Continuous limits of linear and nonlinear quantum walks

Abstract

In this paper, we consider the continuous limit of a nonlinear quantum walk (NLQW) that incorporates a linear quantum walk as a special case. In particular, we rigorously prove that the walker (solution) of the NLQW on a lattice $\delta \mathbb{Z}$ uniformly converges (in Sobolev space $H^s$) to the solution to a nonlinear Dirac equation (NLD) on a fixed time interval as $\delta \to 0$. Here, to compare the walker defined on $\delta \mathbb{Z}$ and the solution to the NLD defined on $\mathbb{R}$, we use Shannon interpolation.

arXiv comment19 pages, to appear in Reviews in Mathematical Physics

References in corpus (20)

• Exploring Topological Phases With Quantum Walks
• From quantum cellular automata to quantum lattice gases
• Lattice Boltzmann Equation for Quantum Mechanics
• Dirac and Weyl Equations on a Lattice as Quantum Cellular Automata
• Relativistic quantum walks
• Quantum Field as a quantum cellular automaton: the Dirac free evolution in one dimension
• Nonlinear Propagation of Light in One Dimensional Periodic Structures
• The Dirac equation as a quantum walk: higher dimensions, observational convergence
• Nonlinear optical Galton board
• Free Dirac evolution as a quantum random walk
• Discrete-time quantum walks: continuous limit and symmetries
• Bulk-edge correspondence of one-dimensional quantum walks
• From Discrete Time Quantum Walk to Continuous Time Quantum Walk in Limit Distribution
• Quantum walks with an anisotropic coin I: spectral theory
• Quantum walk as a simulator of nonlinear dynamics: Nonlinear Dirac equation and solitons
• Quantum walks with an anisotropic coin II: scattering theory
• Nonlinear Quantum Search Using the Gross-Pitaevskii Equation
• Attractor-repeller pair of topological zero-modes in a nonlinear quantum walk
• Feshbach Resonance Management of Bose-Einstein Condensates in Optical Lattices
• Discrete spacetime, quantum walks and relativistic wave equations

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• A Constructive Approach to Topological Invariants for One-dimensional Strictly Local Operators
• Asymptotic stability of small bound state of nonlinear quantum walks
• Logarithmic expansion of many-body wave packets in random potentials
• Feynman checkers: external electromagnetic field and asymptotic properties
• The Witten index for one-dimensional split-step quantum walks under the non-Fredholm condition
• The Witten Index for One-dimensional Non-unitary Quantum Walks with Gapless Time-evolution