When two nuclear packets can still interfere

Adding probabilities differs from adding amplitudes

Two histories reaching one electronic exit have amplitudes $a(R)$ and $b(R)$. Squaring their sum produces a cross term. Adding probabilities from separately evolved histories removes it.

Contributions in orthogonal electronic states do not interfere directly in total nuclear density. A later coupling region can send both histories into the same electronic state; their amplitudes can then interfere and change branching.

Reading: Curchod

Electronic coherence includes a nuclear overlap

Use fixed orthonormal electronic states and normalized nuclear branches $\chi_0(R)$ and $\chi_1(R)$. Their coefficients are $c_0$ and $c_1$. Tracing over nuclear position multiplies the electronic off-diagonal element by the overlap of the two branches.

If nuclear packets separate, this overlap can approach zero. The full electron–nuclear state may still follow the closed-system Schrödinger equation. The electronic coherence changes as correlation with the nuclear branches develops.

Worked example of two separated Gaussians

Take real Gaussians with no relative momentum and identical position standard deviation $\sigma$. Completing the square in their product gives overlap $\exp[-d^2/(8\sigma^2)]$ for center separation $d$.

At two standard deviations the overlap is $e^{-1/2}\simeq0.607$; at four it is $e^{-2}\simeq0.135$. Doubling separation quadruples the magnitude of the exponent. Electronic populations can remain one half throughout this loss of coherence.

A momentum difference introduces spatially varying relative phase and can reduce the overlap further. Density separation is only part of the story.

Equal populations can have different purity

For equal electronic weights and nuclear overlap $S$, the electronic matrix has diagonal entries $1/2$ and off-diagonal magnitude $|S|/2$. Squaring and tracing gives purity $\operatorname{Tr}(\rho_e^2)=(1+|S|^2)/2$.

Identical phase-matched nuclear branches have $|S|=1$ and electronic purity one. Orthogonal branches have $S=0$ and purity one half. This converts the distinction between equal populations and different coherence into a numerical check.

Phase averaging can wash out fringes

Repeated preparations can have different relative phases $\varphi$. Each run may be coherent, yet averaging their densities multiplies interference by $\langle e^{i\varphi}\rangle$. A phase uniformly distributed around a full turn gives zero.

Separate coherent single-system evolution from a statistical mixture of preparations. More samples estimate the specified mixture more precisely; they do not recover phases already averaged out. Unequal arm intensities can also reduce fringe contrast and require a control.

Decoherence corrections are additional method choices

Standard FSSH electronic amplitudes can retain a coherent superposition after nuclear branches have separated. Decoherence corrections damp inactive components using different prescriptions and timescales. Their effect must be checked for the same preparation and observable.

The 2016 interference study compares complete quantum evolution with independently propagated wavefunctions. That comparison isolates a coherent recombination effect. Each corrected surface-hopping implementation requires its own comparison.

Reading: Curchod · Cigrang et al.