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arXiv:2603.22452 · The Journal of Chemical Physics

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

Eric R. Bittner

quant-phcond-mat.stat-mechSubmitted Mar 23, 2026• Announced Mar 25, 2026• 1 citation• Revised Apr 29, 2026 (v2)
v1 · Mar 23, 2026v2 · Apr 29, 2026
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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. \erbedit{For thermal stationary states at fixed temperature, the curvature vanishes, reflecting the integrability of the work one-form.} 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.

Journal reference
J. Chem. Phys 164, 194116 (2026)
DOI
10.1063/5.0335651

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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∼∫ΩW \sim \int ΩW∼∫Ω, defined by the parametric response of the stationary state, establishing an open-system analog of classical thermodynamic area laws. \erbedit{For thermal stationary states at fixed temperature, the curvature vanishes, reflecting the integrability of the work one-form.} 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.
Classical thermodynamics admits a geometric formulation in which work is associated with areas enclosed by cycles in state space. Whether an analogous structure persists in driven, dissipative quantum systems remains an open question. Here we show that quasistatic work in open quantum steady states is governed by an emergent geometric curvature in control-parameter space arising from steady-state coherence. For a driven dissipative two-level system, we construct a work one-form whose curvature determines the work produced in cyclic processes. The work vanishes under strong dephasing, identifying coherence as a necessary condition for nontrivial geometry. However, its magnitude is set not by the coherence itself but by the spatial structure of the curvature: cycles enclosing comparable areas produce different work depending on their location in parameter space. Reversing the cycle orientation reverses the sign of the work, confirming its geometric origin. These results establish a geometric framework for open quantum thermodynamics and identify curvature as the organizing principle of thermodynamic response, with direct implications for driven light--matter systems in cavity quantum electrodynamics.
We develop a response-geometric framework for open quantum systems by decomposing the stationary-state response tensor into symmetric and antisymmetric sectors. The symmetric sector defines a metric-like response tensor governing local susceptibility, while the antisymmetric sector defines a curvature two-form associated with nonreciprocal response and geometric work. We show that the work curvature is precisely the antisymmetric component of the response tensor, establishing a fluctuation--response relation that extends the geometric structure of equilibrium thermodynamics to nonequilibrium steady states, and reveals a response geometry with both metric and symplectic sectors. In equilibrium, the antisymmetric sector vanishes by reciprocity, recovering the familiar metric geometry of thermodynamic response. Open quantum systems admit a broader structure in which reciprocal and nonreciprocal responses coexist on the same control manifold. We illustrate this with a driven dissipative qubit under pure dephasing, where finite curvature emerges from the misalignment between the Hamiltonian eigenbasis and the pointer basis selected by the environment, without requiring strong driving or engineered reservoirs. Comparison with the Bures metric shows that response geometry and information geometry characterize distinct properties of the stationary-state manifold: the former governs susceptibility and geometric work, the latter quantifies state distinguishability. Geometric work thus emerges as a measurable signature of nonreciprocal response in open quantum systems.
Complementarity relations constrain the distribution of coherence, predictability, and openness in quantum systems. Here we show that, in open quantum systems, these local constraints acquire a geometric interpretation through quasistatic transport. For a driven dissipative qubit, the complementarity variables define cylindrical coordinates on the Bloch sphere, while openness appears geometrically as a radial deficit associated with reduction from a larger Hilbert space. Quasistatic driving induces a work connection on the resulting steady-state manifold whose curvature determines the cyclic response. Hamiltonian-aligned dissipation produces an exact work connection and vanishing cyclic work, whereas fixed pointer-basis dissipation generates non-integrable transport, finite curvature, and holonomic response. The resulting curvature admits a phase-resolved representation on the triality manifold and develops perturbatively with pointer--Hamiltonian mismatch. In the weak-mismatch limit, the curvature is governed by a competition between coherence-preserving and pure-dephasing channels, producing symmetry-related positive- and negative-curvature sectors. These results establish a direct connection between complementarity, dissipation, and geometric thermodynamic response, and show that cyclic quasistatic work provides an operational probe of nonequilibrium quantum geometry.
