cases/case07/task.md

Nonlinear radiation and field-state dynamics in a quantized two-level model

Background

High-order harmonic generation is usually formulated with a classical driving field. A fully quantized driver can also carry signatures of its interaction with matter, so the field state itself may contain information about how the nonlinear radiation is produced. A useful account must connect this quantum phase-space dynamics to the emitted harmonic response without relying on the rotating-wave approximation.

Research problem

Develop and assess a quantum phase-space account of high-order harmonic generation in the two-level, single-mode system below. Determine how the atom-field interaction reorganizes the initially coherent field, how the different physical terms in the full Hamiltonian contribute to the reduced- field dynamics and harmonic radiation, and how the resulting high-harmonic response changes with the initial field amplitude.

Use calculated field-state and harmonic evidence to support the answer. The derivation, numerical realization, diagnostic design, choice of figures and structure of the scientific argument are yours. Do not assume the proposed physical picture before testing it.

Physical inputs

Use the standard electric-dipole quantum Rabi model for a two-level system coupled to one quantized bosonic mode. The matter transition frequency is $\omega_0$, the field frequency is $\omega$, and the dimensionless dipole coupling magnitude is

$$g=\mid \frac{\mathrm{\Omega }}{\omega }\mid \mathrm{.}g=\backslash left|\backslash frac\{\backslash Omega\}\{\backslash omega\}\backslash right|.$$

Do not make a rotating-wave approximation. Use

$$\frac{{\omega }_0}{\omega }=2.2,T=\frac{2\pi }{\omega }\mathrm{.}\backslash frac\{\backslash omega\_0\}\{\backslash omega\}=2.2,\; \backslash qquad\; T=\backslash frac\{2\backslash pi\}\{\backslash omega\}.$$

The atom begins in its lower energy state $|g\rangle$. The field begins in a positive-real coherent state $|\alpha_0\rangle$. Investigate all four cases:

Case	$\alpha_0$	$g$	Required physical interval
A	$10$	$0.5$	first optical cycle
B	$10^5$	$10^{-5}$	20 optical cycles
C	$2\times10^5$	$10^{-5}$	20 optical cycles
D	$4\times10^5$	$10^{-5}$	20 optical cycles

State the atomic-basis phase, field-quadrature orientation, Fourier-sign, harmonic-order and display conventions used in your work. Globally equivalent branch phases, label exchanges and phase-space reflections are accepted after consistent alignment. Absolute harmonic-spectrum normalization is not a grading target.

Method and scope

Use a moving von Neumann lattice of coherent states to treat the complete atom-field dynamics. Here this method name refers to a nonorthogonal coherent- state representation whose field labels are drawn from

$${\alpha }^{(mn)}=\sqrt{\pi }(m+in),m,n\in \mathbb{Z},\backslash alpha^\{(mn)\}=\backslash sqrt\{\backslash pi\}\backslash ,(m+in),\backslash qquad\; m,n\backslash in\backslash mathbb\; Z,$$

with an appropriate finite subset chosen by the investigator. Standard coherent-state and quantum phase-space identities may be derived or cited. No particular auxiliary construction is prescribed.

The final research account must be supported by a computed phase-space description of the reduced field state and by high-harmonic radiation obtained from the material dipole response. All numerical choices not fixed above are yours. Check that the reported signals are sufficiently reliable and reproducible to support the conclusions; no particular lattice size, solver, time step, tolerance, phase-space grid, Fourier treatment, display range or refinement sequence is prescribed.

Submission

Submit one self-contained scientific report and an offline reproduction package. The report must answer the central research question and include enough quantitative evidence to audit every case-dependent or mechanistic claim. The selection and organization of that evidence are yours.

Include a provenance note listing every offline source or software document actually consulted and the method basis for items supplied from general knowledge.

Include the derivation actually used, source code, an executable command or script, machine-readable data underlying the reported evidence, plotting or analysis code, references, and concise run and environment information. No fixed file tree, filename scheme, complete exploratory history, named control, or examiner-selected intermediate plot 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.