Nonlinear flip-flop quantum walks through potential barriers

The dynamics of nonlinear flip-flop quantum walk with amplitude-dependent phase shifts with pertubing potential barrier is investigated. Through the adjustment between uniform local perturbations and a Kerr-like nonlinearity of the medium we find a rich set of dynamic profiles. We will show the existence of different Hadamard quantum walking regimes, including those with mobile soliton-like structures or self-trapped states. The latter is predominant for perturbations with amplitudes that tend to $\phi \to \pi /2$. In this system, the qubit shows an unusual behavior as we increase the amplitudes of the potential barriers and displays a monotonic decrease in the self-trapping ${\phi }_c$ with respect to the nonlinear parameter. A chaotic-like regime becomes predominant for intermediate nonlinearity values. Furthermore, we show that, by changing the quantum coins ($\theta$) a nontrivial dynamic arises, where coins close to Pauli-$X$ drives the system to a regime with predominant soliton-like structures, while the chaotic behavior are restricted to a narrow region in the $\chi -\phi$ plane. We believe that is possible to implement and observe the proprieties of this model in an integrated photonic system.

Figure 1

Snapshot of the space-time evolution of the probability density $P_n$ for some representative configurations between parameters $\chi$ and $\phi$. The left column represents the case without potential barriers, $\phi =0.0$. The right column displays a new dynamic when the quantum walker senses potential barriers with amplitudes $\phi =\pi /4$. The time ($t$) is given in the number of time steps.

Figure 2

Time evolution of the participation function $\left[\xi \right(t\left)\right]$ and survival probability $\left[\mathrm{\Omega }\right(t\left)\right]$ for three nonlinearity setting: (a,b) $\chi =0.0$, (c,d) $\chi =0.2,$ and (e,f) $\chi =0.4$. In the absence of nonlinearity, the participation function displays an increasing behavior with time for the two values of $\phi =0;\pi /4$ and a survival probability declines with a scaling behavior $\mathrm{\Omega }\sim t^{-1}$. This scenario changes dramatically when we add nonlinearity. In (e) the participation function saturates in finite values, indicating the existence of localized states for the two configurations of potential barriers. The self-trapped quantum walk regime emerges for $\phi =\pi /4$. The distinction between self-trapping walk and walking soliton-like structures is made through panel (f) where $\mathrm{\Omega }\sim t^{-1}$ when $\phi =0.0$, while the self-trapped states at the initial position $n_0$ presents $\mathrm{\Omega }\sim t^0$. The time ($t$) is given in the number of time steps.

Figure 3

Long-time average of the participation function [$\overline{\xi }\left(t_{\infty }\right)$] and survival probability [$\overline{\mathrm{\Omega }}\left(t_{\infty }\right)$] versus the potential barrier parameter $\phi \in [0,\pi /2]$ for nonlinearity strength $\chi =0.0$, 0.2, 0.4. In the absence of nonlinearity, the qubit wave packet spreads through the chain irrespective of the amplitudes of the barrier potentials, see (a,b). In the presence of nonlinearity ($\chi \ne 0$) the dynamics of the quantum walker is significantly changed. $\phi$ is given in radians.

Figure 4

Phase diagram in the $\chi$ vs. ${\phi }_c$ showing the transition between two regimes from the soliton-like to self-trapped. A monotonic decreasing of the critical value ${\phi }_c$ as we increase the nonlinear parameter $\chi$. $\phi$ is given in radians.

Figure 5

Plot of $\chi$ versus $\phi$ for the long-time average of the participation function (a) $\overline{\xi }\left(t_{\infty }\right)$ and (b) survival probability $\overline{\mathrm{\Omega }}\left(t_{\infty }\right)$. We note the absence of trapped structures for sufficiently small $\phi$ values, even for a strong nonlinear parameter $\chi$. However, for $\chi >0.6$ a chaotic-like regime is found. $\phi$ is given in radians.

Figure 6

$\chi$ versus $\phi$ diagram for the long-time average of the survival probability $\overline{\mathrm{\Omega }}\left(t_{\infty }\right)$ and also the long-time average participation function for three representative quantum coins (a,b) $\theta =\pi /6$, (c,d) $\theta =\pi /3$, and (e,f) $\theta =\pi /2$. The soliton-like phase suppresses the chaotic-like phase with the increase of $\theta$ and limits it to higher values of nonlinearity and in the low values an extended phase is observed in high $\theta$, as shown in (f). The self-trapping phase boundaries have little change in its length. $\phi$ is given in radians.

Figure 7

Universal $2\times 2$ optical gate illustration. Two coupled-fiber loops of different lengths are connected by a fiber couple. In (c) the gate can be tuned between bar, cross, and partial states.