广义Fibonacci时间准周期量子行走波包扩散的动力学特性

Dynamic behaviors of spreading in generalized Fibonacci time quasiperiodic quantum walks

本文通过数值计算的方法研究了一维离散时间准周期量子行走的动力学特性,主要研究了两个自旋空间C 算符按照广义Fibonacci准周期排列的量子行走,发现对两类广义Fibonacci准周期序列,波包扩散都是超扩散（即标准方差σ约为tγ,0.5〈γ〈1）,而且在给定的两个 C 算符下,第二类广义Fibonacci准周期序列的幂指数γ大于第一类广义Fibonacci准周期序列.通过对波包扩散的概率分布情形和标准方差的研究发现,第一类广义Fibonacci准周期序列的波包扩散更接近于经典随机行走（γ=0.5）,而第二类广义Fibonacci准周期序列的波包扩散更接近于均匀量子行走（γ=1）,这与两类广义Fibonacci准周期量子自旋链中量子相变时的特性相反.

Quantum walk （QW）, the quantum mechanical counterpart of classical random walk, has recently been studied in various fields. The evolution of the discrete time quantum walk can be described as follows： the walker changes its spin state by the coin operator C, then takes one step left or right according to its spin state. For homogeneous quantum walk, the coin operator is independent of time and the standard deviation of the position grows linearly in time. It is quadratically faster than that in the classical random walk. In this work, we numerically study the dynamical behaviors of spreading in a one-dimensional discrete time quasiperiodic quantum walk （DTQQW）. The DTQQW is that the coin operator is dependent on time and takes two different coins C（α） and C（β） arranged in generalized Fibonacci （GF） sequences. The GF sequences are constructed from A by the recursion relation： A → AmBn, B → A, for m, n are positive integers. They can be classified into two classes according to the wandering exponentω. Forω〈0, they belong to the first class, and for ω 〉 0, they belong to the second class. For one dimensional system, the behaviors of two classes of GF systems are different either for the electronic spectrum of an electron in quasiperiodic potentials or for the quantum phase transitions of the quasiperiodic spin chains. In this paper, we discuss the cases of two different C operators （C（α）, C（β）） arranged in GF sequences and find that the spreading behaviors are superdiffusion （the standard deviation of the positionσ～tγ, 0.5〈γ〈1） for the two classes of GF DTQQW. For the second class of GF DTQQW, the exponent valuesγ are larger than those of the first class of GF DTQQW in the case of two identical C operators. By exploring the probability distribution in the real space, we find that for the first class of GF DTQQW, the probability distributions are almost the same for different initial states and are similar to the classical Gaussian distribution. For the probability distributions of the second class of GF DTQQW, there are two peaks at the two edges and the height of the two peaks can be different for different initial states. They are similar to the ballistic distribution of the homogeneous quantum walk. Therefore, we conclude that for the first class of GF DTQQW, the spreading behaviors are close to those of the classical random walk （γ = 0.5） while for the second class of GF DTQQW, they are close to those of the homogeneous quantum walk （γ = 1）. This result is quite different from the characteristics of the quantum phase transitions in two classes of GF quasiperiodic quantum spin chains.