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arXiv:2210.11661 · Physical Review A

Nonlinear flip-flop quantum walks through potential barriers

F. S. Passos, A. R. C. Buarque

Published Dec 7, 2022• 9 citations
arXivHTMLPDF

Abstract

The dynamics of nonlinear flip-flop quantum walk with amplitude-dependent phase shifts with pertubing potential barrier is investigated. Through the adjustment between uniform local perturbations and a Kerr-like nonlinearity of the medium we find a rich set of dynamic profiles. We will show the existence of different Hadamard quantum walking regimes, including those with mobile soliton-like structures or self-trapped states. The latter is predominant for perturbations with amplitudes that tend to $\ensuremath{\varphi}\ensuremath{\rightarrow}\ensuremath{\pi}/2$. In this system, the qubit shows an unusual behavior as we increase the amplitudes of the potential barriers and displays a monotonic decrease in the self-trapping ${\ensuremath{\varphi}}_{c}$ with respect to the nonlinear parameter. A chaotic-like regime becomes predominant for intermediate nonlinearity values. Furthermore, we show that, by changing the quantum coins ($\ensuremath{\theta}$) a nontrivial dynamic arises, where coins close to Pauli-$X$ drives the system to a regime with predominant soliton-like structures, while the chaotic behavior are restricted to a narrow region in the $\ensuremath{\chi}\ensuremath{-}\ensuremath{\varphi}$ plane. We believe that is possible to implement and observe the proprieties of this model in an integrated photonic system.

DOI
10.1103/physreva.106.062407

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The dynamics of nonlinear flip-flop quantum walk with amplitude-dependent phase shifts with pertubing potential barrier is investigated. Through the adjustment between uniform local perturbations and a Kerr-like nonlinearity of the medium we find a rich set of dynamic profiles. We will show the existence of different Hadamard quantum walking regimes, including those with mobile soliton-like structures or self-trapped states. The latter is predominant for perturbations with amplitudes that tend to \ensuremathφ\ensuremath→\ensuremathπ/2\ensuremath{\varphi}\ensuremath{\rightarrow}\ensuremath{\pi}/2\ensuremathφ\ensuremath→\ensuremathπ/2. In this system, the qubit shows an unusual behavior as we increase the amplitudes of the potential barriers and displays a monotonic decrease in the self-trapping \ensuremathφc{\ensuremath{\varphi}}_{c}\ensuremathφc​ with respect to the nonlinear parameter. A chaotic-like regime becomes predominant for intermediate nonlinearity values. Furthermore, we show that, by changing the quantum coins (\ensuremathθ\ensuremath{\theta}\ensuremathθ) a nontrivial dynamic arises, where coins close to Pauli-XXX drives the system to a regime with predominant soliton-like structures, while the chaotic behavior are restricted to a narrow region in the \ensuremathχ\ensuremath−\ensuremathφ\ensuremath{\chi}\ensuremath{-}\ensuremath{\varphi}\ensuremathχ\ensuremath−\ensuremathφ plane. We believe that is possible to implement and observe the proprieties of this model in an integrated photonic system.
We study the existence and characterization of self-trapping phenomena in discrete-time quantum walks. By considering a Kerr-like nonlinearity, we associate an acquisition of the intensity-dependent phase with the walker while it propagates on the lattice. Adjusting the nonlinear parameter $\ensuremath{\chi}$ and the quantum gates $\ensuremath{\theta}$, we will show the existence of different quantum walking regimes, including those with traveling solitonlike structures or localized by self-trapping. The latter scenario is absent for quantum gates close enough to the Pauli-XXX gate. It appears for intermediate configurations and becomes predominant as quantum gates get closer to the Pauli-ZZZ gate. By using $\ensuremath{\chi}$ versus $\ensuremath{\theta}$ diagrams, we will show that the threshold between quantum walks with delocalized or localized regimes exhibits an unusual aspect in which an increment of the nonlinear strength can induce the system to transition from a localized to a delocalized regime.
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 $\ensuremath{\propto}{t}^{\ensuremath{-}1/2}$. 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.
Abstract We propose a nonlinear quantum walk model inspired in a photonic implementation in which the polarization state of the light field plays the role of the coin-qubit. In particular, we take profit of the nonlinear polarization rotation occurring in optical media with Kerr nonlinearity, which allows to implement a nonlinear coin operator, one that depends on the state of the coin-qubit. We 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 to gain some insight into the existence of different solitonic structures, such as bright and dark solitons. We illustrate several properties of these solitons with numerical calculations, including the effect on them of an additional phase simulating an external electric field.
We investigate the interplay between nonlinearity and inhomogeneities in discrete-time quantum walks on one-dimensional lattices. Nonlinear effects are introduced through a Kerr-like, intensity-dependent local phase, while spatial and temporal inhomogeneities are implemented via random variations of the quantum gate operations. By analyzing typical quantities, such as the return probability and the participation function, we identify distinct quantum walking regimes as the nonlinear parameter χχχ and the quantum gate parameter θθθ are varied. Spatial inhomogeneities weaken nonlinear self-trapping and constrict the region of robust localization. In this process, partially localized regimes emerge, characterized by the coexistence of a confined core and dispersive wave-packet components. In contrast, temporal inhomogeneities act as time-dependent perturbations that continuously disrupt the phase coherence required for self-trapped excitation, thereby enhancing dispersive emission and promoting delocalization. By using χχχ versus θθθ diagrams, we display a comprehensive characterization of how inhomogeneities modify the stability and extent of prevailing dynamical regimes, elucidating the competition between nonlinearity and inhomogeneities in discrete-time quantum walks.
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.
The sensitivity of quantum systems to external disturbances is a fundamental problem for the implementation of functional quantum devices, quantum information, and computation. Based on remarkable experimental progress in optics and ultracold gases, we study the consequences of a short-time (instantaneous) noise while an intensity-dependent phase acquisition is associated with a qubit propagating on an NNN cycle. By employing quantum coherence measures, we report emerging unstable regimes in which quantum walks arise, such as self-focusing and breathing dynamics. Our numerical and analytical results unveil appropriate settings which favor the stable regime, with the asymptotic distribution surviving for weak nonlinearities and disappearing in the thermodynamic limit with 1/N1/N1/N. The diagram showing the threshold between different regimes reveals the quantum gates close to Pauli-Z as more noise tolerant. As we move towards the Pauli-X quantum gate, such aptness dramatically decreases and the threshold to the self-focusing regime becomes almost unavoidable. Quantum gates close to Hadamard exhibit an unusual aspect, in which an increment of the nonlinear strength can remove the dynamics from the self-focusing regime.