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

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.

💡 Research Summary

This paper establishes a rigorous bridge between three sophisticated frameworks: bi‑Hamiltonian Poisson‑Lie geometry, homogeneous symplectic realizations leading to contact dynamics, and the Gorini‑Kossakowski‑Sudarshan‑Lindblad (GKSL) description of open quantum systems. Starting from a Poisson‑Lie group (G) equipped with two compatible multiplicative Poisson tensors (\pi_{0}) and (\pi_{1}), the authors construct a Poisson pencil (\pi_{\lambda}=\pi_{1}-\lambda\pi_{0}). The associated bi‑Hamiltonian vector field (X) admits a recursion operator and generates an infinite hierarchy of commuting Hamiltonians, guaranteeing Liouville integrability on symplectic leaves.

To capture dissipation in a geometrically canonical way, the paper requires an exact, homogeneous symplectic realization ((M,\theta)) of ((G,\pi_{0})). The Liouville vector field (\Delta) satisfies (L_{\Delta}\theta=-\theta), and a Hamiltonian (H) homogeneous of degree one ((L_{\Delta}H=H)) lifts the bi‑Hamiltonian flow to ((M,\omega=-d\theta)). Selecting a hypersurface (C\subset M) transverse to (\Delta) yields a contact manifold ((C,\alpha=\theta|{C})) with contact Hamiltonian (h=H|{C}). The contact bracket ({\cdot,\cdot}{\alpha}) is isomorphic to the symplectic Poisson bracket, so conserved quantities on (M) project to “dissipated quantities’’ (I{k}) on (C) satisfying ({I_{k},h}_{\alpha}=0).

Quantization is performed by a map (Q_{\hbar}:C^{\infty}(M)\to B(\mathcal H)) assigning to each classical observable a bounded operator on a finite‑dimensional Hilbert space (\mathcal H). In the semiclassical limit (\hbar\to0), commutators reproduce the underlying Poisson brackets, and after restriction to (C) they reproduce the contact brackets. The authors then introduce a class of GKSL generators \