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arXiv:2608.20132

SAKE: Spectral Autodiff Kernel Expansion for Geometric Liouvillian Transport. A Differential-Geometric Framework for Response Transport in Quantum Dynamical Systems

Eric R. Bittner, Carlos Silva-Acuna, Hao Li, Simon Paiva-Ortega

quant-phphysics.chem-phSubmitted Aug 20, 2026• Announced Aug 21, 2026• Revised Aug 21, 2026 (v2)
v1 · Aug 20, 2026v2 · Aug 21, 2026
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Abstract

We introduce the Spectral Autodiff Kernel Expansion (SAKE), a differentiable computational framework for transporting nonlinear spectroscopic response between neighboring quantum dynamical models. Rather than recomputing multidimensional spectra independently for each Hamiltonian or Liouvillian, SAKE constructs local transport expansions about a reference model by combining forward-mode automatic differentiation with Duhamel transport theory. Automatic differentiation generates first-, second-, and third-order derivatives of the parameter-dependent Liouvillian, which are assembled into a pathway transport operator that maps the nonlinear response of a reference model onto neighboring systems. The framework is validated for a four-level excitonic dimer possessing an $su(2)\times su(2)$ symmetry by comparing second- and third-order transported pathway operators with exact projected transport matrices obtained from direct calculations. The third-order expansion accurately reproduces the projected transport operator and its associated pathway mixing. Beyond providing an efficient computational strategy, the transport operator reveals how coherent and dissipative perturbations redistribute amplitude among double-sided Feynman pathways, exposing mechanistic information that is not directly apparent from the nonlinear spectrum. SAKE thereby establishes a differentiable computational framework for nonlinear spectroscopy that supports efficient local parameter exploration, sensitivity analysis, and future inverse-design applications.

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We establish an exact realization of the structural drift construction of the Negative-Entropy framework in a standard two-level quantum system. The abstract constraint operator J, its kernel L=\ker J, and the associated orthogonal projector \Pi_L are realized precisely by the spectral eigenspace structure of the two-level Hamiltonian, with the identification J_-=H-E_-I=2d\,\Pi_+. The projector-drift identity of the earlier framework reduces exactly to \dot\Pi_L=\dot\Pi_-, and the resulting operator chain J_-\to\Pi_-\to\dot\Pi_-\to K\to(g_{ij},F_{ij})\to\Gamma_J connects the abstract construction to quantum geometric data through the Kato parallel-transport generator, the quantum metric, and the Berry curvature. We further prove a non-equivalence result within the original K_3 network-flow model: at a fixed active-set region where the structural projection is locally constant, the structural drift strength vanishes identically, while the multiplicative dynamical feedback strength remains strictly positive. This shows that structural drift is not a necessary mechanism for dynamical feedback in the original K_3 construction, and no linear operator factorizes the feedback response through the directional structural-change operator. The paper makes no empirical claims about consciousness, and does not assert a unified theory. The exact realization is an operator-level identification within the two-level spectral geometry. It does not introduce a new quantum dynamical law, nor does it establish a physical equivalence between the quantum geometric response and the network-flow feedback strength. The paper establishes a mathematically verifiable bridge between the abstract structural framework and a standard physical system, together with a rigorous boundary on where the bridge does not extend. Note on notation. The constraint matrix of the K_3 network-flow model is denoted \mathsf{B}, while the scalar entry B of the dynamical matrix M_q retains the unadorned symbol B. Note on operator conventions. The framework is developed in finite-dimensional real or complex Hilbert spaces. The adjoint of a linear operator A is denoted A^\dagger; in the real case this coincides with the transpose A^\top. Projectors are orthogonal projectors with respect to the Hilbert-space inner product.

