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

Leonardo Colombo, Asier López-Gordón

TL;DR

This work fuses bi-Hamiltonian Poisson–Lie geometry with contact integrability and GKSL open quantum dynamics to build quantum evolutions whose semiclassical limit reproduces the dissipative contact flows. By lifting Poisson–Lie systems to exact symplectic realizations and projecting to Liouville-transverse hypersurfaces, it identifies dissipated quantities that form Jacobi-commutative algebras, quantizes them to invariant commutative C*-subalgebras, and designs contact-compatible Lindblad generators with an Egorov-type limit. The authors introduce bi-Lindblad pencils—convex combinations of GKSL generators sharing the same quantum integrals—mirroring bi-Hamiltonian pencils, and demonstrate the framework with an explicit Euler-top-type PN pencil that yields pure dephasing aligned with the classical invariants. The results offer a principled approach to open quantum systems where dissipation preserves integrable structure, yielding decoherence into a pointer sector defined by classical invariants and enabling semiclassical consistency with contact dynamics. The explicit Euler-top example serves as a concrete, low-dimensional testbed for the interplay among integrability, dissipation, and decoherence in the semiclassical regime.

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.

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

TL;DR

This work fuses bi-Hamiltonian Poisson–Lie geometry with contact integrability and GKSL open quantum dynamics to build quantum evolutions whose semiclassical limit reproduces the dissipative contact flows. By lifting Poisson–Lie systems to exact symplectic realizations and projecting to Liouville-transverse hypersurfaces, it identifies dissipated quantities that form Jacobi-commutative algebras, quantizes them to invariant commutative C*-subalgebras, and designs contact-compatible Lindblad generators with an Egorov-type limit. The authors introduce bi-Lindblad pencils—convex combinations of GKSL generators sharing the same quantum integrals—mirroring bi-Hamiltonian pencils, and demonstrate the framework with an explicit Euler-top-type PN pencil that yields pure dephasing aligned with the classical invariants. The results offer a principled approach to open quantum systems where dissipation preserves integrable structure, yielding decoherence into a pointer sector defined by classical invariants and enabling semiclassical consistency with contact dynamics. The explicit Euler-top example serves as a concrete, low-dimensional testbed for the interplay among integrability, dissipation, and decoherence in the semiclassical regime.

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 -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.
Paper Structure (22 sections, 11 theorems, 255 equations, 1 figure)

This paper contains 22 sections, 11 theorems, 255 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Bi-Hamiltonian geometry and contact integrability
  3. Bi-Hamiltonian structures
  4. Exact symplectic manifolds and homogeneity
  5. Contact Hamiltonian systems and integrability
  6. Symplectization and the contact--homogeneous correspondence
  7. Bi-Hamiltonian Poisson--Lie systems and contact realizations
  8. Bi-Hamiltonian structures on Poisson--Lie groups
  9. Exact symplectic realizations and homogeneous lifts
  10. Contact realizations from homogeneous lifts
  11. Example: a Poisson--Nijenhuis bi-Hamiltonian system with two nontrivial contact integrals
  12. Lindblad dynamics compatible with contact geometry
  13. GKSL generators and Heisenberg picture
  14. Invariant commutative subalgebras and constants of motion
  15. Contact-compatible Lindblad generators
  16. ...and 7 more sections

Key Result

Proposition 2.8

Let $(C,\alpha,h)$ be a contact Hamiltonian system and $(C^\Sigma,\theta,H^\Sigma)$ its symplectization as above. Then:

Figures (1)

  • Figure 1: From a bi-Hamiltonian Poisson--Lie system on $(G,\pi_0,\pi_1)$ to a contact integrable system $(C,\alpha,h)$ via an exact, homogeneous symplectic realization $(M,\theta,H)$ and a Liouville-transverse hypersurface $C$.

Theorems & Definitions (58)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8: Contact--homogeneous correspondence
  • Remark 2.9
  • Remark 2.10
  • ...and 48 more