etent le pi de rétroEdi'usion ohérenteD etD en prtiulierD nous trouvonsD de fçon inttendueD une onstrution inomplète du pique ux temps ourtsF hns le hpitre TD nous prenons une diretion di'érente et onsidérons un modèle détermiE niste @quoique hotiqueAD le kik rotorF xous montrons que des modèles de kiks rotors o'rent des perspetives intéressntes pour l9étude de lolistion d9endersonF h9une prtD ve de fortes moE tivtions expérimentles et théoriquesD nous présentons des éléments pronts en fveur d9un kik rotor sns spin dns l9ensemle sympletiqueF h9utre prtD le réexmen de kiks rotors modulés qusiEpériodiquementD ommunément étudiésD révèle des résultts intrigntsF in(nD le hpitre U résume nos résultts et donne quelques perspetives pour de futurs trvuxF

ges trois nnées de thèse furent à l fois plisntes et enrihissntesD je le dois à de nomreuses personnes que je souhiteris remerier iiF te voudris d9ord remerier henis fsko et eter hlghek pour voir epté le rôle de rpporE teurF te les remerie ussiD insi que les utres memres du juryD vetii guglindoloD inent tosse et trizi ignoloD pour leurs questions pertinentes lors de l soutenneF t9i eu l hne d9être endré pendnt es trois nnées pr deux direteurs de thèse investisF v thèse doit euoup à l pssion de hominique et l rigueur de xiolsF te leur suis in(niment reonnissnt pour leurs onseilsD nos nomreuses disussions et plus générlement pour l9mine qu9ils ont su réer u sein du groupeF t9i pssé trois exellentes nnées u vufD pour el je remerie en prtiulier mes oEureux suessifs qui ont toujours illuminé le ureu PHTF weri nyotisD njiD wihelD ierreEilieD hiult et mrF te remerie ussi tous eux ve qui j9i prtgé mes déjeunersD en prtiulier les memres de l9équipe gsimirD les métrologistes et ierreEprnçoisF te remerie vuile d9voir epté le rôle de mrrineD et de l9voir pleinement ssumé @de l meilleure mnièreD en m9invitnt régulièrement u resturntAF te suis ussi reonnissnt à vetitiD ominD hierryD wonique et xor pour leur ienveillne et leur e0ité Y ve le reul je suis surpris de l filité ve lquelle tous les spets dministrtifs ont été tritésF weri ussi ux personnes qui se sont suédées u servie info et à fintou et ennikF weri ussi à eux que j9i eu l hne de renontrer pendnt l thèseD glémentD pélixD himothéeD homsD srénéeD ghrlieD xiolD wloD renryD tnF einsi qu9à eux que j9i eu plisir à retrouver ii ou làD hiultD pélixD voïD et tous les ollègues d9yrsyF in(nD je souhiteris remerier mes mis et m fmilleF weri ux mis de ghrlotteD à éverine pour les meilleurs runhs que j9i onnuD et ux mis de phrms et sdriss ve qui on toujours plisir à prtger un verre @ou deuxAF ux mis d9yrsyD et 0liésD pour toutes es soiréesD tous es ons momentsD à vureD wëlD wrgotD tfD gorine et ierreD pour votre hospitlité sns (nD 9est toujours un plisir de séjourner à euzeville @et illeursAD ux mis de toujoursD que j9i toujours une grnde joie à retrouverD pour votre mitié indéfetileD à l fmilleD pour votre générositéD pour votre soutienD et pour e que vous êtesD à ghrlotteD d9être làD dns l sérénité omme dns l9ppréhensionD pour tout le onheur que tu m9s pportéD et que tu ontinues à me prourerF i Resumé hns ette thèseD nous étudions théoriquement di'érents e'ets liés à l lolistion d9endersonF xous nous onentrons sur le ontexte des tomes froidsD dns lesquels sont simulés des potentiels désordonnés ou des systèmes hotiquesF ge résumé long en frnçis suit l struture de l thèse en sept hpitresF eprès une introdution à l lolistion d9endersonD présentnt à l fois des spets théoriques et expérimentux et onstitunt le hpitre ID nous présentons une théorie pprohée de l distriution d9énergie des nuges tomiques dns les potentiels speklesF gette étude est menée en deux étpesF v première est présentée dns le hpitre PD où nous onsidérons les propriétés sttistiques des potentiels optiques de type spekleD en prtiulier elles des minim d9intensitéF hns le hpitre QD nous présentons l deuxième prtie de l9étudeD une méthode de lul de l fontion spetrle et de l densité d9éttsD vlle dns l limite de désordre fortF in utilisnt les résultts otenus u hpitre PD nous ppliquons ette méthode u s unidimensionnelF hns le hpitre RD nous rtérisons le mouvement du entre de msse de pquets d9ondes lnés ve une vitesse (nie dns un potentiel létoireF gette étude dévoile une mnifesttion nouvelle et inttendue de l lolistion d9enderson X près un mouvement initil listiqueD le entre de msse du pquet présente une rétroEré)exion et revient lentement à s position initileF xous vons nommé e phénomène l9e'et de oomerng quntiqueF ve ÷ur du hpitre R onsiste en une desription nlytique de l9e'et de oomerng quntique en une dimensionF hns le hpitre SD les intertions tomeEtome sont introduites dns les simultions numériquesD u niveu hmp moyen @qrossEitevskiiAF veur e'et sur le phénomène de oomerng quntique est étudiéF v9e'et des intertions est ussi étudié dns l9étlement en énergie d9ondes plnes évolunt dns des potentiels létoires tridimensionnelsF in(nD nous disutons l fçon dont les intertions Introduction v thèse ommene pr une introdution à l physique de l lolistion d9endersonF v loliE stion d9enderson été prédite en IWSV pr endersonF hepuisD le sujet été l9ojet de nomreux trvuxF gei furent ussi ien théoriques qu9expérimentuxF ve sujet est ussi intéressnt du point de vue mthémtiqueD des preuves mthémtiques de l lolistion ont pu être otenuesF sl ser vin d9essyer de rendre ompte de tous es développement dns une thèseF iii gonernnt les spets théoriquesD nous vons don hoisi de présenter dns les grndes lignes deux théories populires de l lolistionF v premièreD l théorie utoEohérente de ollhrd et öl)eD est une théorie mirosopique de l lolistionF v9idée est de déomposer l distriution de densité à un point r et à un temps t omme une somme interférentielle de heminsF rmi es di'érents heminsD seuls deux lsses sont prises en ompteF ve hoix de es deux lsses étnt sé sur des rguments physiquesF v première lsse rssemle tous les termes nonEinterférentielsD 9est le hi'usonD le proessus ssoié est un proessus de mrhe létoireD ien dérit pr une éqution de di'usion ux temps longs et sur des éhelles de distne su0smment grndesF v deuxième lsseD ppelée gooperonD orrespond à e qui est ommunément ppelé l lolistion fileF gette ontriE ution interférentielle est en e'et responsle d9une rédution du trnsport ssoié u hi'usonF v lolistion d9enderson survient qund le gooperon devient importntF g9est le s en sse dimenE sions @d ≤ 2A et à désordre fort en hute dimensionsF v théorie utoEohérente permet de triter le premier sD sous une hypothèse de hmp moyenF gette hypothèse empêhe l desription du s tridimensionnelD où l présene d9un point ritiqueD s9ompgnnt de )ututions importntesD est inomptile ve le hmp moyenF xous montrons ussi ertines limites de ette théorie en une dimensionD pr l omprison ve une théorie exteD l théorie de ferezinskiiF yn oserve en prtiulier des dévitions à l théorie utoEohérente dns les iles de l distriution de densitéF và enoreD es dévitions sont ssoiées à des )ututions importntesD qui violent l9hypothèse de hmp moyenF v théorie d9éhelle de l lolistion propose une pprohe di'érenteF edoptnt un point de vue mrosopiqueD ette dernière onsiste à étudier omment l ondutne g d9un éhntillon vrie ve s longueurF gette étude est menée à trvers une fontion qellEwnnEvow βD qui enode l vrition du logrithme de l ondutne ve le logrithme de l tille du systèmeF gette théorie se se sur deux hypothèsesF v première est que l fontion β onsidérée ne dépend que de l ondutne elleEmême @d9où le nom de théorie d'échelle à un paramètreAF v deuxième hypothèse est l ontinuité et l régulrité de l fontion βF v onnissne des régimes symptotiques @à grnd et petit gA lôt le risonnementF in e'etD omment nous le montrons dns l thèseD le omportement qulittif de l fontion β se déduit diretement de es trois éléments @un seul prmètre pertinentD régulrité de l fontion β et onnissne des régimes symptotiquesAF he e omportement qulittifD on peut tirer deux informtions d9importne X l théorie d9éhelle à un prmètre prédit que l lolistion prévut en sse dimensions @d ≤ 2A et qu9une trnsition entre lolistion et di'usion est ttendue en hute dimensions @d > 2AF in prtiulierD l possiilité d9une trnsition en deux dimensions longtemps étit déttueD l prédition de l théorie d9éhelle à un prmètreD en fveur de l lolistion de tous les éttsD est ujourd9hui ommunément eptéeF eprès es disussions théoriquesD nous donnons dns l thèse un ref ompte rendu des nomreux trvux expérimentux ssoiés à l lolistion d9endersonF eu déprtD le sujet s9est développé utour d9expérienes de mtière ondenséeF ges expérienes ont donné lieu à de nomreux dévelopE pementsD mis l rihesse de l physique de l mtière ondensée onstitue une di0ulté qunt à l9oservtion direte de l lolistion d9endersonF in prtiulierD les intertions életronEéletron ou le ouplge életronEphonon rendent l9oservtion de l lolistion d9enderson pure très di0ileF r l suiteD de nomreuses expérienes ont été rélisées dns le ut d9oserver l lolistion d9enderson dns des ontextes di'érentsF in e'etD l lolistion d9enderson est vnt tout un phénomène d9interférenesD il peut don être oservé à priori ve tout type d9ondesF ges expérienes ont permis l9oservtion de l lolistion d9ondes très vriéesD prmi lesquelles les miroEondesD les ondes élstiquesD l lumière et les ondes sonoresF lus réemmentD l lolistion d9enderson d9ondes de mtière été oservée ve des tomes froidsF ves plteformes expérimentles d9tomes froids o'rent des possiilités très intéressntes pour l9étude de l lolistionF h9une prtD es plteformes o'rent un hut niveu de ontrôleF in prtiE v ulierD les intertions interEtomiques peuvent être mitriséesD et il est possile d9tteindre un file ouplge tomeEenvironnementF h9utre prtD les expérienes utilisnt le ouplge lumièreEmtière permettent l9oservtion des tomes u sein même du milieu désordonné et de suivre leur évolutionF ges possiilités lés ont donné lieu à un renouvellement de l9intérêt porté à l lolistion d9enE dersonF in e'etD es possiilités o'rent d9intéressntes perspetives expérimentles et théoriquesF r exempleD les expérienes d9tomes froids sés sur un protoole de trempe impose de prendre en ompte l distriution d9énergie des tomes @u lieu de onsidérer uniquement l9énergie de permiAF v9introdution se termine pr une disussion de l9e'et des intertions sur l lolistion d9enE dersonF heux grndes lignes de reherhe sont distinguéesF in premier lieu est disuté l9e'et des intertions u niveu hmp moyenD où de nomreux sénrios ont été étudiésF ge point de vueD dopté pr l suite dns le hpitre S de l thèseD est ontrsté pr une ourte disussion sur l lolistion à x orpsF gette thémtique de reherhe explore l possiilité d9étendre l lolistion d9enderson à des prolèmes à x orpsD pour lesquels l lolistion est prédite dns l9espe de pokF Propriétés statistiques des potentiels speckles ve deuxième hpitre de l thèse onerne les propriétés sttistiques des potentiels speklesF ve hpitre ommene pr une introdution à l physique des tomes froidsF hns les expérienes d9tomes froidsD on utilise l9intertion entre les tomes et un ryonnement lumineux monohromtique pour induire un potentiel e'etifF in prtiqueD sous ertines onditionsD l dynmique des tomes plongés dns le ryonnement peut être dérite pr une dynmique hmilE tonienneD dns lquelle le potentiel extérieur est proportionnel à l9intensité lumineuseF hns une première setion de e hpitreD nous montrons omment l9on dérive ette évolution hmiltonienneF v dynmique des tomes plongés dns le ryonnement est ien dérite pr une pproximtion semiElssique où les tomes sont trités quntiquement et le hmp életrique omposnt le ryonE nement est trité lssiquementF ous l9pproximtion d9un tome à deux niveuxD l9hmiltonien de l9tome est lors omposé de trois termesF ve premier est ssoié à l9énergie inétique de l9tomeD le deuxième à l9intertion dipolire entre l9tome et le ryonnement et le troisième est onsré à l9étt interne de l9tomeF e prtir de et rmiltonienD on peut luler l9éqution d9évolution de l9opérE teur densitéF ous l9hypothèse d9un file nomre d9tomes exitésD l9éqution pour l projetion de l9opérteur densité sur l9étt interne fondmentl est ferméeF sl su0t lors d9intégrer le mouvement sur les miroEosilltions du entre de msse de l9tome pour otenir qu9en e'etD l dynmique de l9tome à deux niveux se réduit à une dynmique hmiltonienneD et on trouve que le potentiel extérieur est proportionnel à l9intensité lumineuseF e prtir de es résulttsD on se rend ompte que réer un potentiel désordonné se réduit à généE rer un pro(l d9intensité lumineuse désordonnéF gelui i est générlement otenu expérimentlement en trnsmettnt un lser à trvers une plque rugueuseF hns l thèseD nous disutons ette posE siilitéF xous introduisons le shém expérimentl orrespondnt et dérivons le pro(l d9intensité insi otenuF gette dernière étpe onsiste en l9expression du hmp omplexe en un point donné omme l superposition du hmp di'usé pr hque moreu de l plque rugueuse @prinipe de ruygensEpresnelAF sl s9en suitD pr le théorème de l limite entrleD que le hmp omplexe une stE tistique gussienneD dont on déduit l sttistique du potentiel résultnt omme étnt exponentielleF yn onsidère ensuite les orréltions sptiles du potentiel résultntF gelles i se trouvent liés à l trnsformée de pourier de l distriution d9intensité u sein de l plque rugueuseF ves orréltions peuvent insi être ontrôlées pr l9pplition de msques sur l plque rugueuseF vi hns une troisième prtie du hpitreD nous onsidérons des propriétés plus spéi(ques des potenE tiels speklesD elles des minim d9intensitéF lus préisémentD nous nous intéressons à l distriution jointe de l profondeur et de l ourure des dits minim d9intensitéF ve lul de l distriution jointe se fitD en suivnt qoodmnD pr l9introdution du hmp omplexeD dont le potentiel est le module rréF gomme nous l9vons vuD e hmp omplexe est gussienF e prtir de e hmp omE plexe et de ses dérivéesD elles ussi gussienneD on otient l distriution reherhée pr hngement de vrileF gette distriution trois propriétés remrqulesD importntes pour l suite de l thèseF remièrement elles présentent une très file proilité de trouver un minim dont l ourure est fileF gette dernière u ontrire tendne à prendre une vleur typiqueD donnée dns l thèse en terme des propriétés du potentielF v dernière propriété remrqule est l présene de l pluprt des minim à sses énergiesF Fonction spectrale et densité d'états semi-classiques dans les potentiels speckles hns e hpitreD on s9intéresse ux propriétés spetrles des tomes dns les potentiels de type spekleF ges propriétés sont très importntes dns le ontexte des expérienes d9tomes froidsF in e'etD l proédure de trempeD générlement utilisée dns e type d9expérieneD peuple toute une distriution d9énergieF ves quntités physiques @lire prours moyenD longueur de lolistionD FFFA dépendnt de l9énergieD l dynmique totle résulte de l superposition des omposntes d9énergie du pquet d9ondesF hns une première prtie de e hpitreD nous onsidérons l limite de désordre fileD où des luls perturtifs sont possilesF yn hoisit pour mener à ien es luls d9introduire le formlisme des fontions de qreenF e et e'etD on introduit d9ord formellement l fontion de qreenD puis son développement perturtif ssoiéF xous introduisons ensuite le onept de selfEénergieD d9ord omme une simpli(tion du développement perturtif de l fontion de qreenF xous disutons ensuite son sens physique X l prtie réelle déple l9énergie lors que l prtie imginire donne un temps de vieF r l suiteD nous disutons le sens à donner à l terminologie 4désordre file4D en prtiulier dns le s des potentiels de type spekleF pinlementD nous introduisons l fontion spetrleD l distriution d9énergie moyenne des ondes plnes dns le potentiel désordonnéF hns une deuxième prtie du hpitreD nous motivons l9étude de l fontion spetrleF onnisE sne est en e'et primordile pour dérire l distriution d9énergie de pquets d9ondes ritriresF it l distriution d9énergie joue un rôle très importntF xous donnons quelques exemples pour lesquels l distriution d9énergie joue un rôle ruilF hns un premier tempsD nous onsidérons une sitution rélisée expérimentlementD où un pE quet d9ondes initilement étroit évolue dns un potentiel spekle unidimensionnelF ge spekle une rtéristique intrignte X à file désordreD le oe0ient de di'usion est fortement énergieE dépendentD présentnt des hngements rusques lorsque l9énergie dépsse des vleurs spéi(quesF v9évolution totleD intégrée sur toute les omposntes d9énergies du pquet d9ondesD peut don di'érer de l9évolution d9une omposnte énergétique individuelleF in e'etD à grnde distneD l lolistion exponentielle @hituellement ttendueA est trnsformée en une lolistion lgériqueF À l9inverseD un pquet d9ondes omposé uniquement d9tomes di'usifs peut pprître prohe de lolisé en rison de l fontion spetrleF in e'etD même si pour une énergie donnéeD le noyu est di'usifD l9évolution totle peut pprître sousEdi'usive en rison de l dépendne énergétique spéi(que du oe0ient de di'usionF v9évolution totle dns e s ressemle à l sitution où une seule omposnte d9énergie ve un noyu sousEdi'usif est en jeuD imitnt insi l9pprition de l lolistion d9endersonF vii v onnissne préise de l distriution d9énergie est églement néessire pour l rtérisE tion de l trnsition d9enderson en trois dimensionsF in e'etD dns e sD les omposntes d9énergie se réprtissent des deux ôtés du seuil de moilitéD l dynmique omplète est lors une superposition de omportements lolisés et di'usifsF v onnissne de l distriution d9énergie est néessire pour extrire des quntités physiquement pertinentes @pr exemple l position du seuil de moilitéAF v troisième prtie du hpitre est onsrée à l limite de désordre fortF gomme on le montre dns l thèseD ette limite est ien dérite pr des pproximtions semiElssiquesF xous ommençons pr l limite lssiqueD en négligent l nonEommuttivité de p et rF xous omprons le résultt insi otenu à des simultions numériques extes pour trois distriutions de potentiel di'érentes X l distriution gussienneD l distriution spekle rouge et l distriution spekle leuF yn trouve que pour l distriution gussienneD l fontion spetrle est ien dérite pr s limite lssiqueF ve s d9une distriution spekle est plus délitD en prtiulier prohe de s disontinuité où l limite lssique est omplètement fusseF our ller plus loinD on propose une expnsion en puissne de de l9opérteur d9évolutionD à prtir de lquelle on déduit des orretions systémtiques à l limite lssique de l fontion spetrleF gette méthode présente deux étpes priniplesD dns une premier temps une expnsion en ommutteur de l9opérteur d9évolution est présentéeF e prtir de ette expnsionD l moyenne sur le désordre est lulée pr une expnsion en umulntF gette dernière s9ppuie sur l représenttion du potentiel spekle omme le module rré d9un hmp omplexeD insi d9un théorème de veonov et hiryevD qui permet de mener à ien le lulF gette tehnique nous permet de luler une orretion u premier ordre non nul en de l fontion spetrleF gette orretion 0ne l desription de l fontion spetrle dns le s du potentiel gussien et des potentiels spekle loin de leur disontinuitéF v prolème prohe de l disontinuité des potentiels spekles demeureF our luler l fontion spetrle des potentiels spekles prohe de leur disontinuitéD nous développons une nouvelle pprohe semiElssiqueD ette fois i sée sur une pproximtion de phse sttionnireF sl pprit qu9à l9pproximtion de phse sttionnire et ux énergies d9intérêts @pour lesquelles l9pprohe préédente ne su0t psAD il est possile d9pproximer le potentiel spekle pr des osillteurs hrmoniques @inversésA isolésF in utilisnt les propriétés sttistiques des potentiels spekles lulées dns le hpitre PD on rrive à une très onne desription de l fontion spetrle prohe des disontinuités des potentiels speklesF in onnetnt les deux méthodesD une desription stisfisnte de l9ensemle du spetre énergéE tique est possileF xotre desription semiElssique fournit en outre une interpréttion physique de rtéristiques intrigntes de l fontion spetrleF in prtiulierD pour le potentiel spekle leuD nous vons montré que le pique de l fontion spetrle à file énergie est essentiellement ssoié à l9étt fondmentl d9un tome dns un puits de potentielD lors que l osse seondire est ssoE iée ux étts exitésF xous vons églement souligné qu9en dépit de leur symétrieD spekle rouge et leu ont des rtéristiques remrqulement di'érentes dns le régime semiElssiqueD vennt de l nture des trjetoires lssiques impliquées ux énergie prohe de zéro X pour le potentiel leuD es trjetoires lssiques se trouvent dns des puits potentiels profondsD lors que pour le potentiel rougeD elles sont u voisinge du sommet de puits inversésF ge trvil donné lieu à une pulition dns hysil eview eF ne suite logique de e trvil serit de onsidérer le s tridimensionnelD impliqué dns des questions importntes liées à l lolistion d9endersonF gette tâhe semle ependnt di0ileD en rison de l9existene de oures le long desquelles le potentiel est nulD rendnt l9pplition de l méthode développée dns ette thèse moins évidenteF viii Eet de boomerang quantique hns e hpitreD on s9intéresse à une sitution prohe de l9expériene rélisée en PHHV à liseuF hns ette expérieneD un pquet d9ondes étroit est lâhé dns un potentiel désordonné et l9évolution de son pro(l de densité est enregistrée u ours du tempsF xous proposons de reproduire l même expérieneD en donnnt en plus une vitesse initileF hns l thèseD nous donnons une desription nlytique et numérique omplète du mouvement du entre de msse du pquet d9ondes en une dimensionF hns un premier tempsD on onsidère une pprohe lssique du prolèmeF gelleEi utilise le théorème d9ihrenfest pour relier le entre de msse à l vitesse moyenneF v dynmique de l vitesse moyenne est ensuite dérite pr deux équtions de foltzmnn oupléesF v solution de es équtions prédit un mouvement du entre de msse reltivement simple X près un mouvement listiqueD le entre de msse sture u lire prours moyenF gei s9interprète très simplement pr l9isotropistion de l distriution de vitesse ux temps ourtsF ne fois l distriution de vitesse isotropeD il n9y plus de mouvement du entre de msseD elui reste à un lire prours moyenD qu9il prouru vnt que l distriution de vitesse ne devienne isotropeF xous présentons dns un deuxième temps une pprohe numérique permettnt de simuler e0E ement e prolèmeF gette méthode utilise les polynômes de gheyshev pour otenir une représenE ttion de l9opérteur d9évolution ien ppropriée à une implémenttion numériqueF xous donnons quelques détils sur ette méthode et sur son implémenttionF ille est ensuite utilisée pour simuler le mouvement du pquet d9ondes initilement doté d9une vitesse initileF yn trouve queD u lieu de sturer u lire prours moyen omme ttenduD le entre de msseD près son mouvement listiqueD retourne lentement à l9origineF xous ppelons et e'et l9e'et de foomerng quntiqueF hns un troisième tempsD nous pportons une preuve simple de l9e'et de foomerng quntiqueD utilisnt une expnsion en modes du pquet d9ondesF gette démonstrtion donne l forme (nle du pquet d9ondesD prfitement symétrique pr rpport à l9origineF in étudint numériquement l dynmique du pquet d9ondesD on oserve que eluiEi suit en e'et une symétristion guheE droite u ours du tempsF lutôt qu9à un mouvement rigide et glol du pquet d9ondesD l9e'et de foomerng quntique est en fit ssoié à ette symétristionF our ller plus loinD et luler le entre de msse ux temps (nisD nous utilisons une tehnique digrmmtiqueF gette tehniqueD exte dns l limite de désordre fileD été introduite pr ferezinskii en IWUQF hns un premier tempsD nous relions le entre de msse u produit de deux fontions de qreenD e qui nous permet d9utiliser l tehnique de ferezinskii pour luler le entre de msseF v tehnique de ferezinskii est sée sur une expnsion perturtive du produit des deux fonE tions de qreenD qui est ensuite resomméeD permettnt insi une desription non perturtiveF v9exE pnsion perturtive est exprimée en termes de digrmmesD eux même onstitués de vertex reliés pr des lignesF our e'etuer l resommtion de ses digrmmesD un point lé est l possiilitéD en une dimensionD d9inlure les fteurs ssoiés ux lignes dns les vertexF n digrmme est insi le produit des fteurs ssoiés à ses vertexF ves lignes se réduisent à des ontrintes dns le hoix des vertexF e prtir de ette représenttion en termes de vertexD il est possile de luler les digrmmes en étudint omment ei hngent lorsque le point initil et le point (nl sont déplésF yn déduit de e risonnement des équtions pour le entre de msseF ges équtions demeurent ompliquéesD leur solution générle n9est ps onnueF ne possiilité est lors de les résoudre ux temps longsD e qui simpli(e l the et rend possile l9otention du retour symptotique du pquet d9ondes à l9origineF ge résultt est en très on ord ve nos simultions ix numériques ux temps longsF our ller plus loinD et dérire le entre de msse à tout tempsD nous vons développé une nouvelle méthode de résolution des équtions de ferezinskiiF in e'etD nous vons montré qu9il est possile de résoudre es équtions sous l forme d9une expnsion de ylorF gette dernière peut être lulée de fçon systémtique à l9ide d9un ordinteurF in lulnt les IHH premiers termes de l9expnsion de ylorD on se rend vite ompte que l série orrespondnte à un ryon de onvergene (ni @que nous estimons à qutre temps de di'usionAF sl est ependnt possile de dépsser ette limite pr une resommtion de déF gette dernière o're une exellente desription du entre msse à tout tempsD même qund l9pproximnt de dé est d9ordre reltivement sF v méthode de ferezinskii développée dns e hpitre est en prinipe limitée u s d9un potentiel gussien non orréléF xous rgumentons dns l thèse que le entre de msse suit l même oure pour des potentiels nonEgussien et orrélésF es rguments sont on(rmés pr des simultions numériques qui montrent que le entre de msse est indépendnt de l distriution du potentielF our (nirD nous montrons qu9une reltion simple relie le entre de msse et l lrgeur du pquet d9ondesF gette reltion se trouve être à l se de notre lul du mouvement du entre de msseF ges trvux ont donné lieu à l rédtion d9une lettre soumise à hysil eview vettersF ve risonnement présenté dns l thèse et expliqunt l9e'et de oomerng quntique en termes de modes lolisés réels s9pplique en dimension quelonqueF v9e'et de oomerng quntique est don ttendu églement en dimensions supérieuresD à ondition que l dynmique soit loliséeF gei o're des perspetives intéressntes pour de futurs trvuxF r exempleD en prolongement de trvux réents sur le pique de rétroEdi'usionD il serit très intéressnt d9e'etuer une nlyse numérique (ne de l9e'et de oomerng quntique en trois dimensionsF ne utre piste de reherhe intéressnte serit de herher des phénomènes similires dns d9utres lsses de symétrie @pr exemple dns l lsse unitire où l ompréhension qulittive de l9e'et de oomerng quntique en termes de modes lolisés réel ne s9pplique psAF Paquets d'ondes interagissant faiblement hns e hpitreD on prend en ompte les intertions tomeEtomeF yn se ple dns le ontexte des gz dilués de osons ondensésD qui dérit ien les expérienes mises en ÷uvre à liseu et ploreneF hns une première prtieD nous donnons une dérivtion simple de l9éqution de qrossEitevskiiF gette dernière dérit l9évolution de l fontion d9onde du ondenstF gette desription néglige les tomes nonEondensésD et l possiilité pour les tomes de sortir du ondenstF xénmoinsD l9éqution de qrossEitevskii explique très ien de nomreux résultts expérimentuxF port de ette simpli(tionD nous introduisons ensuiteD dns une deuxième prtieD un méthode nuE mérique permettnt de simuler l9évolution du gz de osonsD régit pr l9éqution de qrossEitevskiiF gette méthode s9ppuie sur l méthode développée u hpitre préédent pour intégrer numériqueE ment l9éqution de hrödingerF xous montrons que l9inlusion de l nonElinérité revient à jouter des phses vnt et près l9pplition de l méthode linéireF gette méthode est symptotiquement exte qund le temps est in(niment disrétiséF in prtiqueD on peut s9ppuyer sur le fit que l9erreur à temps (xé déroit omme le rré du ps en tempsF gette méthode est prtiulièrement dptée à l limite de files intertionsF ille permet en e'et de triter e0ement l prtie linéire de l9éqution de qrossEitevskiiD de file intertions n9imposnt que des ps temporels risonnlesF x sl reste nénmoins que l9on souhite tteindre des temps reltivement longs dns nos simultions numériquesF xous souhitons en e'et éder à des régimes de temps où l nonElinérité se fit sentirF our ette risonD nous doptons dns ertines setions un modèle réseuD plus file à simuler numériquementF yn noter que dns l limite linéire et onernnt l lolistion d9endersonD il n9y essentiellement ps de di'érene entre l physique des réseux et elle du ontinuF yn peut don espérer dérireD u moins qulittivementD l physique du ontinuF our vlider notre méthode numériqueD nous onsidérons une sitution qui fit l9ojet de nomreuses étudesF ille onsiste à pler un pquet d9ondes à un point d9un réseu désordonné et à le lisser l9étlerF yn oserve lors que l nonElinérité vient perturer l lolistion d9endersonD qui prédit une sturtion de l9étlementF in e'etD une dynmique sousEdi'usive est ttendueF xous oservons ien e omportement sousEdi'usif dns nos simultions ve un exposnt de sousEdi'usion en ord ve de préédents trvuxF hns une troisième prtieD nous nous tournons lors vers un prolème nouveuD elui de l9e'et de l nonElinérité sur l9e'et de oomerng quntiqueF xous étudions ette sitution numériquementD et oservons que l nonElinérité à l9ir d9interrompre l9e'et de oomerng quntiqueF ve oomerng quntique nonElinéire semle ne ps revenir à l9origineF our omprendre un peu mieux e phénomèneD nous le omprons à un phénomène de déoE héreneF ge dernier est étudié numériquement pr l9introdution dns les simultions numérique d9un potentiel désordonné supplémentireD dont l9mplitude vrie dns le tempsF our e'etuer es simultionsD on utilise l même méthode numérique que pour le s nonElinéireF yn trouve que de fçon surprennteD il est possile de reproduire presque à l9identique les oures du mouvement du entre de msse nonElinéire ve e modèle de déohéreneF ge résultt suggère que l nonElinérité git ii omme une soure de déohéreneF yn peut lors ssoier à l nonElinérité un temps de déohéreneF ge dernier est otenu en onsidérnt l di'usion ssoié u modèle de déohérene dont le mouvement du entre de msse reproduit elui otenu ve l9éqution de qrossEitevskiiF yn trouve lors qu9à l nonElinérité est ssoiée un temps de déohérene inversement proporE tionnel à l fore de l nonElinéritéF hns une qutrième prtieD nous onsidérons l dynmique de l distriution d9énergie du pquet d9ondesF in e'etD ontrirement u s linéire pour lequel l distriution d9énergie n9évolue psD l nonElinérité est suseptile d9induire des ollisions inélstiques et don un hngement de l distriution d9énergieF

