| # 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 |
|
|
| $$ |
| \frac{1}{2}\frac{\partial^2\psi}{\partial t^2} |
| +\gamma\frac{\partial\psi}{\partial t} |
| =\psi-|\psi|^2\psi+(\boldsymbol{\nabla}-i\mathbf A)^2\psi . |
| $$ |
|
|
| Time is measured in inverse Higgs-mode frequency and distance in coherence |
| lengths. The system starts in the uniform equilibrium state, |
|
|
| $$ |
| \psi(x,y,0)=1,\qquad \partial_t\psi(x,y,0)=0, |
| $$ |
| |
| and approaches that equilibrium state far from the illuminated region. |
| |
| Use a continuous resonant, linearly polarized Gaussian field, |
| |
| $$ |
| \mathbf A(x,y,t)=\hat{\mathbf x}\cos t |
| \sum_j A_j |
| \exp\!\left[-\frac{(x-x_j)^2+y^2}{w_0^2}\right]. |
| $$ |
| |
| 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](../../submission_requirements.md) for the common submission entry points. |
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|
