Nonlinearity-Inhomogeneity Competition in Discrete-Time Quantum Walks

Abstract

We investigate the interplay between nonlinearity and inhomogeneities in discrete-time quantum walks on one-dimensional lattices. Nonlinear effects are introduced through a Kerr-like, intensity-dependent local phase, while spatial and temporal inhomogeneities are implemented via random variations of the quantum gate operations. By analyzing typical quantities, such as the return probability and the participation function, we identify distinct quantum walking regimes as the nonlinear parameter $χ$ and the quantum gate parameter $θ$ are varied. Spatial inhomogeneities weaken nonlinear self-trapping and constrict the region of robust localization. In this process, partially localized regimes emerge, characterized by the coexistence of a confined core and dispersive wave-packet components. In contrast, temporal inhomogeneities act as time-dependent perturbations that continuously disrupt the phase coherence required for self-trapped excitation, thereby enhancing dispersive emission and promoting delocalization. By using $χ$ versus $θ$ diagrams, we display a comprehensive characterization of how inhomogeneities modify the stability and extent of prevailing dynamical regimes, elucidating the competition between nonlinearity and inhomogeneities in discrete-time quantum walks.

Figures (4)

• Figure 1: (Color on-line) Time evolution of the position-space probability density of a quantum walker on one-dimensional nonlinear lattices. The left (right) column corresponds to $\theta_0=\pi/4$, $\chi=0.3$ ($\theta_0=\pi/3$, $\chi=0.6$). The homogeneous scenarios [(a),(b)] exhibit solitonlike and stationary self-trapped regimes, in full agreement with the results reported in Refs. PhysRevA.75.062333PhysRevA.101.023802. Spatial [(c),(d)] and temporal [(e),(f)] inhomogeneities suppress the nonlinear coherent dynamics responsible for the formation of traveling and stationary self-trapped states.
• Figure 2: (Color on-line) Time evolution of the return probability $\mathrm{R_0}(t)$ and the participation function $\mathrm{PR}(t)$ for the same configurations as in Fig. \ref{['fig:1']}. Spatial [(c),(d)] and temporal inhomogeneities [(e),(f)] of the quantum gates suppress the coherent nonlinear dynamics underlying travelling and stationary self-trapped regimes.
• Figure 3: (Color on-line) In the absence of inhomogeneities, the position-space probability density exhibits emerging self-trapped profiles that become extremely sensitive to minor variations of the nonlinear strength $\chi$ [(a),(b)]. The corresponding return probability $\mathrm{R_0}(t)$ (c) and participation function $\mathrm{PR}(t)$ (d) reveal wave packets either remaining confined and oscillating around the origin or eventually escaping from the initial position. Spatial and temporal inhomogeneities [(e),(f)] suppress this chaotic-like dynamics, promoting localized and extended regimes, respectively.
• Figure 4: (Color on-line) Long-time averaged return probability (top panels) and participation function (bottom panels) mapped in the $(\chi,\theta_0)$ plane for (a,b) homogeneous lattices and for lattices with (c,d) spatial and (e,f) temporal inhomogeneities in the quantum gates. In the homogeneous scenario, stationary self-trapping is favored as $\theta_0$ approaches $\pi/2$, in agreement with Ref. PhysRevA.101.023802. Spatial inhomogeneities substantially reduce the parameter region associated with strong stationary self-trapping, leading to the emergence of partially localized states. In contrast, temporal inhomogeneities entirely suppress localized regimes, promoting extended asymptotic dynamics.