xous ommençons ette prtie pr un réexmen du prolème de l9étlement du pquet d9ondesD qui montre l9importne de l dynmique de l distriution d9énergieF in e'etD nous fisons une expériene numérique reltivement simpleD qui onsiste à omprer deux sénrios très prohesF hns les deux sD nous onsidérons l9étlement d9un pquet d9ondesD simplement dns le premier s l9énergie initile du pquet d9ondes se trouve à un endroit ritrire du spetre lors que dns le seond sD l9énergie du pquet d9ondes est hoisie telle qu9elle orresponde à l9énergie pour lquelle l longueur de lolistion est mximleF in e'etD si dns le premier s le pquet d9ondes est suseptile d9explorer des régions du spetre d9énergie où l longueur de lolistion est grndeD dns le seond sD le pquet d9ondes ne peut qu9explorer des régions où l longueur de lolistion est reltivement petiteF ves simultions numériques révèlent que ette di'érene est fondmentleD ve un trnsport euoup moins e0e dns le s où le pquet d9ondes est dès le déprt u mximum de l longueur de lolistionF fien que les simultions du prgrphe préédent soient fite en une dimensionD nous onsidérons ensuite le s tridimensionnelD plus simple en e qu9il permet de négliger les e'ets de lolistionF v dynmique de l distriution d9énergie peut lors être luléeF xous omprons e lul à des simultions numériquesD et trouvons un on ordF xous pouvons lors onlure que l distriution d9énergie évolue sur un temps rtéristique proportionnel u rré de l9inverse de l fore de l xi nonElinéritéF xous oservons ependnt dns les simultions numériques une dynmique de l distriution d9énergie sur une éhelle de temps euoup plus ourteD elle du temps de di'usionF xous ttriuons ette dynmique à deux e'ets de temps ourtsD liés à l dynmique du pquet d9ondes sur ette éhelle de tempsF xous onsidérons (nlement dns une dernière prtie l9e'et des intertions sur le pique de rétroE di'usion ohérenteF he l même fçon que pour l distriution d9énergieD les simultions révèlent deux temps rtéristiquesF in premier lieuD ux temps ourts @de l9ordre du temps de di'usionAD on oserve une onstrution inomplète du pique de rétroEdi'usion ohérenteF insuiteD sur des temps plus longsD l9mplitude du pique diminueF xous postulons que es deux e'ets sont liés ux deux e'ets oservés sur l distriution d9énergieF fien que nous yons souligné les ménismes physiques à l9÷uvreD notre trvil été essenE tiellement numériqueF ne rtéristion nlytique des e'ets introduits dns e hpitre devrit permettre d90ner leur ompréhension et o're des perspetives intéressntes pour de futurs trvuxF Le kick rotor, un simulateur paradigmatique de la localisation d'Anderson hns e hpitreD nous hngeons quelque peu de thémtiqueF xous onsidérons di'érentes vE rintes de kiks rotorsD qui sont des modèles déterministesF v dynmique lssique de es modèles est ependnt générlement hotiqueD et leur dynmique quntique essentiellement identique à elle de modèles désordonnésF ge hpitre s9ouvre pr une setion d9introdution à l physique du kik rotorF xous y disutons notmment les rélistions expérimentles ve des tomes froidsD et introduisons quelques dé(nitions utiles pour l suiteF insuiteD dns une deuxième setionD on introduit le onept des propriétés spetrles universelles et les ensemle de mtries létoires de hysonF xous disutons notmment du rôle entrl joué pr les symétries pr renversement du tempsF einsiD nous introduisons les trois ensemles de hysonD pr une rève dérivtion des propriétés sttistiques de mtries invrintes pr renversement du tempsF xous montrons notmment que les mtries pprtennt à l9ensemle sympletique sont rtérisées pr une dégénéresene de urmersF xous donnons ensuite les formes pproximées @igner surmiseA des distriutions d9espement de niveu dns hun des trois ensemlesF uite à quoiD nous disutons du rôle des symétries pr renversement du temps pour les opérteurs de ploquetF xous disutons ussi du lien entre les ensemles de hyson et les lsses d9universlité des systèmes désordonnésF eprès es prties introdutivesD nous exminons l possiilité qu9un kik rotor sns spin soit dns l lsse sympletiqueF yn noter qu9il est ommunément epté qu9une telle possiilité est exlueF our trouver un kik rotor sns spin dns l lsse sympletiqueD nous proposons de ré)éhir utour du phénomène de lolistion fileF ge dernier se mnifeste en e'et très di'éremment dns l lsse sympletique puisqu9il est trnsformé en anti-localisationF v9idée est lors de hnger le signe de l lolistion fileF gomme nous le montrons dns l thèseD ei est possile pr introdution d9un potentiel disontinu et d9une lternne de deux kiks d9mplitude di'érentesF gomme nous le montrons dns l thèseD e modèle reproduit ien les signtures de l9anti-localisationD pourvu que l disontinuité soit plée le long d9une diretion supplémentireD omposée d9un petit nomre de xii sitesF our ller plus loinD on se tourne vers les propriétés spetrles introduites dns l setion préE édenteF yn retrouve les rtéristiques de l9ensemle sympletiqueD ve l présene de l dégénéE resene de urmers et l sttistique des espements de niveux ttendue dns et ensemleF xous montrons ensuiteD qu9en e'etD l9opérteur de ploquet ommute ve un opérteur de renversement du tempsD et que e dernier est de rré -1F our ller plus loinD nous proposons de onsidérer le s tridimensionnelD pr l9introdution de fréquenes inommensurlesF gette stueD permettnt de simuler des prolèmes désordonnés de dimensions entières ritrires à l9ide de kiks rotors modulés dns le tempsD n9 jusque là été ppliquée qu9à des systèmes pprtennt à l lsse orthogonlF evnt de se lner dns l9étude du modèle sns spin supposé sympletiqueD nous véri(ons que l9stue s9pplique ussi u s sympleE tiqueF xous e'etuons à et e'et des simultions numériques de kik rotor ve spinF ves résultts suggèrent que ette stue ne s9pplique ps dns l lsse sympletiqueD l9exposnt ritique oservé est en e'et distint de elui ttendu pour un modèle désordonné équivlentF xous sommes don forés de reporter l9étude du modèle sns spin supposé sympletiqueF our essyer de omprendre pourquoi l9stue usuelleD onsistnt à moduler l9mplitude des kiks pr des produits de fontions trigonométriques pour simuler des dimensions supplémentiresD ne fontionne ps dns l lsse sympletiqueD nous nous sommes intéressés à une nouvelle modultionF lus préisémentD nous vons essyé d9identi(er les propriétés importntes des séquenes issues de produits de fontions trigonométriques pour simuler des prolèmes désordonnés en dimensions non entièreF wise à prt l ompréhension des kiks rotors modulésD ette étude est ussi motivée pr les perspetives intéressntes o'ertes pr l simultion de prolèmes désordonnés en dimensions non entièresD telle que l9identi(tion de l dimension ritique inférieure dns l9ensemle sympletique @prédite stritement supérieur à I et stritement inférieur à PA et d9explorer les onséquenes de ette dimension ritique inférieure non entièreF e et e'etD nous vons proposé une nouvelle séquene qusiEpériodiqueF gette dernière est otenue de fçon reltivement similire à l tehnique hituelle onsistnt à éhntillonner des produits de fontions trigonométriquesF lus préisémentD nous proposons une fontion dont l9éhntillonnge pour des rguments entiers génère une série qusiEpériodiqueF ves résultts sont surprenntsD ve di'érents régimes de sousEdi'usion dépendnt de l9mplitude des kiksF fien que l9ojetif de simuler des prolèmes désordonnés en dimensions non entières ne soit ps tteintD nous pensons que les résultts sont prometteursF pinlementD nous vons montré que le kik rotor onstitue une exellente plteEforme pour E rtériser divers spets de l lolistion d9endersonD et vons identi(é trois diretions de reherhe prometteuses pour de futurs trvuxF out d9ordD dé(nt une hypothèse ommunément eptéeD nous vons présenté des éléments pronts en fveur d9un kik rotor sns spin dns l lsse sympleE tiqueF heuxièmementD nous vons oservé que l9stue des séquenes qusi périodiques de gsti et al. semle éhouer dns l lsse sympletiqueF roisièmeD nous vons exploré l possiilité d9utiliser de nouveux types de séquenes qusiEpériodiquesF xous pensons que notre étude préliminire dns ette diretion montre des résultts prometteurs et ouvre des perspetives pssionnntes pour de futurs trvuxF Abstract his thesis theoretilly investigtes severl e'ets relted to enderson loliztionD fousing on the ontext of disordered nd hoti oldEtomi systemsF sn oldEtomi systemsD optil spekle ptterns re often used to rete the disorderF he resulting potentils felt y the toms di'er from qussin rndom potentilsD ommonly ssumed in the desription of ondensedEmtter systemsF sn the (rst prt of the thesisD we disuss their spei(itiesD with n emphsis on the spetrl properties of toms in suh potentilsF sn prtiulrD we derive severl pproximtions for the spetrl funtionF etomEoptis experiments o'er interesting possiilitiesD suh s the possiility to diretly proe the toms inside the disordered potentilF sn view of these possiilitiesD we onsider in the seond prt of the thesis the spreding of mtter wve pkets initilly lunhed with nonEzero veloityF e (nd tht fter n initil llisti motionD the pket enterEofEmss experienes retrore)etion nd slowly returns to its initil positionD mimiking oomerngF e show tht this unexpeted quntum oomerng e'et is onsequene of enderson loliztionD nd desrie it oth numerilly nd nlytilly in dimension IF etomEtom intertions re then introdued in third prtF e onsider dilute ondensed osoni gsesD nd tret the intertions t the menE(eld @qrossEitevskiiA levelF rious situtions re studied numerillyD in prtiulr the quntum oomerng senrioD nd the dynmil spreding ! oth in momentum nd energy ! of mtter wves prepred s plne wvesF sn the lst prtD we show tht hoti models o'er interesting prospets for the study of enderson loliztionF yn the one hndD going ginst ommon wisdomD we present strong evidenes in fvor of spinless kiked rotor in the sympleti ensemleF yn the other hndD seond look t ommonly studied qusiEperiodilly modulted kiked rotor revels intriguing resultsF xiii Contents 1 Introduction 1

