The simplest 2D quantum walk detects chaoticity

TL;DR

The paper asks whether the simplest 2D quantum walk can detect chaoticity in bounded billiards. It studies a spin-1/2 particle performing alternating horizontal and vertical steps inside rectangular (regular) and Bunimovich stadium (chaotic) domains, with evolution $\hat{Q_w}=\hat{W_n}\hat{C}_2\hat{W_m}\hat{C}_1$. Spectral analysis of the evolution operator shows non-Poissonian level statistics in the stadium, with a Brody parameter in the range $δ \approx 0.07$–$0.15$, and a tendency toward Poisson-like statistics in the rectangle; eigenfunctions exhibit localization (typical $PR \approx 1150$ in the stadium vs $PR \approx 1500$ in the rectangle) and visible scarring on unstable periodic orbits. The findings indicate that a diffusion-based QW can encode chaoticity in a manner not identical to Hamiltonian systems and offer a simple, experimentally relevant probe of quantum chaos.

Abstract

Quantum walks are at present an active field of study in mathematics, with important applications in quantum information and statistical physics. In this paper, we determine the influence of basic chaotic features on the walker behavior. For this purpose, we consider an extremely simple model consisting of alternating one-dimensional walks along the two spatial coordinates in bidimensional closed domains (hard wall billiards). The chaotic or regular behavior induced by the boundary shape in the deterministic classical motion translates into chaotic signatures for the quantized problem, resulting in sharp differences in the spectral statistics and morphology of the eigenfunctions of the quantum walker. Indeed, we found for the Bunimovich stadium -- a chaotic billiard -- level statistics described by a Brody distribution with parameter $δ\simeq 0.1$. This indicates a weak level repulsion, and also enhanced eigenfunction localization, with an average participation ratio (PR) $\simeq$ 1150) compared to the rectangular billiard (regular) case, where the average PR $\simeq$ 1500. Furthermore, scarring on unstable periodic orbits is observed. The fact that our simple model exhibits such key signatures of quantum chaos, e.g., non-Poissonian level statistics and scarring, that are sensitive to the underlying classical dynamics in the free particle billiard system is utterly surprising, especially when taking into account that quantum walks are diffusive models, which are not direct quantizations of a Hamiltonian.

Figures (11)

• Figure 1: (Color online) Desymmetrized Bunimovich stadium billiard. Some integers $(m,n)$ specifying the particle position in a grid inside the billiard, and the shape functions of Eqs. (\ref{['eq:A7']}) and (\ref{['eq:A8']}) used in the evolution operator definition are shown for reference. The evolution of the walker: coin 1 toss, horizontal step (horizontal arrow), coin 2 toss, vertical step (vertical arrow), … is schematically indicated, inside the domain. See Sec. \ref{['sec:Model']} for details.
• Figure 2: (Color online) Time evolution of the central position eigenstate (see main text for details) for $t$ = 38 (a), 76 (b), 152 (c), 232 (d) in the rectangular billiard. A $(m_R,n_U)=(150, 75)$ position grid, and $\alpha=\beta=\pi/4$ have been used in the calculations. The color scale bar at the right represents the probability of the wavefunction.
• Figure 3: (Color online) Same as Fig. \ref{['Fig2']} for the Bunimovich stadium billiard.
• Figure 4: (Color online) Unfolded level spacing distributions $P(s)$ vs. $s$ for the Bunimovich stadium billiard with coin angles $\alpha = \beta = \pi / 4$ (a) and $\alpha = \pi / 4$$\beta = \pi / 3$ (b), and for the rectangular billiard with coin angles $\alpha = \beta = \pi / 4$ (c) and $\alpha = \pi / 4$$\beta = \pi / 3$ (d). The corresponding Wigner surmise $P_{\rm W}$ of Eq. (\ref{['eq:Wigner']}), best fitting Brody distribution $P_{\rm B}$ of Eq. (\ref{['eq:Brody']}) (with the parameters and errors reported in Table \ref{['tab:freq']}), and the Poisson distributions of Eq. (\ref{['eq:Poisson']}), are also shown, for comparison, as blue (black) dashed lines, red (gray) solid lines and green (light gray) dotted lines, respectively.
• Figure 5: (Color online) Distribution of the participation ratios of Eq. (\ref{['eq:PPR']}),$P(\rm{PR})$ vs. PR, for the Bunimovich stadium with $\alpha = \beta = \pi \ / 4$ (a), and $\alpha = \pi \ / 4$, $\beta = \pi \ / 3$ (b); and the rectangular billiard with $\alpha = \beta = \pi \ / 4$ (c) and $\alpha = \pi \ / 4$, $\beta = \pi \ / 3$ (d).