| # Harmonic radiation from driven few-level systems |
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|
| ## Background and research question |
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|
| Discrete bound-state dynamics can contribute to harmonic radiation below and |
| around an electronic transition. For the controlled few-level systems below, |
| determine what transition-dependent information can be recovered from the |
| emitted radiation and whether the system containing orthogonal transitions |
| provides a useful polarization-based signature. |
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|
| Design the model construction, theoretical analysis, computation, tests, |
| evidence and scientific argument needed to answer this question. Do not assume |
| in advance that the radiation contains a usable signature. |
|
|
| ## Physical systems |
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| Use atomic units. System A is an isolated two-level system with energy |
| separation $\omega_{10}=0.1$, real transition dipole $d=-1.5$, and its lower |
| level initially populated. Drive it with peak field amplitude $E_0=0.015$. |
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|
| Use a finite field |
|
|
| $$ |
| E(t)=E_0\,\operatorname{env}(t)\cos(\omega_L t), |
| $$ |
|
|
| where $\operatorname{env}(t)$ is a Gaussian centred at |
| $t_{\mathrm{end}}/2$, with |
| |
| $$ |
| t_{\mathrm{end}}=\frac{120\pi}{\omega_L}, |
| \qquad |
| \sigma=\frac{t_{\mathrm{end}}}{16}, |
| $$ |
|
|
| and the finite record is brought smoothly to zero at its boundaries. Choose a |
| photon-frequency and radiation-frequency domain sufficient to determine the |
| transition-dependent structures on which your conclusion relies. |
|
|
| System B contains two uncoupled two-level transitions with orthogonal dipole |
| axes. Their energy separations are $0.1$ along $x$ and $0.126$ along $y$, and |
| both transition dipoles are $-1.5$. Initially populate the two lower levels |
| with equal coherent amplitudes and zero relative phase. Drive both transitions |
| simultaneously with the same finite-pulse convention, peak amplitude |
| $E_0=0.015$, carrier frequency $\omega_L=0.1/7.5$, and a variable linear- |
| polarization direction $\alpha$ measured from the $x$ axis. Choose an angular |
| and radiation-frequency domain sufficient to test the claimed polarization |
| signature. |
|
|
| ## Research boundary |
|
|
| Use a standard few-level TDSE treatment. All other theoretical, numerical, |
| diagnostic and presentation choices are yours. Establish that the evidence |
| supporting each retained claim is sufficiently reliable and reproducible. |
| Absolute spectral intensity is not a grading target. |
|
|
| ## Submission and resources |
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|
| Submit one self-contained scientific report and an offline reproduction |
| package containing the producing code, numerical data underlying the retained |
| evidence, 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 scientific figure set, file tree or exploratory history |
| is required. |
|
|
| The active examination resources and limits are defined in the companion |
| `resource_limits.json`. All retained production calculations must run |
| within that fixed allocation and active wall time. More expensive modelling |
| does not receive additional credit. |
|
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| ## Reference and integrity policy |
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|
| 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. |
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|
| See [submission requirements](../../submission_requirements.md) for the common submission entry points. |
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