A Nonrecursive Lindblad Quantization of Dissipative Polynomial Dynamics
Abstract
We present a direct and nonrecursive construction that maps an arbitrary planar polynomial dissipative flow α̇=h(α,α*) 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 |α|∼ S→∞, 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.
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BibTeX
@misc{li2026nonrecursive,
title = {{A Nonrecursive Lindblad Quantization of Dissipative Polynomial Dynamics}},
author = {Tingfei Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15632},
eprint = {2609.15632},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15632},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Nonrecursive Lindblad Quantization of Dissipative Polynomial Dynamics
AU - Tingfei Li
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15632
UR - https://arxiv.org/abs/2609.15632
AB - We present a direct and nonrecursive construction that maps an arbitrary planar polynomial dissipative flow α̇=h(α,α^*) 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 |α|∼ S→∞, 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.
ER -