Egorov-Type Semiclassical Limits for Open Quantum Systems with a Bi-Lindblad Structure

TL;DR

This paper establishes a connection between classical bi-Hamiltonian systems and quantum open systems with Lindblad dynamics, introducing a class of contact-compatible Lindblad generators that preserve classical invariants and admit a semiclassical limit.

Contribution

It introduces a novel framework linking classical bi-Hamiltonian structures with quantum Lindblad evolutions, including explicit models demonstrating semiclassical limits and integrability.

Findings

Development of contact-compatible Lindblad generators preserving classical invariants

Construction of a semiclassical limit for quantum dissipative dynamics

Explicit example using an Euler-top-type Poisson-Lie pencil

Abstract

This paper develops a bridge between bi-Hamiltonian structures of Poisson-Lie type, contact Hamiltonian dynamics, and the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) formalism for quantum open systems. On the classical side, we consider bi-Hamiltonian systems defined by a Poisson pencil with non-trivial invariants. Using an exact symplectic realization, these invariants are lifted and projected onto a contact manifold, yielding a completely integrable contact Hamiltonian system in terms of dissipated quantities and a Jacobi-commutative algebra of observables. On the quantum side, we introduce a class of contact-compatible Lindblad generators: GKSL evolutions whose dissipative part preserves a commutative $C^\ast$-subalgebra generated by the quantizations of the classical dissipated quantities, and whose Hamiltonian part admits an Egorov-type semiclassical limit to the contact dynamics. This construction provides a mathematical mechanism compatible with the semiclassical limit for pure dephasing, compatible with integrability and contact dissipation. An explicit Euler-top-type Poisson-Lie pencil, inspired by deformed Euler top models, is developed as a fully worked-out example illustrating the resulting bi-Lindblad structure and its semiclassical behavior.