Evolution of hybrid quantum-classical wavefunctions

Abstract

A gauge-invariant wave equation for the dynamics of hybrid quantum-classical systems is formulated by combining the variational setting of Lagrangian paths in continuum theories with Koopman wavefunctions in classical mechanics. We identify gauge transformations with unobservable phase factors in the classical phase-space and we introduce gauge invariance in the variational principle underlying a hybrid wave equation previously proposed by the authors. While the original construction ensures a positive-definite quantum density matrix, the present model also guarantees the same property for the classical Liouville density. After a suitable wavefunction factorization, gauge invariance is achieved by resorting to the classical Lagrangian paths made available by the Madelung transform of Koopman wavefunctions. Due to the appearance of a phase-space analogue of the Berry connection, the new hybrid wave equation is highly nonlinear and it is proposed here as a platform for further developments in quantum-classical dynamics. Indeed, the associated model is Hamiltonian and appears to be the first to ensure a series of consistency properties beyond positivity of quantum and classical densities. For example, the model possesses a quantum-classical Poincaré integral invariant and its special cases include both the mean-field model and the Ehrenfest model from chemical physics.

New version. 35 pages in total

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Cited by in corpus (12)

• Efficient quantum amplitude encoding of polynomial functions
• Hybrid quantum-classical dynamics of pure-dephasing systems
• The wave operator representation of quantum and classical dynamics
• 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
• Koopmon trajectories in nonadiabatic quantum-classical dynamics
• Hybrid quantum-classical systems: statistics, entropy, microcanonical ensemble and its connection to the canonical ensemble
• Entropy functionals and equilibrium states in mixed quantum-classical dynamics
• Madelung transform and variational asymptotics in Born-Oppenheimer molecular dynamics
• Lévy-Leblond Equation and Eisenhart-Duval lift in Koopman-von Neumann Mechanics
• Heisenberg dynamics of mixed quantum-classical systems