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## Background and central research question
Strong-field excitation in a dense material is often represented by a single
active electron with phenomenological loss of coherence. A single dephasing
time is convenient, but the physical regime in which it predicts the final
population reliably is not evident.
Investigate the same question with an explicit bosonic environment. Determine
whether, and under what physically justified bath conditions, the environment
enhances or suppresses the post-pulse conduction-band population. Establish
the physical origin of any regime change, assess when a single relaxation-time
description is an adequate surrogate, and state the strongest conclusion
supported by your evidence and its scope. Do not assume the outcome in advance.
Design the task-specific derivation, computation, evidence and scientific
argument needed to answer this one research question. The organization of the
argument and the number and form of the numerical outputs are yours.
## Physical model and indispensable inputs
Use a one-dimensional two-band model along a symmetry direction of a ZnO-like
semiconductor. In the valence/conduction basis, couple the driven electron
longitudinally to a thermal bosonic bath:
$$
H(t)=-\frac{1}{2}{\cal E}(K_t)\sigma_z
+\frac{\hbar\Omega(K_t,t)}{2}\sigma_x
+\sum_q\hbar\omega_q b_q^\dagger b_q
+\sigma_z\sum_q g_q\left(b_q+b_q^\dagger\right),
$$
where
$$
K_t=K+\frac{eA(t)}{\hbar},\qquad
-\frac{dA}{dt}=E(t),\qquad
\Omega(K_t,t)=\frac{2e}{\hbar}d(K_t)E(t),
$$
and $e>0$. Use
$$
{\cal E}(K)=E_g+\Delta\,[1-\cos(Ka)],
\qquad
d(K)=\frac{E_gd_0}{{\cal E}(K)},
$$
with
$$
E_g=0.129E_h,\qquad
\Delta=0.17E_h,\qquad
a=5.32a_0,\qquad
d_0=3.64a_0.
$$
The field is linearly polarized and has the form
$$
E(t)=E_0\exp(-t^2/\tau^2)\cos(\omega_0t),
$$
with
$$
\lambda_0=\frac{2\pi c}{\omega_0}=3.2\ \mu{\rm m},
\qquad
\tau=20\ {\rm fs},
\qquad
E_0=1.5\times10^9\ {\rm V\,m^{-1}}.
$$
Initially the valence band is occupied, the conduction band is empty, and the
bath is in thermal equilibrium. For positive-frequency bath modes, use the
spectral-density normalization
$$
J_+(\omega)=2\sum_q
\left|\frac{g_q}{\hbar\omega_q}\right|^2
\delta(\omega-\omega_q),\qquad \omega>0.
$$
In any all-frequency bath integral, $J(\omega)$ denotes the odd extension of
$J_+$: $J(\omega)=J_+(\omega)$ for $\omega>0$,
$J(\omega)=-J_+(-\omega)$ for $\omega<0$, and $J(0)=0$. Under this convention,
$J$ has units of inverse angular frequency. Represent the bath by the Ohmic
form
$$
J(\omega)=j_0\frac{\omega}{\omega_c^2}
\exp\!\left(-\frac{|\omega|}{\omega_c}\right),
$$
where $j_0$ is dimensionless. This normalization is part of the physical case;
an alternative common convention is acceptable only after an explicit
conversion that leaves the same bath influence.
The final comparison observable is
$$
\eta=\frac{n_c(j_0\ne0,t\rightarrow\infty)}
{n_c(j_0=0,t\rightarrow\infty)},
$$
where $n_c$ is the conduction population integrated over the one-dimensional
Brillouin zone. Make clear how the mathematical observable used in your code
implements this definition.
The mandatory comparison set consists of all eight combinations
$$
T\in\{300\ {\rm K},\ 2.0\times10^4\ {\rm K}\},\qquad
j_0\in\{0.1,\ 5\},\qquad
\frac{\omega_c}{\omega_0}\in\{0.1,\ 3.4\}.
$$
Within this commission, these paired values define the labels low/high
temperature, weak/strong coupling, and lower/higher cutoff. They are case
anchors rather than universal regime boundaries. You may add other bath cases
when they materially test a conclusion, but no unlisted source sweep is
required.
Determine from the calculation how temperature, coupling and cutoff affect
the answer; no direction, ordering or magnitude is stipulated. Conclusions
about a dynamic material temperature, electron-electron scattering, damage or
experiment must be limited to what the stated equilibrium bosonic-bath model
supports.
## Method boundary and numerical freedom
Work within the bath-traced second-order Dyson treatment of the driven
spin-boson model, or an algebraically and physically equivalent formulation
with the same second-order content. The representation, transformations,
bath kernel and numerical realization are for you to derive or source from
eligible standard knowledge. A bare replacement of the bath by an imposed
relaxation time is not a substitute for the bath-resolved calculation.
The time and momentum domains, quadrature or propagation algorithm, grid,
step size, tolerances, any optional parameter sampling, controls, diagnostics
and evidence representation are student choices. Define enough conventions to
make the calculation reconstructible. No fixed plot, panel sequence, scan
range, display convention or author-specific numerical setting is prescribed.
## Submission and resources
Submit one self-contained scientific report and an auditable offline
calculation package containing producing code, reconstructible numerical data
for material claims, a provenance note listing every offline source or software document actually consulted and the method basis for items supplied from general knowledge, and concise run and environment
information. Evidence must be sufficiently reliable for the claims made, but
no particular validation procedure is required.
Resource limits are supplied exclusively by the accompanying
`resource_limits.json`. Its fixed CPU allocation, candidate-active
wall time, network boundary and temporary-storage allowance are binding.
No separate aggregate CPU-hour limit applies.
## 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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