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Quantum walk with a local spin interaction

Manami Yamagishi, Naomichi Hatano, Kohei Kawabata, Chusei Kiumi, Akinori Nishino, Franco Nori, Hideaki Obuse

Published Jan 1, 2026• 0 citations
Publisher
DOI
10.2139/ssrn.6660638

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Abstract We present an overview of two models of quantum random walk. In the first model, the discrete quantum random walk, we present the explicit solution for the recurring amplitude of the quantum random walk on a one-dimensional lattice. We also introduce a new method of solving the problem of random walk in the most general case and use it to derive the hitting amplitude for quantum random walk on the hypercube. The second is a special model based on a local interaction between neighboring spin-1/2 particles on a one-dimensional lattice. We present explicit results for the relevant quantities and obtain an upper bound on the speed of convergence to limiting probability distribution.
We investigate continuous-time quantum walks of two fermionic atoms loaded in one-dimensional optical lattices with on-site interaction and subjected to a Zeeman field. The quantum walks are accompanied by spin-flipping processes. We calculate the time-dependent density distributions of the two fermions with opposite spins which are initially positioned at the center site by means of an exact numerical method. Besides the usual fast linear expansion behavior, we find an interesting spin-flipping-induced localization in the time evolution of density distributions. We show that the fast linear expansion behavior can be restored by simply ramping up the Zeeman field or further increasing the spin-flipping strength. The intrinsic origin of this exotic phenomenon is attributed to the emergence of a flat band in the single-particle spectrum of the system. Furthermore, we investigate the effect of on-site interaction on the dynamics of the quantum walkers. The two-particle correlations are calculated and the signal of localization is also shown therein. A simple potential experimental application of this interesting phenomenon is proposed.
We introduce a model of a quantum walk on a graph in which a particle jumps between neighboring nodes and interacts with independent spins sitting on the edges. Entanglement propagates with the walker. We apply this model to the case of a one-dimensional lattice to investigate its magnetic and entanglement properties. In the continuum limit, we recover a Landau-Lifshitz equation that describes the precession of spins. A rich dynamics is observed, with regimes of particle propagation and localization, together with spin oscillations and relaxation. Entanglement of the asymptotic states follows a volume law for most parameters (the coin rotation angle and the particle-spin coupling).