PDEs in Mixed Quantum-Classical Dynamics and the Koopmon Method

To alleviate the computational cost associated with fully quantum dynamics, a novel mixed quantum-classical (MQC) particle method – the Koopmon method – has recently been introduced [1]. Unlike conventional MQC models, which often suffer from inconsistencies such as violations of Heisenberg’s principle, this new approach resolves these issues by blending Koopman’s formulation of classical mechanics on Hilbert spaces with tools from symplectic geometry.

The resulting continuum model, which can be regarded as a correction to the Ehrenfest PDE, retains both a variational and a Hamiltonian structure, while its nonlinear character calls for suitable closure schemes. Exploiting the underlying action principle, we introduce a regularization procedure that enables a singular-solution ansatz, thereby defining the trajectories of computational particles – the so-called Koopmons.

Numerical experiments demonstrate the capability of the Koopmon method to reproduce nonadiabatic quantum-classical transitions and to capture Rashba-type spin dynamics in quantum nanowire systems. We conclude by proposing a novel strategy for a hybrid Ehrenfest–Koopman implementation that integrates the advantages of both approaches.

[1] Bauer, W.; Bergold, P.; Gay-Balmaz, F., and Tronci, C. Koopmon trajectories in nonadiabatic quantum-classical dynamics. SIAM Multiscale Model. Simul., 22(4):1365-1401 (2024)
[2] Gay-Balmaz, F.; Tronci, C. Evolution of hybrid quantum-classical wavefunctions. Phys. D 440, 133450 (2022)
[3] Gay-Balmaz, F.; Tronci, C. Koopman wavefunctions and classical states in hybrid quantum-classical dynamics. J. Geom. Mech. 14, n. 4, 559-596 (2022)