Contact manifolds and dissipation, classical and quantum

TL;DR

This paper generalizes contact Hamiltonian systems to describe dissipative dynamics in both classical and quantum systems, highlighting their emergence in Lagrangian mechanics and quantum state evolution.

Contribution

It introduces a new framework for modeling dissipative systems using contact geometry, extending previous approaches to include non-exact contact manifolds and quantum dynamics.

Findings

Unified geometric description of dissipative classical and quantum systems

Application to nonlinear evolutions on quantum pure states

Connection with Lagrangian mechanics in dissipative contexts

Abstract

Motivated by a geometric decomposition of the vector field associated with the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) equation for finite-level open quantum systems, we propose a generalization of the recently introduced contact Hamiltonian systems for the description of dissipative-like dynamical systems in the context of (non-necessarily exact) contact manifolds. In particular, we show how this class of dynamical systems naturally emerges in the context of Lagrangian Mechanics and in the case of nonlinear evolutions on the space of pure states of a finite-level quantum system.