Electronic decoherence along a single nuclear trajectory

TL;DR

This work develops a single-trajectory, non-unitary electronic dynamics framework based on exact factorization (EF) to describe decoherence without tracing out the environment. By representing the full electron-nuclear wave function as $\Psi(r,R,t)=\chi(R,t)\phi(r|t,R)$ and taking the nuclei to follow a single classical path, the electronic subsystem evolves with a non-Hermitian, norm-preserving equation driven by non-adiabatic couplings, yielding trajectory-local decoherence. Through a perturbative expansion in the electron-nuclear mass ratio $\mu$, the authors show that non-adiabatic correlations induce non-unitarity in the trajectory-bound electronic evolution, and quantify coherence as $\Gamma(t)=\int d\boldsymbol{R}\,\chi_{1}^{*}(\boldsymbol{R},t)\chi_{0}(\boldsymbol{R},t)$ with EF giving $\Gamma(t)\approx C_{1}^{*}(t,\boldsymbol{R}_{t}^{c})C_{0}(t,\boldsymbol{R}_{t}^{c})$. Comparing EF with Ehrenfest dynamics across an avoided-crossing model reveals that single-trajectory EF captures decoherence trends that Ehrenfest fails to reproduce, highlighting a trajectory-native mechanism for decoherence. The results suggest broad applicability to other composite systems and potential relevance for solids, where single-trajectory, non-unitary electronic dynamics can encode decoherence without ensemble averaging.

Abstract

We describe a novel approach to subsystem decoherence without the usual tracing-out of the environment. The subsystem of focus is described entirely by a pure state evolving non-unitarily along a single classical trajectory of its environment. The approach is deduced from the exact factorization framework for arbitrary systems of electrons and nuclei. The non-unitarity of the electronic dynamics arises exclusively fromnon-adiabatic correlations between electrons and nuclei. We demonstrate that the approach correctly describes the coherence gain and the subsequent decoherence for the example of a nuclear trajectory passing through an avoided crossing, the prototypical case where single-trajectory Ehrenfest dynamics fails to produce decoherence.

Figures (3)

• Figure 1: Distinct regimes exhibited by the avoided-crossing system used here. Upper panel: the regime of weak non-adiabatic transition (regime I). The inter-BO-surface population transfer is almost negligible (red-solid/black dashed for upper/lower-surface population). The blue dashdot line in the upper right plot indicates unity population for eye guide. Lower panel: strong non-adiabatic transitions with moderate transfer (regime IIA) and nearly 50$\%$ transfer (regime IIB). Here we demonstrate these regimes by setting the model parameters to $M=20$, $\Delta=1.0$, $P^{c}_{0}=1.04$, $R^{c}_{0}=-0.1$ for Regime I and $M=15$,$\Delta=3.0$, $P^{c}_{0}=15$, $R^{c}_{0}=-1$ Regime IIA while Regime IIB uses $M=5$ with other parameters being the same as Regime IIA.
• Figure 2: Coherence magnitude $\left\vert\Gamma(t) \right\vert$ are shown by solid green lines on the first row. The second/third row takes snapshots of nuclear densities at the time indicated by the black/blue arrow on the time axis of the first row. The upper-surface densities $\left\vert\chi_{1}(t) \right\vert^{2}$ are red solid curves calibrated by the right vertical axis and the lower-surface densities $\left\vert\chi_{0}(t) \right\vert^{2}$ are black dashed lines calibrated by the left-vertical axis.
• Figure 3: The first row show trajectories from exact solution (cyan solid), EF-based single trajectory method (green dashed) and the Ehrenfest method (red dotted). The exact coherence dynamics is replicated by cyan solid lines in the second and the third row. The Ehrenfest coherence with self-consistently obtained trajectories of the first row is shown as the red dashed lines while that obtained by input exact trajectories is shown as the black dotted lines. The third row shows coherence from the EF-based method with self-consistently obtained trajectories (green dashed) and with exact trajectories (blue dotted). Despite slight deviations in these two trajectories in Regime IIB, the coherence from the EF-based method associated with these two trajectories agree with each other.