IFI prom wek to strong loliztionX the selfEonsistent pproh F F F F F F F F F F F F F I IFIFI ek loliztion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F I IFIFP trong loliztionX the selfEonsistent pproh F F F F F F F F F F F F F F F F F Q IFP ferezinskii digrmmti tehnique F F F F F F F F F F F F F F F F F F F F F F F F F F F F Q IFQ ling theory of loliztion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F R IFR ixperimentl oservtions of enderson loliztion F F F F F F F F F F F F F F F F F F F T IFRFI irly ondensedEmtter experiments F F F F F F F F F F F F F F F F F F F F F F F T IFRFP e wve phenomenon F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F T IFS enderson loliztion nd intertions F F F F F F F F F F F F F F F F F F F F F F F F F F U IFSFI wenE(eld e'ets F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F U IFSFP wnyEody e'ets F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F V IFT yutline of the thesis F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F V 2 Statistical properties of speckle patterns 9 PFI hipolr potentil for oldEtoms F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F W PFIFI hipolr tomElser intertion F F F F F F F F F F F F F F F F F F F F F F F F F F F W PFIFP hipolr potentil F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IH PFIFQ king dvntge of sorption or omplex internl strutures F F F F F F F F F IP PFP pekle ptternsX generlities F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IP PFPFI presnel integrl F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IQ PFPFP ynEsite intensity distriution F F F F F F F F F F F F F F F F F F F F F F F F F F F IR PFPFQ gorreltion funtions F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IR PFPFR xumeril implementtion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IS PFQ ttistis of intensity minim F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IT PFQFI toint distriution P (V, ω) F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IU PFQFP hensity of minim F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IW PFQFQ woEdimensionl se F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F PH PFR gonlusion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F PH
3 Semiclassical spectral function and density of states in speckle optical potentials 21

QFI ek disorderD perturtive lultions F F F F F F F F F F F F F F F F F F F F F F F F F PI QFIFI erturtive tretment with qreen funtions F F F F F F F F F F F F F F F F F F PP QFIFP he selfEenergy F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F PQ QFIFQ ttering men free time nd sttering men free pth F F F F F F F F F F F F PR QFIFR he spetrl funtion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F PS QFP smportne of the spetrl funtion in the dynmis of oldEtoms in rndom potentils PT QFQ trong disorderD semilssil regime F F F F F F F F F F F F F F F F F F F F F F F F F F F PU QFQFI he(nitions nd methods F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F PV QFQFP mooth quntum orretions F F F F F F F F F F F F F F F F F F F F F F F F F F F PW xv xvi gontents QFQFQ retment of low energies F F F F F F F F F F F F F F F F F F F F F F F F F F F F F QP QFQFR petrl funtion nd density of sttes in one dimensionX results F F F F F F F QS QFQFS lidity of the hrmoniEosilltor pproximtion F F F F F F F F F F F F F F F F QU QFQFT lidity of the inverted hrmoniEosilltor pproximtion F F F F F F F F F F F QU QFQFU righer dimensions F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F QV QFR gonlusion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F QV
QFS ertileX Semiclassical spectral function and density of states in speckle potentials F F F QW 4 Quantum boomerang eect in one-dimensional random potentials 57

RFI snitil ondition F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F SU RFP glssil pproh F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F SV RFQ xumeril solution F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F SW RFQFI gheyshev polynomils F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F TH RFQFP ixpnsion of the evolution opertor over gheyshev polynomils F F F F F F F TH RFQFQ rtil implementtion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F TI RFQFR esults for x F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F TI RFR gonvergene of the density to its in(niteEtime limit F F F F F F F F F F F F F F F F F F F TP RFS genterEofEmss motion from ferezinskii digrmmti tehnique F F F F F F F F F F F F TR RFSFI genter of mss in terms of qreen funtions F F F F F F F F F F F F F F F F F F F TR RFSFP higrmmtis F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F TR RFSFQ iqutions for the enter of mss F F F F F F F F F F F F F F F F F F F F F F F F F TW RFSFR esults F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F UQ RFT qenerl se F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F UQ RFTFI gse of nonEqussin potentil F F F F F F F F F F F F F F F F F F F F F F F F F UQ RFTFP gse of orrelted potentil F F F F F F F F F F F F F F F F F F F F F F F F F F F UR RFTFQ gse of rod initil wve pket F F F F F F F F F F F F F F F F F F F F F F F F US RFU e simple reltion etween x nd x 2 F F F F F F F F F F F F F F F F F F F F F F F F F F UT RFV gonlusion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F
F F F F F F F F F F F F F F F F F F F F WI SFIFI wnyEody rmiltonin F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F WI SFIFP fogoliuov pproximtion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F WP SFIFQ woEody low energy ollisions nd sttering potentil F F F F F F F F F F F F WP SFIFR gomments on the qrossEitevskii eqution F F F F F F F F F F F F F F F F F F F WQ SFP xumeril integrtion of the qrossEitevskii eqution F F F F F F F F F F F F F F F F F WQ SFPFI xumeril sheme nd error estimte F F F F F F F F F F F F F F F F F F F F F F F WQ SFPFP sn prtie F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F WR SFPFQ xonlinerity nd spreding F F F F F F F F F F F F F F F F F F F F F F F F F F F F WS SFQ xonliner quntum oomerng F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F WT SFQFI xumeril experiment F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F WT SFQFP gomprison with deoherene F F F F F F F F F F F F F F F F F F F F F F F F F F F WU SFR xonlinerity nd energy distriution F F F F F F F F F F F F F F F F F F F F F F F F F F F WV SFRFI wotivtionsX seond look t spreding F F F F F F F F F F F F F F F F F F F F F WV SFRFP xumeril study of the energy distriution dynmis F F F F F F F F F F F F F F IHH SFRFQ hortEtime @t ≈ τ A pitureX shifting nd sreening F F F F F F F F F F F F F F F F IHI SFRFR sntermediteEtime pitureX kineti eqution F F F F F F F F F F F F F F F F F F F IHP gontents xvii SFRFS hensity of sttes F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IHQ SFRFT gomprison with numeris F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IHQ SFRFU sn(niteEtime equilirium F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IHR SFS xonlinerity nd the oherent k sttering pek F F F F F F F F F F F F F F F F F F F IHS SFSFI goherent k sttering pek F F F F F F F F F F F F F F F F F F F F F F F F F F IHS SFSFP xumeris F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IHT SFSFQ hysil piture F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IHT SFT gonlusion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IHU eppendix SFe epproximtions of the density of sttes F F F F F F F F F F F F F F F F F F F F IHV SFeFI hensity of sttes in term of selfEenergy F F F F F F F F F F F F F F F F F F F F F F IHV SFeFP epproximtions of the selfEenergy F F F F F F F F F F F F F F F F F F F F F F F F F IHW 6
The kicked rotor, a paradigmatic simulator for Anderson localization 111

TFI prom hos to disorder F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F III TFIFI glssil hoti dynmis F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IIP TFIFP untum evolutionX ploquet opertor F F F F F F F F F F F F F F F F F F F F F F IIP TFIFQ untum evolutionX pseudoErndom kiked rotor F F F F F F F F F F F F F F F F IIQ TFIFR prom lssil di'usion to loliztion F F F F F F F F F F F F F F F F F F F F F F IIQ TFP niverslity lsses nd rndom mtrix theory F F F F F F F F F F F F F F F F F F F F F IIR TFPFI prom perturtive perspetive F F F F F F F F F F F F F F F F F F F F F F F F F IIR TFPFP niversl properties nd rndom mtrix theory F F F F F F F F F F F F F F F F F IIS TFPFQ ime reversl symmetry for ploquet models F F F F F F F F F F F F F F F F F F F IIV TFPFR gritil exponents F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IIW TFQ e spinless kiked rotor in the sympleti lssc F F F F F F F F F F F F F F F F F F F F F IIW TFQFI ek ntiloliztion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IIW TFQFP iigenvlue sttistis F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IPP TFQFQ imeEreversl invrine F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IPR TFQFR gonlusion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IPS TFR prom inommensurte frequenies to higher dimensions F F F F F F F F F F F F F F F F F IPS TFRFI qenerlities F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IPS TFRFP snommensurte frequenies in the sympleti lss F F F F F F F F F F F F F F F IPT TFRFQ feyond the inommensurte frequenies F F F F F F F F F F F F F F F F F F F F F IPU TFS gonlusion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IQH Chapter 1

