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

$$
H_0=
\begin{pmatrix}
0&t_0&0&0\\
t_0&0&t_1&0\\
0&t_1&0&t_0\\
0&0&t_0&0
\end{pmatrix}.
$$

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

$$
\mathbf e(\alpha)=(\cos\alpha,\sin\alpha),\qquad
\mathbf R_{\mathrm A}=-\frac d2\mathbf e(\alpha_{\mathrm{inter}}),\qquad
\mathbf R_{\mathrm B}=+\frac d2\mathbf e(\alpha_{\mathrm{inter}}),
$$

and place the sites at

$$
\begin{aligned}
\mathbf r_0&=\mathbf R_{\mathrm A}-\frac l2\mathbf e(\alpha_{\mathrm{mol}}),
&
\mathbf r_1&=\mathbf R_{\mathrm A}+\frac l2\mathbf e(\alpha_{\mathrm{mol}}),\\
\mathbf r_2&=\mathbf R_{\mathrm B}-\frac l2\mathbf e(\alpha_{\mathrm{mol}}),
&
\mathbf r_3&=\mathbf R_{\mathrm B}+\frac l2\mathbf e(\alpha_{\mathrm{mol}}).
\end{aligned}
$$

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

$$
\mathbf A(t)=\frac{E_0}{\omega_0}
\begin{pmatrix}\cos\phi\\ \sin\phi\end{pmatrix}
\sin^2\!\left(\frac{\omega_0 t}{2n_{\mathrm{cyc}}}\right)
\cos(\omega_0 t),
\qquad
0\leq t\leq \frac{2\pi n_{\mathrm{cyc}}}{\omega_0},
$$

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

$$
H(t)=H_0+E_x(t)X+E_y(t)Y.
$$

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

$$
\mathbf r_{\mathrm{adia\text{-}intra}}(t)
=2\mathbf r_{00}(t)+2\mathbf r_{11}(t),
\qquad
\mathbf r_{mn}(t)=
\langle\mu_m(t)\lvert (X,Y)\rvert\mu_n(t)\rangle .
$$

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](../../submission_requirements.md) for the common submission entry points.