Semiclassical Ehrenfest paths

Lucien Lemmens

Xiao-Kan Guo

We study the semiclassical Ehrenfest trajectories in open quantum systems. We first give a self-contained derivation of the Fokker--Planck equation that governs the time evolution of the mixing measure for a Gaussian mixture. Then, we embed the generalized Ehrenfest theorem recently obtained for open quantum systems into this phase-space picture to study the time evolution of the expectations of observables with respect to the Gaussian mixture. We show how the coherent and irreversible contributions are separated in the frictionless harmonic regime relevant to the semiclassical approximation. Our work provides a transparent phase-space interpretation of the emergence of classical trajectories in open quantum dynamics.

Erich Joos

Parmeet Nijjar, Joanna Jankowska, Oleg V. Prezhdo

We present a semiclassical approach for nonadiabatic molecular dynamics based on the Ehrenfest method with corrections for decoherence and detailed balance. Decoherence is described via a coherence penalty functional that drives dynamics away from regions in Hilbert space characterized by large values of coherences. Detailed balance is incorporated by modification of the off-diagonal matrix elements with a quantum correction factor used in semiclassical approximations to quantum time-correlation functions. Both decoherence and detailed balance corrections introduce nonlinear terms to the Schrödinger equation. At the same time, the simplicity of fully deterministic dynamics and a single trajectory for each initial condition is preserved. In contrast, surface hopping is stochastic and requires averaging over multiple realization of the stochastic process for each initial condition. The Ehrenfest-decoherence-detailed-balance (Ehrenfest-DDB) method is adapted to the classical path approximation and ab initio time-dependent density functional theory and applied to an experimentally studied nanoscale system consisting of a fluorophore molecule and an scanning tunneling microscopy tip and undergoing current-induced charge injection, cooling, and recombination. Ehrenfest-DDB produces time scales that are similar to those obtained with decoherence induced surface hopping, which is a popular nonadiabatic molecular dynamics technique applied to condensed matter. At long times, Ehrenfest-DDB dynamics slows down considerably because the detailed balance correction makes off-diagonal elements go to zero on approach to Boltzmann equilibrium. The Ehrenfest-DDB technique provides efficient means to study quantum dynamics in large systems.