Introduction

enderson loliztionD the sene of wve di'usion due to destrutive interferene etween prtil wves multiply sttered y disordered potentilD ws predited theoretilly in IWSV IF enderson formulted the prolem of loliztion under the form of perturtive expnsionD for whih he ws le to derive onvergene riterion nd thus ondition for loliztion to ourF vter onD vrious theories of enderson loliztion emergedF he sujet is lso of interest to mthemtiinsD who were le to give mthemtilly rigorous proofs of loliztion PF sn this hpterD we introdue severl desriptions of enderson loliztion tht will e useful in the rest of the thesisF e strt with the selfEonsistent theory of loliztion in setion IFIF his theory will llow us to gin intuition on the phenomenon of enderson loliztionD nd to introdue some importnt oneptsF he selfEonsistent theoryD whih is kind of menE(eld pproximtionD is then ompred with more rigorous pproh in oneEdimensionD ferezinskii digrmmti tehniqueD in setion IFPF he sling theory of loliztion is then introdued in setion IFQF fy dopting mrosopi point of viewD the ltter o'ers di'erent perspetive on enderson loliztionF enderson loliztion hs een experimentlly oserved in vrious ontextsF sn setion IFRD we give rief ount of these experimentl developmentsD nd disuss the spei(ities of oldE tom pltforms in the (eld of enderson loliztionF sn setion IFSD we rie)y disuss the role of intertionsD often unvoidle in experimentsF e onlude this hpter with n outline of the thesis in setion IFTF 1.1 From weak to strong localization: the self-consistent approach the density distriutionF yn the one hndD the soElled lssil ontriution desries rndom wlk proess where the vrious pths do not interfere nd their intensity simply dd upF he other quntum ontriutionD on the other hndD ontins ll possile interferene e'etsF he pirs of pths responsile for the quntum ontriution ome with phse ftor of order k F rere k is relted to the he froglie wve length of the prtile λD through k = 2π/λD nd is the sttering men free pthD the verge distne trveled y the toms etween suessive sttering eventsF sf now we impose tht k 1 @this de(nes wek disorder riterionD see setion QFI for detilsAD the quntum ontriution is negligile on vergeF nder this pproximtionD the prtile experienes lssil di'usion proess t lrge slesD with di'usion oe0ient D 0 X

A i 2 = paths i A i A * i classical + paths i =j A i A * j quantum . @IFIA
(∂ t -D 0 ∇ 2 r )n cl (r, t) = δ(r -r )δ(t) ⇔ n cl (r, t) = dω 2π dq (2π) d e iq.(r-r )-iωt 1 -iω + D 0 q 2 , @IFPA
where n cl (r, t) is the disorder verged lssil densityF xote tht (-iω + D 0 q 2 )n cl (q, ω) = 1 is the di'usion eqution in pourier speF his desription is however inompleteF sndeedD some refully designed pirs of pths turn out to survive the disorder vergeD provided eh pth umultes the sme phse during its propgtionF hen timeEreservl invrine holdsD this is preisely the se for pth nd its timeEreserved ounterprtF r r @A hi'uson r r @A gooperon pigure IFI ! wo types of pirs of sttering pths ontriuting to the verged density distriutionX @A the hi'usonX oth pths follow the sme sequene of stterers in the sme diretionD nd @A the gooperonX the two pths follow the sme sequene of stterersD ut in opposite diretionsF

Cooperon loop

his rgument leds us to keep two types of pirs of pthsD the lssil ones @lled hi'usonA nd their timeEreserved ounterprt @lled gooperonAF hey re depited in (gure IFIF tritly spekingD hi'uson nd gooperon re symmetri upon timeEreversl symmetry only when r = r F his property mkes the gooperon ply singulr roleX it enhnes the proility of oserving the prtile ner its strting pointF prom the two uilding loks in (gure IFID we n uild more omplited pths y hining hi'usons nd gooperons @see (gure oppositeAF he gooperons n only pper s losed loopsD their ourrene during the propgtion leds to redution of the di'usion oe0ient1 X

n WL (r, t) = dω 2π dq (2π) d e iq.(r-r )-iωt 1 -iω + D(ω)q 2 1 D(ω) = 1 D 0 + 1 πρD 0 dQ (2π) d 1 -iω + D 0 Q 2 =n cl (r ,ω) , @IFQA
where ρ is the disorderEverged density of sttes per unit volumeF he redution of the di'usion oe0ient y interferene e'ets is known s wek loliztionD nd hs een the sujet of extensive reserhD in prtiulr in ondensedEmtter physisD where it ws trked through its interply with tempertureD mgneti (eldD spinEorit oupling or mgneti impurities @see eFgF the review R nd the more reent experiment SAF xote tht one n lterntively oserve the gooperon ontriuE tion s oherent enhnement of the return proility T or under the form of oherent k sttering pek UD VF yne sees tht the redution of the di'usion oe0ient D 0 in eqution @IFQA is proportionl to the lssil return proility n cl (r , ω)F es it turns outD this orretion eomes signi(nt in low dimensions @d ≤ 2A s well s t strong disorder in higher dimensions QF 1.1.2 Strong localization: the self-consistent approach sn ftD if the quntum orretions @the gooperon loopsA eome more importntD more omplex sttering pths pE perD s shown in the (gure on the rightF hese sttering pths my inlude loops nested into loopsF he pproximte solution to this prolem hs een provided y ollhrdt nd öl)e in IWVH WD who suggested to renormlize the di'uE sion oe0ient in the return proility itselfX

n SCTL (r, t) = dω 2π dq (2π) d e iq.(r-r )-iωt 1 -iω + D(ω)q 2 1 D(ω) = 1 D 0 + 1 πρD 0 dQ (2π) d 1 -iω + D(ω)Q 2 =n SCTL (r ,ω)
, @IFRA ollhrdt nd öl)e selfEonsistent theory hs enjoyed quite some suess @see IH for reent reviewAF st is however inurte in the viinity of ritil pointsD where its kind of menE(eld nture2 prevents urte determintions of ritil exponents IIF iqution @IFRA n e fully solved in Ih IPD where the resulting in(niteEtime density pro(le tkes the form

n SCTL (x, t = ∞)| 2 = e -|x|/2 4 . @IFSA

Berezinskii diagrammatic technique

ferezinskii digrmmti tehnique is rigorous pproh to loliztion restrited to the oneE dimensionl @IhA seF st ws developed y ferezinskii in IWUQ IQ nd is disussed in detils in hpter RF es the selfEonsistent theoryD this method is sed on the resummtion of digrmmti expnsionF roweverD in ontrst with the selfEonsistent theoryD the hoie of relevnt digrms is rigorously ontrolled nd their summtion is extF e prtiulrly interesting resultD otined with this tehnique y qogolin IRD is the in(niteEtime verge density pro(le rehed y n initilly nrrow wve pket spreding in disordered potentil 3

D n G (x, t = ∞) = ∞ 0 dηπ 2 32 η 1 + η 2 2 sinh(πη)e -(1+η 2 )|x|/8 [1 + cosh(πη)] 2 .
@IFTA iqution @IFTA n e interpreted s n verge over exponentilly lolized pro(les ∝ exp(-|x|/ξ)D the integrl over η ounting for the distriution of loliztion lengths ξF es visile in (gure IFPD eqution @IFTA somewht di'ers from the result otined using the selfEonsistent theory of lolizE tionD eqution @IFSAF he mjor di'erene ours in the wingsD where the ext result @IFTA deys s exp(-|x|/8 )F his di'erene lies in the lrge )ututions present in the wingsD whih mke the verge pro(le dominted y rre events not ptured y the selfEonsistent theoryD whih only desries the typil pro(le ISF sn the enter of the pro(leD )ututions re muh smller ITD IUD nd the selfEonsistent theory works wellF 2000 1000 0 1000 2000

x/ pigure IFP ! gomprison etween the ext qogolin pro(le for the verge density n(x, t → ∞) @eqution @IFTAD red urveA nd two simple exponentil formsF sn the min plotD the omprison is done with exp(-|x|/8 )D shown s dshed lue lineF he inset presents zoom on the smll x prtD where the omprison is done with exp(-|x|/2 )D shown s dshed green lineF he ltter is the predition of the selfEonsistent theoryF 1.3 Scaling theory of localization he seemingly simple ide of onsidering how the properties of system hnge when its size hnges n e very fruitful for desriing omplex systemsF snspired y the ides developed in the ontext of ritil phenomen in sttistil physisD erhmsD endersonD viirdello nd mkrE ishnn introdued sling theory of enderson loliztion in disordered system of (nite size IVF sn the present setionD we give rief ount of their ontriutionF he ide is to desrie how the dimensionless ondutne g of disordered smple sles with its size L d F gonretely spekingD one onsiders the qellEwnnEvow β funtionD de(ned s

β = d ln(g) d ln(L) . @IFUA
he key ingredients llowing to hrterize the β funtion re the followingX @iA g is the only relevnt vrileD so tht β depends only on g @hene the nme one-parameter scaling theoryAD @iiA the β funtion is ontinuous nd regulr nd @iiiA the knowledge of the β funtion in its symptoti regimesF yn the one hndD the smll g symptoti follows from the exponentil dey of the ondutneD expeted in the loliztion regimeD nd perturtion theory round this ehvior IVD xote tht the β funtion @IFUA n e lulted from the selfEonsistent theory introdued in setion IFIF sts ehvior is onsistent with the generl predition of the sling theory PHF ht onlusions n we drw from (gure IFQc yn the one hndD for smll dimensions @d ≤ 2AD the β funtion is lwys negtiveD iFeF the ondutne dereses with inresing system sizeD nd s one moves long the urve of the β funtionD one ends up in the loliztion regime β(g) ∼ ln(g) for lrge systemsF yn the other hndD for higher dimensions @d > 2A the β funtion rosses HF he point g c suh tht β(g c ) = 0 seprtes two drstilly di'erent sling ehviorsF por g < g c D one reovers the smll dimension sitution nd loliztion eventully previls for lrge enough systemsF por g > g c in ontrstD the ondutne inreses with inresing system sizeD signling regime of lssil di'usionF g c de(nes the soElled moility edgeD seprting lolized phse from delolized phse for in(nite systemsF he ssoited trnsition is lled enderson trnsitionD nd ws lredy predited y enderson in his seminl pper IF snterestinglyD the twoEdimensionl @PhA se ppers very sensitive to the sign of the orretion in 1/g t lrge g sign of a in @IFWAF sn the se of spinless timeEreversl invrint systems onsidered so frD the wek loliztion orretion ws responsile for smll negtive orretion to β(g 1) = 0F es it turns outD this perturtive orretion strongly depends on the symmetries of the rmiltoninF essuming wek mgneti (eld @thus reking timeEreversl invrineAD the 1/g term of eqution @IFWA disppers @a = 0AD ut the β funtion is not qulittively 'etedD s the next order termD in 1/g 2 D ers negtive orretion s well PIDPPF sn shrp ontrstD ounting for the eletron spin through spinEorit term in the rmiltonin turns the wek loliztion orretion into wek antilocalization orretion nd led to trnsition in two dimensions PPF his is ut one mehnism leding to ritility in Ph disordered systems IIF 1.4 Experimental observations of Anderson localization 1.4.1 Early condensed-matter experiments yne of the initil motivtion tht led to the theoretil predition of loliztion ws the underE stnding of spin trnsport in doped semiondutorsD nd in prtiulr the oservtions of slow spin trnsport y peher nd qere PQD PRF sn the dedes following enderson9s disovery of loliztionD lot of experimentl e'ort were devoted to the sujet @see PS for reviewAF ixperiments were performed with solidEstte smples suh s thin metlli (lms or semiondutorsF gorrespondinglyD theoretil works were primrily onerned with the ondutivity @or relted oservlesA of these systems @see review PSAF he rihness of solidEstte system my however lso onstitute di0ultyF por instneD the e'et of eletronEeletron intertions is a priori importnt in this ontextD their interply with loliztion ws intensively studied @see PT for reviewAF he possiility of other mehnisms inhiiting trnsport @eFgF mnyEody e'ets for wott insultorsA mkes the de(nite oservtion of enderson loliztion di0ult PUF purthermoreD the oupling of eletrons with externl degrees of freedom @eFgF phononsA mkes the loliztion phenomenon di0ult to oserve in pure onditionsF 1.4.2 A wave phenomenon es we hve seen in setion IFID enderson loliztion is mnifesttion of interferenes eE tween prtil wves multiply sttered y the disorderF sn other wordsD it is intrinsilly wve phenomenon PVF es suhD it ws oserved with vrious kinds of wves inluding mirowves PWD ending wves QHD light QID QP @see however QQD QRA nd ultrsound wves QSD to give the min exmplesF eentlyD enderson loliztion of tomi mtter wves hs lso een oserved QT!RHF e mjor dvntge of these setups is the possiility to ontrol tomEtom intertions @throughD eFgFD peshh resonnesAD nd to hieve wek oupling to the environmentF purthermoreD tomEoptis experiments o'er the possiility to diretly proe loliztion phenomen inside the tomi systemD s well s to follow their evolution in the ourse of time RID RPF IFSF enderson loliztion nd intertions U hese key possiilities hve led to n enrihment of enderson loliztion phenomenology with the disovery of the oherent forwrd sttering pek RQD whih n e used to hrterize enE derson trnsitions RRF sn dditionD oldEtom experiments sed on quenh protool QU!RHD RS impose to ount for rod spetrum of energy omponents RPD RT @s opposed to ondensedE mtter experiments where only eletrons ner the permi energy ontriute to trnsportAF et lstD the spei(ities of spekle optil potentilsD often used to rete the disorder QU!RHDRSD led to peE ulir feturesD suh s the existene of series of pprent moility edges etween energy regions where loliztion lengths di'er y orders of mgnitude RUF 1. 5 Anderson localization and interactions he question of intertions is nturlly importntD s they re often unvoidle in experimentsF gonerning this issueD two min trends hve reently emergedF yn the one hndD lrge ody of work tkles the question from the perspetive of wekly interting osoni gsesD nd tret intertions t the menE(eld levelD possily inluding perturtive orretions to the menE(eld solutionF yn the other hndD growing ommunity hs emrked into the sujet of mnyEody loliztionD n extension of enderson loliztion for mnyEody systemsF he present thesis is only onerned with the former point of viewD whih is introdued in the next susetionF e hve hosen to rie)y disuss few spets of mnyEody loliztion right fterwrdsD to o'er ontrsting viewF 1.5.1 Mean-eld eects et the menE(eld levelD the dynmis of old osoni gses is desried y nonliner equtionF his nonlinerity mkes the physis very rihF sndeedD one expets the nonlinerity to hve di'erent e'ets depending on the initil onditionsD s the superposition priniple no longer holdsF pour situtions hve ttrted lot of ttentionF pirstD the interply etween enderson loliztion nd intertions ws trked in the spreding of wve pkets @see RVD RW for reent reviewsAF st ws found tht enderson loliztion is destroyed in fvor of sudi'usion t long times s result of trdeEo' etween destrution of loliztion nd dilution of the nonlinerityF he destrution of enderson loliztion in fvor of sudi'usion ws oserved experimentlly SHF udi'usion is however predited to rekdown t extremely long timesD repled y slower spreding @if nyA @see SID SP for rigorous proofs nd SQD SR for possile physil mehnismsAF eondD the e'ets of inelsti ollisions nd the ensuing dynmis of the energy distriution ws studied SS! SW s well s the eventul thermliztion THF hirdD the e'et of intertions on the oherent ksttering pek ws studied in n tomElserElike on(gurtionD where stti tomi em is re)eted from nonliner disordered medium TID TPF st ws found tht intertions my trnsform the oherent ksttering pek into dipF xote tht one of the motivtion for this work me from the optil ontextD where the e'et of intertions on the oherent ksttering pek hd een previously onsidered TQ!TTF pourthD in similr tomElserElike on(gurtionsD the trnsmission of foseEiinstein ondenstes through disordered regions ws onsidered TU!TWD reporting destrution of enderson loliztion in fvor of super)uidity for smll ondenste veloitiesF et lrge veloitiesD enderson loliztion previls for smll disordered regions @stillD lrger thn the loliztion lengthA nd n instility of the menE(eld solution is expeted for lrger disordered regionsF he physis of perturtive orretions to the menE(eld solution ws lso onsidered in the frmework of fogoliuov theory UH!UTD through the trunted igner method UUD nd with digrmmti theory UVD UWF 1.5.2 Many-body eects he (eld of mnyEody loliztion emerged fter the pioneering work of fskoD eleiner nd eltshuler VHD who onsidered the possile loliztion of mnyEody wve funtion in the pok spe of endersonElolized sttesF prom this formultion of the prolemD they were le to uild resoning somewht similr to enderson9s originl rguments for loliztionF his seminl pE per generted lrge ody of worksD see VID VP for reent reviewsF hese works on(rmed the phenomenon of mnyEody loliztion nd ssoited to it whole phenomenologyD withD in prtiE ulrD the sene of thermliztion nd the logrithmi growth of entnglement entropyF he ltter property should e ontrsted with oneEprtile enderson loliztionD for whih the entnglement entropy is ounded VQF wnyEody loliztion hs een reently oserved experimentlly VRD VSF xote tht the mnyEody ounterprt of the gooperon @(gure IFIA is predited to e visile in pok spe VTF wnyEody loliztion hs lso found pplitions in guge theories VU nd time rystls VVF he sujet is lso of interest to mthemtiins VW!WIF 1.6 Outline of the thesis he generl gol of this thesis is to theoretilly investigte severl interesting e'ets relted to enderson loliztionD fousing on the ontext of oldEtomi systemsF he study flls into three min tegoriesF e (rst onsider generl spets of oldEtom setups feturing spekle optil potentilsD with n emphsis on the spetrl properties of the tomsF he seond fet of this work onerns new spets of enderson loliztion tht n e explored with oldEtom setupsF he third theme disusses the e'et of intertions on vrious phenomen ssoited to enderson loliztionD from the perspetive of ondensed osonsF he thesis is orgnized s followsF e (rst present theoretil pproh to the energy distriE ution of tomi louds in spekle potentilsF his study onsists of two stgesF he (rst one is presented in hpter PD where we onsider sttistil properties of spekle optil potentilsD with n emphsis on intensity minimF sn hpter QD we present the seond stgeD novel theoretil method for the lultion of the spetrl funtion nd the density of sttes in the strong disorder limitF sing the results otined in hpter PD we pply this method to the Ih seF he Ph se is treted in our pulished pperD reprodued in setion QFSF sn hpter RD we ome to the seond fet of the workD with the hrteriztion of the enterEofE mss motion of wve pkets lunhed with (nite veloity in rndom potentilsF his study unveils novel nd unexpeted mnifesttion of enderson loliztionX fter n initil llisti motionD the pket enterEofEmss experienes retrore)etion nd slowly returns to its initil positionF e dued this phenomenon the quntum oomerng e'etF he ore of hpter R onsists in the nlytil desription of the quntum oomerng e'et in one dimensionF sn hpter SD tomEtom intertions re introdued in the numeril simultionsD t the menE (eld @qrossEitevskiiA levelF heir e'ets on the quntum oomerng phenomenon re (rst disE ussedF he e'et of intertions is further investigted in the energy spreding of plne wves evolving in threeEdimensionl @QhA rndom potentilsF pinllyD we lso disuss how they 'et the oherent k sttering pekD nd in prtiulr revel n unexpetedly smll rise of the pek t short timesF sn hpter TD we tke di'erent diretion nd onsider deterministi @leit hotiA modelD the kiked rotorF e show tht suh kikedErotor models o'er interesting prospets for the study of enderson loliztionF yn the one hndD with strong experimentl nd theoretil motivtionsD we present ompelling evidenes in fvor of spinless kiked rotor in the sympleti ensemleF yn the other hndD seond look t qusiEperiodilly modulted kiked rotors revels intriguing resultsF pinllyD hpter U summrizes our (ndings nd gives some perspetivesF Chapter 2

