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## Background
A strongly driven two-level emitter can radiate while ending in either of its
bare states. The same radiation may also be discussed in terms of the part
that is phase coherent across an ensemble of identical emitters and the part
associated with quantum fluctuations. These are different physical
resolutions of one emitted state. Their relation under strong driving is the
object of this task.
## Research problem
Develop a quantum-optical calculation that resolves the radiation from the
driven two-level system both by the emitter's final state and by radiation
coherence. Determine how the two descriptions are related and whether, and if
so how, their relationship can be controlled by changing the driving strength.
The derivation, numerical construction, choice of scientific evidence and
organization of the argument are yours. Establish the conclusion from your
calculation rather than assuming a fixed correspondence between the two
descriptions.
## Physical inputs
The emitter has bare states $|g\rangle$ and $|e\rangle$, energy separation
$\hbar\omega_a$, and only the off-diagonal electric-dipole matrix element
$\Omega=\langle e|\hat d|g\rangle$ (with its conjugate). It starts in
$|g\rangle$. The quantized radiation modes start in vacuum.
Treat the classical drive nonperturbatively. Its electric field is
$$
E_L(t)=E_0\sin^2\!\left(\frac{\omega_0t}{2N_c}\right)
\sin(\omega_0t),
\qquad 0\le t\le \frac{2\pi N_c}{\omega_0},
$$
and is zero outside this interval. Use the same sine-squared envelope as the
turn-on and turn-off factor in the coupling to the quantized radiation modes.
Treat that coupling to first order; reabsorption and higher orders in the
emitted field are outside the task.
Use
$$
\omega_0=\frac{\omega_a}{7},\qquad N_c=20,
$$
and the following cases:
| Case | $E_0|\Omega|/\omega_0$ |
|---|---:|
| A | 11.23 |
| B | 13.21 |
| C | 13.87 |
You may set $\hbar=\omega_a=|\Omega|=1$ or use another consistent unit
system. State all phase, Fourier and basis conventions.
The graded spectral object is the squared modulus of the first-order emission
amplitude before any optional frequency-dependent radiative,
density-of-modes or detector prefactor. Declare any constant normalization
used in a comparison; absolute spectral magnitude is not a grading target.
## Method boundary
Use standard two-level dynamics and first-order quantum-optical emission
theory. You must derive the task-specific quantities used to resolve the two
descriptions and their relationship. No particular propagator, quadrature,
grid, transform implementation, plotting layout or validation procedure is
prescribed.
You are responsible for checking that the computed evidence is sufficiently
reliable and reproducible for the claims made. A state or spectral comparison
must be backed by reconstructible numerical data from the corresponding case.
## Submission
Submit one self-contained scientific report and an offline reproduction
package containing the derivation, executable code, numerical data underlying
the reported emitter-state and radiation-coherence evidence, references
actually used, and concise run and environment information. The choice and
number of figures are yours. No fixed file tree, filename scheme, exploratory
history or examiner-selected validation sequence is required.
Include a provenance note listing every offline source or software document actually consulted and the method basis for items supplied from general knowledge.
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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