Nonlinear three-state quantum walks

P. R. N. Falcão, J. P. Mendonça, A. R. C. Buarque, W. S. Dias, G. M. A. Almeida, and M. L. Lyra

Phys. Rev. A 106, 042202 – Published 3 October 2022

Abstract

The dynamics of a three-state quantum walk with amplitude-dependent phase shifts is investigated. We consider two representative inputs whose linear evolution is known to display either full dispersion of the wave packet or intrinsic localization on the initial position. The nonlinear counterpart presents much more involved dynamics featuring self-trapping, solitonic pulses, radiation, and chaoticlike behavior. We show that nonlinearity leads to a metastable self-trapped wave-packet component that radiates in the long-time regime with a survival probability $\propto t^{- 1 / 2}$ . A sudden dynamical transition from such a metastable state to the point when the radiation process is triggered is found for a set of parameters.

Received 6 July 2022
Accepted 15 September 2022

DOI:https://doi.org/10.1103/PhysRevA.106.042202

Physics Subject Headings (PhySH)

Quantum simulation Quantum walks

Chaos & nonlinear dynamics

Quantum Information, Science & TechnologyNonlinear Dynamics

Authors & Affiliations

P. R. N. Falcão ^1,*, J. P. Mendonça ^1,2, A. R. C. Buarque^1,3, W. S. Dias¹, G. M. A. Almeida¹, and M. L. Lyra ¹

¹Instituto de Física, Universidade Federal de Alagoas, 57072-900 Maceió, Alagoas, Brazil
²Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland
³Secretaria de Estado da Educação de Alagoas, 57055-055 Maceió, Alagoas, Brazil

^*pedro.falcao@fis.ufal.br

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 106, Iss. 4 — October 2022

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Time evolution of the probability density $| ψ_{n} |^{2}$ in the position space for some representative values of the nonlinear parameter $χ$ . The coin component $| c 〉$ of the input is set as (a)–(d) $| σ^{+} 〉$ and (e)–(h) $| σ_{1}^{-} 〉$ . From top to bottom, $χ = 0$ , 0.2, 0.6, 1.0. Note that nonlinearity brings the localized component to a metastable state and induces some degree of localization when $| c 〉 = | σ_{1}^{-} 〉$ .
Reuse & Permissions
Figure 2
Snapshots of the probability density $| ψ_{n} |^{2}$ after $t = 5000$ steps. (a) Localized component around the initial position for linear ( $χ = 0$ ) and nonlinear ( $χ = 0.6$ ) cases. In the first case when $| ψ (0) 〉 = | σ^{+} 〉$ , the wave function is exponentially localized, with $| ψ_{n} |^{2} = 12 {(- 5 + 2 \sqrt{6})}^{2 | n |}$ for $n \neq 0$ and $| ψ_{0} |^{2} = 15 - 6 \sqrt{6}$ in the asymptotic long-time limit (dashed-dotted line) [39]. In the nonlinear regime, the localized component is well described by a hyperbolic secant function of the form $| ψ_{n} |^{2} \propto {sech}^{2} [a n]$ (dashed line), with $a$ being a fitting parameter, for either coin input states $| σ^{+} 〉$ and $| σ_{1}^{-} 〉$ . This is what we refer to as the self-trapped state. (b) Solitonlike pulse propagating outwards, also fitted by the hyperbolic secant function (dashed line).
Reuse & Permissions
Figure 3
Time evolution of the participation ratio for the coin input states (a) $| σ^{+} 〉$ and (b) $| σ_{1}^{-} 〉$ . This quantity offers a clear signature of the radiation dynamics when nonlinear effects are taken into account.
Reuse & Permissions
Figure 4
Survival probability vs time steps for the usual coin inputs, $| σ^{+} 〉$ and $| σ_{1}^{-} 〉$ , and different values of $χ$ . We note that the radiation dynamics of the localized pulse approximately follows a power law.
Reuse & Permissions
Figure 5
Duration of the metastable state $τ_{c}$ as a function of the nonlinear strength $χ$ for both coin inputs. Note the pronounced peaks for specific values of $χ$ when $| c 〉 = | σ^{+} 〉$ . To determine $τ_{c}$ , we first get the participation ratio data for a long-time frame. Then we make every consecutive set of 20 numbers bundled up into a representative value obtained by their mean. This is done so that the sharp increase of these values can indicate the onset of the radiation process. $τ_{c}$ is finally defined whenever the difference between two consecutive representative values of the participation ratio is largest.
Reuse & Permissions
Figure 6
Phase portraits $SP \times d (SP) / d t$ for the coin inputs $| σ^{+} 〉$ (left column) and $| σ_{1}^{-} 〉$ (right column), and distinct values of $χ$ . The color gradient is to give a sense of direction. Note that the localized (self-trapped) component becomes unstable in the presence of nonlinearity.
Reuse & Permissions

Physical Review A

covering atomic, molecular, and optical physics and quantum information