Heisenberg dynamics of mixed quantum-classical systems

Abstract

We consider the dynamics of interacting quantum and classical systems in the Heisenberg representation. Unlike the usual construction in standard quantum mechanics, mixed quantum-classical systems involve the interplay of unitary operators acting on the quantum observables and the Lagrangian trajectories sweeping the classical degrees of freedom. This interplay reflects an intricate structure which is made particularly challenging by the backreaction excerpted on the classical trajectories by the quantum degrees of freedom. While the backreaction is underestimated in the common Ehrenfest model, more recent methodologies succeed in capturing this important effect by resorting to Koopman wavefunctions in classical mechanics. Luckily, both Ehrenfest and Koopman models enjoy a variational framework which is exploited here to unfold the geometric structure underlying quantum-classical coupling. A special role is played by the action of the diffeomorphic Lagrangian paths on a non-Abelian pure-gauge potential which comprises statistical correlations. After presenting the treatment in the simple case of Ehrenfest dynamics, we move on to the Koopman model and present the role of the backreaction terms therein. Finally, we compare both models in the context of pure-dephasing systems.

22 pages, 1 figure. Submitted as a conribution to the Focus Point "Mathematics and Physics at the Quantum-Classical Interface" in the European Physical Journal Plus

References in corpus (18)

• Koopman wavefunctions and classical-quantum correlation dynamics
• On Koopman-von Neumann Waves
• Mixing quantum and classical mechanics and uniqueness of Planck's constant
• Madelung transform and probability densities in hybrid classical-quantum dynamics
• Ehrenfest dynamics is purity non-preserving: a necessary ingredient for decoherence
• Coupling Quantum Matter and Gravity
• Geometry and symmetry of quantum and classical-quantum variational principles
• Evolution of hybrid quantum-classical wavefunctions
• Hybrid quantum-classical dynamics of pure-dephasing systems
• Euler-Poincaré approaches to nematodynamics
• Overcoming positivity violations for density matrices in surface hopping
• Effective nonlinear Ehrenfest hybrid quantum-classical dynamics
• Hybrid Geometrodynamics: A Hamiltonian description of classical gravity coupled to quantum matter
• Dynamics of mixed quantum-classical spin systems
• Lagrangian trajectories and closure models in mixed quantum-classical dynamics
• Complex fluid models of mixed quantum-classical dynamics
• Hybrid Koopman C*-formalism and the hybrid quantum-classical master equation
• Koopmon trajectories in nonadiabatic quantum-classical dynamics