cases/case42/task.md

Collective response of optically structured superconducting order

Physical setting

Work with the dimensionless complex superconducting order parameter $\psi(x,y,t)$. Its inertial dynamics obey

$$\frac{1}{2}\frac{{\mathrm{\partial }}^2\psi }{\mathrm{\partial }{t}^2}+\gamma \frac{\mathrm{\partial }\psi }{\mathrm{\partial }t}=\psi -\mathrm{\mid }\psi {\mathrm{\mid }}^2\psi +(\mathbf{\nabla }-i\mathbf{A}{)}^2\psi \mathrm{.}\backslash frac\{1\}\{2\}\backslash frac\{\backslash partial^2\backslash psi\}\{\backslash partial\; t^2\}\; +\backslash gamma\backslash frac\{\backslash partial\backslash psi\}\{\backslash partial\; t\}\; =\backslash psi-|\backslash psi|^2\backslash psi+(\backslash boldsymbol\{\backslash nabla\}-i\backslash mathbf\; A)^2\backslash psi\; .$$

Time is measured in inverse Higgs-mode frequency and distance in coherence lengths. The system starts in the uniform equilibrium state,

$$\psi (x,y,0)=1,{\mathrm{\partial }}_t\psi (x,y,0)=0,\backslash psi(x,y,0)=1,\backslash qquad\; \backslash partial\_t\backslash psi(x,y,0)=0,$$

and approaches that equilibrium state far from the illuminated region.

Use a continuous resonant, linearly polarized Gaussian field,

$$\mathbf{A}(x,y,t)=\widehat{\mathbf{x}}\cos t\sum _j{A}_j\exp \text{\hspace{0.17em}⁣}[-\frac{(x-x_j{)}^2+y^2}{w_0^2}]\mathrm{.}\backslash mathbf\; A(x,y,t)=\backslash hat\{\backslash mathbf\; x\}\backslash cos\; t\; \backslash sum\_j\; A\_j\; \backslash exp\backslash !\backslash left[-\backslash frac\{(x-x\_j)^2+y^2\}\{w\_0^2\}\backslash right].$$

Here $A_j$ is the peak dimensionless vector-potential amplitude of one beam. Investigate the following two physical cases:

case	beam centres $x_j$	$A_j$	$w_0$	$\gamma$
S	$0$	$1.0$	$10$	$0.05$
P	$-40,+40$	$2.0$	$20$	$0.10$

The second-order inertial equation and the two field configurations define the physical problem. You may choose a stable spatial representation, boundary realization, time propagator, sampling strategy, transform convention and evidence layout, provided that your choices preserve this problem and do not materially contaminate the claims.

Research problem

Determine the physical origin and classification of the collective response generated by the two specified optical configurations. Establish which conclusions about the response are supported across the S and P cases, and what material alternatives remain compatible with the computed dynamics.

Base the argument on auditable calculations of the complex order parameter and student-chosen observables with enough discriminating force to support the classification. State clearly which claims are established by these two cases and which broader claims are not.

The solver, grid, box, boundary implementation, sampling interval, spectral window, momentum convention, plotting range, normalization and number or order of figures are not prescribed. Declare the conventions that materially affect your interpretation.

Submission and resources

Submit one self-contained scientific report and an auditable offline calculation package containing the task-specific derivation, code, numerical data underlying every material claim, 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. No fixed file tree, filename scheme, panel count or complete exploration history is required.

Resource limits are supplied exclusively by the accompanying resource_limits.json. Its fixed CPU allocation, candidate-active wall time, network boundary are binding. No separate aggregate CPU-hour limit applies. Greater computational expense receives no 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.