cases/case23/task.md

Can a bosonic environment enhance or suppress strong-field ionization?

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:

$$H(t)=-\frac{1}{2}\mathcal{E}(K_t){\sigma }_z+\frac{\mathrm{\hslash }\mathrm{\Omega }(K_t,t)}{2}{\sigma }_x+\sum _q\mathrm{\hslash }{\omega }_q{b}_q^{†}{b}_q+{\sigma }_z\sum _q{g}_q(b_q+b_q^{†}),H(t)=-\backslash frac\{1\}\{2\}\{\backslash cal\; E\}(K\_t)\backslash sigma\_z\; +\backslash frac\{\backslash hbar\backslash Omega(K\_t,t)\}\{2\}\backslash sigma\_x\; +\backslash sum\_q\backslash hbar\backslash omega\_q\; b\_q^\backslash dagger\; b\_q\; +\backslash sigma\_z\backslash sum\_q\; g\_q\backslash left(b\_q+b\_q^\backslash dagger\backslash right),$$

where

$$K_t=K+\frac{eA(t)}{\mathrm{\hslash }},-\frac{dA}{dt}=E(t),\mathrm{\Omega }(K_t,t)=\frac{2e}{\mathrm{\hslash }}d(K_t)E(t),K\_t=K+\backslash frac\{eA(t)\}\{\backslash hbar\},\backslash qquad\; -\backslash frac\{dA\}\{dt\}=E(t),\backslash qquad\; \backslash Omega(K\_t,t)=\backslash frac\{2e\}\{\backslash hbar\}d(K\_t)E(t),$$

and $e>0$. Use

$$\mathcal{E}(K)=E_g+\mathrm{\Delta }[1-\cos (Ka)],d(K)=\frac{E_g{d}_0}{\mathcal{E}(K)},\{\backslash cal\; E\}(K)=E\_g+\backslash Delta\backslash ,[1-\backslash cos(Ka)],\; \backslash qquad\; d(K)=\backslash frac\{E\_gd\_0\}\{\{\backslash cal\; E\}(K)\},$$

with

$$E_g=0.129E_h,\mathrm{\Delta }=0.17E_h,a=5.32a_0,d_0=3.64a_0\mathrm{.}E\_g=0.129E\_h,\backslash qquad\; \backslash Delta=0.17E\_h,\backslash qquad\; a=5.32a\_0,\backslash qquad\; d\_0=3.64a\_0.$$

The field is linearly polarized and has the form

$$E(t)=E_0\exp (-t^2\mathrm{/}{\tau }^2)\cos ({\omega }_{0}t),E(t)=E\_0\backslash exp(-t^2/\backslash tau^2)\backslash cos(\backslash omega\_0t),$$

with

$${\lambda }_0=\frac{2\pi c}{{\omega }_0}=3.2\mu \mathrm{m},\tau =20\mathrm{f}\mathrm{s},E_0=1.5\times {10}^9\mathrm{V}{\mathrm{m}}^{-1}\mathrm{.}\backslash lambda\_0=\backslash frac\{2\backslash pi\; c\}\{\backslash omega\_0\}=3.2\backslash \; \backslash mu\{\backslash rm\; m\},\; \backslash qquad\; \backslash tau=20\backslash \; \{\backslash rm\; fs\},\; \backslash qquad\; E\_0=1.5\backslash times10^9\backslash \; \{\backslash rm\; V\backslash ,m^\{-1\}\}.$$

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

$$J_{+}(\omega )=2\sum _q{\mid \frac{g_q}{\mathrm{\hslash }{\omega }_q}\mid }^2\delta (\omega -{\omega }_q),\omega >0.J\_+(\backslash omega)=2\backslash sum\_q\; \backslash left|\backslash frac\{g\_q\}\{\backslash hbar\backslash omega\_q\}\backslash right|^2\; \backslash delta(\backslash omega-\backslash omega\_q),\backslash qquad\; \backslash omega>0.$$

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

$$J(\omega )=j_0\frac{\omega }{{\omega }_c^2}\exp \text{\hspace{0.17em}⁣}(-\frac{\mathrm{\mid }\omega \mathrm{\mid }}{{\omega }_c}),J(\backslash omega)=j\_0\backslash frac\{\backslash omega\}\{\backslash omega\_c^2\}\; \backslash exp\backslash !\backslash left(-\backslash frac\{|\backslash omega|\}\{\backslash omega\_c\}\backslash right),$$

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

$$\eta =\frac{n_c(j_0\mathrm{\ne }0,t\to \mathrm{\infty })}{n_c(j_0=0,t\to \mathrm{\infty })},\backslash eta=\backslash frac\{n\_c(j\_0\backslash ne0,t\backslash rightarrow\backslash infty)\}\; \{n\_c(j\_0=0,t\backslash rightarrow\backslash infty)\},$$

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

$$T\in \{300\mathrm{K},2.0\times {10}^4\mathrm{K}\},j_0\in \{0.1,5\},\frac{{\omega }_c}{{\omega }_0}\in \{0.1,3.4\}\mathrm{.}T\backslash in\backslash \{300\backslash \; \{\backslash rm\; K\},\backslash \; 2.0\backslash times10^4\backslash \; \{\backslash rm\; K\}\backslash \},\backslash qquad\; j\_0\backslash in\backslash \{0.1,\backslash \; 5\backslash \},\backslash qquad\; \backslash frac\{\backslash omega\_c\}\{\backslash omega\_0\}\backslash in\backslash \{0.1,\backslash \; 3.4\backslash \}.$$

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 for the common submission entry points.