Quantum Intermittency for Sparse CMV Matrices with an Application to Quantum Walks on the Half-Line

Damanik, David; Erickson, Jon D.; Fillman, Jake; Hinkle, Gerhardt; Vu, Alan

doi:10.48550/arxiv.1507.02041

Cited by 3 publications

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“…Additionally, the process of averaging can transform dynamical quantities in particularly pleasant ways. For example, the exponential averages considered in this paper recast the a's into averages of the Poisson kernel against spectral measures; this point of view is explored in more detail in [9]. To be completely proper, the normalizing factor in (2.2) should really be 1 − e −2/K so that a is also a probability distribution on Z.…”

Section: Precise Statementsmentioning

confidence: 99%

Spreading estimates for quantum walks on the integer lattice via power-law bounds on transfer matrices

Damanik

Fillman

Ong

2016

Journal de Mathématiques Pures et Appliquées

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We discuss spreading estimates for dynamical systems given by the iteration of an extended CMV matrix. Using a connection due to Cantero-Grünbaum-Moral-Velázquez, this enables us to study spreading rates for quantum walks in one spatial dimension. We prove several general results which establish quantitative upper and lower bounds on the spreading of a quantum walk in terms of estimates on a pair of associated matrix cocycles. To demonstrate the power and utility of these methods, we apply them to several concrete cases of interest. In the case where the coins are distributed according to an element of the Fibonacci subshift, we are able to rather completely describe the dynamics in a particular asymptotic regime. As a pleasant consequence, this supplies the first concrete example of a quantum walk with anomalous transport, to the best of our knowledge. We also prove ballistic transport for a quantum walk whose coins are periodically distributed.

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Section: Precise Statementsmentioning

confidence: 99%

Spreading estimates for quantum walks on the integer lattice via power-law bounds on transfer matrices

Damanik

Fillman

Ong

2016

Journal de Mathématiques Pures et Appliquées

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“…One can use spectral methods to deduce bounds on spreading [31,32]. On the other hand, estimates of a dynamical nature can be turned into estimates on spectral quantities, for example, quantitative regularity of the spectral measures [30].…”

Section: Introductionmentioning

confidence: 99%

Singular continuous Cantor spectrum for magnetic quantum walks

et al. 2020

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In this note, we consider a physical system given by a two-dimensional quantum walk in an external magnetic field. In this setup, we show that both the topological structure as well as its type depend sensitively on the value of the magnetic flux Φ: while for Φ/(2π) rational the spectrum is known to consist of bands, we show that for Φ/(2π) irrational the spectrum is a zero-measure Cantor set and the spectral measures have no pure point part.arXiv:1908.09924v1 [quant-ph]

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“…Extended CMV matrices arise as a natural playground for spectral theoretic techniques, as they furnish canonical unitary analogs of Schrödinger operators and Jacobi matrices. Moreover, they are interesting in their own right, since they naturally arise in connection with quantum walks in one dimension [3,6,7,8], the classical Ising model [9,16], and orthogonal polynomials on the unit circle [36,37]. More specifically, given a sequence α = (α n ) n∈Z with α n ∈ D = {z ∈ C : |z| < 1}, the corresponding extended CMV matrix is given by E = LM, where…”

Section: Introductionmentioning

confidence: 99%

Purely singular continuous spectrum for Sturmian CMV matrices via strengthened Gordon Lemmas

Fillman

2016

Proc. Amer. Math. Soc.

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The Gordon Lemma refers to a class of results in spectral theory which prove that strong local repetitions in the structure of an operator preclude the existence of eigenvalues for said operator. We expand on recent work of Ong and prove versions of the Gordon Lemma that are valid for CMV matrices and which do not restrict the parity of scales upon which repetitions occur. The key ingredient in our approach is a formula of Damanik-Fillman-Lukic-Yessen which relates two classes of transfer matrices for a given CMV operator. There are many examples to which our result can be applied. We apply our theorem to complete the classification of the spectral type of CMV matrices with Sturmian Verblunsky coefficients; we prove that such CMV matrices have purely singular continuous spectrum supported on a Cantor set of zero Lebesgue measure for all (irrational) frequencies and all phases. We also discuss applications to CMV matrices with Verblunsky coefficients generated by general codings of rotations.

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Quantum Intermittency for Sparse CMV Matrices with an Application to Quantum Walks on the Half-Line

Cited by 3 publications

References 0 publications

Spreading estimates for quantum walks on the integer lattice via power-law bounds on transfer matrices

Spreading estimates for quantum walks on the integer lattice via power-law bounds on transfer matrices

Singular continuous Cantor spectrum for magnetic quantum walks

Purely singular continuous spectrum for Sturmian CMV matrices via strengthened Gordon Lemmas

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