cases/case04/task.md

Quantum radiation from a strongly driven two-level system

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

$$E_L(t)=E_0{\sin }^2\text{\hspace{0.17em}⁣}\left(\frac{{\omega }_{0}t}{2N_c}\right)\sin ({\omega }_{0}t),0\le t\le \frac{2\pi N_c}{{\omega }_0},E\_L(t)=E\_0\backslash sin^2\backslash !\backslash left(\backslash frac\{\backslash omega\_0t\}\{2N\_c\}\backslash right)\; \backslash sin(\backslash omega\_0t),\; \backslash qquad\; 0\backslash le\; t\backslash le\; \backslash frac\{2\backslash pi\; N\_c\}\{\backslash omega\_0\},$$

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

$${\omega }_0=\frac{{\omega }_a}{7},N_c=20,\backslash omega\_0=\backslash frac\{\backslash omega\_a\}\{7\},\backslash qquad\; N\_c=20,$$

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 for the common submission entry points.