Discrete time quantum walk of locally interacting walkers

TL;DR

This work addresses how phase-controlled local interactions between two quantum walkers modify their spatial distributions and quantum correlations in a discrete-time quantum walk. They introduce a general on-site interaction V conditioned on co-location, parameterized by a phase θ in the simple V0(θ) form and analyze the resulting dynamics using joint probabilities, coarse-grained partitioning, and the von Neumann entropy of the reduced density matrix. The results show that increasing θ induces central localization and strong bunching, while the entanglement between walkers exhibits a 2π-periodic dependence on θ and saturates near θ ≈ 3π/2 before decreasing again. The framework unifies prior models, offers a versatile platform for quantum simulation and sensing, and suggests directions for extending to more complex multi-walker interactions.

Abstract

In this work, we introduce a general form of a two-parameter family of local interactions between quantum walkers conditioned on the internal state of their coins. By choosing their particular case, we systematically study the impact of these interactions on the dynamics of two initially localized and noncorrelated walkers. Our general interaction framework, which reduces to several previously studied models as special cases, provides a versatile platform for engineering quantum correlations with applications in quantum simulation, state preparation, and sensing protocols. It also opens up the possibility of analyzing many-body interactions for larger numbers of walkers.

Figures (5)

• Figure 1: The two-particle density distribution $\mathrm{P}(x_1,x_2;t)$ after $t = 100$ time steps for non-interacting walkers ($\theta=0$). The initial state is taken to be $\ket{\Psi(0)} = \ket{+} \otimes \ket{+} \otimes \ket{0} \otimes \ket{0}$.
• Figure 2: The two-particle density distribution $\mathrm{P}(x_1,x_2;t)$ after $t = 100$ time steps for different interaction parameter $\theta$. By comparison with Fig. \ref{['Fig1']} we investigate the impact of interactions to the dynamics. The limits for the colorbars are set to be $[0 \; \;0.006]$.
• Figure 3: Density distribution of a single walker $n(x,t)$ after $t = 100$ time steps as a function interaction parameter $\theta$.
• Figure 4: Partial probabilities \ref{['PartProb']} of finding both walkers in different parts of the lattice obtained from the two-particle density distribution $\mathrm{P}(x_1,x_2;t)$ for $t=100$ as functions of the interaction parameter $\theta$. Different colors correspond to different areas of the lattice. For clarity, a schematic division of the probability distribution into different zones is shown below the plot.
• Figure 5: Entanglement entropy $\mathcal{E}(t)$ between two quantum walkers as a function of time and interaction parameter $\theta$, respectively on (top) and (bottom) panel. The initial state of the composite system is given by Eq. \ref{['eq:initialstate']}.