodynamic response in classical systems and by basis misalignment between the Hamiltonian eigenbasis and the environment-selected pointer basis in open quantum systems.

Figures (4)

• Figure 1: Geometric structure of quasistatic thermodynamics for a driven qubit on the control manifold $(\omega,g)$. (a) Vector field of generalized forces $(\langle \sigma_z \rangle, \langle \sigma_x \rangle)$ derived from the stationary state, showing relaxation toward the instantaneous ground-state direction. Solid and dashed red loops indicate two circular thermodynamic cycles of different size. (b,c) Curvature $\Omega_W = d\mathcal{A}_W$ governing geometric work for two inverse temperatures $\beta$. The numerical values of the cycle work, obtained from $W_{\mathrm{cyc}}=\iint_\Sigma \Omega_W$, are annotated for each loop. At higher temperature the curvature is broader and more weakly localized, whereas at lower temperature it is concentrated near $\epsilon=\sqrt{\omega^2+g^2}\approx 0$. The figure illustrates directly that the work is determined by the flux of curvature through the enclosed region and depends on both the size of the cycle and the temperature-dependent geometry of the stationary manifold.
• Figure 2: Normalized cycle work $W_{\mathrm{cyc}}(\epsilon)/W_\infty(\beta)$ for circular protocols defined by $\omega=\epsilon\cos\theta$ and $g=\epsilon\sin\theta$. The curves show how the enclosed curvature—and hence the work—accumulates as the cycle probes increasing energy scales. Larger $\beta$ leads to rapid saturation due to curvature localization near $\epsilon\approx 0$, while smaller $\beta$ distributes the response over a broader range.
• Figure 3: Normalized cycle work $W_{\mathrm{cyc}}(\phi)$ for a temperature-modulated protocol. The sinusoidal dependence reflects the alignment between the temperature modulation and regions of large curvature along the cycle, demonstrating that the work can be tuned through the bath trajectory even for a fixed geometric loop.
• Figure 4: Comparison of the thermal and coherent curvature fields on the control manifold $(\omega,g)$. (a) Thermal curvature $\Omega_W^{\rm th}$, which is isotropic and strictly positive. (b) Coherent curvature $\Omega_W^{\rm coh}$ for the fixed-basis Lindblad model in Eq. (\ref{['eq:coherent_lindblad']}). The curvature is anisotropic and changes sign, reflecting the mismatch between the Hamiltonian eigenbasis and the environment-selected pointer basis. The solid red loop encloses positive and negative regions nearly symmetrically, leading to cancellation of the net work, while the dashed loop samples predominantly one sign and produces finite work.