# Causal time-domain inference in a driven three-state system ## Background Attosecond transient-absorption spectra can encode the causal electronic response of a laser-dressed system. Recovering the corresponding motion in the time domain would make the spectrum a direct probe of ultrafast dynamics, but an experimental attosecond probe has a finite duration rather than the ideal impulsive form often assumed in such an interpretation. ## Research problem Develop and assess a reproducible procedure, within causal linear-response theory, for recovering the time-dependent dipole motion of the infrared-dressed system below from its transient-absorption signal. Determine how faithfully the relevant motion can be recovered for attosecond-probe durations of 100, 300 and 500 as, why the finite probe matters, and what physical claim is supported by the calculated evidence. The derivation, numerical realization, signal processing, diagnostic design, choice of figures and structure of the scientific argument are yours. Assess the reconstruction from computed evidence rather than assuming that it works. ## Physical inputs Use atomic units for the dynamical calculation. The model contains a ground state $|g\rangle$, a dipole-allowed excited state $|a\rangle$ and an infrared-coupled state $|f\rangle$. The only nonzero transition dipoles in the model are $$ \mu_{ga}=0.34,\qquad \mu_{af}=2.72, $$ with energy differences $$ E_a-E_g=21.057\ \mathrm{eV},\qquad E_f-E_a=-0.82\ \mathrm{eV}. $$ The system is initially in $|g\rangle$ before either field is appreciable. Use electric-dipole coupling and the following linearly polarized fields. The attosecond probe is $$ E_X(t)=E_{X0}e^{-\lambda |t|}\cos(\omega_X t), \qquad \tau_X=\frac{2\ln 2}{\lambda}, $$ with photon energy $21\ \mathrm{eV}$, peak intensity $10^{10}\ \mathrm{W\,cm^{-2}}$, and $\tau_X=100$, 300 or 500 as. Define the probe centre as $t=0$. The infrared field is $$ E_{\mathrm{IR}}(t,\tau)=E_0 A(t-\tau) \cos[\omega_{\mathrm{IR}}(t-\tau)], $$ where $$ A(u)= \begin{cases} \cos^2\!\left(\dfrac{\omega_{\mathrm{IR}}u}{2N}\right), & |u|\le N\pi/\omega_{\mathrm{IR}},\\[4pt] 0,&\text{otherwise}, \end{cases} $$ $N=10$, $\hbar\omega_{\mathrm{IR}}=0.82\ \mathrm{eV}$ and the peak intensity is $5\ \mathrm{TW\,cm^{-2}}$. Set the infrared-pulse centre five infrared optical cycles after the probe centre. Obtain field amplitudes from a standard, consistently reported intensity conversion. For this task, the source-model decoherence is imposed by the following author-supplied policy. First solve the driven three-state problem and obtain its physical real dipole $d_0(t)$. Then define $$ d_{\mathrm{phys}}(t)= \begin{cases} d_0(t),&t\le 0,\\ d_0(t)e^{-t/(50\ \mathrm{fs})},&t>0. \end{cases} $$ Apply this operation before constructing the absorption signal. Do not put it in the Hamiltonian, reinterpret it as population or amplitude damping, or apply an additional Fourier apodization window. State the time, frequency, Fourier-sign, dipole-sign and display conventions used in your work. Absolute arbitrary-unit vertical scale is not a grading target. ## Method and scope Use the standard three-level time-dependent Schrödinger equation for the stated couplings and a causal linear-response or Kramers-Kronig reconstruction from the positive-frequency transient-absorption signal. You may choose and cite standard formulations consistent with these method families. Do not add levels, ionization channels or nonlinear probe response to the graded model. All numerical choices not fixed above are yours. Check that the reported signals are sufficiently reliable and reproducible to support your claim; no particular grid, record length, propagator, transform algorithm, regularizer, comparison metric or refinement sequence is prescribed. ## Submission Submit one self-contained scientific report and an offline reproduction package. The report must answer the research question from your calculated time-dependent reconstruction and must contain enough quantitative evidence to audit every probe-duration-dependent 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 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.