Quantum walk as a simulator of nonlinear dynamics: Nonlinear Dirac equation and solitons

Quantum walk (QW) provides a versatile tool to study fundamental physics and also to make a variety of practical applications. We here start with the recent idea of nonlinear QW and show that introducing nonlinearity to QW can lead to a wealth of remarkable possibilities, e.g., simulating nonlinear quantum dynamics, thus enhancing the applicability of QW above the existing level for a universal quantum simulator. As an illustration, we show that the dynamics of a nonlinear Dirac particle can be simulated on an optical nonlinear QW platform implemented with a measurement-based-feedforward scheme. The nonlinear evolution induced by the feed-forward introduces a self-coupling mechanism to (otherwise linear) Dirac particles, which accordingly behave as a soliton. We particularly consider two kinds of nonlinear Dirac equations, one with a scalar-type self-coupling (Gross-Neveu model) and the other with a vector-type one (Thirring model), respectively. Using their known stationary solutions, we confirm that our nonlinear QW framework is capable of exhibiting characteristic features of a soliton. Furthermore, we show that the nonlinear QW enables us to observe and control an enhancement and suppression of the ballistic diffusion.

Figure 1

Optical setup for the nonlinear quantum walk (NQW) using feed-forward scheme. The schematic shown here is part of the entire setting and the blobs imply that the same components repeat in the corresponding directions. The polarization of each beam is rotated by a wave plate implementing the coin operator in Eq. (5) and the beam is split into two by a polarized beam splitter (shift operator), which implements the linear QW evolution. Each WP in the figure is actually a combination of quarter and half WPs but we refer to it collectively by WP for brevity. On the other hand, the intensity of each split beam is measured and used to control the succeeding WP (scalar type NQW) or phase shifter (vector type NQW), respectively, which generates a nonlinear evolution.

Figure 2

(a) Charge $Q$ (intensity distribution) of a stationary solution of NDE on NQW platform for scalar type self-interaction. (b) Charge $Q$ of the superposition of oppositely moving solutions on NQW platform for scalar type self-interaction.

Figure 3

(a)–(d) Snapshot of the charge distribution at $t=200$ of a walker starting from the origin with coin configuration $\left|{\phi }_{+}\right.〉=\left|\uparrow \right.〉+i\left|\downarrow \right.〉$ (a) $[\left|{\phi }_{-}\right.〉=\left|\uparrow \right.〉-i\left|\downarrow \right.〉$ (c)] and the corresponding growth of the position width $\sigma \left(t\right)$ with time (b) [(d)] until $t=200$. (e) Ballistic diffusion speed $\sigma \left(t\right)/t$ of linear QW, scalar-type NQW, and vector-type NQW for 12 482 initial states of the form $\left|x=0\right.〉\otimes [cos(\theta /2)\left|\uparrow \right.〉+sin(\theta /2)e^{i\varphi }\left|\downarrow \right.〉]$. Note that the data points are arranged in ascending order of the speed value of linear QW for comparison. The gray curve denotes the case of linear QW, the blue curve the case of scalar-type NQW, and the red curve the case of vector-type NQW.

Figure 4

Snapshot of the charge distribution at $t=200$ with the nonlinear-coupling strength $g=2$ (a) [(c)] and the corresponding growth of the position width $\sigma \left(t\right)$ (b) [(d)] for the case of $\left|{\phi }_{+}\right.〉=\left|\uparrow \right.〉+i\left|\downarrow \right.〉[\left|{\phi }_{-}\right.〉=\left|\uparrow \right.〉-i\left|\downarrow \right.〉]$. Ballistic diffusion speed $\sigma \left(t\right)/t$ as a function of coupling strength $g$ for $\left|{\phi }_{+}\right.〉$ (e) and for $\left|{\phi }_{-}\right.〉$ (f), respectively. In each plot the gray curve denotes the case of linear QW, the blue curve the case of scalar-type NQW, and the red curve the case of vector-type NQW.

Figure 5

Snapshot of the charge distribution at $t=200$ in the case of $\mathrm{\Theta }=\pi /6$ (a) [(c)] and the corresponding growth of the position width $\sigma \left(t\right)$ (b) [(d)] for the case of $\left|{\phi }_{+}\right.〉\left[\left|{\phi }_{-}\right.〉\right]$. (e) Ballistic diffusion speeds of $\left|{\phi }_{-}\right.〉$ at $t=200$ as functions of $\mathrm{\Theta }$. (f) Ballistic diffusion speeds of linear QW (yellow) and scalar type case (violet) of $\left|{\phi }_{-}\right.〉$ as functions of coupling constant $g$ and coin angle $\mathrm{\Theta }$. Except for the case of (f), the color legends of each curve are the same as in Fig. 4.