Geometric Curvature Governs Work in Open Quantum Steady States

Abstract

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.

Figures (3)

• Figure 1: Curvature $\mathcal{F}_{\Delta\Omega}$ in the $(\Delta,\Omega)$ plane. Two representative cycles are shown: loop A in a weak-curvature region and loop B in a high-curvature region. The curvature is strongly localized near resonance.
• Figure 2: Cycle work $W_{\rm cyc}$ as a function of dephasing for loops A, B, and C. Loop B produces significantly larger work due to its location in a region of larger curvature. Loop C spans regions of opposite-sign curvature and yields $W_{\rm cyc}=0$ due to cancellation of the net curvature flux. All responses decay to zero with increasing dephasing, reflecting the suppression of steady-state coherence and the associated flattening of the curvature landscape.
• Figure 3: Orientation dependence of the cycle work. Reversing the direction of traversal reverses the sign of the work, confirming its geometric origin.