A loop changes an interference fringe

Geometric Phase Around a Conical Intersection

Why can a circuit around a conical intersection turn a bright fringe dark? An adiabatic electronic state changes sign around a conical intersection. When two coherent paths reunite, this relative phase shifts their interference pattern. A path that leaves the intersection outside its loop gives a useful control.

Move a guided nuclear circuit across two electronic energy surfaces. Compare the recombined probability with the electronic geometric phase included and omitted.

Prescribed nuclear guides; height is electronic energy. Cyan and violet are the two arms.

After the paths recombine

A relative sign moves the fringes.

Geometric phase includedSame arms, phase omitted

Both curves show output port A of an ideal balanced recombiner. The shaded central window runs from minus 0.3 to 0.3 in the dimensionless output coordinate.

Electronic loop phase

\pi

Smallest electronic gap

4

Probability in the central window

0.02

Ideal fringe contrast

1

Model, normalization and downloadable data

This is an adiabatic two-arm reduction of a two-coordinate linear vibronic-coupling Hamiltonian. Two imposed semicircular guides split and recombine a coherent nuclear amplitude on the lower electronic surface. Reflection symmetry makes their electronic energies and path lengths equal, so their dynamical phases match.

H_{\mathrm{el}}=x\sigma_z+y\sigma_x,\quad (x,y)=(c+R\cos\theta,R\sin\theta),\quad V_\pm=\pm\sqrt{x^2+y^2}

The electronic loop phase is the gauge-invariant product of 512 normalized neighbouring eigenvector overlaps, including the closing link. Circuits with an electronic gap below 0.24 are outside the displayed adiabatic model.

p_{A,B}(q)=\frac{G(q)}2[1\pm2\sqrt{w(1-w)}\cos(2kq+\phi+\gamma)],\quad G(q)=\frac{e^{-q^2/(2s^2)}}{\sqrt{2\pi}s},\quad s=0.9,\ k=1.8

These are output probability densities from a normalized coherent amplitude model. The Gaussian output modes, balanced recombiner and transverse phase gradient are imposed. Motion along the guides is an illustration of transport; nuclear forces, nonadiabatic transfer, environmental decoherence and a particular molecular reaction are omitted.

Integrated probability, both output ports

1

Start with the essentials

Focus question: Why can a circuit around a conical intersection turn a bright fringe dark?
One-sentence intuition: A real electronic eigenstate transported around one conical intersection returns with the opposite sign. The relative sign shifts the interference pattern.

Core mathematical model

The two-coordinate electronic model

H_{\mathrm{el}}(x,y)=x\sigma_z+\kappa y\sigma_x,\qquad V_\pm=\pm\sqrt{x^2+\kappa^2y^2}

The horizontal axes are nuclear configuration coordinates. Energy is the surface height; all quantities are dimensionless. Mixing is fixed to one in the main controls.

Following the lower electronic state

W(C)=\prod_j\frac{\langle u_-(\mathbf R_j)\mid u_-(\mathbf R_{j+1})\rangle}{|\langle u_-(\mathbf R_j)\mid u_-(\mathbf R_{j+1})\rangle|}=e^{i\gamma}

The last point connects to the first. Independent sign choices for the electronic vectors cancel in the product. A loop around the isolated intersection has a geometric phase of pi.

Probability after a balanced recombiner

p_{A,B}(q)=\frac{G(q)}2\left[1\pm2\sqrt{w(1-w)}\cos(2kq+\phi+\gamma)\right]

The Gaussian envelope is normalized, the upper arm has probability w, and the two outgoing modes have opposite transverse wave numbers. Both output ports together have total probability one.

Common difficulties

A population is not a phase measurement

Typical misconceptionIf both arms stay on the lower surface, they must give the same output.

Better mental modelPopulation records the size of an amplitude. Interference also depends on its sign or phase.

The imposed guides

Typical misconceptionThe paths are predicted reaction trajectories.

Better mental modelA dimensionless, adiabatic two-arm reduction of a two-coordinate linear vibronic-coupling model. Symmetric imposed guides have equal dynamical phases. An ideal coherent recombiner produces the displayed output densities. The guide animation is prescribed; nuclear forces, surface transfer and a particular molecule are not simulated.

Run the experiment

01
Predict the central fringe
Compare the cyan curve with the gold phase-omitted reference in the first preset.
What to observe: At the centre, equal arms cancel when the loop encloses the intersection.
02
Move the whole circuit
Drag the gold handle or use Move the circuit. Stop on either side of the crossing warning.
What to observe: A loop that misses the intersection has zero holonomy phase, and the central fringe becomes bright.
03
Give the arms unequal weights
Choose One arm dominates, then change Extra arm phase.
What to observe: A weak arm cannot cancel a strong arm completely. The fringe contrast falls even though the electronic geometric phase is unchanged.

Geometric Phase Around a Conical Intersection

A relative sign moves the fringes.