Lo Gullo N, Ambarish C V, Busch Th, Dell'Anna L, Chandrashekar C M
Department of Physics, Univeritá degli Studi di Milano, Via G. Celoria, 16, 20133 Milano, Italy.
The Institute of Mathematical Sciences, C. I. T. Campus, Taramani, Chennai 600113, India.
Phys Rev E. 2017 Jul;96(1-1):012111. doi: 10.1103/PhysRevE.96.012111. Epub 2017 Jul 6.
We investigate the role of different aperiodic sequences in the dynamics of single quantum particles in discrete space and time. For this we consider three aperiodic sequences, namely, the Fibonacci, Thue-Morse, and Rudin-Shapiro sequences, as examples of tilings the diffraction spectra of which have pure point, singular continuous, and absolutely continuous support, respectively. Our interest is to understand how the order, intrinsically introduced by the deterministic rule used to generate the aperiodic sequences, is reflected in the dynamical properties of the quantum system. For this system we consider a single particle undergoing a discrete-time quantum walk (DTQW), where the aperiodic sequences are used to distribute the coin operations at different lattice positions (inhomogeneous DTQW) or by applying the same coin operation at all lattice sites at a given time but choosing different coin operation at each time step according to the chosen aperiodic sequence (time dependent DTQW). We study the energy spectra and the spreading of an initially localized wave packet for different cases, finding that in the case of Fibonacci and Thue-Morse tilings the system is superdiffusive, whereas in the Rudin-Shapiro case it is strongly subdiffusive. Trying to understand this behavior in terms of the energy spectra, we look at the survival amplitude as a function of time. By means of the echo we present strong evidence that, although the three orderings are very different as evidenced by their diffraction spectra, the energy spectra are all singular continuous except for the inhomogeneous DTQW with the Rudin-Shapiro sequence where it is discrete. This is in agreement with the observed strong localization both in real space and in the Hilbert space. Our paper is particularly interesting because quantum walks can be engineered in laboratories by means of ultracold gases or in optical waveguides, and therefore would be a perfect playground to study singular continuous energy spectra in a completely controlled quantum setup.
我们研究了不同非周期序列在离散时空下单量子粒子动力学中的作用。为此，我们考虑三个非周期序列，即斐波那契序列、图厄 - 摩尔斯序列和鲁丁 - 夏皮罗序列，作为平铺的示例，其衍射谱分别具有纯点支撑、奇异连续支撑和绝对连续支撑。我们感兴趣的是理解由用于生成非周期序列的确定性规则内在引入的序如何反映在量子系统的动力学性质中。对于这个系统，我们考虑一个经历离散时间量子行走（DTQW）的单个粒子，其中非周期序列用于在不同晶格位置分布硬币操作（非均匀DTQW），或者在给定时间在所有晶格位点应用相同的硬币操作，但根据所选非周期序列在每个时间步选择不同的硬币操作（时间相关DTQW）。我们研究了不同情况下的能谱以及初始局域化波包的扩散情况，发现对于斐波那契和平铺和图厄 - 摩尔斯平铺，系统是超扩散的，而在鲁丁 - 夏皮罗情况下，它是强次扩散的。为了从能谱角度理解这种行为，我们将存活振幅视为时间的函数。通过回波，我们提供了强有力的证据表明，尽管这三种排序方式在其衍射谱中表现出非常不同，但除了具有鲁丁 - 夏皮罗序列的非均匀DTQW（其能谱是离散的）外，所有能谱都是奇异连续的。这与在实空间和希尔伯特空间中观察到的强局域化一致。我们的论文特别有趣，因为量子行走可以在实验室中通过超冷气体或光波导来实现，因此将是在完全可控的量子设置中研究奇异连续能谱的理想场所。