β(g) ∼ g→0 ln(g) [1 + αg] , @IFVA with α > 0F
Statistical properties of speckle patterns sn oldEtom experimentsD one tkes dvntge of the intertion etween lser nd the toms for vrious purposesF yf prtiulr interest to us in this hpter is the possiility to shpe the potentil felt y the tomsF sndeedD under ertin onditionsD to e disussed in setion PFID the lser ts for the toms s potentil whih is simply proportionl to the lser intensityF sn low dimensionsD the ommon route to generte rndom potentils is to shine lser on rough plteF he resulting potentil is then the omplex di'rtion pttern of the plteD nd the rndomness origintes from the rndom lol roughness of the plteF e preise understnding of the sttistis of the resulting potentil is of utmost importne for the nlysis of suh experimentsF his hpter is divided in three prtsF pirstD we give rief introdution to the oldEtom frmeE work in setion PFIF eondD setion PFP gives n introdution to the sttistis of spekle potentilsD nd is onerned with generl propertiesD suh s the onEsite distriution nd orreltion funtionsF his introdutory setion follows the ook y qoodmn WPF st will lso e the osion to explin how spekle ptterns re generted in the numeril simultionsF henD in setion PFQD we dive into more spei( sttistil propertiesD the sttistil properties of intensity minimF he distriutions onsidered there were derived during the thesisD nd used to lulte spetrl properties of tomi louds in spekle potentilsD whih is the ojet of hpter QF 2.1 Dipolar potential for cold-atoms sn this setionD we show how one n tke dvntge of the intertion etween lser nd the internal struture of n tom to in)uene the dynmis of its external degrees of freedomF his introdutory setion follows the ourses y hlird t gollège de prne WQF por the ske of simpliityD throughout this setion we onsider twoElevel tom interting with lser of frequeny ωD s illustrted in (gure PFIF he ground stte |g nd the exited stte |e of the tom desrie its internal degrees of freedomD typilly |g is the oritl wve funtion of the outermost oupied shell in the ground stte nd |e is n unoupied oritl wve funtionF sn oldE tom experimentsD one tkes dvntge of the internl struture of the tom to indue the dynmis of its externl degrees of freedom in ontrolled wyF he intertion etween lser nd toms o'ers mny possiilitiesF e detil only one of them in this mnusriptD the possiility to tilor the externl potentil felt y the tomsF yther exmples re rie)y mentioned in susetion PFIFQF

V d = -d.E(r), @PFPA
where the dipole opertor writes d = -e r i D with e the elementry eletri hrge nd r i the position opertor of the eletron in the referene frme of the tomF rere r is the tom enterEofEmssF st is useful to express V d in the twoElevel sisD

V d = -
V d (r, t) = -d 0 E(r)( σ + + σ -) cos(ωt + φ(r)). @PFTA

Dipolar potential

Statement of the problem nder ertin onditionsD to e spei(ed in this susetionD the dipolr tomE(eld intertion mimiks n externl potentilX the enter of mss of the tom in the eletri (eld experienes n rmiltonin dynmisD the dipolr intertion tking the form of n externl potentilF he gol of this susetion is to derive the orresponding e'etive rmiltoninF o tht endD we strt from the full rmiltonin

H = p 2 2m + V d (r, t) + ω 0 |e e| , @PFUA
where the (rst term ounts for the externl kineti degrees of freedom of the tomD the seond term desries its oupling to the eletri (eld t the dipolr pproximtionD nd the lst term is ssoited with the internl struture of the tom @we hve hosen to ount the energy strting from the ground stte energyAF e hve lredy negleted spontneous emissionD whih intrinsilly nnot e desried y n rmiltonin dynmisD nd thus ssumed smll numer of toms in the exited stteF he stte of the tom is desried y the density opertor ρD desriing oth its internl nd externl degrees of freedomF yf prtiulr interest to us is its projetion on the ground stte of the tomD ρ gg = g| ρ|g F Ground-state dynamics he evolution of ρ gg is governed y the reisenerg eqution

d ρ gg dt = 1 i H, ρ gg = 1 i p 2 2m , ρ gg -cos(ωt -φ)iΩ( r) ρ eg + cos(ωt -φ) ρ ge iΩ( r), @PFVA
where nturlly ρ eg = e| ρ|g nd ρ ge = ρ † eg F e hve lso introdued the i frequeny Ω( r) = -d 0 E( r)/ F he evolution of ρ gg is oupled to the evolution of ρ eg tht oeys

d ρ eg dt = 1 i H, ρ eg = 1 i p 2 2m , ρ eg -iω 0 ρ eg -cos(ωt -φ)iΩ( r) ρ gg + cos(ωt -φ) ρ ee iΩ( r), @PFWA
where ρ ee = e| ρ|e F o simplify this equtionD we mke use of two ssumptionsF pirstD we ssume tht negligile frtion of toms re exitedD iFeF ρ ee ≈ 0F hen we ssume tht internl degrees of freedom vry muh fster thn externl onesD nd thus neglet the kineti prtF nder this two ssumptionsD eqution @PFWA provides

ρ eg = Ω( r) 2
e -i(ωt-φ) ωω 0 -e i(ωt-φ) ω + ω 0 ρ gg , @PFIHA whih we plug in eqution @PFVA to (nd losed eqution for

ρ gg D d ρ gg dt = 1 i p 2 2m , ρ gg + Ω 2 ( r) 2i cos(ωt -φ) e -i(ωt-φ) ω -ω 0 - e i(ωt-φ) ω + ω 0 ρ gg -ρ gg Ω 2 ( r) 2i cos(ωt -φ) e i(ωt-φ) ω -ω 0 - e -i(ωt-φ) ω + ω 0 , @PFIIA
we hve used ρ gg = ρ † gg F he optil frequeny ω is very lrgeF yn time sles lrger thn 1/ωD we n neglet the miroEosilltions of the tomi enterEofEmss indued y the rpidly osillting terms @rotting wve pproximtionAF nder this pproximtionD the dynmis is governed y

d ρ gg dt = 1 i p 2 2m + V ( r), ρ gg , @PFIPA withD in the limit ω + ω 0 |ω -ω 0 |D V ( r) d 2 0 E 2 ( r) 4 1 ω -ω 0 . @PFIQA
o mke more diret onnetion with the properties of the twoElevel tomD we express the redued tomi dipole in terms of the dey rte of the exited stte Γ WQX

d 2 0 = 3hc 3 0 Γ 2ω 0 , @PFIRA
where c is the speed of lightF lugging @PFIRA in eqution @PFIQAD nd introduing the lser intensity I(r) = c 0 E 2 ( r)/2D we otin the following expression for the potentil

V ( r) 3πc 2 Γ 2ω 0 I(r) δ , @PFISA
where δ = ωω 0 is the lser detuning with respet to the twoElevel trnsitionF sn onlusionD the ground stte of the tom oeys n rmiltonin dynmis nd feels n externl potentil V ( r) proportionl to the lser intensityF hepending on the lser frequeny detuning with respet to the twoElevel trnsition onsideredD the sign of the potentil n e positive or negtiveF he physil piture underlying the ove lultion is the followingX the inoming (eld polrizes the tomD whih in turn interts with itF nder the onditions detiled in the present susetionD this intertionD verged over the rpid )ututions of the eletri (eldD ts s potentil for the externl degrees of freedom of the tomF Assumptions sn deriving eqution @PFIPAD we hve mde vrious ssumptionsD whih we now summrize nd disussF o egin withD we hve onsidered twoElevel tomF his is good model for lklineEerth metls @when they nuler spin is vnishingAD ut it needs to e extended for lkli metlsF por more omplited internl struturesD the pproh is somewht similr nd the onlusion is the smeX for the tomD the lser mimiks n externl potentilF enother importnt ssumption is tht few toms get exited y the lserD whih is only possile if the lser frequeny is frEenough detuned with respet to the twoElevel trnsitionF xote thtD in prinipleD it does not set limit on the strength of the potentil felt y the tomF sndeedD it is possile with lrge lser intensity nd orrespondingly lrge detuning to rete strong potentils while preserving smll numer of exittionsF et lstD we hve ssumed tht internl degrees of freedom vry muh fster thn externl onesD ondition generlly met in oldEtom experimentsF 2.1.3 Taking advantage of absorption or complex internal structures o frD we hve only seen photon sorption or more omplex internl strutures s nnoynes without whih we would e etter o'F sn ftD photon sorption is the key to ool toms down to very low temperturesD step whih preludes ll the present disussionF yn the other hndD one n use omplex internl strutures to mimik guge (eldsF por more detils out the ooling posE siilities o'ered y tomElser intertion see WRF he reder interested in the genertion of guge (elds in oldEtom experiments n onsult the review WSF edditionllyD one n tke dvntge of the eemn e'et to tune tomEtom intertions y pplition of mgneti (eldD yet gin formidle sujet we do not im t overing in the present mnusriptD see WT for reent reviewF he oldEtom tehnology is now widely usedD s demonstrted y WUD nd hs llowed progress in mny res of quntum physisD s emphsized in WVD WWF 2.2 Speckle patterns: generalities sn the light of setion PFID the genertion of rndom potentil oils down to the genertion of su0iently disordered intensity pttern I(r) = c 0 E 2 ( r)/2F he ommon wy to hieve this pigure PFP ! hemti representtion of n experimentl setup used to generte spekle pttern in oldEtom experimentsF e lser is shined on rough plteD tht stters the eletroEmgneti wveF he resulting interferene pttern is oserved in the oservtion plneD t distne D from the plteF e lens is pled right fter the rough plteD its fol plne orresponding to the oservtion plneD thus mimiking fr (eld on(gurtion with resonle DF gol onsists in reting spekle ptternD otined y di'rtion of rough plteD s illustrted in (gure PFPF he resulting spekle pttern is oserved in the fr (eld nd orresponds to the di'rtion pttern of the plteF sn prtieD one ples lens right fter the rough plte nd uses the fol plne s oservtion plneD thus mimiking frE(eld on(gurtion with resonle distne etween the plte nd the oservtion plneF he eletri (eld polriztion does not ply n importnt roleD we ssume slr (eld for the ske of simpliityF 2.2.1 Fresnel integral he spekle pttern is hrterized y positionEdependent intensity I(x, y)F o ompute I(x, y)D we follow ruygensEpresnel priniple nd rek down the rough plte into mny independent itsD eh it ehving s seondry soureF he oserved spekle pttern is then the interferene pttern of the eletroEmgneti @iwA wve sttered y ll the itsF qiven it t (α, β)D nd its orresponding omplex mplitude a(α, β) @in whih we lso inlude the phse umulted when going through the lensAD the resulting omplex eletri (eld t (x, y) writes

E α,β (x, y) = a(α, β)e ikr iλr , @PFITA with r = (x -α) 2 + (y -β) 2 + D 2 D λ = 2π
/k the wve length of the lser nd D the distne etween the rough plte nd the oservtion plneF sn the limit D x, y, α, βD one n resort to the presnel pproximtionX

r = D 1 + (x -α) 2 + (y -β) 2 D 2 D + (x -α) 2 + (y -β) 2 2D . @PFIUA
he totl omplex (eld t (x, y) is otined y integrtion over the its onstituting the rough plteX

E(x, y) = dαdβA α,β (x, y) = e ikD iλD e ik 2D (x 2 +y 2 ) dαdβe ik 2D (α 2 +β 2 ) e -ik D (xα+yβ) a(α, β). @PFIVA
he iw wve intensity follows from I ∝ |E|2 F sn this setionD the proportionlity onstnt is irreleE vntD we set it to I nd use I = |E| 2 to lighten the nottionsF he rough plte nd its ssoited lens re desried y the funtion a(α, β)D whih ontins the positionEdependent sttering properties of the plte nd the phse umulted when going through the lensF e model it y rndom funtionF por the present purposesD we do not need the full distriution of a(α, β)D ut only its orE reltion funtionF essuming tht the oservtion region is not lrge enough to llow the resolution of the detils of the rough plteD we n neglet sptil orreltions of a(α, β)1 X a(α, β)a * (α , β ) = I rp (α, β)δ (2) (αα , ββ ). @PFIWA I rp (α, β) hrterizes the verge intensity trnsmission of the rough plte t (α, β)F sn experimentsD it n e tuned y pplition of mskF 2.2.2 On-site intensity distribution e hve lid out ll the neessry ingredients for the sttistil hrteriztion of spekle potentilsF vet us strt with the intensity distriution t given point (x, y)F por n ritrry spekle ptternD the omplex eletri (eld t (x, y) is given y eqution @PFIVA s the sum over ontriutions from very lrge numer of seondry souresF prom the entrl limit theoremD it thus oeys qussin lw 2 X

P e(E), sm(E) = 1 2πσ 2 E exp - e(E) 2 + sm(E) 2 2σ 2 E @PFPHA
o otin the distriution of the intensityD I = |E| 2 D we introdue the hnge of vriles e(E) = I cos(θ), @PFPIA sm(E) = I sin(θ), @PFPPA the toin is simply 1/2F he joint distriution of intensity nd phse followsX

P (I, φ) = θ(I) 4πσ 2 E exp - I 2σ 2 E .
@PFPQA where θ is the reviside thet funtionF he phse is thus uniformly distriutedD nd the intensity oeys n exponentil lwX

P (I) = θ(I) I 0 exp - I I 0 , @PFPRA
where we hve introdued the verge intensity

I 0 = I = I 2 -I 2 = 2σ 2 E . @PFPSA

Correlation functions

fesides the onEsite distriutionD it is ruil to ount for the sptil orreltion of the potenE tilF he omplex eletri (eld eing qussin rndom vrile of vnishing men vlueD ll its orreltions funtions re enoded in its seondEorder orreltion funtion

C E (x, y; x , y ) = E(x, y)E * (x ,
C E = e ik 2D (x 2 +y 2 -x 2 -y 2 ) λ 2 D 2 dαdβdα dβ a(α, β)a * (α , β )e ik 2D (α 2 +β 2 -α 2 -β 2 ) e -ik D (xα+yβ-x α -y β ) .
@PFPVA lugging in eqution @PFIWAD we (nd

