利用碎形動力學研究單一原子在光子晶體中的自發輻射

Abstract

原子在光子晶體中的自發輻射具有長時間記憶(long-time memory)的效應。 因此利用記憶核心(memory kernel)我們可以了解原子在等向性(isotropic)及非等向性(anisotropic)光子晶體的自發輻射會導致碎形現象(fractal phenomenon)產生。 此種現象我們可以利用碎形微積分(fractional calculus)來描述。 所以當原子的躍遷頻率(transition frequency)落在光子晶體中的光能隙(photonic band-gap)外時，利用碎形微積分我們不會有多值解的問題，也不會有非物理的狀態出現，而這些問題是在之前的研究[Phy. Rev. A 50, 1764 (1994)]中所會遇到的。 以上所探討的原子在空間中視為一個點(point-like)，不過在實際的情況下，原子是具有尺寸的。 因此在有尺寸的效應之下，需要考慮多極(multi-pole)的貢獻。所以我們利用具有尺寸的記憶核心來探討對原子自發輻射的影響，會發現其偶合常數(coupling constant)會改變。

We use memory kernel to study the spontaneous emission of an atom in a photonic crystal with the isotropic band and anisotropic band. Our studies show that the long-time memory of the spontaneous emission in the photonic crystal induces a fractal phenomenon. Therefore, the fractional calculus is a natural mathematics to describe the fractal phenomenon. When the atomic transition frequency lies within the allowed band, using the fractional calculus we show that there is no multiple-valued problem and no fractionalized steady-state inversion encountered in the previous studies [J. Mod. Opt. 41, 353 (1994)]. On the other hand. We also consider an atom with finite size effect leading to multipole contributions and yield the memory kernel with the same form as a point-like atom. However, the memory kernel including finite-size effect has different coupling constant.