cases/case21/task.md

Can smooth single-lobe fields generate spectral interference?

Background and research question

Momentum-space interference in strong-field particle production is often associated with carrier oscillations, pulse trains or several distinct real-time production events. It is not evident whether such temporal substructure is necessary.

For the two smooth, carrier-free single-lobe fields specified below, determine whether spectral interference can arise without subcycle structure, under what nonadiabatic conditions it becomes physically relevant, and what sets its origin. Assess whether the same explanation remains valid when the problem is transferred from scalar QED to a gapped two-dimensional Dirac material. State the strongest conclusion supported by your computed evidence and its physical scope.

Design the task-specific derivation, calculation, evidence and scientific argument needed to answer this question. Do not assume the outcome in advance.

Physical systems and research boundary

Driving fields

Write the spatially homogeneous, linearly polarized electric field as

$$E(t)=E_{0}e(t\mathrm{/}\tau ),A(t)=-{\int }_0^{t}E(t^{\mathrm{\prime }})d{t}^{\mathrm{\prime }}\mathrm{.}E(t)=E\_0\; e(t/\backslash tau),\; \backslash qquad\; A(t)=-\backslash int\_0^t\; E(t\text{'})\backslash ,dt\text{'}.$$

Compare the Gaussian and weakly deformed profiles

$$e_{\mathrm{G}}(z)=\exp (-z^2),e_{\mathrm{D}}(z)=\exp (-z^2-z^4)\mathrm{.}e\_\{\backslash mathrm\; G\}(z)=\backslash exp(-z^2),\; \backslash qquad\; e\_\{\backslash mathrm\; D\}(z)=\backslash exp(-z^2-z^4).$$

Use the same field and vector-potential conventions consistently in both physical settings.

Scalar-QED setting

Use $\hbar=c=1$, electron mass $m$ and electron charge $e<0$. The peak field is

$$E_0=0.1E_{\mathrm{c}},E_{\mathrm{c}}=\frac{m^2}{\mathrm{\mid }e\mathrm{\mid }}\mathrm{.}E\_0=0.1E\_\{\backslash mathrm\; c\},\; \backslash qquad\; E\_\{\backslash mathrm\; c\}=\backslash frac\{m^2\}\{|e|\}.$$

Consider the longitudinal asymptotic particle spectrum at zero transverse momentum for

$$\tau =5,10,20m^{-1},\gamma =\frac{m}{\mathrm{\mid }e\mathrm{\mid }{E}_0\tau }=2,1,0.5,\backslash tau=5,\backslash \; 10,\backslash \; 20\backslash \; m^\{-1\},\; \backslash qquad\; \backslash gamma=\backslash frac\{m\}\{|e|E\_0\backslash tau\}=2,\backslash \; 1,\backslash \; 0.5,$$

respectively. Production spectra must be obtained from exact numerical scalar mode evolution or a physically equivalent exact kinetic formulation. Complex-time turning-point or WKB methods may be used to investigate the mechanism, but an approximate semiclassical spectrum is not a substitute for the exact production signal.

Gapped-Dirac-material setting

Use the spatially uniform two-dimensional Hamiltonian

$$H_{\mathbf{k}}(t)=v_{\mathrm{F}}\mathit{\sigma }\cdot [\mathrm{\hslash }\mathbf{k}-e\mathbf{A}(t)]+\mathrm{\Delta }{\sigma }_z,H\_\{\backslash mathbf\; k\}(t)=v\_\{\backslash mathrm\; F\}\backslash boldsymbol\{\backslash sigma\}\backslash cdot\; [\backslash hbar\backslash mathbf\; k-e\backslash mathbf\; A(t)]+\backslash Delta\backslash sigma\_z,$$

where $\Delta$ is the Dirac mass term, equal to half the band gap. Take

$$v_{\mathrm{F}}={10}^6\mathrm{m}{\mathrm{s}}^{-1},\mathrm{\Delta }=0.1\mathrm{e}\mathrm{V},E_0=40\mathrm{k}\mathrm{V}\mathrm{c}{\mathrm{m}}^{-1},v\_\{\backslash mathrm\; F\}=10^6\backslash \; \backslash mathrm\{m\backslash ,s^\{-1\}\},\backslash qquad\; \backslash Delta=0.1\backslash \; \backslash mathrm\{eV\},\backslash qquad\; E\_0=40\backslash \; \backslash mathrm\{kV\backslash ,cm^\{-1\}\},$$

and consider the post-pulse longitudinal conduction-band occupation at zero transverse wave vector for

$$\tau =20,30,50\mathrm{f}\mathrm{s},{\gamma }_{\mathrm{D}}=\frac{\mathrm{\Delta }}{\mathrm{\mid }e\mathrm{\mid }{v}_{\mathrm{F}}{E}_0\tau }=1.25,0.833,0.5,\backslash tau=20,\backslash \; 30,\backslash \; 50\backslash \; \backslash mathrm\{fs\},\; \backslash qquad\; \backslash gamma\_\{\backslash mathrm\; D\}=\backslash frac\{\backslash Delta\}\{|e|v\_\{\backslash mathrm\; F\}E\_0\backslash tau\}\; =1.25,\backslash \; 0.833,\backslash \; 0.5,$$

respectively. Use exact numerical Dirac evolution, an adiabatic-basis Bogoliubov formulation, a quantum-kinetic formulation, or another physically equivalent exact evolution with the filled valence band as the incoming state.

The final occupations are asymptotic post-pulse observables. The numerical time boundary must therefore be far enough into the field-free region for the definition used in your calculation to be valid; no particular finite cutoff is prescribed.

The momentum domain, sampling, evolution algorithm, time step, asymptotic truncation, complex-root method, diagnostic construction and evidence representation are yours. Define the conventions needed to make the results reconstructible. No fixed grid, solver, tolerance, plotting range, figure set or author-specific decomposition is prescribed. Absolute occupations are graded when they follow from the stated normalization; comparisons displayed with an additional arbitrary scale must retain the underlying unscaled data.

Submission and resources

Submit one self-contained scientific report and an auditable offline calculation package containing the producing code, reconstructible numerical data for 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 argument and the choice and number of figures are yours. Evidence must be sufficiently reliable and reproducible for the claims made, but no particular validation procedure is required.

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 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.