cases/case18/task.md

Coupling and polarization as controls of dimer high-harmonic emission

Background

Weak electronic coupling between neighbouring molecules can reorganize a strongly nonlinear optical response even when it is much smaller than the intramolecular coupling. A minimal molecular dimer makes it possible to ask whether and how adequately this reorganization is captured by adiabatic motion within instantaneous states.

Research problem

Using the model below, determine how the magnitude of the weak intermolecular coupling and the direction of a linearly polarized drive jointly organize the order-resolved high-harmonic response. Establish the physical origin of the structures you find from your own derivation and computed evidence. Then assess the fidelity and applicability of the stated adia-intra approximation, and the level of conclusion it supports within the weak-coupling model.

How you construct and substantiate the investigation is for you to decide. Do not assume a particular polarization preference or coupling trend in advance.

Physical model and inputs

Use units with (\hbar=|e|=1). The one-electron localized basis contains four sites. Sites (0,1) form one molecule, sites (2,3) form the other, and the field-free Hamiltonian is

Set (t_0=-1), which fixes the energy unit, and write (g=|t_1/t_0|). The sign convention (t_1<0) may be used. Restrict physical claims to the weak-coupling model, (0<g\leq 0.1), and choose the sampling needed to support your conclusions.

Let

$$\mathbf{e}(\alpha )=(\cos \alpha ,\sin \alpha ),{\mathbf{R}}_{\mathrm{A}}=-\frac{d}{2}\mathbf{e}({\alpha }_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}}),{\mathbf{R}}_{\mathrm{B}}=+\frac{d}{2}\mathbf{e}({\alpha }_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}}),\backslash mathbf\; e(\backslash alpha)=(\backslash cos\backslash alpha,\backslash sin\backslash alpha),\backslash qquad\; \backslash mathbf\; R\_\{\backslash mathrm\; A\}=-\backslash frac\; d2\backslash mathbf\; e(\backslash alpha\_\{\backslash mathrm\{inter\}\}),\backslash qquad\; \backslash mathbf\; R\_\{\backslash mathrm\; B\}=+\backslash frac\; d2\backslash mathbf\; e(\backslash alpha\_\{\backslash mathrm\{inter\}\}),$$

and place the sites at

Use

Quantity	Value or condition
Molecular length	(l=5.5)
Distance between molecular centres	(d=9.2)
Molecular-axis angle	(\alpha_{\mathrm{mol}}=90^\circ)
Intermolecular-axis angle	(\alpha_{\mathrm{inter}}=20^\circ)
Filling	Four electrons; the two lowest field-free one-electron eigenstates are each spin-doubly occupied
Reference coupling for the drive frequency	(t_1=-0.02)
Field amplitude	(E_0=0.15)
Pulse length	(n_{\mathrm{cyc}}=15) optical cycles

A common translation of all four sites is equivalent. Define (X=\operatorname{diag}(x_0,x_1,x_2,x_3)) and (Y=\operatorname{diag}(y_0,y_1,y_2,y_3)). For a polarization angle (\phi) measured from (+x), use the vector potential

$$\mathbf{A}(t)=\frac{E_0}{{\omega }_0}\left(\begin{array}{c}\cos \varphi \\ \sin \varphi \end{array}\right){\sin }^2\text{\hspace{0.17em}⁣}\left(\frac{{\omega }_{0}t}{2n_{\mathrm{c}\mathrm{y}\mathrm{c}}}\right)\cos ({\omega }_{0}t),0\le t\le \frac{2\pi n_{\mathrm{c}\mathrm{y}\mathrm{c}}}{{\omega }_0},\backslash mathbf\; A(t)=\backslash frac\{E\_0\}\{\backslash omega\_0\}\; \backslash begin\{pmatrix\}\backslash cos\backslash phi\backslash \backslash \; \backslash sin\backslash phi\backslash end\{pmatrix\}\; \backslash sin^2\backslash !\backslash left(\backslash frac\{\backslash omega\_0\; t\}\{2n\_\{\backslash mathrm\{cyc\}\}\}\backslash right)\; \backslash cos(\backslash omega\_0\; t),\; \backslash qquad\; 0\backslash leq\; t\backslash leq\; \backslash frac\{2\backslash pi\; n\_\{\backslash mathrm\{cyc\}\}\}\{\backslash omega\_0\},$$

