Emergent nonlinear phenomena in discrete-time quantum walks

Quantum walks are important tools for the development of quantum algorithms and carrying out quantum simulations. Recent interest in nonlinear discrete-time quantum walks aims to use it as a shortcut through dynamical regimes hard to obtain using current methods. We introduce a model featuring a modified conditional shift operator to carry dependence on the local occupation probability with a given strength we are able to control. It accounts for a third-order nonlinear contribution which is found in many physical contexts. We find a rich set of dynamical profiles, including solitonlike propagation, self-trapping, and chaos, all these arising from rather simple rules. Our tool set goes beyond unitary transformations, thus broadening the possibilities for controlling quantum dynamics.

Figure 1

(a) Two coupled-fiber loops of different lengths are connected by a 50:50 fiber coupler in order to induce a time delay between them. This arrangement is isomorphic to (b) an optical network spanned by pulse arrival times, taking the role of positions vs number of round trips. $S$ ($L$) stands for short (long) loop, which reduces (advances) the position coordinate by 1. Considering the standard time-multiplexing architecture [47], a single step in our DTQW model must correspond to two consecutive round trips through the coupled-fiber system so as to fulfill Eqs. (4) and (5).

Figure 2

Time evolution of probability $P_x\left(t\right)$ in space for (a) $\alpha =0$ (linear DTQW), (b) $\alpha =0.2$, (c) $\alpha =1$, and (d) $\alpha =50$. $N$ is set large enough to ensure the wave function does not reach the border and the initial state is the symmetric one, $|\mathrm{\Psi }(t=0)\rangle =\left(\right|x_0,\uparrow \rangle +i|x_0,\downarrow \rangle )/\sqrt{2}$ for $x_0=0$.

Figure 3

Single-pulse dispersion ${\sigma }_{\mathrm{sp}}\left(t\right)$ for $\alpha =0.3,0.5$ (top panel) and $\alpha =20,30$ (bottom panel). The stationary regime of ${\sigma }_{\mathrm{sp}}\left(t\right)$ indicates formation of a stable solitonlike pulse. Here we define the soliton formation time $\tau$ as the instant ${\sigma }_{\mathrm{sp}}$ reaches its maximum. The inset of the bottom panel shows the same curves, zoomed in for better visualization of the stationary regime.

Figure 4

Soliton formation time $\tau$ vs $\alpha$ as evaluated via the single-pulse dispersion dynamics (see text). There are two dynamical regimes, one of which is characterized by $\tau \sim {\alpha }^{-2}$ (left side) and the other (right side) by $\tau \sim \alpha$. The latter is where the walker undergoes a transition from a self-trapped state to solitonlike behavior. There is a discontinuity between both regimes represented by the shadowed area in the interval $\alpha \in [1,2)$ where $\tau$ cannot be evaluated precisely.

Figure 5

Time evolution of the single-pulse average position $\langle x\left(t\right)\rangle$ for some representative values of the nonlinearity level $\alpha$. For strong nonlinearities, $\langle x\left(t\right)\rangle$ has a very slow initial increase during the self-trapping time. After that it evolves with a constant velocity. For weak nonlinearities, there is a crossover from $v=1/2$ at short times to a larger velocity after the soliton formation. The inset shows this crossover around the soliton formation time for $\alpha =0.01$.

Figure 6

Asymptotic velocity $v$ of the single-pulse average position $\langle x\left(t\right)\rangle$ against nonlinearity level $\alpha$, obtained via linear regression at long times ($t=8600$ in some cases). The maximum velocity is found to be $v=1/\sqrt{2}$ when $\alpha \to 0$.

Figure 7

Averaged Lyapunov exponent $\lambda$ vs nonlinearity strength $\alpha$. For each $\alpha$ we generated about 350 independent sets of $\left\{{\delta }_x\right\}$ (randomly distributed within $[-{10}^{-6},{10}^{-6}]$) to form $|{\mathrm{\Psi }}_{\delta }\left(t_0\right)\rangle$ and thus evaluate the distance $d\left(t\right)$ (see text for details) starting at $t_0=500$. A linear fit is put in order for $lnd\left(t\right)$ so as to find its slope $\lambda$, conveniently between steps $t=600$ and $t=800$. The system size is fixed to $N=8000$ to avoid boundary effects, and the initial state $|\mathrm{\Psi }(t=0)\rangle =\left(\right|x_0,\uparrow \rangle +i|x_0,\downarrow \rangle )/\sqrt{2}$, with $x_0=0$. Chaotic features begin to set in above ${\alpha }_c\approx 23$. The line is for guiding the eye, and it follows that $0.1(\alpha -{\alpha }_c)/[1+(\alpha -{\alpha }_c)]$ in the chaotic regime. Each point is the average over many neighboring outcomes.

Figure 8

Snapshots of the single pulse distribution for two representative values of the nonlinearity. (a) For $\alpha =10$ the distribution keeps its form while propagating along the chain. (b) For $\alpha =50$ the wave packet develops irregular fluctuations typical of the chaotic regime.

Figure 9

Density plot of the single pulse dispersion as a function of time $t$ and nonlinearity $\alpha$. Peaks in dispersion separate the initial transient regimes from the solitonic regime. The initial transient features spread (self-trapped) distributions for weak (strong) nonlinearities. Dispersion has no peak at intermediate nonlinear strengths. The chaotic regime is characterized by irregular fluctuations of the wave-packet dispersion.