Dynamics and energy spectra of aperiodic discrete-time quantum walks

Abstract

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.

Emergence of anomalous dynamics from the underlying singular continuous spectrum in interacting many-body systems

“…1(c), where we can appreciate a deviation from ballistic spreading at any finite λ. This behavior can be traced back to the critical nature of the eigenfunctions together with the SC nature of the spectrum [5,10,36], and it is shared by other aperiodic structures [39,40].…”

“…[8], and its link to anomalous propagation of correlations and to the spreading of an initially localized wave packet was investigated in Refs. [9,10]. A particularly interesting, exemplary physical model where the nature of the spectrum plays a crucial role is the Aubry-André model (AAM), which describes particle hopping in a one-dimensional quasiperiodic lattice.…”

Quantum Walks in Quasi-Periodic Photonics Lattices

“…Because it is impossible to achieve symmetrical LQWs in spatially randomly disordered systems, the authors proposed temporally disordered operations using multiple quantum coins. Furthermore, the concept of localization due to temporally disordered operations has been later expanded to consider multiple quantum coins with quasiperiodic sequences that include Fibonacci, Thue-Morse, and Rudin-Shapiro sequences [37]. Although the idea of employing symmetrical LQWs for quantum memory is very interesting and worthy of further exploration, we would like to stress that experimental implementation of QWs is not simple even with only one quantum coin; therefore the idea of using multiple quantum coin operations would be extremely difficult in practice.…”

“…On the other hand, LQWs have recently been proposed for secure quantum memory applications [19]. To achieve symmetrically distributed LQWs, the authors of [19,37] have proposed to use temporally disordered operations in spatially ordered systems. However, their approach requires multiple quantum coins for temporally disordered operation which could be extremely difficult in practice.…”

“…At this point, we want to stress that localization of light has been investigated theoretically and experimentally in 1D and 2D arrays of identical waveguides that are constructed with the Fibonacci sequence in distances [38,39]. The quasiperiodic arrays of waveguides considered in those works are defined as the Fibonacci elements in Figure 2; however, the waveguides A and B were defined as the two distances 1 (unit) and τ = 1.618 (golden ratio of the Fibonacci sequence) between identical SM waveguides [37,38]. Since the quasiperiodic properties of these elements are determined by the Fibonacci sequences in spacing of waveguides, they can be classified as PLs with the Fibonacci spacing sequence (FSS).…”