Geometric thermodynamics in open quantum systems: Coherence, curvature, and work

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 ∼∫Ω, defined by the parametric response of the stationary state, establishing an open-system analog of classical thermodynamic area laws. For thermal stationary states at a 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.