cases/case48/task.md

# Benchmarking approximations to quantum light emitted by a driven correlated chain

## Research commission

A finite, half-filled correlated chain is driven by a coherent mid-infrared pulse. Its electronic current generates a weak multimode quantum field. Determine how faithfully three tractable descriptions of that emitted field represent both the radiation spectrum and its nonclassical photon statistics, and assess whether, and if so how, their relative fidelity depends on pulse duration.

Your conclusion must be supported by calculations for both pulse durations and by all three observable families defined below. You are responsible for choosing an economical evidence package and for deciding which comparisons, controls, and representations are needed to sustain or limit your claim. A plot-by-plot recreation of any prior publication is not requested.

## Physical system

Use atomic units. The electronic system is an eight-site one-dimensional Fermi-Hubbard ring with periodic boundary conditions, four spin-up and four spin-down electrons, lattice spacing \(a=7.5589\), nearest-neighbour hopping \(t_0=0.0191\), and on-site interaction \(U=10t_0\). Start from the spin-symmetric, zero-total-crystal-momentum ground state.

For a vector potential polarized along the chain, use

\[
\hat H_{\mathrm{el}}(t)
=-t_0\sum_{j,\mu}
\left[
e^{iaA_{\mathrm{cl}}(t)}\hat c^\dagger_{j,\mu}\hat c_{j+1,\mu}
+\mathrm{H.c.}
\right]
+U\sum_j \hat n_{j,\uparrow}\hat n_{j,\downarrow},
\]

with

\[
A_{\mathrm{cl}}(t)=A_0\sin(\omega_L t+\pi/2)
\sin^2\!\left(\frac{\omega_L t}{2N_c}\right),
\qquad
0\le t\le \frac{2\pi N_c}{\omega_L},
\]

and \(A_{\mathrm{cl}}(t)=0\) outside the pulse. Take

\[
\omega_L=0.005,\qquad A_0=0.194,
\qquad N_c\in\{10,18\}.
\]

The current along the chain is

\[
\hat j(t)=-iat_0\sum_{j,\mu}
\left[
e^{iaA_{\mathrm{cl}}(t)}\hat c^\dagger_{j,\mu}\hat c_{j+1,\mu}
-\mathrm{H.c.}
\right].
\]

Let \(\lvert\phi_m(t)\rangle\) be the semiclassically propagated field-free electronic eigenstates, let \(i\) denote the initial ground-state branch, and define

\[
j_{mn}(t)=\langle\phi_m(t)\rvert\hat j(t)\lvert\phi_n(t)\rangle.
\]

The emitted-field coupling is \(g_0=4\times10^{-8}\). For a retained mode \(\lambda\) of frequency \(\omega_\lambda\) and chain polarization, use

\[
\hat A_{Q,\lambda}(t)=\frac{g_0}{\sqrt{\omega_\lambda}}
\left(\hat a_\lambda e^{-i\omega_\lambda t}
+\hat a_\lambda^\dagger e^{i\omega_\lambda t}\right).
\]

## Case definition for the three photonic reductions

The following equations define the case-specific reductions commissioned in this problem. They fix the physical state represented by each route but do not prescribe a propagator, discretization, truncation strategy, or evidence layout.

For the **ground-transition Fock description**, decouple the radiation modes but retain the electronic branches \(\lvert\chi_\lambda^{(m)}(t)\rangle\) coupled to the initial branch through

\[
i\,\partial_t\lvert\chi_\lambda^{(i)}\rangle
=\hat A_{Q,\lambda}(t)
\sum_m j_{im}(t)\lvert\chi_\lambda^{(m)}\rangle,
\]

\[
i\,\partial_t\lvert\chi_\lambda^{(m)}\rangle
=\hat A_{Q,\lambda}(t)j_{mi}(t)
\lvert\chi_\lambda^{(i)}\rangle,
\qquad m\ne i,
\]

with \(\lvert\chi_\lambda^{(i)}(t_i)\rangle=\lvert0_\lambda\rangle\) and \(\lvert\chi_\lambda^{(m\ne i)}(t_i)\rangle=0\). The retained modes form one branch-correlated multimode object. As an examiner task policy defining that finite object, use

\[
\omega_\lambda/\omega_L=1,3,5,\ldots,59.
\]

Thus, for a one-mode operator \(\hat O_\lambda\), use the globally normalized branch trace

\[
\langle\hat O_\lambda\rangle
=\frac{\displaystyle
\sum_m
\langle\chi_\lambda^{(m)}\rvert\hat O_\lambda
\lvert\chi_\lambda^{(m)}\rangle
\prod_{\kappa\ne\lambda}
\langle\chi_\kappa^{(m)}\vert\chi_\kappa^{(m)}\rangle}
{\displaystyle
\sum_m\prod_\kappa
\langle\chi_\kappa^{(m)}\vert\chi_\kappa^{(m)}\rangle}.
\]