with (\mathbf A(t)=0) outside this interval and (\mathbf E(t)=-\partial_t\mathbf A(t)). The driven one-electron Hamiltonian is

$$H(t)=H_0+E_x(t)X+E_y(t)Y\mathrm{.}H(t)=H\_0+E\_x(t)X+E\_y(t)Y.$$

At the reference coupling, let (\Delta\varepsilon=\varepsilon_2-\varepsilon_1) be the gap between the two middle field-free levels and set (\omega_0=\Delta\varepsilon/6.3). Keep this drive fixed when varying (t_1).

Calculation and observable scope

For the exact branch, propagate the two initially occupied one-electron states under the time-dependent Schrodinger equation for (H(t)), retaining the spin factor of two. For the comparison branch, diagonalize (H(t)) instantaneously, denote its ordered instantaneous eigenstates by (\lvert\mu_m(t)\rangle), and use the adia-intra position

$${\mathbf{r}}_{\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{a}\text{-}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}}(t)=2{\mathbf{r}}_{00}(t)+2{\mathbf{r}}_{11}(t),{\mathbf{r}}_{mn}(t)=⟨{\mu }_m(t)\mid (X,Y)\mid {\mu }_n(t)⟩\mathrm{.}\backslash mathbf\; r\_\{\backslash mathrm\{adia\backslash text\{-\}intra\}\}(t)\; =2\backslash mathbf\; r\_\{00\}(t)+2\backslash mathbf\; r\_\{11\}(t),\; \backslash qquad\; \backslash mathbf\; r\_\{mn\}(t)=\; \backslash langle\backslash mu\_m(t)\backslash lvert\; (X,Y)\backslash rvert\backslash mu\_n(t)\backslash rangle\; .$$

Construct physically reasonable, reconstructible order-resolved HHG evidence from the total dipole or its acceleration. Choose and state the finite-pulse spectral treatment and the reduction used to resolve harmonic order, and keep those choices consistent across every physical comparison they support. The overall spectral scale is arbitrary; any scale or normalization used for comparison must be declared and kept common wherever relative magnitudes support a claim.

Standard TDSE or matrix-propagation methods may be used. All grids, time steps, propagators, derivatives, transforms, interpolation, angular sampling, tie handling, and evidence design not fixed above are your choices. Document them and establish that the reported physical conclusions are numerically reliable. No particular solver, sampling schedule, plot layout, number of figures, or validation procedure is prescribed.

Submission

Submit one self-contained scientific report and an auditable offline calculation package containing the code, numerical data underlying material claims, a provenance note listing every offline source or software document actually consulted and the method basis for items supplied from general knowledge, and concise run and environment information. The organization of the report and the evidence package is yours.

The available CPU allocation, candidate active deadline, node, network, memory and scratch limits are specified in the accompanying resource_limits.json. Complete the derivation, implementation, calculation, validation, interpretation and final-file preparation within these resource limits. More expensive modelling does not receive additional credit.

Reference and integrity policy

This examination is conducted without network access or live literature retrieval. You may rely on your existing general scientific knowledge and, if any are explicitly supplied with the examination, on those offline materials or software documentation. Record every such source actually consulted. For formulas, methods, constants, or conventions used from general knowledge rather than a consulted source, state enough of the method basis, definition, value, units, and convention to make the calculation auditable; do not invent a citation. You are not required to identify or prove a publication date or to retrieve literature during the examination.

The hidden target article and its identifying metadata, figures or captions, supplementary material, author code or data, target-specific reproduction code, solution or scoring material, online solutions, and worked reproductions of this case must not be used. All submitted scientific figures and numerical data must be generated by your own implementation of the stated problem.

See submission requirements for the common submission entry points.