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arXiv:1911.10831 · Physical Review A

Self-trapped quantum walks

A. R. C. Buarque, Wandearley S. Dias

Published Feb 4, 2020• 21 citations
arXivHTMLPDF

Abstract

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-$X$ gate. It appears for intermediate configurations and becomes predominant as quantum gates get closer to the Pauli-$Z$ 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.

DOI
10.1103/physreva.101.023802

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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 \ensuremathχ\ensuremath{\chi}\ensuremathχ and the quantum gates \ensuremathθ\ensuremath{\theta}\ensuremathθ, 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 \ensuremathχ\ensuremath{\chi}\ensuremathχ versus \ensuremathθ\ensuremath{\theta}\ensuremathθ 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.
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.
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-XXX 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.
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.
Quantum walks are important tools for the development of quantum algorithms and carrying out quantum simulations. Recent interest in nonlinear discrete-time quantum walks aims to use it as a shortcut through dynamical regimes hard to obtain using current methods. We introduce a model featuring a modified conditional shift operator to carry dependence on the local occupation probability with a given strength we are able to control. It accounts for a third-order nonlinear contribution which is found in many physical contexts. We find a rich set of dynamical profiles, including solitonlike propagation, self-trapping, and chaos, all these arising from rather simple rules. Our tool set goes beyond unitary transformations, thus broadening the possibilities for controlling quantum dynamics.
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.
We explore a continuous-time quantum walk starting at a single vertex on the discrete path and cycle with a cubic nonlinearity. Such nonlinearities arise in Bose-Einstein condensates described by the Gross-Pitaevskii equation or by nonlinear optical waveguide arrays. When the nonlinearity is sufficiently strong, the walker remains localized at its initial vertex, a phenomenon known as self-trapping. This contrasts with linear quantum walks, which are known for spreading quickly in one dimension. While self-trapping has been known numerically, we introduce an analytical method that proves self-trapping and yields a quantitative relationship between the nonlinearity coefficient and the trapping probability. We propose that this trapping can be used for timing in quantum state transfer, where a qubit is held at a node until it is ready to be transferred, and it can also be held again at the receiving node. This scheme can also be interpreted as a form of quantum memory, with the trap and transfer corresponding to the storage and release of quantum information.