cases/case22/task.md

Can pulse polarity control vacuum-pair acceleration and yield?

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

$$E_{\mathrm{c}}=\frac{m^2}{\mathrm{\mid }e\mathrm{\mid }}\mathrm{.}E\_\{\backslash mathrm\; c\}=\backslash frac\{m^2\}\{|e|\}.$$

The two linearly $x$-polarized pulses propagate in opposite $z$ directions. For $\zeta=+1$ and $-1$, respectively, take

$$E_x^{(\zeta )}(t,z)=E_{0\zeta }\exp \text{\hspace{0.17em}⁣}[-{\left(\frac{{\eta }_{\zeta }}{\sigma }\right)}^2]\sin ({\eta }_{\zeta }+{\phi }_{\zeta }),{\eta }_{\zeta }=\frac{\sigma (t-\zeta z)}{\tau },\sigma =1.E\_x^\{(\backslash zeta)\}(t,z)\; =E\_\{0\backslash zeta\}\backslash exp\backslash !\backslash left[-\backslash left(\backslash frac\{\backslash eta\_\backslash zeta\}\{\backslash sigma\}\backslash right)^2\backslash right]\; \backslash sin(\backslash eta\_\backslash zeta+\backslash varphi\_\backslash zeta),\; \backslash qquad\; \backslash eta\_\backslash zeta=\backslash frac\{\backslash sigma(t-\backslash zeta\; z)\}\{\backslash tau\},\; \backslash qquad\; \backslash sigma=1.$$

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 for the common submission entry points.