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

Abstract

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 {\it nonlinear} QW and show that introducing {\it 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 \emph{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.

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• Continuous limits of linear and nonlinear quantum walks
• Edge states in a two-dimensional quantum walk with disorder
• Stability of topologically protected edge states in nonlinear quantum walks: Additional bifurcations unique to Floquet systems
• Probing coherence and noise tolerance in discrete-time quantum walks: unveiling self-focusing and breathing dynamics
• Nonlinear flip-flop quantum walks through potential barriers
• Quantum walk on a graph of spins: magnetism and entanglement
• A new type of quantum walks based on decomposing quantum states
• Asymptotic stability of small bound state of nonlinear quantum walks
• Weak limit theorem for a nonlinear quantum walk
• Scattering and inverse scattering for nonlinear quantum walks
• The universe as a nonlinear quantum simulation: Large $n$ limit of the central spin model