---
title: "Geometric Phase Around a Conical Intersection Interactive Lab | Sandbox Physics"
description: "Why can a circuit around a conical intersection turn a bright fringe dark? An adiabatic electronic state changes sign around a conical intersection."
canonical_url: "https://sandboxphysics.com/labs/conical-geometric-phase/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/conical-geometric-phase/"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

A 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.

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 included Same 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 $1$

- [M. V. Berry, Quantal phase factors accompanying adiabatic changes (1984), §2–3](https://michaelberryphysics.wordpress.com/wp-content/uploads/2013/07/berry120.pdf)
- [C. A. Mead and D. G. Truhlar, J. Chem. Phys. 70, 2284–2296 (1979)](https://doi.org/10.1063/1.437734)

Physics tutorial

## Geometric Phase Around a Conical Intersection

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

**Why it matters** Two equal paths can return to the same electronic energy and still disagree in phase. Their interference reveals what their separate populations miss.

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 misconception** If both arms stay on the lower surface, they must give the same output.

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

#### The imposed guides

**Typical misconception** The paths are predicted reaction trajectories.

**Better mental model** A 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

1. 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.

2. 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.

3. 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.

Physical conclusion

A 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.

## Related experiments and articles

Nonadiabatic molecular dynamics · Reading path

### From energy surfaces to reaction branches

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Why paths around a conical intersection can cancel**](https://sandboxphysics.com/stories/nonadiabatic-dynamics/geometric-phase-interference/)
2. [Article **When two nuclear packets can still interfere**](https://sandboxphysics.com/stories/nonadiabatic-dynamics/coherence-and-decoherence/)
3. [Article **Reading interference and molecular-benchmark papers**](https://sandboxphysics.com/stories/nonadiabatic-dynamics/reading-nonadiabatic-research/)
4. [Lab **Benchmark Nonadiabatic Reaction Methods** Supply what a paper reader needs to repeat a comparison](https://sandboxphysics.com/labs/nonadiabatic-method-benchmark/)

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)