  Four-wave mixing (FWM) in gas-filled hollow-core capillaries, a nonlinear
optical process that mixes signal and pump photon frequencies to generate idler
frequency photons, offers a method for precise spectral phase transfer from
signal to idler at ultrashort timescales and extreme powers. However, this
regime is challenged by competing linear and nonlinear dynamics, leading to
significant trade-offs between spectral phase transfer and conversion
efficiency. Our computational investigation focuses on upconversion of
femtosecond pulses from the infrared (IR) to the ultraviolet (UV), a range
notoriously difficult to manipulate. We explore an intermediate energy regime
that strikes an optimal balance between FWM-mediated phase-transfer fidelity
and nonlinear conversion efficiency. By adjusting the energy ratios and
spectral phase profiles of the input signal, we achieve conversion efficiencies
of approximately 5%-15% while maintaining an effectively quasi-linear spectral
phase transfer. These findings contribute to establishing first-principles and
scaling laws essential for applications such as high-precision imaging,
spectroscopy, quantum transduction, and distributed entangled interconnects,
facilitating advanced control of ultrafast photonic and electronic wavepackets
in quantum materials with unprecedented spatial and temporal precision.