C E = e ik 2D (x 2 +y 2 -x 2 -y 2 ) λ 2 D 2 dαdβe -ik D ((x-x )α+(y-y )β) I rp (α, β).
@PFPWA e see tht C I @= |C E | 2 D eqution @PFPUAA is the modulus squre of the inverse pourier trnsform of I rp (α, β)D the intensity distriution of the lser in the rough plteD nd depends only on (x-x , y-y )F he intensity distriution of the lser in the rough plte n e experimentlly tuned y pplition of msk on the rough plte @we ssume tht the onverging lens trnsmit uniformly the iw wveAF hysillyD C I orresponds to the di'rtion pttern of the mskF qiven rough plte of size RD the intensity orreltion funtion in the oservtion plne typilly deys over distne D/kRF his distne de(nes σD the orreltion length of the potentilF hree exmples of msks nd their ssoited intensity orreltion funtion re shown in tle PFIF e hve so fr only onsidered Ph spekle ptternsF e Ih spekle n esily e otined from Ph oneD simply y utting slie in the Ph ptternF he genertion of Qh spekle ptterns is more involvedF sndeedD the setup desried here to generte spekle ptterns hs smll ngulr pertureD resulting in spekle grins elongted in the z diretionF eentlyD Qh spekle ptterns were implemented in two experiments QVDRHD y superimposing two Ph spekle ptterns generted long two di'erent diretionsD so tht the resulting Qh spekle pttern remins nisotropiF 2.2.4 Numerical implementation rving hrterized the sttistil properties of spekle potentilsD we re now in position to implement them numerillyF his implementtion serves two purposes in the present thesisF yn the one hndD we use them diretly to study their properties in setion PFQF yn the other hndD they from key ingredient for the numeril simultions of hpters Q nd RF heir implementtion mounts to generting sptilly orrelted rndom funtionF sn prtieD one genertes the omplex mplitude in two stepsF pirstD one genertes n unorrelted grid of points with the desired distriution @eFgF omplex qussinAF eondD one onvolutes the unorrelted grid y the desired orreltion funtionF he onvolution n e onveniently performed in pourier speD where it is simply multiplitionF iventullyD the spekle potentil is otined s the modulus squre of the omplex mplitudeF sn ftD this proedure extly mimiks the experimentl senrio disussed in the previous susetionsD where the intensity orreltion funtion is simply the modulus squre of the inverse pourier trnsform of the intensity distriution in the rough plteD the orreltion funtion is thus imprinted in pourier spe y the rough plte mskF wo reliztions of Ih rndom potentils used in the numeril simultions of hpter Q re shown in (gure PFQF hey orrespond to reliztions of spekle optil potentilsD with the following wsk @I rp (α, β)A qure θ(R -|α|)θ(R -|β|) le PFI ! hree possile msks pplied on the rough plteD nd their resulting intensity orreltion funtionsF α nd β re expressed in ritrry unitsF por the isotropi msks @the irle nd the qussinAD the orreltion funtion is isotropi s wellD however the orreltion funtion ssoited to the squre msk is nisotropiF θ is the reviside thet funtionD sin(x) = sin(x)/x nd J 1 is the (rst order fessel funtion of the (rst kindF e hve introdued σD the orreltion length of the potentilD whih depends on the typil length over whih I rp (α, β) deys RD the wve length of the lser λ @= 2π/kA nd the distne etween the rough plte nd the oservtion plne DF onEsite distriution @we move from the intensity nottion to the potentil one for onsisteny with the rest of the thesisAX

P V (x) = 1 V 0 θ ±V (x) exp ∓ V (x) V 0 , @PFQHA
where θ is the reviside thet funtionF he disorder strength V 0 > 0 enters oth the verge V (x) = ±V 0 nd the vrine V (x) 2 -V (x) 2 = V 2 0 F sn eqution @PFQHAD the upper sign refers to lueEdetuned spekle potentil @ω > ω 0 in eqution @PFIQAAD ounded y zero from elowD nd the lower sign to redEdetuned spekle potentil @ω < ω 0 AD ounded y zero from oveF hey re oth hrterized y qussin orreltion funtionX

V (x)V (x ) -V (x) 2 = V 2 0 exp - |x -x | 2 2σ 2 . @PFQIA

Statistics of intensity minima

he ove sttistil hrteriztion of spekle potentils opens the wy for lulting more spei( sttistil propertiesF e re interested in intensity minimD nd more preisely the joint distriution of their depths nd urvturesD together with the density of minimD in the Ih situtionF e wnt eventully to pply these sttistil properties to the lultion of spetrl properties of toms in hpter QD we thus herefter dopt the lnguge of potentil rther tht of intensityF o set the nottionsD we introdue the potentil onEsite distriutionD pigure PFQ ! xumeril reliztions of redE @leftA nd lueEdetuned @rightA Ih spekle potentilF he onEsite distriution is exponentil eqution @PFQHA nd the twoEpoint orreltion funtion is qussin eqution @PFQIAF where θ is the reviside thet funtionF elong with the onEsite distriutionD the potentil is hrE terized y qussin orreltion funtion eqution @PFQIAF he orreltion funtion is set for de(nitenessD the lultion n e done for ritrry orreltion funtionsF o e onreteD given the expnsion of the potentil ner minim

P V (x) = 1 V 0 θ V (x) exp - V (x) V 0 , @PFQPA
V (x) = V + 1 2 mω 2 x 2 + . . . , @PFQQA
we re interested in the joint distriution of V nd ω @P (V, ω)AD s well s the density of minimF xote tht results on the Ph ounterprt of P (V, ω) n e found in our pulished pperD reprodued in setion QFSF por the ske of revityD we just rie)y omment on these results in susetion PFQFQF hroughout this setionD we use hrteristi sle for ωD

ω 0 = V 0 mσ 2 . @PFQRA 2.3.1 Joint distribution P (V, ω)
sn this susetionD we lulte the joint proility distriution P (V, ω) disussed oveF he distriution P (V, ω) is losely relted to the jointD onditionl proility distriution of V (x) nd its seond derivtive V (x) given tht V (x) = 0 nd V (x) > 0D P (V (x), V (x)|V (x) = 0, V (x) > 0)D tht we propose to lulte (rstF prom here on we use the following revited nottion for the potentil nd its derivtives t point xX

V ≡ V (x), V x ≡ V (x), V xx ≡ V (x).
@PFQSA he ove distriution follows from

P (V, V xx |V x = 0, V xx > 0) = N × lim Vx→0 P (V, V x , V xx ) P (V x ) . @PFQTA
he numeril onstnt N tht ppers in eqution @PFQTA stems from the ft tht only positive urvtures re seleted on the leftEhnd sideD wheres on the rightEhnd side ll possile vlues re understoodF st will e lter determined from the normliztion onditionF sn order to ompute the joint distriution P (V, V x , V xx )D we follow qoodmn WP nd write the potentil s

V = (x) 2 + (x) 2 .
@PFQUA p to onstnt multiplitive ftorD (x) nd (x) respetively desrie the rel nd imginry prt of the lser eletri (eld t point xD from whih the spekle potentil V is uilt onD s explined in setion PFPF es for the potentilD we introdue the following shortEhnd nottions

≡ (x), x ≡ (x), xx ≡ (x) ≡ (x), x ≡ (x), xx ≡ (x). @PFQVA
he motivtion for introduing the (elds nd is tht they re independent qussin vriles with zero men nd equl vrine WPF heir derivtives re likewise qussinD sine ny linE er trnsformtion of qussin retins qussin sttistisF hey lso hve zero menF es onsequeneD the six rndom vriles of interest oey the multiEdimensionl qussin distriution

P ( , , x , x , xx , xx ) = e -u t C -1 u/2 8π 3 det(C) , @PFQWA
where u t is row vetor with entries ( , , x , x , xx , xx )D nd C is the ovrine mtrixF e then introdue in eqution @PFQWA the hnge of vriles

= √ V cos θ, = √ V sin θ, @PFRHA
from whih we lulte the distriution P (V, θ, V x , θ x , V xx , θ xx )D with orresponding toin equl to 1/8F fy expliitly evluting the entries of the C mtrix for the qussin orreltion funtion @PFQIA nd lulting the remining integrls over θD θ x nd θ xx with wthemti IHHD we (nd

P (V, V x , V xx ) = σ 4 4 √ 2πV 3 0 V e - 24V +16Vxxσ 2 +(V 2 x -2V Vxx) 2 σ 4 /V 3 16V 0 (-V 2 x + 2V V xx )V 0 V , ×    I -1 4 (V 2 x -2V V xx ) 2 σ 4 16V 3 V 0 + I 1 4 (V 2 x -2V V xx ) 2 σ 4 16V 3 V 0    , @PFRIA
where I 1/4 nd I -1/4 re the modi(ed fessel funtions of the (rst kindF xote tht this expression is vlid only when V 2

x -2V V xx < 0D ondition ful(lled sine only minim of the potentil re onsidered 3 F he distriution P (V, V x , V xx ) is regulr with respet to the limit V x → 0F sn equE tion @PFQTAD we n thus tke this limit seprtely in numertor nd denomintorD reduing the ltter to numeril onstnt whih n e sored in the normliztion preftor N F prom the joint distriution @PFRIAD we re now in position to ess the proility P (V, V xx |V x = 0, V xx > 0) using eqution @PFQTAF he result is

P (V, V xx |V x = 0, V xx > 0) = N √ V xx V e - 6V 2 +4V Vxxσ 2 +V 2 xx σ 4 4V 0 V   I -1 4 V 2 xx σ 4 4V V 0 + I 1 4 V 2 xx σ 4 4V V 0   . @PFRPA
fy imposing tht the distriution is normlizedD we (nd N = σ 5 /(2cV

5/2 0 )D where c = √ 3Γ 1/4 Γ 5/4 -Γ -1/4 Γ 7/4 3 3/4 √ 2π 1.00685, @PFRQA
whih will e repled y I in the followingF he lst stge of the lultion onsists in onneting P (V, V xx |V x = 0, V xx > 0) to the sought for distriution P (V, ω)F his mounts to hnging the vriles from V x = 0 to x suh tht V x (x) = 0D pigure PFR ! toint distriution P (V, ω) of minim nd potentil urvture round minimD for IhD lueEdetuned spekle potentil with qussin orreltion funtion eqution @PFRRAF nd from V xx to ω suh tht mω 2 = V xx F he ssoited toin is |dV x /dx×dV xx /dω| = 2m 5/2 ω 3 F e (nlly infer

P (V, ω) = 1 V ω 0 ω ω 0 4 e -3 2 V V 0 2 -ω ω 0 2 - V 0 4V ω ω 0 4   I -1 4 V 0 4V ω ω 0 4 + I 1 4 V 0 4V ω ω 0 4   .
@PFRRA he joint distriution is shown in (gure PFRF et given potentil minimum V D we oserve tht it is mximum for ω ∼ ω 0 D t smller ωD the distriution rpidly flls to zeroF vow energy minimD V V 0 D will turn out of speil importne for the spetrl properties of toms in spekle potentilsD it is thus interesting to express the distriution P (V, ω) in the limit

V → 0 IHIX P (V, ω)V 0 ω 0 ∼ V →0 2 π V 0 V ω ω 0 2 e -ω ω 0 2 .
@PFRSA he 1/ √ V divergene t low V shows tht most minim lie t very low V V 0 F he distriution of their ssoited frequeny ω is peked round ω 0 F 2.3.2 Density of minima e re lso interested in the density of minim ρF o evlute itD we follow IHPDIHQ nd onsider the generl identity

dxδ(V (x))f (x) = n 1 |V (x n )| f (x n ), @PFRTA
vlid for ny funtion f F he sum is over ll point x n where V (x) ≡ V x vnishesF sf we hoose f (x) to e |V (x)| ≡ |V xx |D then the integrl is equl to the numer of points t whih V x vnishesF his de(nes the density of extrem per unit length s

δ(V x )|V xx |. @PFRUA
he orresponding density restrited to minim of the potentil is

δ(V x )V xx θ(V xx ), @PFRVA
with θ the reviside funtionF he disorderEverged density of minim then reds

ρ = dV x dV xx P (V x , V xx )δ(V x )V xx θ(V xx ). @PFRWA
sing eqution @PFRIAD we otin ρ = c /σD where c 0.284026F

Two-dimensional case

es we hve shown t the end of susetion PFQFID most minim lie t very low V V 0 in oneE dimensionD with squreEroot singulrity of the distriution of minim t smll V eqution @PFRSAF st turns out tht the twoEdimensionl se is even more singulrD with (nite density of minim exactly t V = 0F he presene of suh minim n e redily understood from the deomposition of V s V = 2 + 2 eqution @PFQUAF sndeedD they orrespond to the intersetions of the lines long whih the qussin rndom funtions nd re vnishingF hese prtiulr minim ply ruil role for the lultion of spetrl properties of toms in spekle potentils in the next hpterF wore preiselyD from the expnsion of V round minimD

V (x, y) = 1 2 mω 2 x x 2 + 1 2 mω 2 y y 2 + . . . , @PFSHA
where the sis if hosen suh tht the rossed term @in xyA is vnishingD the joint distriution of ω x nd ω y is requiredF sts derivtion is it tehnil nd not very enlighteningD we thus prefer to refer to our pulished pperD reprodued in setion QFSD for detiled derivtionF he sitution is somewht similr to the oneEdimensionl se with typil frequenies tking vlues round ω 0 F

Conclusion

sn this hpterD we hve skethed out the genertion of rndom potentils in oldEtom exE periments nd seen how to implement them numerillyF e hve hrterized their sttistil propertiesD s modulus squre of omplex qussin rndom vrilesD with vrious possile sptil orreltionsF prom this hrteriztionD we hve then extrted importnt informtion for hpter QD tht of sttistis of intensity minim in oneEdimensionF he intensity minim essentilly lie t low energiesD V V 0 D nd the distriution of their proper frequeny is peked round

ω 0 = V 0 /(mσ 2 )F Chapter 3
Semiclassical spectral function and density of states in speckle optical potentials es prelude to the study of enderson loliztionD it is importnt to hrterize the spetrl properties of the toms in the rndom potentilF sn onnetion with oldEtom experimentsD we re spei(lly onerned with spekle potentilsD introdued in hpter PF o get (rst hnd on the sujetD we strt with the wek disorder limit @setion QFIAF sn this limitD perturtive tretment of the rndom potentil is possileD llowing for desription of the toms in terms of qusiEprtilesF st is nturl frmework for introduing importnt quntities for the rest of the thesisF emong themD the spetrl funtionD vizF the verge energy distriution of plne wve in the spekle potentilD reeives speil ttentionD nd its importne for the dynmis of toms is disussedF e ompnion of the spetrl funtion is the density of sttesF he ltter is lso n importnt quntityD in prtiulr in the disussion of phses of dirty interting osons THD UID IHRF sn the strong disorder regimeD semilssil methods turn out prtiulrly vlule for luE lting the ove quntitiesF yn the one hndD perturtive expnsion in llows for systemti orretions to the lssil limit in the lrge energy setor IHSF yn the other hndD the low enE ergy prt requires nonEperturtive methodF he perturtive expnsion ws reently rried out in IHSD sed on ignerEeyl formlismF sn ontrstD here we ddress the singulr orretions tht pper t low energyD using new method sed on sttionry phse pproximtionsD nd pplied to the one nd two dimensionl sesF hese lultions re desried nd disussed in detils in pulished pperD reprodued in setion QFSF e hoose here to present oth the perturtive expnsion in @with method di'ering from tht of IHSA nd the nonEperturtive methodD to o'er somewht omplete desription of the semilssil regimeF es the twoEdimensionl se is qulittively similr to the oneEdimensionl oneD we prefer to detil only the former hereD nd refer to the rtile of setion QFS for the ltterF etion QFI is introdutoryD the reder fmilir with the qreen funtion lnguge my prefer to skip itF etion QFP motivtes our study of the spetrl funtionD nd setion QFQ develops our ontriution to the sujetF 3.1 Weak disorder, perturbative calculations sn the wek disorder limit @to e de(ned shortlyAD one n resort to perturtion theory to get n understnding of the physis t plyF e propose to rry out this progrm in the present setionD within the qreen funtion formlismF o lighten the nottionsD we ssume Ih systemF he generliztion of the mthemtis to ny dimension is strightforwrdF e losely follow the PI PP ghpter QF emilssil spetrl funtion nd ho in spekle optil potentils ook y ekkermns nd wontmux QF 3.1.1 Perturbative treatment with Green functions qreen funtions o'er powerful tool to rry out systemti perturtive lultionsF sn this respetD they re widely used in ondensed mtter physis IHTF qiven n rmiltonin

H = H 0 + V, @QFIA
the @retrdedA qreen funtion is de(ned s

G(x 1 , x 2 , t) = θ(t) x 2 |e -iHt/ |x 1 , @QFPA
with θ the reviside thet funtionF sts pourier trnsform in turn writes

G(x 1 , x 2 , ) = ∞ -∞ dte i( +i0 + )t G(x 1 , x 2 , t) = x 2 | G|x 1 , @QFQA
where we hve introdued the qreen opertor GX