•Oct 11, 2026•0 citations
We introduce the Spectral Autodiff Kernel Expansion (SAKE), a differentiable computational framework for transporting nonlinear spectroscopic response between neighboring quantum dynamical models. Rather than recomputing multidimensional spectra independently for each Hamiltonian or Liouvillian, SAKE constructs local transport expansions about a reference model by combining forward-mode automatic differentiation with Duhamel transport theory. Automatic differentiation generates first-, second-, and third-order derivatives of the parameter-dependent Liouvillian, which are assembled into a pathway transport operator that maps the nonlinear response of a reference model onto neighboring systems. The framework is validated for a four-level excitonic dimer possessing an su(2)×su(2)su(2)\times su(2)su(2)×su(2) symmetry by comparing second- and third-order transported pathway operators with exact projected transport matrices obtained from direct calculations. The third-order expansion accurately reproduces the projected transport operator and its associated pathway mixing. Beyond providing an efficient computational strategy, the transport operator reveals how coherent and dissipative perturbations redistribute amplitude among double-sided Feynman pathways, exposing mechanistic information that is not directly apparent from the nonlinear spectrum. SAKE thereby establishes a differentiable computational framework for nonlinear spectroscopy that supports efficient local parameter exploration, sensitivity analysis, and future inverse-design applications.
Liouville pathways provide the conceptual foundation for interpreting multidimensional spectroscopies and are commonly represented as independent contributions to the nonlinear response. In open quantum systems, however, the effective reduced Liouvillian need not preserve this observational pathway decomposition. Environmental interactions can transport amplitude between pathway sectors during the free-evolution intervals, generating measurable signatures in the nonlinear spectroscopic response.} We develop a geometric framework in which pathway transport is governed by a Liouvillian connection, its associated curvature, and the resulting observational holonomy. The framework applies to open quantum systems in which the environment selects a pointer basis distinct from the observational basis used to construct the spectroscopic response. This basis incompatibility induces transport among Liouville pathways, generating characteristic spectral distortions and a nontrivial Liouvillian curvature. Using a Duhamel expansion of the Liouvillian propagator, we derive a reconstruction procedure that identifies the transport operators responsible for the observed redistribution of pathway weight, accurate throughout the full range of basis misalignment. This perspective reframes spectral features as determined not only by which pathways exist but by how amplitude is transported among them. Spectral distortions, peak shifts, and otherwise-forbidden pathway contributions become geometric signatures of a curved Liouville-space manifold rather than phenomenological broadening corrections, identifying pathway geometry as a complementary layer of organization in nonlinear spectroscopy.
We develop a geometry for transporting stationary-state response across the control space of an open quantum system. A physical model is represented by the ordered pair of its stationary state and dynamical generator. Embedding these pairs in a common ambient space induces a metric, a response one-form, and a closed two-form on the control manifold. The ambient space admits a canonical complex structure that exchanges state and generator directions, but the Liouvillian null-state condition restricts physical models to a submanifold that need not preserve this structure. For an amplitude-damped optical Bloch model, the metric and response two-form are compatible at the single point ω=0ω=0ω=0 and g/Γ=1/2g/Γ=1/\sqrt{2}g/Γ=1/2​; the physical manifold is non-Kähler elsewhere. The induced metric defines the Levi--Civita connection, geodesics, and parallel transport without requiring Kähler compatibility. We compute the connection by automatic differentiation through the stationary Liouvillian solve and recover the symbolic result to machine precision. A two-point calculation then shows that the resulting geodesic differs from linear interpolation in control space and follows a shorter path through the family of state--generator models. This geometry supplies the intrinsic derivative and transport structure needed to carry observable response, including multidimensional spectra, between admissible stationary models.
