哪种脉冲能最大化共振非线性转换？
Which pulse maximizes resonant nonlinear conversion?
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中文总结 AI 辅助
该研究求解谐振器中最大化n阶非线性转换的最优脉冲波形，得到最优解为非线性薛定谔方程基态孤子，其性能优于传统脉冲，适用范围广。
中文摘要 AI 辅助
在固定脉冲能量下，何种驱动波形能从谐振器中提取最多的n阶非线性转换（二次谐波对应n=2）？短脉冲与窄谐振耦合差，长脉冲能量分散，无线性规则确定折中方案。我们对振幅衰减率为κ的单模精确求解该问题：消除驱动后，固定入射能量转化为仅对存储场的约束，优化问题成为尖锐的Gagliardo–Nirenberg不等式，其极值为非线性薛定谔方程的基态孤子。最优存储场为sech^(1/(n-1))[(n-1)κt]，由非对称输入维持，该输入以e^(κt)上升、以e^(-(2n-1)κt)下降；最大转换能量以闭式形式给出。上升指数、时间反转方案在n=2时最多保留该界限的79.0%，大n时保留2/e；双速率脉冲保留97%以上。临界耦合推广为n倍过耦合，最优输入耦合为固有损耗率的n倍。该界限适用于微环到超导电路，限制了宽带光子对源的每脉冲亮度。
英文摘要
At fixed pulse energy, which drive waveform extracts the most $n$th-order nonlinear conversion from a resonator ($n=2$ for second harmonic)? A short pulse couples poorly to a narrow resonance, a long one dilutes its energy, and no linear rule fixes the compromise. We solve the problem exactly for a single mode of amplitude decay rate $κ$. Eliminating the drive turns fixed incident energy into a constraint on the stored field alone, and the optimization becomes a sharp Gagliardo--Nirenberg inequality whose extremal is the ground-state soliton of the nonlinear Schrödinger equation. The optimal stored field is $\mathrm{sech}^{1/(n-1)}[(n-1)κt]$, sustained by an asymmetric input that rises as $e^{κt}$ and falls as $e^{-(2n-1)κt}$; the largest converted energy follows in closed form. A rising exponential, the time-reversal recipe, retains at most $79.0\%$ of the bound at $n=2$ and $2/e$ at large $n$; a two-rate pulse retains above $97\%$. Critical coupling generalizes to $n$-fold overcoupling, with optimal input coupling $n$ times the intrinsic loss rate. The bound applies from microrings to superconducting circuits and caps the per-pulse brightness of broadband photon-pair sources.
