A Nonrecursive Lindblad Quantization of Dissipative Polynomial Dynamics

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.