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## Background and research question
Two electromagnetic pulses may carry the same energy while differing in the
time integral of their electric field. Whether that distinction remains
physically consequential in the zeptosecond regime is not evident from pulse
energy alone.
For the two counterpropagating pulses specified below, determine how pulse
polarity, relative phase arrangement and duration control the final
electron-positron momentum distribution and the total pair yield. Establish
whether the equal-energy pulse classes are physically equivalent, whether a
mixed phase arrangement produces directional momentum transfer, and which
duration regime is most productive. Explain the mechanism and state the
strongest conclusion supported by your computed evidence, including its
domain of validity. Do not assume the outcome in advance.
Design the task-specific derivation, calculation, evidence and scientific
argument needed to answer this question.
## Physical system and method boundary
Use natural units $\hbar=c=1$, electron mass $m$, charge $e<0$, Compton time
$\tau_{\mathrm C}=m^{-1}$ and critical field
$$
E_{\mathrm c}=\frac{m^2}{|e|}.
$$
The two linearly $x$-polarized pulses propagate in opposite $z$ directions.
For $\zeta=+1$ and $-1$, respectively, take
$$
E_x^{(\zeta)}(t,z)
=E_{0\zeta}\exp\!\left[-\left(\frac{\eta_\zeta}{\sigma}\right)^2\right]
\sin(\eta_\zeta+\varphi_\zeta),
\qquad
\eta_\zeta=\frac{\sigma(t-\zeta z)}{\tau},
\qquad \sigma=1.
$$
Investigate three visible physical configurations:
- two unipolar pulses, $(\varphi_+,\varphi_-)=(\pi/2,\pi/2)$;
- two bipolar pulses, $(\varphi_+,\varphi_-)=(0,0)$; and
- a mixed pair, $(\varphi_+,\varphi_-)=(0,\pi/2)$.
For the equal-energy comparison between the first two configurations, use the
task-policy amplitude $E_*=0.1E_{\mathrm c}$ for the unipolar pulse and choose the bipolar amplitude
so that each bipolar pulse has the same value of
$\int_{-\infty}^{\infty}E_x^2(t,z)\,dt$ as its unipolar counterpart. For the
mixed configuration use the common amplitude $E_{0+}=E_{0-}=E_*$. Reversing
which member of the mixed pair is bipolar is an equivalent orientation test
when the associated $z$ convention is transformed consistently.
Use the leading non-vanishing perturbative-QED treatment of pair creation by
the two pulses while retaining their spacetime dependence. A derivation that
treats one pulse exactly and the other perturbatively, or treats both at the
same perturbative order, is acceptable when it produces the same leading
observable. A spatially homogeneous Schwinger model is not an equivalent
substitute.
The differential observable is the asymptotic electron number density in
momentum space, reported with a declared continuum normalization. The total
pair number per transverse normalization area is its three-dimensional
momentum integral. Cover sub-Compton and Compton-scale durations sufficiently
to support the global duration claims you make. The momentum domain,
quadrature, sampling, solver, truncation, diagnostic construction, figures and
reliability strategy are yours. Preserve the underlying unscaled numerical
data whenever display rescaling is used.
## Submission and resources
Submit one self-contained scientific report and an auditable offline
calculation package containing the 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. The organization of the argument and the choice and
number of figures are yours. Evidence must be sufficiently reliable for the
claims made, but no particular validation procedure is prescribed.
The available CPU allocation, candidate active deadline, node, network,
memory and scratch limits are specified in the accompanying
`resource_limits.json`. Complete the derivation, implementation,
calculation, validation, interpretation and final-file preparation within these
resource limits. More expensive modelling 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.
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