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arXiv:2603.24557

Geometric Curvature Governs Work in Open Quantum Steady States

Eric R. Bittner

quant-phcond-mat.stat-mechSubmitted Mar 25, 2026• Announced Mar 26, 2026• 0 citations• Revised Mar 26, 2026 (v2)
v1 · Mar 25, 2026v2 · Mar 26, 2026
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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.

AI Summary

The paper asks whether quasistatic work in driven, dissipative quantum steady states can be described geometrically in control-parameter space. It defines the work one-form $\mathcal A=\sum_i\mathrm{Tr}(\rho_{\rm ss}\partial_iH)\,d\lambda_i$ and its curvature $\mathcal F=d\mathcal A$. Stokes’ theorem then expresses cyclic work as the curvature flux, $W_{\rm cyc}=\oint\mathcal A=\int_\Sigma\mathcal F$. The explicit example is a driven two-level system with $H=(\Delta/2)\sigma_z+\Omega\sigma_x$, Lindblad relaxation at rate $\gamma$, and pure dephasing at rate $\gamma_\phi$. Analytical steady-state Bloch components yield a closed expression for $\mathcal F_{\Delta\Omega}$. The reported curvature is nonuniform and concentrated near resonance. Representative cycle calculations illustrate that loops enclosing comparable areas can produce different work because they sample different curvature distributions; opposite-sign contributions can cancel, and reversing a loop reverses the work’s sign. In the strong-dephasing limit, both the dissipative-basis coherence and cyclic work vanish. This is interpreted as evidence that coherence enables the response, while curvature determines its magnitude. The treatment assumes smooth instantaneous steady states and quasistatic control, and its explicit results concern this Markovian two-level model. No experimental demonstration or finite-speed analysis is presented. Many-body systems, correlated environments, and symmetry- or topology-related Liouvillian structures are proposed as extensions. The discussion also identifies effectively one-dimensional Hamiltonian dependence as a possible mechanism for vanishing curvature, illustrated by the SSH Bloch Hamiltonian at $k=\pi$.

Likely largely AI-generatedOverclaimsUnclear noveltyMissing comparisonPossible error

Central claim may not hold: The claimed general necessity of steady-state coherence is false: a diagonal steady state with parameter-dependent populations can produce nonzero work curvature without any coherence. The paper’s explicit formulas give large-dephasing scalings x_ss∝Γ₂⁻², y_ss∝Γ₂⁻¹ and F_ΔΩ∝Γ₂⁻¹ for generic fixed nonzero controls, contradicting its stated curvature scaling Γ₂⁻². The strong-dephasing limit removes dissipative-basis coherence but generally leaves instantaneous-energy-basis coherence, undermining the paper’s interchangeable use of these notions. Antisymmetry of cyclic work under orientation reversal does not establish thermodynamic reversibility or absence of entropy production in a driven nonequilibrium steady state. Cancellation of curvature flux for loop C does not imply that the work one-form is exact or that only an exact component was sampled.

Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.

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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.
The paper asks whether quasistatic work in driven, dissipative quantum steady states can be described geometrically in control-parameter space. It defines the work one-form A=∑iTr(ρss∂iH) dλi\mathcal A=\sum_i\mathrm{Tr}(\rho_{\rm ss}\partial_iH)\,d\lambda_iA=∑i​Tr(ρss​∂i​H)dλi​ and its curvature F=dA\mathcal F=d\mathcal AF=dA. Stokes’ theorem then expresses cyclic work as the curvature flux, Wcyc=∮A=∫ΣFW_{\rm cyc}=\oint\mathcal A=\int_\Sigma\mathcal FWcyc​=∮A=∫Σ​F.
The explicit example is a driven two-level system with H=(Δ/2)σz+ΩσxH=(\Delta/2)\sigma_z+\Omega\sigma_xH=(Δ/2)σz​+Ωσx​, Lindblad relaxation at rate γ\gammaγ, and pure dephasing at rate γϕ\gamma_\phiγϕ​. Analytical steady-state Bloch components yield a closed expression for FΔΩ\mathcal F_{\Delta\Omega}FΔΩ​. The reported curvature is nonuniform and concentrated near resonance. Representative cycle calculations illustrate that loops enclosing comparable areas can produce different work because they sample different curvature distributions; opposite-sign contributions can cancel, and reversing a loop reverses the work’s sign. In the strong-dephasing limit, both the dissipative-basis coherence and cyclic work vanish. This is interpreted as evidence that coherence enables the response, while curvature determines its magnitude.
The treatment assumes smooth instantaneous steady states and quasistatic control, and its explicit results concern this Markovian two-level model. No experimental demonstration or finite-speed analysis is presented. Many-body systems, correlated environments, and symmetry- or topology-related Liouvillian structures are proposed as extensions. The discussion also identifies effectively one-dimensional Hamiltonian dependence as a possible mechanism for vanishing curvature, illustrated by the SSH Bloch Hamiltonian at k=πk=\pik=π.
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.
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.
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.
We extend Nielsen's geometric approach for quantum complexity from closed to open quantum systems, whose dynamics is governed by Lindbladian evolution. In this framework, complexity is defined through an optimal-control problem on the space of mixed states, with a cost assigned to both unitary and non-unitary generators. We show that the resulting geometric structure differs fundamentally from the Riemannian geometry that emerges in the case of unitary evolution. In the open-system setting, the natural geometry is typically sub-Finslerian. Dissipation makes the geodesics non-reversible, while the admissible tangent directions are restricted by the physically allowed controls. We analyze several physically motivated examples, including a single qubit subject to depolarizing and amplitude-damping channels, as well as the damped harmonic oscillator. We show that, similarly to the unitary case, varying the penalty factors in the cost functional modifies the geometric properties through changes in the flag curvature, the Finslerian analog of sectional curvature. Our results provide a geometric framework for quantifying the abstract notion of complexity in dissipative quantum systems, with potential connections to experimentally realizable setups.
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.