--- title: "Why paths around a conical intersection can cancel" description: "Derive a two-state loop phase and connect the electronic eigenvector sign to a controlled two-port interference distribution." canonical_url: "https://sandboxphysics.com/stories/nonadiabatic-dynamics/geometric-phase-interference/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/nonadiabatic-dynamics/geometric-phase-interference/" last_modified: "2026-10-28" --- [Nonadiabatic molecular dynamics](https://sandboxphysics.com/stories/nonadiabatic-dynamics/) L3 · Chapter 6 # Why paths around a conical intersection can cancel Sandbox Physics · 2026-10-28 An electronic state carried around a conical intersection can return with a minus sign. Its own density stays unchanged. Recombining it with another amplitude can turn a bright fringe dark, making the path-dependent phase observable in the model. ## Before you begin Read chapter five. A unit-modulus complex phase preserves an individual amplitude’s density but changes the sum with another amplitude. The path must avoid degeneracy and be traversed slowly enough for adiabatic following. [Review the preceding chapter](https://sandboxphysics.com/stories/nonadiabatic-dynamics/conical-intersections/) - Derive the sign change of a half-angle eigenvector. - Calculate a gauge-invariant loop from neighboring overlaps. - Relate electronic phase to an ideal two-port distribution. ## 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 $-1$. $$ |u_-(\theta)\rangle=\begin{pmatrix}-\sin(\theta/2)\\\cos(\theta/2)\end{pmatrix},\qquad |u_-(\theta+2\pi)\rangle=-|u_-(\theta)\rangle $$ This continuous real gauge is not single-valued after a complete turn. The endpoint relation belongs to the loop calculation. 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. $$ W=\prod_{j=0}^{N-1}\frac{\langle u_j|u_{j+1}\rangle}{|\langle u_j|u_{j+1}\rangle|},\qquad u_N=u_0 $$ The result is positive or negative one for this real model. General complex eigenstates can give any unit-modulus phase. ## 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. $$ p_{A,B}(q)=\frac{G(q)}2\left[1\pm2\sqrt{w(1-w)}\cos(2kq+\varphi+\gamma)\right] $$ The phase-omitted control retains the same paths, weights, envelope and external phase. Two paths meet with a relative phase. Conceptual diagram; positions and heights are schematic. ## 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. ## Try the prediction ### [Move the loop and reverse the central fringe](https://sandboxphysics.com/labs/conical-geometric-phase/) 1. With equal arm weights and zero external phase, predict the center of each port for an enclosing loop. 2. Drag the path or change the center offset; compare phase-retained and phase-omitted outputs. 3. Use the extra-half-turn preset, then change arm weights. Explain fringe position and contrast separately. ## Exercises and explanations 1. What happens if a single sampled eigenvector is multiplied by negative one? Read the explanation Its two neighboring overlaps each reverse sign. Their product and the predicted interference remain unchanged. 2. Can the squared magnitude on one arm reveal its returning minus sign? Read the explanation The overall sign disappears from that density. It becomes detectable through recombination with another amplitude or a closed-loop phase comparison. [Previous: How two surfaces form a conical intersection](https://sandboxphysics.com/stories/nonadiabatic-dynamics/conical-intersections/) [Next: When two nuclear packets can still interfere](https://sandboxphysics.com/stories/nonadiabatic-dynamics/coherence-and-decoherence/) ## Sources and calculation scope 1. [C. A. Mead & D. G. Truhlar, On the determination of Born–Oppenheimer nuclear motion wave functions including complications due to conical intersections and identical nuclei, J. Chem. Phys. 70, 2284–2296 (1979)](https://doi.org/10.1063/1.437734) Published original connecting conical intersections with nuclear-motion phase. Bibliography and the 1983 erratum are recorded by the authors’ group. This is further reading; the guided Lab derives its phase from Berry’s fixed two-state example. 2. [M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. Lond. A 392, 45–57 (1984)](https://michaelberryphysics.wordpress.com/wp-content/uploads/2013/07/berry120.pdf) Published primary derivation: equations (5)–(6) separate dynamical and geometrical phase; section 3 treats a degeneracy. The path experiment is a controlled two-state model, with the reduction and phase comparison specified on its page.