cases/case10/task.md

Harmonic radiation from driven few-level systems

Background and research question

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.

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

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 $E_0=0.015$.

Use a finite field

$$E(t)=E_0\mathrm{env}(t)\cos ({\omega }_{L}t),E(t)=E\_0\backslash ,\backslash operatorname\{env\}(t)\backslash cos(\backslash omega\_L\; t),$$

where $\operatorname{env}(t)$ is a Gaussian centred at $t_{\mathrm{end}}/2$, with

$$t_{\mathrm{e}\mathrm{n}\mathrm{d}}=\frac{120\pi }{{\omega }_L},\sigma =\frac{t_{\mathrm{e}\mathrm{n}\mathrm{d}}}{16},t\_\{\backslash mathrm\{end\}\}=\backslash frac\{120\backslash pi\}\{\backslash omega\_L\},\; \backslash qquad\; \backslash sigma=\backslash frac\{t\_\{\backslash mathrm\{end\}\}\}\{16\},$$

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

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.

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.