© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:1808.04354 · Journal of Physics Communications

Dynamics of solitons for nonlinear quantum walks

Masaya Maeda, Hironobu Sasaki, Etsuo Segawa, Akito Suzuki, Kanako Suzuki

Published Jan 14, 2019• 12 citations

Abstract

Abstract We present some numerical results for nonlinear quantum walks (NLQWs) studied by the authors analytically (Maeda et al 2018 Discrete Contin. Dyn. Syst. 38 3687–3703; Maeda et al 2018 Quantum Inf. Process. 17 215). It was shown that if the nonlinearity is weak, then the long time behavior of NLQWs are approximated by linear quantum walks. In this paper, we observe the linear decay of NLQWs for range of nonlinearity wider than studied in (Maeda et al 2018 Discrete Contin. Dyn. Syst. 38 3687–3703). In addition, we treat the strong nonlinear regime and show that the solitonic behavior of solutions appears. There are several kinds of soliton solutions and the dynamics becomes complicated. However, we see that there are some special cases so that we can calculate explicit form of solutions. In order to understand the nonlinear dynamics, we systematically study the collision between soliton solutions. We can find a relationship between our model and a nonlinear differential equation.

DOI: 10.1088/2399-6528/aafe2c

Related papers

Non-linear Quantum Walks

Carlos Navarrete–Benlloch, Armando Pérez, Eugenio Roldán

We introduce coined Nonlinear Quantum Walks (NLQW) on the lattice. The NLQW is based on the walker acquisition of non-linear (probability dependent) phases at each step of the walk. The most striking result is the formation of non-dispersive pulses in the probability distribution (soliton-like structures). These exhibit a variety of dynamical behaviors, including ballistic motion, dynamical localization, non-elastic collisions and chaotic behavior, in the sense that the dynamics is very sensitive to the nonlinearity strength.

•Apr 12, 2006•0 citations

On nonlinear scattering for quantum walks

Masaya Maeda, Hironobu Sasaki, Etsuo Segawa +2 more

We study large time behavior of quantum walks (QW) with self-dependent coin. In particular, we show scattering and derive the reproducing formula for inverse scattering in the weak nonlinear regime. The proof is based on space-time estimate of (linear) QW such as Strichartz estimate. Such argument is standard in the study of nonlinear Schrödinger equations but it seems to be the first time to be applied to QW. We also numerically study the dynamics of QW and observe soliton like solutions.

arXiv:1711.00620•Nov 2, 2017•1 citation

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

Chang-Woo Lee, Paweł Kurzyński, Hyunchul Nha

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 nonlinear QW and show that introducing 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 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.

arXiv:1512.08358•Physical Review A•Nov 30, 2015•33 citations

Continuous limits of linear and nonlinear quantum walks

Masaya Maeda, Akito Suzuki

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 [Formula: see text] uniformly converges (in Sobolev space [Formula: see text]) to the solution to a nonlinear Dirac equation (NLD) on a fixed time interval as [Formula: see text]. Here, to compare the walker defined on [Formula: see text] and the solution to the NLD defined on [Formula: see text], we use Shannon interpolation.

arXiv:1902.02017•Reviews in Mathematical Physics•Aug 28, 2019•22 citations

Solitons in a photonic nonlinear quantum walk: lessons from the continuum

Andreu Anglés-Castillo, Armando Pérez, Eugenio Roldán

We analyse a nonlinear QW model which can be experimentally implemented using the components of the electric field on an optical nonlinear Kerr medium, which translates into a rotation in the coin operator, with an angle which depends (in a nonlinear fashion) on the state of the walker. This simple dependence makes it easy to consider the space-time continuum limit of the evolution equation, which takes the form of a nonlinear Dirac equation. The analysis of this continuum limit allows us, under some approximations, to gain some insight into the nature of soliton structures, which is illustrated by our numerical calculations. These solitons are stable structures whose trajectories can be modulated by choosing the appropriate initial conditions. We have also studied the stability of solitons when they are subject to an additional phase that simulates an external electric field, and also explored if they are formed in higher dimensional spaces.

arXiv:2308.01014•Aug 2, 2023•0 citations

Scattering and inverse scattering for nonlinear quantum walks

Masaya Maeda, Hironobu Sasaki, Etsuo Segawa +2 more

We study large time behavior of quantum walks (QWs) with self-dependent (nonlinear) coin. In particular, we show scattering and derive the reproducing formula for inverse scattering in the weak nonlinear regime. The proof is based on space-time estimate of (linear) QWs such as dispersive estimates and Strichartz estimate. Such argument is standard in the study of nonlinear Schrödinger equations and discrete nonlinear Schrödinger equations but it seems to be the first time to be applied to QWs.

arXiv:1801.03214•Discrete and Continuous Dynamical Systems•Jan 1, 2018•25 citations

Abstract We present some numerical results for nonlinear quantum walks (NLQWs) studied by the authors analytically (Maeda et al 2018 Discrete Contin. Dyn. Syst. 38 3687–3703; Maeda et al 2018 Quantum Inf. Process. 17 215). It was shown that if the nonlinearity is weak, then the long time behavior of NLQWs are approximated by linear quantum walks. In this paper, we observe the linear decay of NLQWs for range of nonlinearity wider than studied in (Maeda et al 2018 Discrete Contin. Dyn. Syst. 38 3687–3703). In addition, we treat the strong nonlinear regime and show that the solitonic behavior of solutions appears. There are several kinds of soliton solutions and the dynamics becomes complicated. However, we see that there are some special cases so that we can calculate explicit form of solutions. In order to understand the nonlinear dynamics, we systematically study the collision between soliton solutions. We can find a relationship between our model and a nonlinear differential equation.

We introduce coined Nonlinear Quantum Walks (NLQW) on the lattice. The NLQW is based on the walker acquisition of non-linear (probability dependent) phases at each step of the walk. The most striking result is the formation of non-dispersive pulses in the probability distribution (soliton-like structures). These exhibit a variety of dynamical behaviors, including ballistic motion, dynamical localization, non-elastic collisions and chaotic behavior, in the sense that the dynamics is very sensitive to the nonlinearity strength.

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 nonlinear QW and show that introducing 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 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.

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 [Formula: see text] uniformly converges (in Sobolev space [Formula: see text]) to the solution to a nonlinear Dirac equation (NLD) on a fixed time interval as [Formula: see text]. Here, to compare the walker defined on [Formula: see text] and the solution to the NLD defined on [Formula: see text], we use Shannon interpolation.

We analyse a nonlinear QW model which can be experimentally implemented using the components of the electric field on an optical nonlinear Kerr medium, which translates into a rotation in the coin operator, with an angle which depends (in a nonlinear fashion) on the state of the walker. This simple dependence makes it easy to consider the space-time continuum limit of the evolution equation, which takes the form of a nonlinear Dirac equation. The analysis of this continuum limit allows us, under some approximations, to gain some insight into the nature of soliton structures, which is illustrated by our numerical calculations. These solitons are stable structures whose trajectories can be modulated by choosing the appropriate initial conditions. We have also studied the stability of solitons when they are subject to an additional phase that simulates an external electric field, and also explored if they are formed in higher dimensional spaces.