Decoherence and Disorder in Quantum Walks: From Ballistic Spread to Localization

We investigate the impact of decoherence and static disorder on the dynamics of quantum particles moving in a periodic lattice. Our experiment relies on the photonic implementation of a one-dimensional quantum walk. The pure quantum evolution is characterized by a ballistic spread of a photon’s wave packet along 28 steps. By applying controlled time-dependent operations we simulate three different environmental influences on the system, resulting in a fast ballistic spread, a diffusive classical walk, and the first Anderson localization in a discrete quantum walk architecture.

Figure 1

(a) Schematic setup (see text for details). (b) Probability distribution after 28 steps of a symmetric Hadamard walk with initial circular polarization. Stacked bars: Adapted theory split into the two coin states $|V⟩$ (blue, bottom) and $|H⟩$ (red, top). Gray dots show experimental data for vertical polarization, black dots show the sum of both polarizations. Error bars correspond to statistical errors.

Figure 2

Measured probability distribution (front) and respective theory (back, gray bars) of 11 steps of a quantum walk ($\theta =\pi /8$) with static disorder (a), dynamic disorder (b), and in a decoherence free environment [inset (c)]. The insets in (a) and (b) show the measured distribution in semilog scale with linear (a) and parabolic fit (b). (c) Transition of the variance from ballistic quantum walk to diffusive or localized evolution due to dynamic (red squares) and static (green dots) disorder with increasing disorder strength ${\Phi }_{\max }$; dashed lines: theory with adaption for experimental imperfections. The red solid line marks the variance of a classical random walk. (Vertical error is smaller than the dot size). (d) Relative frequency $f(|{\varphi }_V|)$ of the applied phases ${\varphi }_V$ for the signal with interval ${\Phi }_{\max }=(1.02\pm 0.05)\pi$. The dashed line indicates the uniform distribution.

Figure 3

(a) Averaged probability distribution in a slow decoherence scenario with different coin angles $\theta \in [0,\pi /4]$: Measurement (orange, front) and theory (gray, back). (b) Measured trend of the variance up to 12 steps (dots) with respective theoretical simulation (lines). Photons in scenario (i) (blue triangles) and (iv) (orange dots) show a ballistic behavior. In scenario (iii) (red squares) they move diffusively, and in (ii) (green diamonds) they stagnate.