Nonlinear response functions, formulated as multipoint correlation functions or Volterra kernels, encode the dynamical and spectroscopic properties of physical systems and underpin a wide range of nonlinear transport and optical phenomena. However, their evaluation rapidly becomes prohibitive at high orders because of combinatorial (often factorial) scaling or severe numerical errors. Here, we establish a systematic and efficient framework to compute nonlinear response functions directly from real-time dynamics, without explicitly constructing multipoint correlators or relying on numerically unstable finite-difference methods for order-resolved extraction. Our approach is based on the Gateaux derivative with respect to the external field in function space, which yields a closed hierarchy of tangent equations of motion (TEOM). Propagating the TEOM alongside the original dynamics isolates each perturbative order with high accuracy, providing a term-by-term decomposition of physical contributions. The computational cost scales exponentially with response order in the fully general setting and reduces to polynomial complexity when all perturbation directions are identical; both regimes avoid the factorial scaling of explicit multipoint-correlator evaluations. We demonstrate the power of TEOM by computing frequency-resolved fifth-order response functions for a solid-state electron model and by obtaining nonlinear response functions up to the 49th order with controlled accuracy in a classical Duffing oscillator. We further show that our time-evolution formulation allows optical conductivities to be evaluated directly while remaining numerically stable even near zero frequency. TEOM can be incorporated seamlessly into existing real-time evolution methods, yielding a general framework for computing nonlinear response functions in quantum and classical dynamical systems.
We present a projection-based, stability-preserving methodology for computing time correlation functions in open quantum systems governed by generalized quantum master equations with non-Markovian effects. Building upon the memory kernel coupling theory framework, our approach transforms the memory kernel hierarchy into a system of coupled linear differential equations through Mori-Zwanzig projection, followed by spectral projection onto stable eigenmodes to ensure numerical stability. By systematically eliminating unstable modes while preserving the physically relevant dynamics, our method guaranties long-time convergence without introducing artificial damping or ad hoc modifications. The theoretical framework maintains mathematical rigor through orthogonal projection operators and spectral decomposition. Benchmark calculations on the spin-boson model show excellent agreement with exact hierarchical equations of motion results while achieving significant computational efficiency. This approach provides a versatile and reliable framework for simulating non-Markovian dynamics in complex systems.
A central challenge in nonlinear spectroscopy is the order-selective readout of weak higher-order responses that spectrally overlap with dominant lower-order signals. This bottleneck is particularly severe in two-dimensional (2D) spectroscopy, where extending conventional phase-cycling schemes to higher orders rapidly increases measurement and analysis complexity. Here we introduce a computation-assisted strategy that combines rotating-frame acquisition with a frame-shift tracking algorithm to separate signals by their frame-dependent spectral shifts. In a rubidium vapor experiment, we use this approach to isolate a 7th-order nonlinear contribution from coexisting 3rd-order components, enabling direct access to higher-order quantum-coherence dynamics without sacrificing operation at comparatively high pulse intensities. The method is broadly compatible with multidimensional spectroscopy platforms and provides a practical route to probing many-body and collective ultrafast dynamics beyond third order.
We establish an exact realization of the structural drift construction of the Negative-Entropy framework in a standard two-level quantum system. The abstract constraint operator J, its kernel L=\ker J, and the associated orthogonal projector \Pi_L are realized precisely by the spectral eigenspace structure of the two-level Hamiltonian, with the identification J_-=H-E_-I=2d\,\Pi_+. The projector-drift identity of the earlier framework reduces exactly to \dot\Pi_L=\dot\Pi_-, and the resulting operator chain J_-\to\Pi_-\to\dot\Pi_-\to K\to(g_{ij},F_{ij})\to\Gamma_J connects the abstract construction to quantum geometric data through the Kato parallel-transport generator, the quantum metric, and the Berry curvature. We further prove a non-equivalence result within the original K_3 network-flow model: at a fixed active-set region where the structural projection is locally constant, the structural drift strength vanishes identically, while the multiplicative dynamical feedback strength remains strictly positive. This shows that structural drift is not a necessary mechanism for dynamical feedback in the original K_3 construction, and no linear operator factorizes the feedback response through the directional structural-change operator. The paper makes no empirical claims about consciousness, and does not assert a unified theory. The exact realization is an operator-level identification within the two-level spectral geometry. It does not introduce a new quantum dynamical law, nor does it establish a physical equivalence between the quantum geometric response and the network-flow feedback strength. The paper establishes a mathematically verifiable bridge between the abstract structural framework and a standard physical system, together with a rigorous boundary on where the bridge does not extend. Note on notation. The constraint matrix of the K_3 network-flow model is denoted \mathsf{B}, while the scalar entry B of the dynamical matrix M_q retains the unadorned symbol B. Note on operator conventions. The framework is developed in finite-dimensional real or complex Hilbert spaces. The adjoint of a linear operator A is denoted A^\dagger; in the real case this coincides with the transpose A^\top. Projectors are orthogonal projectors with respect to the Hilbert-space inner product.