arXiv:1801.03214 · Discrete and Continuous Dynamical Systems

Scattering and inverse scattering for nonlinear quantum walks

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

Published Jan 1, 2018• 25 citations

Abstract

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.

DOI: 10.3934/dcds.2018159

Related papers

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:1711.00620Nov 2, 20171 citation

Dynamics of solitons for nonlinear quantum walks

Masaya Maeda, Hironobu Sasaki, Etsuo Segawa +2 more

arXiv:1808.04354Journal of Physics CommunicationsJan 14, 201912 citations

A remark on uniform Strichartz estimate for quantum walks on 1D lattice

Aso, Takumi, Masaya Maeda

In this short note, we study quantum walks (QWs) on one dimensional lattice $δ\mathbb{Z}$. Following Hong-Yang, we prove Strichartz estimates for QWs independent of the lattice width $δ$.

arXiv:2112.01000Dec 2, 20210 citations

Non-linear Quantum Walks

Carlos Navarrete–Benlloch, Armando Pérez, Eugenio Roldán

Apr 12, 20060 citations

Continuous limits of linear and nonlinear quantum walks

Masaya Maeda, Akito Suzuki

arXiv:1902.02017Reviews in Mathematical PhysicsAug 28, 201922 citations

Dispersive estimates for quantum walks on 1D lattice

Masaya Maeda, Hironobu Sasaki, Etsuo Segawa +2 more

arXiv:1808.05714Journal of the Mathematical Society of JapanAug 5, 202112 citations

arXiv:1801.03214 · Discrete and Continuous Dynamical Systems

Scattering and inverse scattering for nonlinear quantum walks

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

Published Jan 1, 2018• 25 citations

arXiv HTML PDF

Abstract

DOI: 10.3934/dcds.2018159

Related papers

On nonlinear scattering for quantum walks

Masaya Maeda, Hironobu Sasaki, Etsuo Segawa +2 more

arXiv:1711.00620Nov 2, 20171 citation

Dynamics of solitons for nonlinear quantum walks

Masaya Maeda, Hironobu Sasaki, Etsuo Segawa +2 more

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.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.

arXiv:1808.04354Journal of Physics CommunicationsJan 14, 201912 citations

A remark on uniform Strichartz estimate for quantum walks on 1D lattice

Aso, Takumi, Masaya Maeda

In this short note, we study quantum walks (QWs) on one dimensional lattice $δ\mathbb{Z}$ . Following Hong-Yang, we prove Strichartz estimates for QWs independent of the lattice width $δ$ .In this short note, we study quantum walks (QWs) on one dimensional lattice $δ\mathbb{Z}$. Following Hong-Yang, we prove Strichartz estimates for QWs independent of the lattice width $δ$.

arXiv:2112.01000Dec 2, 20210 citations

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.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, 20060 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.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:1902.02017Reviews in Mathematical PhysicsAug 28, 201922 citations

Dispersive estimates for quantum walks on 1D lattice

Masaya Maeda, Hironobu Sasaki, Etsuo Segawa +2 more

We consider quantum walks with position dependent coin on 1D lattice $\mathbb{Z}$ . The dispersive estimate $\| U^{t} P_{c} u_0\|_{l^{\infty}} \lesssim (1+|t|)^{-1/3} \|u_0\|_{l^1}$ is shown under $l^{1,1}$ perturbation for the generic case and $l^{1,2}$ perturbation for the exceptional case, where $U$ is the evolution operator of a quantum walk and $P_{c}$ is the projection to the continuous spectrum. This is an analogous result for Schrödinger operators and discrete Schrödinger operators. The proof is based on the estimate of oscillatory integrals expressed by Jost solutions.We consider quantum walks with position dependent coin on 1D lattice $\mathbb{Z}$. The dispersive estimate $\| U^{t} P_{c} u_0\|_{l^{\infty}} \lesssim (1+|t|)^{-1/3} \|u_0\|_{l^1}$ is shown under $l^{1,1}$ perturbation for the generic case and $l^{1,2}$ perturbation for the exceptional case, where $U$ is the evolution operator of a quantum walk and $P_{c}$ is the projection to the continuous spectrum. This is an analogous result for Schrödinger operators and discrete Schrödinger operators. The proof is based on the estimate of oscillatory integrals expressed by Jost solutions.

arXiv:1808.05714Journal of the Mathematical Society of JapanAug 5, 202112 citations

In this short note, we study quantum walks (QWs) on one dimensional lattice

δ\mathbb{Z}

. Following Hong-Yang, we prove Strichartz estimates for QWs independent of the lattice width

δ

We consider quantum walks with position dependent coin on 1D lattice

\mathbb{Z}

. The dispersive estimate

\| U^{t} P_{c} u_0\|_{l^{\infty}} \lesssim (1+|t|)^{-1/3} \|u_0\|_{l^1}

is shown under

l^{1,1}

perturbation for the generic case and

l^{1,2}

perturbation for the exceptional case, where

U

is the evolution operator of a quantum walk and

P_{c}

is the projection to the continuous spectrum. This is an analogous result for Schrödinger operators and discrete Schrödinger operators. The proof is based on the estimate of oscillatory integrals expressed by Jost solutions.