Experimental Investigation of Superdiffusion via Coherent Disordered Quantum Walks

Many disordered systems show a superdiffusive dynamics, intermediate between the diffusive one, typical of a classical stochastic process, and the so-called ballistic behavior, which is generally expected for the spreading in a quantum process. We have experimentally investigated the superdiffusive behavior of a quantum walk, whose dynamics can be related to energy transport phenomena, with a resolution which is high enough to clearly distinguish between different disorder regimes. By our experimental setup, the region between ballistic and diffusive spreading can be effectively scanned by suitably setting few degrees of freedom and without applying any decoherence to the quantum walk evolution.

Figure 1

Simulation of the variance behavior for 7 and 20 steps as a function of $p$. The average values of the variance have been calculated on a sample of 1000 different phase maps for each value of $p$. In the vertical axes ${\text{Var}}_1(p)$ for 7 and 20 steps have been normalized to their maximum values. Error bands correspond to the standard deviation of the variance distributions.

Figure 2

Computational simulations for similarity between ordered (solid line) and disordered (dashed line) distributions [respectively, $G(p=0)\equiv G(0)$ and $G(p=1)\equiv G(1)$] and the average distribution for a given value $p$ as a function of $p$. Here, $S(0,p)$ $[S(1,p)]$ stands for $S[G(0),G(p)]$ $\{S[G(1),G(p)]\}$. In the inset a zoom on the intersection region is reported. The crossing point shifts towards lower values of $p$ when the step number increases. Different colors stand for different step numbers of the evolution.

Figure 3

Sketch of the setup used in the experiment. BS, beam splitter; BD, beam displacer; RP, rotating glass plates; MM, moving mirror; SMF, single mode fiber. Each side of the square $\mathrm{SI}$ is 50 cm long. Blue and red beams circulate in opposite directions and impinge on the ${\mathrm{BS}}_1$ in the same horizontal point but at different heights along the $z$ direction, due to the effect of BD.

Figure 4

Experimental values of the variance as a function of the number of steps for different values of $p$ (0,0.05,0.1,0.2,1). Dots correspond to experimental data. For a given value of $p$, the dashed line shows the expected average behavior (Exp) of variance for the three selected phase maps. They were computed by taking into account the actual optical parameters of the setup. Dotted lines show the ideal behavior ($I$) of a perfectly symmetric ${\mathrm{BS}}_1$ in the two limit cases, namely the ordered and disordered QW. Note that, up to the third step, the variance is not affected by the phase thus the red and blue dotted lines are coincident; the same effect occurs for all the dashed lines.

Figure 5

Experimental results for the two photon QW. Dots represent experimental data, dashed lines indicate the expected behavior obtained by taking into account the real parameters of the setup (Exp), dotted lines correspond to the limit cases of the ideal ordered and disordered two photon QW (I). Note that, up to the third step, the variance is not affected by the phase thus the red and blue dotted lines are coincident.