A Nonrecursive Lindblad Quantization of Dissipative Polynomial Dynamics

We present a direct and nonrecursive construction that maps an arbitrary planar polynomial dissipative flow $\dot{\alpha }=h(\alpha ,{\alpha }^{\ast })$ 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 $\mathrm{\mid }\alpha \mathrm{\mid }\sim S\to \mathrm{\infty }$, 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.