Why paths around a conical intersection can cancel

An electronic eigenvector reverses after one loop

Use the dimensionless Hamiltonian $H_e=\begin{pmatrix}x&y\\y&-x\end{pmatrix}$ with energies $E_\pm=\pm\sqrt{x^2+y^2}$. Parameterize a centered circle by $x=R\cos\theta$ and $y=R\sin\theta$, with $R>0$ and increasing counterclockwise angle.

A continuous real lower-state eigenvector uses the half-angle below. Advancing the angle through a full turn reverses both components. Energy returns to its original value, but the state acquires phase $\pi$ and factor $-1$.

Reading: M. V. Berry · C. A. Mead & D. G. Truhlar

Include every overlap around the circuit

Let $|u_j\rangle$ be lower states sampled along the path. Normalize each neighboring overlap to unit magnitude and multiply around the full circuit, including the closing link.

A local eigenvector phase appears in two adjacent overlaps with opposite signs, so it cancels. This gauge-invariant product is negative for a loop enclosing this real degeneracy and positive for a nonenclosing loop. A path through the origin has an undefined adiabatic eigenstate and requires another dynamical treatment.

Control dynamical phase with symmetric arms

Prescribe paths $(x,y)=(c+R\cos\theta,R\sin\theta)$. Upper and lower arms travel from left to right along mirror semicircles. Their energy histories and travel times match, so their dynamical phases agree. Offset $c$ and radius $R$ determine whether the combined loop surrounds the origin.

This is a guided adiabatic two-arm amplitude reduction with prescribed nuclear paths. It does not propagate the full two-dimensional nuclear wavefunction. The dimensionless output coordinate $q$ has normalized Gaussian envelope $G(q)$, fringe wave number $k$, external relative phase $\varphi$ and upper-arm weight $w$.

The recombiner adds or subtracts the arm amplitudes and divides each by the square root of two. The upper amplitude is proportional to $\sqrt w\,e^{ikq}$ and the lower to $\sqrt{1-w}\,e^{-i(kq+\varphi+\gamma)}$. Their cross term supplies the cosine.

An ideal balanced recombiner yields the two output densities below. Their sum is the original envelope. Geometric phase $\gamma$ redistributes this probability between the ports.

Worked example of a bright-to-dark central fringe

Set $w=1/2$, $\varphi=0$ and $q=0$. A nonenclosing loop has $\gamma=0$, giving central density $G(0)$ at port A and zero at B. An enclosing loop gives $\gamma=\pi$ and swaps them.

An additional external phase $\pi$ restores the bright center at A. Unequal weights $w=0.92$ give visibility $2\sqrt{0.92\times0.08}\simeq0.543$. Destructive interference then leaves a background because the two amplitudes have different magnitudes.

Check the conditions for interpreting the phase

Moving the loop from enclosing to nonenclosing requires passing through a path that touches the degeneracy. Its minimum gap vanishes and adiabatic following can fail. The experiment marks the small-gap region instead of assigning it a valid adiabatic cycle.

Different enclosing shapes can share the same loop sign. Travel time changes dynamical phase, controlled here by mirrored arms. A freely evolving molecular packet can also change arm weights, widths and arrival times, which must enter a molecular prediction.