G = 1 -H + i0 + , @QFRA
i0 + eing n in(nitesiml imginry prt ensuring the onvergene of the pourier trnsform t long timesF he onveniene of the qreen funtion ppers when one expresses the totl qreen opertor G in terms of the free qreen opertor

G 0 = 1/( -H 0 + i0 + )X G = G 0 + G 0 V G, @QFSA
whih is esily otined from the de(nitions of G nd G 0 F iqution @QFSA is lled the vippmnn! hwinger equtionF sterting it genertes systemti perturtive expnsion of GD the soElled forn seriesX

G = G 0 + G 0 V G 0 + G 0 V G 0 V G 0 + . . . @QFTA
o frD H 0 nd V n e ritrry opertorsF sn this thesisD we re primrily onerned with toms sujeted to disorderF st is thus nturl to hose H 0 s the kineti prt of the rmiltonin nd V s the potentil modeling the disorderF H 0 is digonl in momentum speD so is G 0 D whih writes

G 0 (k, k , ) = dxdx e -ikx+ik x G 0 (x, x , ) = 1 -k + i0 + δ k,k , @QFUA
where k is the dispersion reltionD k = 2 k 2 /2m for free toms of mss mF G 0 (k, k , ) eing digonlD we introdue the shortEhnd nottion

G 0 (k, ) = G 0 (k, k, ) = 1 -k + i0 + .
@QFVA yne expets the full qreen funtion for given disorder reliztion to e very omplitedD nd generlly imprtilF e more pertinent ojet is the distriution of the qreen funtionF he distriution my however remins di0ult ojet to hndleD so tht one usully onsider only the disorderEverged qreen funtionF eording to eqution @QFTAD the ltter n e expressed s

G(x, x , ) =G 0 (x, x , ) + dx 1 G 0 (x, x 1 , )V (x 1 )G 0 (x 1 , x , ) + dx 1 dx 2 G 0 (x 1 , x 2 , )V (x 1 )G 0 (x 1 , x 2 , )V (x 2 )G 0 (x 2 , x , ) + . . . @QFWA
QFIF ek disorderD perturtive lultions PQ ghoosing the origin of energies t V D we n set the men vlue of V to H in eqution @QFWAF prom here on we ssume tht V = 0D nd de(ne the potentil orreltion funtion s

V (x 1 )V (x 2 ) = B(x 1 -x 2 )
. @QFIHA ith these de(nitionsD eqution @QFWA redues to

G(x, x , ) = G 0 (x, x , ) + dx 1 dx 2 G 0 (x, x 2 , )B(x 1 -x 2 )G 0 (x 1 , x 2 , )G 0 (x 2 , x , ) + . . . @QFIIA
rnsltionl invrine fter disorder verging suggests to express the verge qreen funtion in pourier speD where it is digonlX

G(k, ) = d(x -x )e -ik(x-x ) G(x, x , ) = G 0 (k, ) + G 0 (k, ) dq 2π B(q)G 0 (k + q, )G 0 (k, ) + . . .
@QFIPA his expression of the verge qreen funtion provides nturl strting point for perturtive lE ultions in rndom potentilsF fefore emrking into spei( lultionsD it is useful to rerrnge the ove seriesD whih is the ojet of the next susetionF

The self-energy

por simpliityD let us ssume qussin rndom potentil for the momentD suh tht the nE orreltion funtions ppering in eqution @QFWA redue to produts of B through pplition of ik theoremF snterestinglyD t order 2n in V D the ove proedure genertes term of the form

G 0 (k, ) dq 2π B(q)G 0 (k + q, )G 0 (k, ) n .
@QFIQA sn ftD eqution @QFIQA is ut one exmple of high order ontriutions @here of order 2nA tht n e otined y hining lower order ontriutions @here n identil seond order ontriutionsAF his property suggests to introdue quntityD the soElled selfEenergy Σ(k, )D gthering ll ontriutions seprle in lower order ontriutionsF he verge qreen funtion then writes

G(k, ) = G 0 (k, ) + G 0 (k, ) ∞ n=1 Σ(k, )G 0 (k, ) n . @QFIRA
where Σ(k, ) ontins only those ontriutions to G(k, ) tht nnot e ftorizedF pon summing the geometri seriesD eqution @QFIRA redues to the soElled hyson eqution

G(k, ) = G 0 (k, ) + G 0 (k, )Σ(k, )G(k, ) ⇔ G(k, ) = 1 -k -Σ(k, )
. @QFISA et this pointD the physil mening of the selfEenergy strts to pperF sndeedD we hve rested the verge qreen funtion under form reminisent of the free qreen funtion eqution @QFVAD the selfEenergy enoding the di'erene etween the twoF o mke the role of the selfEenergy more lerD it is useful to go k to timeD

G(k, t) = d 2π e -i t/ 1 ( -Σ ) -k -iΣ @QFITA
where we hve split the rel nd imginry prts of the selfEenergy Σ = Σ + iΣ F o perform the integrtionD one needs to know how Σ depends on F nder the ssumption tht Σ is smooth funtion of energyD we n use the residue theorem to (nd

G(k, t) = θ(t)e -i( k +Σ ( k ,k))t/ e Σ ( k ,k)t/ , @QFIUA
where we hve used the ft tht the imginry prt of the selfEenergy must lwys e negtive to gurntee uslityF sn ftD the sene of poles in the upper hlf plneD whih gurntees uslity 1 D lso implies urmersEuronig reltions etween the rel nd imginry prts of G(k, )F iqution @QFIUA should e ompred to the de(nition of G(k, t) @whih follows from eqution @QFPA through pourier trnsformAX G(k, t) = θ(t) k|e -iHt/ |k . @QFIVA Σ thus simply enodes n energy shiftF Σ D on the other hndD enodes the lifetime of plne wves in the disordered potentilF xote tht the onept of selfEenergy is not restrited to the qussin potentils onsidered hereF por exmpleD in the se of spekle potentilsD the introdution of the omplex qussin (eld E @V = ±|E| 2 A llows for similr introdution of the selfEenergy s simpli(tion of the forn series @see IHU for more detils out this proedureAF fy studying the properties of the selfEenergy seriesD one n de(ne wek disorder riterionD whih should e stis(ed for the selfEenergy to e desried y the (rst few terms of its forn seriesF por spekle potentilsD one (nds IHU η kσ, @QFIWA where η = V 0 /E σ F rere V 0 nd E σ re hrteristis of the spekle potentil onsideredF V 0 orresponds to the mplitude of the )ututions nd E σ = 2 /mσ 2 is the hrteristi energy ssoited to the sptil orreltion length σ @see hpter P for detilsAF sn some ses @eFgF when the potentil distriution does not llow for the pplition of ik theoremAD it n e useful to introdue the selfEenergy simply through @QFISA @s opposed to through the forn seriesAF 3.1.3 Scattering mean free time and scattering mean free path he imginry prt of the selfEenergy @Σ A enodes the lifetime of plne wves in disordered potentilF his lifetime is usully lled sttering men free timeD nd denoted y τ F enother pproh to lulting this lifetime is the permi golden ruleX

τ = dk | k|V |k | 2 δ( k -k ), @QFPHA
whih writes in our ontext

τ = dk B(k -k )δ( k -k ). @QFPIA
he permi golden rule @QFPIA turns out to give n pproximtion of the lifetime t lowest order in V D whih should e ompred with the lultion of the imginry prt of the selfEenergy t lowest order in V D given y eqution @QFIPA

Σ = sm dk 2π B(k -k )G 0 (k , ) . @QFPPA
o simplify this equtionD we (rst use the prity of B(x

) to infer tht B(k) is relD nd then useX sm G 0 (k , ) = sm 1 -k + i0 + = -iπδ( -k ). @QFPQA his leds to Σ (k, ) = - 1 2 dk B(k -k )δ( -k ). @QFPRA
his results oinides with the permi golden rule @QFPIAF he ftor P stems from the selfEenergy deling with |k D while the permi golden rule dels with | |k | 2 F he sttering men free time is thus given y τ = -/(2Σ (k, k ))F e ompnion of the sttering men free time is the sttering men free pthF o introdue itD we onsider the qreen funtion in rel speD

G(x, x , ) = dk 2π e ik(x -x) G(k, ) = dk 2π e ik(x -x) 1 ( -Σ ) -k -iΣ .
@QFPSA e ssume gin tht the selfEenergy simply shifts the poleD suh tht

G(x, x , ) = G 0 (x, x , -Σ )e -|x-x |/2 , @QFPTA with = - 2 k ˜ 2mΣ (k ˜ , ˜ ) ˜ = -Σ k ˜ = √ 2m˜ / . @QFPUA
he verge qreen funtion n e viewed s free qreen funtion deying over distne F he prtile @desried y the modulus squre of the wve funtionA thus typilly propgtes freely over distne efore eing stteredD hene the nme sttering men free pth for F sn wek disordered potentilD the plne wves hve (nite lifetime τ D during whih they typilly trvel distne D with τ nd set y the imginry prt of the selfEenergyF sn the present ontextD these plne wves nd ssoited selfEenergy form the qusiEprtiles whih re ommonly used s uilding loks for the desription of oherent e'ets @see eFgF WAF snterestinglyD one n generlly rest the wek disorder riterionD eqution @QFIWA for spekle potentilsD under the form k 1 IHUF ixpressed in terms of the wve length λ = 2π/k D the wek disorder riterion eomes λ D whih orresponds to well seprted sttering events IHVF

The spectral function

he two e'ets of the selfEenergy re lerly visile in the spetrl funtionD de(ned s

A k ( ) = - 1 π sm G(k, ) = |Σ | π 1 ( -Σ -k ) 2 + Σ 2 .
@QFPVA sn the free seD Σ = 0D the spetrl funtion is delt funtion entered t k F he e'et of the disorder is twofoldF yn the one hndD the rel prt of the selfEenergy @Σ A shifts the mximum of the spetrl funtionF yn the other hndD the imginry prt of the selfEenergy @Σ A rodens the pekF hese two e'ets re illustrted in (gure QFIF he rodening e'et is lso visile in (gure QFID whih ompres numerilly omputed spetrl funtion @detils in the ption of (gure QFIA nd eqution @QFPVA with the selfEenergy lulted t lowest order in V @forn pproximtionAX

Σ = dq 2π B(q)G 0 (k + q, ) = dq 2π B(q -k) -q + i0 + . @QFPWA
es visile in (gure QFID t wek disorderD the rel prt of the selfEenergy hs smll e'etF et stronger disorderD preise ount of the rel prt of the selfEenergy n e ruilD in prtiulr for the lultion of the moility edge IHWF 0

( -k ) A k ( ) |Σ |

Σ

Free case Weak disorder 0.5 0.0 0.5

( -k )/V 0 0 1 2 3 4 5 A k ( )V 0 |Σ | Numerics Born approx.

a) b)

pigure QFI ! A hemti representtion of the e'ets of wek disorder on the spetrl funtionD through the selfEenergy ΣF he selfEenergy Σ is deomposed into its rel prt Σ @shift of the pekA nd its imginry prt Σ @width of the pekAF A petrl funtion s funtion of energyD for wekD deltEorrelted qussin potentil @V (x)V (x ) = V 2 0 δ(xx )/kD with V 0 0.076 2 k 2 /mAF he red line shows the perturtive resultD eqution @QFPVAD with Σ evluted t the forn pproximE tion eqution @QFPWAX Σ -iV 2 0 m/ 2 k 2 iΣ F his result is ompred with numeril simultionsD otined y propgting plne wveD |k D in system of size 5000π/kD disretized with UVSRH grid pointsF he results re verged over PSHH disorder reliztions nd periodi oundry onditions re usedF he physil mening of the spetrl funtion ppers ler if one reEexpresses @QFPVA s

A k ( ) = - 1 π sm G(k, ) = - 1 π sm k| 1 -H + i0 + |k = k| δ( -H) |k . @QFQHA
A k ( ) is the verge energy distriution of plne wve |k in the presene of the disorderF et wek disorderD the spetrl funtion is shrply pekedD so tht energies nd moment re generlly used interhngelyF he spetrl funtion is uiquitous quntity for the hrteriztion of the oldEtoms in rndom potentilsF sts role in the desription of the dynmis of toms in rndom potentils is the ojet of the next setionF xote tht it hs very reently een experimentlly mesured in Qh spekle potentils IIHF

Importance of the spectral function in the dynamics of coldatoms in random potentials

vet us onsider ommon experimentl senrio in the ontext of enderson loliztionX the spreding of wve pketF sn suh n experimentD one trks mnifesttions of enderson lolE iztion in the freezing of the verge density pro(leF ht is relly oserved n however e more sutleD for the wve pket is generlly mde of severl energy omponents F o see thisD we introE due entrl ojetD P (xx , t)D the verge proility for prtile of energy to go from x to x given time spn tF he dynmis of the wve pket results from oth the integrtion over the wve pket energy distriution nd the onvolution of P (xx , t) with the initil sptil pro(leF he wve pket energy distriution is essentilly given y the spetrl funtion A k ( )D whih hs to PU e weighted y the initil momentum distriution of the wve pketF o express the wve pket densityD we introdue the joint positionEmomentum distriutionD referred to s igner distriutionD

W (x, p, t) = 1 2π ∞ -∞ dyψ * (x + y/2, t)ψ(x -y/2, t)e -iyp/ .
@QFQIA he verge density t time t is then written s

|ψ(x, t)| 2 = W (x , k, t = 0)A k ( )P (x -x , t)dx d dk. @QFQPA
xote tht this somewht nturl deomposition of the verge density n e derived using the digrmmti formlisms disussed in susetion IFI RP nd setion IFP IIIF sn eqution @QFQPAD one identi(es dkW (x , k, t = 0)A k ( ) s the joint energyEposition initil distriutionF he wve pket density t time t then follows from the propgtion of eh energy omponentD strting from the initil sptil pro(leF sn the ove deompositionD P (xx , t) plys prominent role s it ontins ll the physis ssoited with enderson loliztionF xonethelessD forgetting ompletely out the rest would e dngerousF sndeedD the energy distriution n hve drsti e'ets on the dynmisD even t resonE ly wek disorderF gonretely spekingD let us onsider the sitution experimentlly relized in QUD where n initilly nrrow wve pket evolves in Ih spekle potentilF his prtiulr spekle poE tentil hs n intriguing fetureX t wek disorderD the sttering men pth is strongly energy dependentD exhiiting shrp rossovers when the energy goes through spei( vlues RUF pollowing eqution @QFQPAD the totl evolutionD integrted over ll energy omponents of the wve pketD my thus di'er from the evolution of n individul energy omponentF sndeedD t lrge distnesD the @usully expetedA enderson exponentil loliztion is turned into lgeri loliztion IIPF gonverselyD s ws pointed out in RTDIIQD wve pket mde of only di'usive toms n pper lose to lolized euse of the spetrl funtionF sndeedD even if for given energy D P (xx , t) is di'usive kernelD the totl evolution my pper sudi'usive due to the spei( energy dependene of the di'usion oe0ientF he totl evolution in this se looks s if single energy omponent with sudi'usive kernel ws t plyD thus mimiking the onset of enderson loliztionF es demonstrted in QVD RHD RSD preise ount of the energy distriution is lso required in the hrteriztion of the enderson trnsition in three dimensionsF sndeedD in this se the energy distriution spreds on oth sides of the moility edgeD mking the omplete dynmis superposition of lolized nd di'usive ehviorsF he knowledge of the energy distriution is then needed to extrt physilly relevnt quntities @eFgF the position of the moility edgeA from the omplete dynmis IIRD IISF ell these resons motived us to hve loser look t A k ( )F his is the entrl question ddressed in this hpterF 3.3 Strong disorder, semiclassical regime e hve seen in setions QFI nd QFP tht the spetrl funtion is n importnt quntity to hrterize the dynmis of wve pkets in rndom potentilsF e hve een le to otin it perturtively in the wek disorder limitF he present setion is devoted to the strong disorder limit @to e de(ned shortlyAD for whih semilssil pproximtions turn out to o'er vlule toolsF 3.3.1 Denitions and methods e rell the de(nition of the spetrl funtion eqution @QFQHAX

A k ( ) = k| δ ( -H) |k . @QFQQA
sntroduing the pourier representtion of the hir delt funtion in eqution @QFQQAD it follows tht

A k ( ) = ∞ -∞
dt 2π e i t/ k| e -iHt/ |k , @QFQRA whih estlishes the onnetion with the evolution opertor e -iHt/ F he spetrl funtion is relted to the density of sttes per unit volume @hoAD ν( ), through the reltion

ν( ) = 1 L d r δ ( -H) = d d k (2π) d A k ( ).
@QFQSA he mplitude of the potentil )ututions @V 0 A de(nes nturl energy sle in the prolemF fesides its )ututionsD the potentil is hrterized y its orreltion length @σAD whih de(nes nother importnt energy sleD E σ = 2 /mσ 2 F he rtio of these two energy slesD

η = V 0 E σ = mσ 2 V 0 2 , @QFQTA
is of speil importneF sn the followingD we fous on the soElled semiclassical regime hrterized y the ondition η 1. @QFQUA entiipting on the results presented elowD we hve represented pitorilly the semilssil limit in (gure QFPF

