CAMTP-Center for Applied Mathematics and Theoretical Physics, University of Maribor, Mladinska 3, SI-2000 Maribor, Slovenia, European Union.
Phys Rev E. 2019 Dec;100(6-1):062208. doi: 10.1103/PhysRevE.100.062208.
We study the quantum localization in the chaotic eigenstates of a billiard with mixed-type phase space [J. Phys. A: Math. Gen. 16, 3971 (1983)JPHAC50305-447010.1088/0305-4470/16/17/014; J. Phys. A: Math. Gen. 17, 1049 (1984)JPHAC50305-447010.1088/0305-4470/17/5/027], after separating the regular and chaotic eigenstates, in the regime of slightly distorted circle billiard where the classical transport time in the momentum space is still large enough, although the diffusion is not normal. This is a continuation of our recent papers [Phys. Rev. E 88, 052913 (2013)PLEEE81539-375510.1103/PhysRevE.88.052913; Phys. Rev. E 98, 022220 (2018)2470-004510.1103/PhysRevE.98.022220]. In quantum systems with discrete energy spectrum the Heisenberg time t_{H}=2πℏ/ΔE, where ΔE is the mean level spacing (inverse energy level density), is an important timescale. The classical transport timescale t_{T} (transport time) in relation to the Heisenberg timescale t_{H} (their ratio is the parameter α=t_{H}/t_{T}) determines the degree of localization of the chaotic eigenstates, whose measure A is based on the information entropy. We show that A is linearly related to normalized inverse participation ratio. The localization of chaotic eigenstates is reflected also in the fractional power-law repulsion between the nearest energy levels in the sense that the probability density (level spacing distribution) to find successive levels on a distance S goes like ∝S^{β} for small S, where 0≤β≤1, and β=1 corresponds to completely extended states. We show that the level repulsion exponent β is empirically a rational function of α, and the mean 〈A〉 (averaged over more than 1000 eigenstates) as a function of α is also well approximated by a rational function. In both cases there is some scattering of the empirical data around the mean curve, which is due to the fact that A actually has a distribution, typically with quite complex structure, but in the limit α→∞ well described by the beta distribution. The scattering is significantly stronger than (but similar as) in the stadium billiard [Nonlin. Phenom. Complex Syst. (Minsk) 21, 225 (2018)] and the kicked rotator [Phys. Rev. E 91, 042904 (2015)PLEEE81539-375510.1103/PhysRevE.91.042904]. Like in other systems, β goes from 0 to 1 when α goes from 0 to ∞. β is a function of 〈A〉, similar to the quantum kicked rotator and the stadium billiard.
我们研究了混合相空间中的弹道混沌本征态中的量子局域化[J. Phys. A: Math. Gen. 16, 3971 (1983)JPHAC50305-447010.1088/0305-4470/16/17/014; J. Phys. A: Math. Gen. 17, 1049 (1984)JPHAC50305-447010.1088/0305-4470/17/5/027]，在动量空间中经典输运时间仍然足够大的情况下，对稍微失真的圆形弹道进行了分离，尽管扩散不是正常的。这是我们最近的论文[Phys. Rev. E 88, 052913 (2013)PLEEE81539-375510.1103/PhysRevE.88.052913; Phys. Rev. E 98, 022220 (2018)2470-004510.1103/PhysRevE.98.022220]的延续。在具有离散能谱的量子系统中，Heisenberg 时间 t_{H}=2πℏ/ΔE，其中 ΔE 是平均能级间隔（能级密度的倒数），是一个重要的时间尺度。在与 Heisenberg 时间 t_{H}（它们的比值是参数 α=t_{H}/t_{T}）相关的经典输运时间 t_{T}（输运时间）中，确定了混沌本征态的局域化程度，其度量 A 基于信息熵。我们表明 A 与归一化参与比呈线性关系。混沌本征态的局域化也反映在最近能级之间的分数幂律排斥中，即在小距离 S 上找到连续能级的概率密度（能级间隔分布）遵循∝S^{β}，其中 0≤β≤1，并且 β=1 对应于完全扩展的状态。我们表明，能级排斥指数 β 是经验上 α 的有理函数，而平均值〈A〉（在 1000 多个本征态上平均）作为 α 的函数也很好地由有理函数近似。在这两种情况下，经验数据都围绕着平均值曲线有一些散射，这是由于 A 实际上具有分布，通常具有相当复杂的结构，但在 α→∞的极限下可以很好地由β分布描述。散射比体育场台球[Nonlin. Phenom. Complex Syst. (Minsk) 21, 225 (2018)]和踢转器[Phys. Rev. E 91, 042904 (2015)PLEEE81539-375510.1103/PhysRevE.91.042904]强得多，但相似。β从 0 增加到 1，而 α 从 0 增加到 ∞。β是〈A〉的函数，类似于量子踢转器和体育场台球。
