复杂势场量子弹球中疤痕态的量子化条件

Quantization condition of scarring states in complex soft-wall quantum billiards

量子疤痕是波函数在经典不稳定周期轨道周围反常凝聚的一种量子或波动现象.人们对疤痕态的量子化条件进行了大量研究,对深入理解半经典量子化起到了一定的促进作用.之前大部分研究工作主要集中在硬墙量子弹球上,即给定边界形状的无穷深量子势阱系统.本文研究具有光滑复杂势场的二维量子弹球系统,考察疤痕态的量子化条件及其重复出现的规律,得到了与硬墙弹球不一样的结果,对理解这类现象是一个有益的补充.这些结果将有助于理解具有无规长程杂质分布的二维电子系统的态密度谱和输运行为.

Quantum scar is an intriguing phenomenon in quantum or wave dynamics that the wavefunction takes an exceptionally large value around an unstable periodic orbit. It has attracted much attention and advances the understanding of the semiclassical quantization. Most of previous researches involving quantum scars focus on hard-wall quantum billiards. Here we investigate the quantum billiard with a smooth confinement potential which possesses complex classical dynamics. We demonstrate that the semiclassical quantization approach works well for both the stable and unstable classical periodic orbit, besides the fact that the shape of the orbits varies as the energy increases or even the stability switches. The recurrence rule of the quantum scars in this complex solf-wall billiard differs from that of the hard-wall nonrelativistic quantum billiard, such as being equally spaced in energy instead of being equally spaced in the square root of energy. These results implement the previous knowledge and may be used for understanding the measurements of density of states and transport properties in two-dimensional electron systems with random long-range impurities.