| # Gap and drive dependence of high-harmonic emission in two quantum models |
|
|
| ## Background |
|
|
| The excitation gap and light-matter coupling are both central scales in |
| off-resonant high-harmonic generation. Their joint influence can be studied in |
| a dissipative two-level model and then compared with a two-band semiconductor |
| model that includes dispersion and momentum aggregation. |
|
|
| ## Research problem |
|
|
| Determine how the excitation gap and peak light-matter coupling jointly |
| organize the polarization-based high-harmonic response of the two model |
| families below. Establish the physical origin of the gap dependence found in |
| the two-level model. Then assess which parts of that |
| explanation transfer to the semiconductor calculation and state the supported |
| scope of the conclusion. |
|
|
| How you construct and substantiate the investigation is for you to decide. |
| Ground every material claim in calculated evidence, and do not assume the |
| direction of the gap dependence or the outcome of the transfer test in advance. |
|
|
| ## Physical models and inputs |
|
|
| For the two-level model, use units with \(e=\hbar=d=1\), where \(d\) is the |
| constant transition dipole. Take the energy unit to be |
| \(\hbar\omega_u=1~\mathrm{eV}\). |
| |
| For the two-level branch, use |
| |
| $$ |
| H(t)= |
| \begin{pmatrix} |
| -\Delta/2 & -\Omega(t)\\ |
| -\Omega(t) & \Delta/2 |
| \end{pmatrix}, |
| \qquad |
| \Omega(t)=\Omega_0\cos(\omega_0 t)\exp(-t^2/\tau^2). |
| $$ |
| |
| Begin in the lower field-free level. Include phenomenological longitudinal |
| relaxation toward that level at rate \(\gamma_L\) and transverse decay of the |
| coherence at rate \(\gamma_T\). Use |
| |
| | Quantity | Value or condition | |
| |---|---| |
| | Carrier frequency | \(\omega_0=0.3\) | |
| | Pulse width | \(\tau=8.5\pi\) | |
| | Reference relaxation rates | \(\gamma_L=2.0\), \(\gamma_T=1.0\) | |
| | Excitation regime | Keep the study and its claims in the off-resonant regime, \(\omega_0\ll\Delta\) | |
| | Gap and coupling study | Sample a domain broad enough to compare coupling below, near, and above the gap and to support every global claim made | |
| |
| The population difference and coherence may be derived from the density-matrix |
| equation. The radiating polarization is the expectation value of the dipole, |
| which is proportional to the real part of the coherence under this convention. |
| |
| For the transfer study, use units with \(e=\hbar=a=1\), where \(a\) is the |
| lattice constant. Work at low doping on the two-dimensional square Brillouin |
| zone \(-\pi\leq k_x,k_y\leq\pi\), and use equal electron and hole dispersions |
| |
| $$ |
| E_{\mathbf k}^{e/h}=\frac{\Delta}{2}+\frac{k^2}{2m}, |
| \qquad m=2.0. |
| $$ |
|
|
| Let \(\theta_{\mathbf k}=\operatorname{atan2}(k_y,k_x)\), and introduce the |
| Nambu spinor |
| \(\Psi_{\mathbf k}=(\alpha_{\mathbf k},\beta^\dagger_{-\mathbf k})^T\). |
| Up to an irrelevant scalar term, use the single-\(\mathbf k\) Hamiltonian |
|
|
| $$ |
| h_{\mathbf k}(t)= |
| \begin{pmatrix} |
| E_{\mathbf k}^{e}+\Omega(t)\cos\theta_{\mathbf k} |
| & |
| i\Omega(t)\sin\theta_{\mathbf k} |
| \\ |
| -i\Omega(t)\sin\theta_{\mathbf k} |
| & |
| -E_{\mathbf k}^{h}-\Omega(t)\cos\theta_{\mathbf k} |
| \end{pmatrix}. |
| $$ |
| |
| Here \(\alpha_{\mathbf k}\) annihilates a conduction-band electron and |
| \(\beta_{-\mathbf k}\) annihilates a hole. Begin with an empty conduction band, |
| no holes, and zero interband polarization. Derive the corresponding standard |
| semiconductor Bloch equations for the electron and hole populations and one |
| complex interband polarization \(P_{\mathbf k}\). Use \(P_{\mathbf k}\) |
| consistently in the dynamics and observable. Add longitudinal population decay |
| and transverse polarization decay with \(\gamma_L=2.0\) and \(\gamma_T=1.0\). |
| Define the macroscopic polarization by |
| |
| $$ |
| p(t)=-\sum_{\mathbf k} |
| \left\langle\frac{\partial h_{\mathbf k}(t)} |
| {\partial\Omega(t)}\right\rangle. |
| $$ |
| |
| For this branch use the same Gaussian carrier form with \(\tau=16\pi\) and |
| \(\omega_0=0.3\). State all basis, phase, and momentum-sum conventions used. |
|
|
| ## Observable and study boundary |
|
|
| Construct the HHG spectrum from the Fourier transform of the corresponding |
| polarization. Use the fifth-harmonic spectral intensity as the common |
| quantitative reference between models. Other evidence may be introduced when |
| it has a clear discriminating function. |
|
|
| The overall intensity scale is arbitrary. Declare the transform, |
| normalization, and comparison conventions, and keep them common wherever |
| relative magnitudes support a claim. Absolute source-scale intensity is not a |
| target. |
|
|
| All momentum quadratures, time records, grids, propagators, transform details, |
| windowing choices, parameter sampling, plotting conventions, and convergence |
| or stability tests are yours. Document them and establish that the reported |
| physical conclusions are reliable. No particular solver, grid, diagnostic, |
| 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. |
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|