For the **Markov-type Fock description**, formally eliminate the branches \(m\ne i\) from the same ground-transition system and retain the leading local-in-time term. With the same vacuum initial condition, the resulting one-mode state is defined by

\[
i\,\partial_t\lvert\chi_\lambda^{(i)}(t)\rangle
=\hat A_{Q,\lambda}(t)
\left[
j_{ii}(t)
-i\sum_{m\ne i}j_{im}(t)
\int_{t_i}^{t}dt'\,
\hat A_{Q,\lambda}(t')j_{mi}(t')
\right]
\lvert\chi_\lambda^{(i)}(t)\rangle.
\]

For the **Markov-state description**, define

\[
\hat j_H(t)=\hat U_{\mathrm{el}}^\dagger(t)\hat j(t)\hat U_{\mathrm{el}}(t),
\qquad
\hat W_\lambda(t)=\int_{t_i}^{t}dt'\,
\hat A_{Q,\lambda}(t')
\left[\hat j_H(t')-j_{ii}(t')\right],
\]

\[
\beta_{ii}^{(\lambda)}(t)
=-i\frac{g_0}{\sqrt{\omega_\lambda}}
\int_{t_i}^{t}dt'\,j_{ii}(t')e^{i\omega_\lambda t'},
\]

where \(\hat U_{\mathrm{el}}\) is the propagator of \(\hat H_{\mathrm{el}}\), and \(\langle\cdot\rangle_{\mathrm{el}}\) below is taken in the initial electronic state. Neglect higher-order commutators in the current-correlation contribution and retain terms through second order in \(g_0\):

\[
\lvert\chi_{\lambda,\mathrm{MS}}(t)\rangle
=\hat D_\lambda\!\left[\beta_{ii}^{(\lambda)}(t)\right]
\exp\!\left[-\frac12
\langle\hat W_\lambda^2(t)\rangle_{\mathrm{el}}\right]
\lvert0_\lambda\rangle.
\]

Here \(\hat D_\lambda\) is the single-mode displacement operator. Derive the moment relations needed to evaluate the observables below, state the approximation made at each reduction, and use the same electronic dynamics and physical conventions in all three routes. A complete recurrence and the expanded moment formulas are intentionally not supplied.

## Observables

For each retained description, construct the emitted spectrum, Mandel statistic, and optimized single-mode quadrature squeezing:

\[
S(\omega_\lambda)=
\frac{\omega_\lambda^3}{g_0^2(2\pi)^2c^3}
\langle\hat n_\lambda\rangle,
\]

\[
Q_\lambda=
\frac{\langle\hat n_\lambda^2\rangle
-\langle\hat n_\lambda\rangle^2}
{\langle\hat n_\lambda\rangle}-1,
\]

and

\[
\eta_\lambda=-10\log_{10}\!\left[
4\min_{0\le\theta<\pi}
\left(\Delta\hat X_\lambda(\theta)\right)^2
\right],
\qquad
\hat X_\lambda(\theta)=
\frac{\hat a_\lambda e^{-i\theta}
+\hat a_\lambda^\dagger e^{i\theta}}{2}.
\]

Interpret \(Q_\lambda\) only where the mean photon number is resolved relative to your numerical uncertainty. Declare and justify the validity rule you use; do not base a physical conclusion on isolated small-denominator spikes.

## Numerical freedom and evidence standard

You may choose the propagator, time step, symmetry reduction, Fock cutoff, frequency sampling away from the stated physical mode set, numerical optimization method, and display representation. Report the material choices and show that the evidence supporting your claims is stable under appropriate refinement. Distinguish a physical radiation-mode set from points added only for interpolation or display.

Assess whether conclusions suggested by any one observable remain supported when tested against the other two. Use the calculated signals to make a bounded scientific judgment about the hierarchy for the two pulse durations, including what your evidence does and does not establish. Physically equivalent normalizations and representations are acceptable when their relation is stated and they preserve the claim-relevant structures.

## Submission

Submit one self-contained scientific report, an auditable offline calculation package, underlying numerical data for every material claim, a provenance note listing every offline source or software document actually consulted and the method basis for items supplied from general knowledge, and concise environment and run information. The report should make the model, approximation definitions, numerical reliability, evidence, and inference independently checkable without prescribing a fixed number or order of figures.

Resource limits are supplied exclusively by the accompanying
`resource_limits.json`. Its fixed CPU allocation, candidate-active
wall time, network boundary are binding. No separate aggregate CPU-hour limit applies. Greater computational expense
receives no 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.