Transport regimes of biphoton-state quantum walks in ordered and disordered arrays of nonlinear waveguides

Abstract

It is well known that a quantum walk in an ordered medium exhibits ballistic propagation. A related process is the driven quantum walk, in which the number of walkers varies over time. In this work, we show that a driven quantum walk in an array of nonlinear waveguides does not propagate ballistically, but instead exhibits two distinct transport regimes: superballistic and superdiffusive. Moreover, we study the effects of off-diagonal and diagonal disorder. In addition to the observation of Anderson localization, both types of disorder lead to the disappearance of the superballistic regime.

Figures (12)

• Figure 1: Schematic of a typical ANW with $N$ waveguides. Both linear coupling and nonlinear effects take place only within the ANW region, which begins at $z=0$. The effective propagation constant of waveguide $j$ is denoted by $\beta_j$ and $C_j$ corresponds to the linear coupling constant between waveguides $j$ and $j+1$. The output state, at the end of the ANW region, could be the input of a subsequent step.
• Figure 2: Parameter $\gamma$ as a function of the propagation distance $C_0z$ for a 71-waveguide array. The upper (lower) row corresponds to when the center (corner) waveguide is pumped. The left (right) column shows a small (large) propagation distance. For each plot, the vertical dashed gray line indicates the transition between a superballistic and a superdiffusive transport regime. For the right column, the red shaded region corresponds to where border effects are significant. The insets on the right column depict the normalized photon number along the propagation (the lighter the color, the larger the values).
• Figure 3: Standard deviation $\sigma$ and parameter $\gamma$ for small propagation distances when the center waveguide of a 51-waveguide array is pumped. The color of each line represents the off-diagonal disorder strength, which is quantified by $\kappa_C$.
• Figure 4: Color map showing the combinations of the disorder strengths, $\kappa_c$ and $\kappa_\beta$, for which the superballistic regime is present (white pixels) or absent (black pixels). The left (right) column corresponds to an injection in the center (corner) of a 51-waveguide array.
• Figure 5: Standard deviation $\sigma$ as a function of disorder strength $\kappa$. The left (right) column depicts $\sigma$ at the values $C_0z=\{5, 10, 10\}$ ($C_0z=\{10, 20, 30\}$) when the center (corner) waveguide is pumped in a 51-waveguide array. The blue, and orange lines represent, respectively, only off-diagonal, or diagonal disorder. For the case of an injection in the corner waveguide, the values of $C_0z$ were doubled since it takes a longer distance to reach similar values of $\sigma$ compared to an injection at the center waveguide.