cases/case35/task.md

Testing phase-space descriptions of quantum-light high-harmonic generation

Background and physical inputs

High-harmonic generation (HHG) driven by a nonclassical optical state need not be characterized by a classical field with one fixed amplitude. The validity of a phase-space treatment depends on whether its representation of the optical state preserves the physically relevant coherence.

Work in atomic units with a one-dimensional single-active-electron model

$$H_0=\frac{p^2}{2}+V(x),V(x)=-\frac{1}{\sqrt{x^2+1}}\mathrm{.}H\_0=\backslash frac\{p^2\}\{2\}+V(x),\; \backslash qquad\; V(x)=-\backslash frac\{1\}\{\backslash sqrt\{x^2+1\}\}.$$

For every quantum-light component and matched classical control, initialize the electron in the normalized field-free ground state $|\psi_g\rangle$ of $H_0$.

Represent each quantum optical input on the square generalized von Neumann lattice

$$\mathrm{\mid }{\alpha }_{mn}⟩,{\alpha }_{mn}=\frac{\sqrt{\pi }}{2}(m+in),m,n\in \mathbb{Z}\mathrm{.}|\backslash alpha\_\{mn\}\backslash rangle,\; \backslash qquad\; \backslash alpha\_\{mn\}=\backslash frac\{\backslash sqrt\{\backslash pi\}\}\{2\}(m+in),\; \backslash qquad\; m,n\backslash in\backslash mathbb\; Z.$$

On the retained finite lattice, write the specified optical state as

$$\mathrm{\mid }\chi ⟩=\sum _i{C}_i\mathrm{\mid }{\alpha }_i⟩,G_{ij}=⟨{\alpha }_i\mathrm{\mid }{\alpha }_j⟩,C^{†}GC=1,|\backslash chi\backslash rangle=\backslash sum\_i\; C\_i|\backslash alpha\_i\backslash rangle,\; \backslash qquad\; G\_\{ij\}=\backslash langle\backslash alpha\_i|\backslash alpha\_j\backslash rangle,\; \backslash qquad\; C^\backslash dagger\; G\; C=1,$$

where the single index $i$ labels retained pairs $(m,n)$. The coefficients must represent the stated optical state to the accuracy needed for the claims. Their construction, lattice extent, and numerical solution are part of the work.

For each coherent component, propagate the same initial electronic state and let $j_i(\Omega)$ denote its complex HHG response amplitude at angular frequency $\Omega$, in a consistently declared dipole, current, or acceleration convention. Define

$$u_i(\mathrm{\Omega })=C_i{j}_i^{\ast }(\mathrm{\Omega })\mathrm{.}u\_i(\backslash Omega)=C\_i\; j\_i^*(\backslash Omega).$$

The required production response is the full nonorthogonal quadratic form

$$S_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}(\mathrm{\Omega })\propto u^{†}(\mathrm{\Omega })Gu(\mathrm{\Omega })\mathrm{.}S\_\{\backslash mathrm\{full\}\}(\backslash Omega)\backslash propto\; u^\backslash dagger(\backslash Omega)G\backslash ,u(\backslash Omega).$$

The corresponding diagonal-only comparator is obtained by deleting every $i\ne j$ term from the same form,

$$S_{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}}(\mathrm{\Omega })\propto \sum _i{G}_{ii}\mathrm{\mid }{u}_i(\mathrm{\Omega }){\mathrm{\mid }}^2,S\_\{\backslash mathrm\{diag\}\}(\backslash Omega)\backslash propto\; \backslash sum\_i\; G\_\{ii\}|u\_i(\backslash Omega)|^2,$$

with $G_{ii}=1$ for normalized coherent states. An equivalent nonorthogonal coordinate convention is acceptable if it preserves the same metric quadratic form. Any other positive-weight diagonal phase-space mixture is an additional approximation: state its weights and keep it distinct from $S_{\mathrm{full}}$.

For a coherent component with carrier frequency $\omega$, bind its classical field amplitude and phase to its lattice amplitude by

$$E_0=2g_0\sqrt{\omega }\mathrm{\mid }\alpha \mathrm{\mid },\varphi =-\arg \alpha \mathrm{.}E\_0=2g\_0\backslash sqrt\{\backslash omega\}\backslash ,|\backslash alpha|,\; \backslash qquad\; \backslash phi=-\backslash arg\backslash alpha.$$

For a squeezed vacuum use $\alpha_r=\sinh r$, and for a Fock state use $\alpha_N=\sqrt N$, when calibrating $g_0$ to the stated effective intensity. For the single-mode cases, obtain the electronic response from the standard one-dimensional TDSE driven by a ten-cycle trapezoidal electric field with a two-cycle turn-on, six-cycle plateau, and two-cycle turn-off. For uniqueness in the two-colour case, use the same common temporal envelope, with its total duration defined by ten cycles of the 800 nm fundamental; this common-envelope choice is task policy.

Investigate the following physical cases together with classical fields having the same carrier frequencies, intensities, phase settings, pulse envelope, and electronic initial state:

case	quantum optical input	carrier and effective intensity
single mode	squeezed vacuum with real $r=4$	$800\ \mathrm{nm}$, $1.0\times10^{13}\ \mathrm{W,cm^{-2}}$
single mode	Fock state $\lvert N=10000\rangle$	$800\ \mathrm{nm}$, $1.0\times10^{14}\ \mathrm{W,cm^{-2}}$
two colour	an $800\ \mathrm{nm}$ coherent state with real amplitude and a $400\ \mathrm{nm}$ squeezed vacuum with $r=4$ whose extended quadrature lies on the imaginary-amplitude axis	$1.0\times10^{14}$ and $1.0\times10^{13}\ \mathrm{W,cm^{-2}}$, respectively

The primary observable is the emitted single-atom HHG spectrum. A common normalization may be used for comparisons; absolute spectral intensity is not a grading target.

Research problem

Determine whether a diagonal phase-space description is sufficient to characterize the HHG response of the stated nonclassical optical fields, and establish how the conclusion depends on photon-state structure and on the two-colour configuration. Base the conclusion on your own calculated radiation signals, matched classical controls, and evidence that connects the optical-state representation to the observed response. State the physical scope within which your conclusion is supported.

The task-specific derivation, numerical implementation, evidence architecture, spectral domain, diagnostics, and report organization are for you to decide. No particular number or layout of figures is prescribed. Finite lattice extent, coefficient cutoff, overlap storage, linear solver, TDSE grid and propagator, absorbing boundary, time step, transform convention, spectral estimator, normalization, display range, and reliability strategy are student choices. State the choices material to your claims and provide sufficiently reliable and reconstructible evidence.

The required production calculation is $S_{\mathrm{full}}$. Clearly separated diagonal approximations are permitted and are needed only insofar as they help answer the research question.

Submission and resources

Submit one self-contained scientific report and an auditable offline calculation package containing the task-specific derivation, code, numerical data underlying all 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. 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 for the common submission entry points.