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Noname manuscript No will be inserted by the editor

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Noname manuscript No. (will be inserted by the editor) Dynamical Localization for d-Dimensional Random Quantum Walks Alain Joye Received: date / Accepted: date Abstract We consider a d-dimensional random quantum walk with site-dependent random coin operators. The corresponding transition coefficients are charac- terized by deterministic amplitudes times independent identically distributed site-dependent random phases. When the deterministic transition amplitudes are close enough to those of a quantum walk which forbids propagation, we prove that dynamical localization holds for almost all random phases. This in- stance of Anderson localization implies that all quantum mechanical moments of the position operator are uniformly bounded in time and that spectral lo- calization holds, almost surely. Keywords Random quantum walks · Dynamical localization · Fractional moments estimates 1 Introduction Quantum walks defined on a lattice, deterministic or random, have become a popular topic of research recently, due the interest they have for different scientific communities, see the reviews [25], [28], [3]. Quantum walks can be used to provide an effective description of the dynamics of certain types of physical quantum systems and belong, in this sense, to the broader category of quantum network models, [1], [32], [12], [9]. In particular, quantum walks accurately describe the dynamics of atoms trapped in certain time dependent optical lattices, as was recently demonstrated experimentally, [23], [40].

  • almost surely

  • dynamical localization

  • unitary operator

  • pure point

  • random quantum

  • transition amplitude

  • spectrum

  • quantum walks


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NonamemanuscriptNo.(willbeinsertedbytheeditor)DynamicalLocalizationford-DimensionalRandomQuantumWalksAlainJoyeReceived:date/Accepted:dateAbstractWeconsiderad-dimensionalrandomquantumwalkwithsite-dependentrandomcoinoperators.Thecorrespondingtransitioncoefficientsarecharac-terizedbydeterministicamplitudestimesindependentidenticallydistributedsite-dependentrandomphases.Whenthedeterministictransitionamplitudesarecloseenoughtothoseofaquantumwalkwhichforbidspropagation,weprovethatdynamicallocalizationholdsforalmostallrandomphases.Thisin-stanceofAndersonlocalizationimpliesthatallquantummechanicalmomentsofthepositionoperatorareuniformlyboundedintimeandthatspectrallo-calizationholds,almostsurely.KeywordsRandomquantumwalksDynamicallocalizationFractionalmomentsestimates1IntroductionQuantumwalksdefinedonalattice,deterministicorrandom,havebecomeapopulartopicofresearchrecently,duetheinteresttheyhavefordifferentscientificcommunities,seethereviews[25],[28],[3].Quantumwalkscanbeusedtoprovideaneffectivedescriptionofthedynamicsofcertaintypesofphysicalquantumsystemsandbelong,inthissense,tothebroadercategoryofquantumnetworkmodels,[1],[32],[12],[9].Inparticular,quantumwalksaccuratelydescribethedynamicsofatomstrappedincertaintimedependentopticallattices,aswasrecentlydemonstratedexperimentally,[23],[40].Quan-tumwalksalsoofferafieldofinvestigationforprobabilistssincethequantuminterpretationofthewavefunctiontogetherwiththerelativesimplicityofthedynamicsofquantumwalkersgiverisetogeneralizationsofclassicalrandomPartiallysupportedbytheAgenceNationaledelaRecherche,grantANR-09-BLAN-0098-01AlainJoyeUJF-Grenoble1,CNRSInstitutFourierUMR5582,Grenoble,38402,France