A Nonrecursive Lindblad Quantization of Dissipative Polynomial Dynamics

arXiv:2609.15632v1 Announce Type: new Abstract: We present a direct and nonrecursive construction that maps an arbitrary planar polynomial dissipative flow $\dot{\alpha}=h(\alpha,\alpha^*)$ to an open quantum system in Gorini--Kossakowski--Sudarshan--Lindblad form. Given the homogeneous components of the classical drift, the corresponding Hamiltonian and collapse-operator blocks are obtained algebraically and independently at each degree. No recursive cancellation of lower-order terms is required: in the large-amplitude limit $|\alpha|\sim S\to\infty$, ordering-generated lower-degree terms are parametrically suppressed, while mean-field closure yields the prescribed leading $O(S)$ drift for semiclassically localized states. This provides a simple and systematic route from dissipative classical dynamics to explicit open-quantum-system realizations, without claiming a unique microscopic quantization of the classical flow. We demonstrate the construction for stable fixed points, a Hopf bifurcation, and a bistable-ring flow. The corresponding Liouvillian spectra recover the classical relaxation exponents, the radial Floquet exponent and neutral phase direction of a limit cycle, and the separation between local relaxation and inter-ring switching in the bistable case. These examples show that the construction can be used not only to analyze a given quantum model, but also to design open quantum systems with prescribed classical dynamical structures in the semiclassical limit.

Abstract:We present a direct and nonrecursive construction that maps an arbitrary planar polynomial dissipative flow $\dot{\alpha}=h(\alpha,\alpha^*)$ to an open quantum system in Gorini--Kossakowski--Sudarshan--Lindblad form. Given the homogeneous components of the classical drift, the corresponding Hamiltonian and collapse-operator blocks are obtained algebraically and independently at each degree. No recursive cancellation of lower-order terms is required: in the large-amplitude limit $|\alpha|\sim S\to\infty$, ordering-generated lower-degree terms are parametrically suppressed, while mean-field closure yields the prescribed leading $O(S)$ drift for semiclassically localized states. This provides a simple and systematic route from dissipative classical dynamics to explicit open-quantum-system realizations, without claiming a unique microscopic quantization of the classical flow. We demonstrate the construction for stable fixed points, a Hopf bifurcation, and a bistable-ring flow. The corresponding Liouvillian spectra recover the classical relaxation exponents, the radial Floquet exponent and neutral phase direction of a limit cycle, and the separation between local relaxation and inter-ring switching in the bistable case. These examples show that the construction can be used not only to analyze a given quantum model, but also to design open quantum systems with prescribed classical dynamical structures in the semiclassical limit.

15 pages. This is a preliminary note; comments and corrections are welcome

Quantum Physics (quant-ph); Chaotic Dynamics (nlin.CD)

arXiv:2609.15632 [quant-ph]

(or arXiv:2609.15632v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.15632 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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A Nonrecursive Lindblad Quantization of Dissipative Polynomial Dynamics

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