| # 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 |
| |
| $$ |
| g=\left|\frac{\Omega}{\omega}\right|. |
| $$ |
| |
| Do not make a rotating-wave approximation. Use |
| |
| $$ |
| \frac{\omega_0}{\omega}=2.2, |
| \qquad T=\frac{2\pi}{\omega}. |
| $$ |
|
|
| 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 |
|
|
| $$ |
| \alpha^{(mn)}=\sqrt{\pi}\,(m+in),\qquad m,n\in\mathbb Z, |
| $$ |
|
|
| 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](../../submission_requirements.md) for the common submission entry points. |
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|
