--- title: "When two nuclear packets can still interfere" description: "Connect amplitude addition to reduced electronic coherence and distinguish spatial separation, phase averaging and trajectory decoherence corrections." canonical_url: "https://sandboxphysics.com/stories/nonadiabatic-dynamics/coherence-and-decoherence/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/nonadiabatic-dynamics/coherence-and-decoherence/" last_modified: "2026-10-28" --- [Nonadiabatic molecular dynamics](https://sandboxphysics.com/stories/nonadiabatic-dynamics/) L3 · Chapter 7 # When two nuclear packets can still interfere Sandbox Physics · 2026-10-28 Interference at a common exit requires overlap and a reproducible relative phase. Electronic populations may remain unchanged as nuclear branches separate. Integrating out the nuclei can then make the electronic subsystem lose coherence even under unitary evolution of the full state. ## Before you begin Read chapters four and six. You need complex conjugation and integration. A density matrix records populations on its diagonal and phase-sensitive relations off the diagonal. [Review the preceding chapter](https://sandboxphysics.com/stories/nonadiabatic-dynamics/geometric-phase-interference/) - Identify the interference term in a common channel. - Relate electronic coherence to nuclear overlap. - Distinguish sampling noise, phase averaging and a dynamical approximation. ## 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. $$ |a+b|^2=|a|^2+|b|^2+2\operatorname{Re}(a^*b) $$ Specify the common channel before identifying an interference contribution. 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. $$ \Psi=c_0\chi_0|0\rangle+c_1\chi_1|1\rangle,\qquad (\rho_e)_{01}=c_0c_1^*\langle\chi_1|\chi_0\rangle $$ This expression uses a fixed electronic basis. Coordinate-dependent electronic states require their spatial relation to be included consistently. ## 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. $$ \chi_j(R)\propto e^{-(R-R_j)^2/(4\sigma^2)},\qquad \langle\chi_1|\chi_0\rangle=e^{-d^2/(8\sigma^2)} $$ The width here is the standard deviation of position probability, which fixes all factors of two. ## 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. $$ \rho_e=\frac12\begin{pmatrix}1&S\\S^*&1\end{pmatrix} $$ Phase convention can change the phase of off-diagonal entries; purity depends only on overlap magnitude. ## 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. ## Try the prediction ### [Separate fringe displacement from contrast loss](https://sandboxphysics.com/labs/conical-geometric-phase/) 1. Keep an enclosing loop and predict how external phase affects fringes and envelope. 2. Hold that phase fixed and move arm weight away from one half; inspect the remaining minimum. 3. Explain that this contrast loss comes from unequal amplitudes. Modeling decoherence would additionally require a phase or environment model. ## Exercises and explanations 1. Do equal electronic populations imply unchanged coherence? Read the explanation Populations are diagonal matrix elements. Off-diagonal coherence also depends on the nuclear overlap and can decrease while the diagonals stay fixed. 2. Does overlap of two density envelopes guarantee complete interference recovery? Read the explanation Phase, momentum, electronic channel and environmental correlations also matter. The spatial phases also enter the complex overlap, even when the density profiles agree. [Previous: Why paths around a conical intersection can cancel](https://sandboxphysics.com/stories/nonadiabatic-dynamics/geometric-phase-interference/) [Next: What makes a dynamics comparison convincing](https://sandboxphysics.com/stories/nonadiabatic-dynamics/reproducible-benchmarks/) ## Sources and calculation scope 1. [Curchod, Agostini & Gross, An Exact Factorization Perspective on Quantum Interferences in Nonadiabatic Dynamics, arXiv:1604.02351v2](https://arxiv.org/html/1604.02351v2) Fixed 30 June 2016 author version; published in J. Chem. Phys. 145, 034103 (2016), DOI 10.1063/1.4958637. Reading scope: equations (5)–(6), (14)–(18), and Figures 3–7. Its Shin–Metiu electron–ion calculation differs from the Tully and conical teaching models. 2. [Cigrang et al., Roadmap for Molecular Benchmarks in Nonadiabatic Dynamics, arXiv:2502.14569v3](https://arxiv.org/pdf/2502.14569v3) Fixed 4 July 2025 author version associated with the published Perspective, DOI 10.1021/acs.jpca.5c02171. Sections II A–F discuss systems, dynamics methods, electronic structure, initial conditions, observables and experimental comparisons. This is a benchmark roadmap, not a completed universal ranking.