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arXiv:2606.22517

Geometric Response in Open Quantum Systems: Coherence, Curvature, and Susceptibility

Eric R. Bittner, Carlos Silva-Acuña

quant-phSubmitted Jun 21, 2026• Announced Jun 23, 2026• 0 citations
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Abstract

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.

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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.
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.
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.
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 theory of adiabatic response for open systems governed by Lindblad evolutions. The theory determines the dependence of the response coefficients on the dephasing rates and allows for residual dissipation even when the ground state is protected by a spectral gap. We give quantum response a geometric interpretation in terms of Hilbert space projections: For a two level system and, more generally, for systems with suitable functional form of the dephasing, the dissipative and non-dissipative parts of the response are linked to a metric and to a symplectic form. The metric is the Fubini-Study metric and the symplectic form is the adiabatic curvature. When the metric and symplectic structures are compatible the non-dissipative part of the inverse matrix of response coefficients turns out to be immune to dephasing. We give three examples of physical systems whose quantum states induce compatible metric and symplectic structures on control space: The qubit, coherent states and a model of the integer quantum Hall effect.
Quantum geometry characterizes the variation of wavefunctions in momentum space through their overlaps and relative phases, providing a general framework for understanding many transport and optical properties. It is generally formulated in terms of interband matrix elements, which, entering the response functions, allow obtaining experimental access to the quantum geometric tensor. Recently, it has been emphasized that quantum geometry can also be interpreted in terms of quantum dipole fluctuations in the ground state driven by interband mixing. Here, we extend this picture to include contributions from many-body collective fluctuations, in which propagators and response vertices are dressed dynamically by the interaction with collective modes. Focusing on the Berry curvature, we show that contributions from collective fluctuations can be experimentally distinguished from bare band-geometric contributions, via specific antisymmetric channels in inelastic scattering spectra. We further identify the non-commutative properties of transverse quantum fluctuations as well as non-local-time interactions as the generators of this dynamical curvature in the susceptibility response.
We develop a theory of adiabatic response for open systems governed by Lindblad evolutions. The theory determines the dependence of the response coefficients on the dephasing rates and allows for residual dissipation even when the ground state is protected by a spectral gap . We give the quantum response a geometric interpretation in terms of Hilbert space projections: for a two-level system and, more generally, for systems with a suitable functional form of the dephasing, the dissipative and non-dissipative parts of the response are linked to a metric and to a symplectic form. The metric is the Fubini–Study metric and the symplectic form is the adiabatic curvature. When the metric and symplectic structures are compatible , the non-dissipative part of the inverse matrix of response coefficients turns out to be immune to dephasing . We give three examples of physical systems whose quantum states induce compatible metric and symplectic structures on control space: qubit, coherent states and a model of the integer quantum Hall effect.