Localization and fractality in inhomogeneous quantum walks with self-duality

We introduce and study a class of discrete-time quantum walks on a one-dimensional lattice. In contrast to the standard homogeneous quantum walks, coin operators are inhomogeneous and depend on their positions in this class of models. The models are shown to be self-dual with respect to the Fourier transform, which is analogous to the Aubry-André model describing the one-dimensional tight-binding model with a quasiperiodic potential. When the period of coin operators is incommensurate to the lattice spacing, we rigorously show that the limit distribution of the quantum walk is localized at the origin. We also numerically study the eigenvalues of the one-step time evolution operator and find the Hofstadter butterfly spectrum which indicates the fractal nature of this class of quantum walks.

Figure 1

(Color online) Probability distribution in the inhomogeneous QW after the 1000 time steps. Here, we take the initial coin state $\left(|L⟩+|R⟩\right)/\sqrt{2}$ and the inverse period $\alpha =\pi /2$, which is an irrational number. Note that, the probability outside the plot range is zero within the numerical accuracy.

Figure 2

Time evolution of the inhomogeneous QW around the site $\pm mQ$ at which the quantum walker is reflected. The coin operator $C$ and shift operator $W$ act on the wave function $|{\psi }_n\left(t\right)⟩={\left[{\psi }_t\left(n,L\right),{\psi }_t\left(n,R\right)\right]}^{\mathbf{T}}$ at each site $n$. The wave function around the reflection point $\pm mQ$ is indicated from the $\left(t_r-1\right)\text{th}$ step to the $\left(t_r+1\right)\text{th}$ step.

Figure 3

(Color online) The self-similar and fractal structure in the inhomogeneous QW. Arguments of the eigenvalues of $CW$ (vertical axis) are plotted as a function of the parameter $\alpha =\frac{P}{4Q}$ (horizontal axis) with $Q\le 50$. Note that, $P$ and $Q$ must be relatively prime.

Figure 4

(Color online) Recurrence property of the inhomogeneous QW with the initial coin state $\left(|L⟩+|R⟩\right)/\sqrt{2}$ and the inverse period $\alpha =\pi /2$. The numerically obtained probabilities at the origin are shown up to 1000 time steps. Note that the probability at the origin must be zero when the step $t$ is odd.