Nuclear interference by electronic de-orthogonalisation

Abstract

Interference is a universal consequence of superposition, yet in composite quantum systems it can encode correlations between subsystems. We show that in coupled electron-nuclear dynamics, interference in the nuclear density can arise dynamically even when it is initially absent. Starting from a superposition of orthogonal Born-Oppenheimer electronic states, we demonstrate within the exact factorisation framework that genuine non-adiabatic electron-nuclear correlations induce de-orthogonalisation of the electronic factors, thereby generating interference terms in the nuclear density. Such interference has no counterpart in adiabatic evolution. Unlike conventional nuclear wave-packet interference or interference that merely reflects electronic coherence in a chosen basis, the effect identified here is a manifestation of the compositeness of the full electron-nuclear state. Nuclear density interference thus emerges as a direct dynamical signature of correlated quantum motion in composite systems.

Figures (4)

• Figure 1: Illustration contrasting adiabatic evolution, which preserves the orthogonality of electronic states (left), with non-adiabatic evolution, which induces de-orthogonalisation (right). Two initially orthogonal electronic wavefunctions are depicted as blue and red crescents with no overlap, and their orthogonality is further visualised by opposite green state vectors, in analogy with antipodal points on a Bloch sphere. Under adiabatic dynamics, the electronic states remain orthogonal throughout the evolution, yielding no interference contribution to the nuclear density. Under non-adiabatic dynamics, orthogonality is not preserved, the electronic states acquire finite overlap, and an interference contribution emerges. Solid black curves denote Born-Oppenheimer energy surfaces, while dashed arrows indicate the schematic direction of evolution.
• Figure 2: BO energy landscape (left, in unit of $g_{0}$) and the NAC profile (right, in unit of $g_{0}/\sigma$) whose quantum dynamics is numerically studied here as an example. The NAC profile shows two distinct regions strong couplings separated by a region of larger BO gap.
• Figure 3: The concurrent growth of electronic de-orthogonalisation (upper rows) and nuclear interference (lower rows) under weaker (left column) and stronger (right column) non-adiabatic conditions represented by $J_{L}$ (see legends). The red solid/black dashed curves are all with $\kappa=1.0$/$\kappa=2.5$. For each $(J_{L},\kappa)$, $R_{0}$ is chosen as the coordinate at which $\left|\left\langle \phi_{0}(t,R)\middle|\phi_{1}(t,R)\right\rangle\right|$ reaches its maximum within the simulation time window. For the left column, the red solid/black dashed curves are with $R_{0}=0.075\sigma$/$R_{0}=0.05\sigma$. For the right column, the red solid/black dashed curves are with $R_{0}=0.1\sigma$/$R_{0}=0.075\sigma$. Here $\omega_{B}t$ presents time as a dimensionless parameter in which we use $\omega_{B}=19.25g_{0}$/$\omega_{B}=20.15g_{0}$ for the left/right column as the estimated energy width of the fully correlated system.
• Figure 4: Snapshots of the profiles for the interference contribution $n_{01}(R,t)$ (upper panel with the real/imaginary part on the left/right column) and for the non-interference contributions $n_{0}(R,t)$ and $n_{1}(R,t)$ (lower panel) to the nuclear density for $J_{L}=0.1\,g_{x}$ and $\kappa=2.5$ (parameters used by the black dashed curves in the right column of Fig. \ref{['de-ort-evo']}). Discrete data points at $\omega_{B}t=5,\,10,\,20$ are shown as red circles, black crosses, and blue stars, respectively. Smooth solid, dashed, and dotted curves are drawn through the data as guides to the eye.