In this work, we present a geometrical formulation of quantum thermodynamics based on contact geometry and principal fiber bundles. The quantum thermodynamic state space is modeled as a contact manifold, with equilibrium Gibbs states forming a Legendrian submanifold that encodes the fundamental thermodynamic relations. A principal fiber bundle over the manifold of density operators distinguishes the quantum state structure from thermodynamic labels: its fibers represent non-equilibrium configurations, and their unique intersections with the equilibrium submanifold enforce thermodynamic consistency. Quasistatic processes correspond to minimizing geodesics under the Bures-Wasserstein metric, leading to minimal dissipation, while the divergence of geodesic length toward rank-deficient states geometrically derives the unattainability aspect of the third law. Non-equilibrium extensions, formulated through pseudo-Riemannian metrics and connections on the principal bundle, introduce curvature-induced holonomy that quantifies a geometric source of irreversibility in cyclic processes. In this framework, the thermodynamic laws in the quantum regime emerge naturally as geometric consequences.
A central open question in quantum thermodynamics is the identification of thermodynamic entropy away from equilibrium. Although thermodynamic entropy coincides with the von Neumann entropy for a Gibbs state at equilibrium, the thermodynamic role of quantum coherence generated during nonequilibrium evolution remains unclear. Here, we show that von Neumann entropy does not, in general, represent the thermodynamic entropy governing heat exchange in coherent nonequilibrium quantum systems. By integrating the resource theory of quantum coherence with thermodynamic principles, we develop a general framework for nonequilibrium quantum thermodynamics. We identify thermodynamic entropy with the energy entropy derived from energy-measurement statistics and introduce a dynamical temperature extending the equilibrium concept to nonequilibrium states. Consequently, we obtain an exact entropy-balance relation separating the entropy flux associated with heat exchange from the coherence-loss contribution to entropy production, establishing an operational connection between quantum coherence, heat flow, and total entropy change. Unlike approaches relying on equilibrium reference states or stationary Gibbs ensembles, our framework uses instantaneous dynamical quantities and remains valid throughout transient nonequilibrium evolution. It provides consistent dynamical definitions of temperature, work, heat, and free energy. Applying the formalism to an exactly solvable open quantum system, we show that the first and second laws hold throughout transient evolution and that equilibrium thermodynamics emerges dynamically in the weak-coupling long-time limit. Our results establish a unified framework for coherent nonequilibrium quantum thermodynamics and suggest opportunities for controlling heat dissipation through coherence engineering in quantum thermal technologies.
The study of non-equilibrium quantum dynamics receives attentions recently. However, the nature and effects of nonequilibirumness such as detailed-balance breaking and the relationship to the underlying intrinsic geometry is still unclear. In this study, we uncover that a gauge field will be induced by the nonequilibriumness in the coherence representation and furthermore its internal geometrical curvature is directly related to the degree of detailed balance breaking. The nonequilibriumness of the quantum system induces an intrinsic geometric curvature which can enhance the quantum coherence, leading to the possibility of the space time origin for the non-local quantum correlations or the possibility of curved space time emergence from nonequilibrium quantum dynamics. We also uncovered that the internal curvature of the gauge field provides a bridge to connect the generalized quantum fluctuation dissipation theorem to the fluctuation theorem and time irreversibility of the quantum dynamics. The quantum time irreversibility is due to the path dependent factor along any particular path in an internal curved space, which is analogous to the Wilson lines (or Wilson loops) in Abelian gauge theory. We also found that the steady state quantum coherence disappears when the non-trivial internal curvature vanishes for the quantum system coupled with environments. When the curvature is relatively small indicating weak detailed balance breaking, the coherence increases as curvature increases. The internal curvature can provide a general and direct quantitative measure of the detail-balance breaking for any quantum/classical non-equilibrium systems, even without knowing the underlying steady state distribution or the steady state flux. Using an example of two harmonic oscillators coupled to two environments with different temperatures, we explicitly show the dependence of the internal curvature and quantum coherence.