Wave Packet Spreading with Disordered Nonlinear Discrete-Time Quantum Walks

We use a novel unitary map toolbox—discrete-time quantum walks originally designed for quantum computing—to implement ultrafast computer simulations of extremely slow dynamics in a nonlinear and disordered medium. Previous reports on wave packet spreading in Gross-Pitaevskii lattices observed subdiffusion with the second moment $m_2\sim t^{1/3}$ (with time in units of a characteristic scale $t_0$) up to the largest computed times of the order of $10^8$. A fundamental and controversially debated question—whether this process can continue ad infinitum, or has to slow down—stands unresolved. Current experimental devices are not capable to even reach $1/{10}^4$ of the reported computational horizons. With our toolbox, we outperform previous computational results and observe that the universal subdiffusion persists over an additional four decades reaching “astronomic” times $2\times {10}^{12}$. Such a dramatic extension of previous computational horizons suggests that subdiffusion is universal, and that the toolbox can be efficiently used to assess other hard computational many-body problems.

Figure 1

A schematic representation of a general discrete-time quantum walk. The vertical arrows indicate the quantum coin action within each two level system, while the horizontal ones show the action of the transfer operator.

Figure 2

Wave packet density profiles ${\rho }_n$ for linear $g=0$ (orange solid lines) and nonlinear $g=3$ (blue solid lines) DTQWs at the time $t_f={10}^8$ (linear case) and $t_f=2\times {10}^{12}$ (nonlinear case). (a) lin-lin plot. (b) log-normal plot. The black dashed lines indicate exponential decay with the corresponding localization length $e^{-2|n-n_0|/\xi }$.

Figure 3

(a) $m_2(t)$ versus time in log-log scale for the linear $g=0$ (orange solid lines) and nonlinear $g=3$ (blue sold lines) case from Fig. 2. (b) The derivative $\alpha (t)$ versus time for the data of the nonlienar run from (a) (blue line). Black dashed line corresponds to $\alpha =1/3$.

Figure 4

(a) The geometric average ${\overline{m}}_2$ versus time for 108 disorder realizations up to time $t={10}^8$. The nonlinear parameter $g$ varies from bottom to top as 0.5 (blue), 1 (orange), 1.5 (green), 2 (red), 2.5 (purple), and 50 (brown). (b) The derivative $\alpha (t)$ versus time for the data from (a) and same color codes. The horizontal dashed line corresponds to $\alpha =1/3$. (c),(d) Same as in (a),(b) but for $g=2.5$ (blue) and $g=0.5$ (orange) and up to time $t={10}^{10}$. Shaded areas indicate the statistical error.