V 0 / √ η V 0 / √ η V 0 @A righ energies V 0 V 0 / √ η V 0 @A vow energies
pigure QFP ! itoril representtion of the semilssil limit for lueEdetuned spekle potentilF he semilssil ondition η 1 implies the existene of mny sttes in typil minim of the potentil @the typil lowest hrmoniEstte lying t ω 0 = V 0 / √ ηD see setion PFQ for more detilsAF et higher energies @AD n expnsion in powers of is possileF his is the ojet of susetion QFQFPF et low energiesD on the ontrryD orretions to the lssil limit re singulrF e present in susetion QFQFQ semilssil pprohD sed on sttionry phse pproximtionsD llowing us to desrie these singulr quntum orretionsF sn the lssil limit η → ∞D the nonEommuttion etween position nd momentum n e negletedD so tht k| e -iHt/ |k ≈ e -i k 2 t/2m e -iV (r)t/ F ith in ddition e -iV (r)t/ = dV P (V )e -iV t/ = 1 1 ± itV 0 / , @QFQVA where P (V ) is the spekle onEsite potentil distriution eqution @PFQHAD eqution @QFQRA yields

A cl k ( ) = ∞ -∞ dt 2π e i( -k )t/ 1 ± itV 0 / = P ( -k ) , @QFQWA
with k = 2 k 2 /(2m)F xote tht in equtions @QFQVA nd @QFQWAD V 0 enodes the vrine of the potentilF xote lso tht the slow dey in time of @QFQVA is rooted in the disontinuity of the spekle onEsite distriution in ontrstD the ounterprt of @QFQVA for qussin potentil deys s exp(-αt 2 )F sn the lssil limitD the spetrl funtion thus mimis the onEsite distriution IHSF ith this result in hndD the lssil ho follows from eqution @QFQSAX

ν cl ( ) = ∞ 0 d k ν 0 ( k )P ( -k ), @QFRHA
where ν 0 is the freeEspe ho IHSD IITF he lssil limit for the spetrl funtion eqution @QFQWA is ompred with numeril simuE ltions in (gure QFQ for three di'erent onEsite distriutionsX qussin @(gure QFQAD redEdetuned @(gure QFQA nd lueEdetuned spekle @(gure QFQAF emrklyD the lssil limit desries reE sonly well the qussin seD whih is smoothD ut is ompletely indequte ner the disontinuity of the spekle distriutionF his lredy suggests tht perturtion theory round the lssil limit should e su0ient for the desription of the spetrl funtion when the distriution is smoothF sn ontrstD ompletely di'erent strting point is proly more suited ner the disontinuity of the spekle distriutionF @A flueEdetuned spekle pigure QFQ ! petrl funtion s funtion of energy in Ph rndom potentilD for three di'erent onEsite distriutionsF sn ll three ses η = 4F he lssil limit @QFQWAD shown s lk urveD is systemtilly ompred with numeril results shown s olored squresF he onEsite distriutions re A qussinD A redEdetuned spekle nd A lueEdetuned spekleF he numeril results re tken from IHSF yn the one hndD one oserves thn the spetrl funtion is resonly well desried y the lssil limit for the smooth qussin distriutionF yn the other hndD the ext spetrl funtion di'ers widely from its lssil limit ner the disontinuity of the spekle distriutionF

Smooth quantum corrections

o go eyond the lssil limitD eqution @QFQWA for the spetrl funtion nd eqution @QFRHA for the hoD it is nturl to look for expnsion in powers of D where the leding order would e the lssil limit nd the next orders would ring orretions to itF his expnsion n e worked out in two wysF yne possiility is to work diretly in the energy dominD nd use ignerEeyl formlism IIUF he other possiility is to strt from the representtion of the spetrl funtion in terms of the evolution opertorD eqution @QFQRAD nd proeed with ommuttor expnsion of the ltterD followed y umulnt expnsionF yf ourseD either wy leds to the sme resultF sn this thesisD we touh upon the min steps of the ommuttorEumulnt expnsionD nd refer to our pulished pperD reprodued in setion QFSD for more detilsF he reder interested in the (rst wyD sed on the ignerEeyl formlismD n (nd the lultion in IHSF xote tht while the results presented elow for the spetrl funtion extend diretly to the hoD through eqution @QFQSAD we fous in this susetion on the spetrl funtion onlyF es the ho diretly follows from the spetrl funtion in eqution @QFQSAD we felt tht spei( disussion on the ho would e redundntF his feeling is strengthened y the di0ulty of getting relile numeril results t lrge energy for the hoD preventing quntittive omprison with ext numeril simultionsF foth the expnsion in powers of @present susetionA nd the method sed on sttionry phse pproximtion @susetion QFQFQA pply to potentils with ritrry orreltion funtionsF por the ske of onretenessD we onsider spekle potentils with qussin orreltion funtionX

V (r)V (r ) = V 2 0 exp - |r -r | 2 2σ 2 . @QFRIA
Commutator-cumulant expansion of the evolution operator e here present n expnsion in powers of of the evolution opertor expettion vlue k| e -i(p2 /2m+V )t/ |k . @QFRPA sn the limit → 0D p 2 nd V ommute nd we hve

k| e -i(p 2 /2m+V )t/ |k ---→ →0 e -i 2 k 2 2m
t k| e -iV t/ |k . @QFRQA gorretions to this limit involve ommuttors of p 2 nd V F his is enoded in ssenhus forE mul IIVX

e t(X+Y ) = e tX e tY e - t2 2 [X,Y ] e t3 6 (2[Y,[X,Y ]]+[X,[X,Y ]]) + O( t4 ). @QFRRA
o revel the smll prmeter of the ommuttor expnsion of @QFRPAD we introdue t = tV 0 / D the time expressed in the nturl units of the prolemF ith this de(nitionD one nturl reovers the lssil limit @QFRQA in the ommuttor expnsion of @QFRPA when → 0F he following lultion is performed with the nturl time tD to leding order in F he expnsion of @QFRPA involves few ommuttorsF heir lultion is tedious ut strightforwrdD let us skip it nd diretly move on to the disorder vergeF everging over disorder n e done y men of the following umulnt expnsionX

exp (X) = exp   ∞ n=1 κ n (X) n!   , @QFRSA
where κ n (X) denotes the n th umulnt of XF sn the present ontextD X is sum of rndom vrilesF gumulnts of sum of rndom vriles n e onveniently expressed with joint umulnts IIWD whih re silly generliztion of the notion of umulnt to severl vriles @K 2 (X) eomes K joint (X, X)D llowing to onsider K joint (X, Y ) s wellAF xote tht even t lowest order in D n in(nite numer of umulnts must e keptF he present expnsion genertes welth of terms whih we do not wnt to show hereD we prefer to restrit ourselves nd only give the )vor of the lultionF he si ide is to write the potentil s 2

V = ± E 2 1 + E 2 2 , @QFRTA
with the + @respF -A sign for lueE@respF redEAdetuned speklesF ine E 1 nd E 2 re independent qussin vriles with zero men nd equl vrineD the deomposition @QFRTA is onvenient when working with umulntsF he rest of the lultion is entered round theorem due to veonov nd hiryev IPHD whih llows to disentngle the joint umulnts ontining powers of the E 1 or E 2 into joint umulnts ontining only E 1 or E 2 t (rst orderD t the prie of some omintorisF sndeedD fter writing the expnsion in terms of joint umulnts ontining only E 1 or E 2 t (rst orderD the lultion is not di0ult to rry out in pourier speF iventullyD it oils down to the summtion of geometri nd logrithmi seriesF e (nd k| e

-i p 2 2m +V t |k = e -i k t/V 0 1 ± i t × 1 + di t3 12η(1 ± i t) + t4 12η(1 ± i t) k V 0 + . . . , @QFRUA
where d is the spe dimensionF e rell tht t = tV 0 / F sn the limit k = 0D the quntum orretion is smll when t = t/ √ η = tV 0 / √ η 1F st de(nes the nturl frequeny unit in this ontextX

ω 0 = V 0 √ η = V 0 mσ 2 , @QFRVA
whih is the typil osilltion frequeny in potentil well of height V 0 nd size σF he ondition of vlidity of eqution @QFRUA simply reds ω 0 t 1. he pourier integrl over time is then well pproximted ifX

| | ω 0 = V 0 √ η . @QFRWA
his disussion is put on more solid ground in the next susetion through omprison with ext numeril simultionsF por the ske of lrityD the limit @QFRWA will e referred to s high energy limit in the followingF he opposite low energy limit orresponding to

| | ω 0 = V 0 / √ ηF

Results and validity

he present susetion is lrgely sed on IHS where the sme results for the spetrl funtion hve een otined using ignerEeyl formlismF he (rstEorder orretion to the lssil limitD eqution @QFRUAD is ompred with ext numeril simultions in the insets of (gure QFRF he results on(rm the disussion of susetion QFQFPD nmely the lssil limit plus the smll orretion desrie very well the highEenergy limitD wheres s pprohes ω 0 = V 0 / √ ηD the spetrl funtion strts to devite strongly from the lssil pproximtion @min plots of (gure QFRA nd the expnsion loses its ury @insets of (gure QFRAF sn ftD for spekle potentils the lowEenergy region is non trivilF hile the lssil spetrl funtionD eqution @QFQWAD hs disontinuity tk , the ext spetrl funtion is widely di'erentX for lueEdetuned spekleD it rigorously vnishes elow = 0 ndD for k = 0D it rpidly inreses etween = 0 nd ∼ V 0 / √ ηF o get some understnding on the low energy prtD it is useful to repet the lultion for qussin potentilD iFeF to trde the spekle onEsite distriutionD eqution @PFQHAD to

P Gaussian V (r) = 1 V 0 √ 2π exp - V (r) 2 2V 2 0 . @QFSHA
he lultion is muh simpler in this seD s the qussin nture of the potentil helps to rry out the umulnt expnsionF he result is k| e

-i p 2 2m +V t |k = e -i k t/V 0 e -t2 × 1 + di t3 12η + t4 12η k V 0 + . . . . @QFSIA -4 -3 -2 -1 0 1 0.0 0.2 0.4 0.6 0.8 1.0 -4 -3 -2 -1 0.0 0.1 E/|V | A 0 (E)|V | ∆A 0 (E) V 2 /E ζ V /E ζ = -2 V /E ζ = -4 V /E ζ = -8 V /E ζ = -16 V /E ζ = -∞ @A edEdetuned spekleF 0 1 2 3 4 0 1 2 3 2 3 4 5 6 -0.01 -0.005 0 0.005 E/V A 0 (E)V ∆A 0 (E)V 2 /E ζ V /E ζ = 2 V /E ζ = 4 V /E ζ = 8 V /E ζ = 16 V /E ζ = ∞ @A flueEdetuned spekleF
pigure QFR ! pigures from IHS showing the spetrl funtion A k=0 ( ) s funtion of energy in two dimensionsD @A redE nd @A lueEdetunedD spekle potentil with qussin orreltion funtionF xote the following orrespondenes with our nottionsD

E = D |V /E ζ | = η nd |V | = V 0 F snsetX fous on the orretion to the lssil limitD ∆A k ( ) = A k ( ) -A cl k ( )D
he predition @QFRUA is shown s the solid ornge urveF he result exhiits remrkle di'erene with its ounterprt for spekle potentilsD eqution @QFRUAX the lssil prt deys exponentilly with timeD wheres it deys only s 1/t for spekle potentilsF his di'erene hs importnt mnifesttions in the spetrl funtionD dedued from equtions @QFRUA nd @QFSIA y pourier trnsform eqution @QFQRAF por spekle potentilsD the quntum orretions generte unphysil singulrities @delt funtions nd derivtivesAF sn ontrstD for qussin potenE tilsD they remin smll nd the lssil limit lredy o'ers n deent pproximtion of the spetrl funtionD s visile in (gure QFSF es shown in (gures QFS nd QFSD ounting for the quntum orretions only re(nes the desription of the spetrl funtionF sn onlusionD the expnsion in powers of is perfetly suited to desrie the high energy limit for spekle potentils nd fils t low energiesD with in prtiulr unphysil singulrities @delt funtions nd derivtivesAF hese di0ulties re sent for qussin potentilsD nd re rooted in the disontinuity of the potentil onEsite distriutionF

Treatment of low energies

Harmonic-oscillator approximation e now would like to desrie the quntum orretions to the lssil limit in the lowEenergy region ∼ V 0 / √ η for spekle potentilsF es we hve seen in susetion QFQFPD n expnsion in powers of is not suited to tkle this regimeD the quntum orretions to the lssil expressions eing singulrF e nonEperturtive method is lled forF e propose n pproh inspired of qutwiller theory IPID for whih we here sketh the essentil idesF he strting point is the n lek form of the propgtorD vlid in the semilssil regime IPPD IPQX r| e -iHt/ |r α (. . . ) e iS α (r,r ,t)/ , @QFSPA where the sum runs over lssil trjetories leding from r to r during the time spn tF S α (r, r , t) is the lssil tion ssoited with the lssil trjetory αF e do not give the expression of the preftors hereF heir ext vlue is not importnt for the present preliminry disussionD where we remin t qulittive level nd wnt only to disuss whih lssil trjetories give

-3 -2 -1 0 1 2 3 0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 -4 -2 0 2 4 -0.2 -0.1 0.0 0.1 0.2 E/V V A 0 (E) V 2 ∆A k (E)/E ζ (E -T k )/V T k = 0 T k = V a b c V /E ζ = 2 V /E ζ = 4 V /E ζ = ∞
pigure QFS ! pigures from IHS showing the spetrl funtion A k ( ) s funtion of energy in Ph qussin potentil with qussin orreltion funtionF xote the following orrespondenes with our nottionsD

E = D V = V 0 D E ζ = E σ D T k = k F pigure @A
shows the spetrl funtionD while (gures @A nd @A fous on the orretion to the lssil limitD ∆A k ( ) = A k ( ) -A cl k ( )F he predition @QFSIA is shown s the solid green urveF the most importnt ontriutionsF he sum over ll lssil trjetories is very omplited oneD oviously di'erent for eh disorder reliztionD mking the verging a priori rther omplexF yne n nevertheless onvine oneself tht the sttistil properties of the potentil will hve strong in)ueneF por instneD for lueEdetuned spekle t low energyD there will e essentilly short trjetories trpped in the potentil wellsD so tht the peulir distriution of energy minim will ply ruil roleF he spetrl funtion is relted to the propgtor @QFSPA through the reltion

A k ( ) = dt 2π d d ∆r L d e i t/ -ik•∆r r| e -iHt/ |r , @QFSQA
where ∆r = rr F he integrl over time n e performed y sttionry phse pproximtionD whih restrits the ontriuting lssil trjetories to those with energy IPQF et the low energies < ω 0 we re trgetingD suh lssil trjetories lie in potentil wells @respF inverted potentil wellsA for lueEdetuned @respF redEdetunedA speklesF e propose to pproximte these @invertedA wells y independent @invertedA hrmoni osilltors3 F nder this pproximtionD the sttionry phse pproximtion eomes ext so one n simply reple the propgtor @QFSPA y the known propgtor of n @invertedA hrmoni osilltor IPTF Blue-detuned speckle ithin the hrmoni osilltor pproximtion desried oveD eqution @QFSQA simply redues to sum of spetrl funtions of in(nitely mny rndom hrmoni osilltors i whose minim V i re entered t x i F por the se of IhD lueEdetuned spekle potentilD this reds

A k ( ) 1 L x i ∞ n=0 |ψ i n (k)| 2 δ -i n , @QFSRA
where ψ i n (k) is the eigenfuntion of the Ih i th osilltor in k speX

ψ i n (k) = (2π) 1/2 √ 2 n n! πmω i 1/4 e -k 2 2mω i +ikx i H n   k 2 mω i   , @QFSSA normlized ording to dk/(2π)|ψ i n (k)| 2 = 1 nd with ssoited eigenenergy i n = V i + ω i (n + 1/2)F
e now mke use of the ssumption tht the hrmoni wells re sttistilly independentD whih llows us to tke the sum over x i out of the disorder vergeF he ltter is then over the rndom frequeny ω i nd the potentil minimum V i of single osilltor onlyF o lighten the nottionsD we drop the now unneessry i indexF fy introduing the joint distriution P (V, ω) of these two rndom vrilesD we rewrite eqution @QFSRA s

A k ( ) = ρ ∞ n=0 dV dωP (V, ω)|ψ n (k)| 2 δ( -n ), @QFSTA
where ρ is the verge density of potentil minimF ρ nd the distriution P (V, ω) hve een derived in setion PFQ for spekle potentilsF xote tht eqution @QFSTA is only justi(ed if typil hrmoni well ommodtes mny sttesF sn the semilssil regime @QFQUAD this is indeed the seX s the typil frequeny of the osilltor will e ω 0 nd the typil depth of potentil well V 0 , the numer of sttes ontined in the well is

∼ V 0 / ω 0 = √ η 1F
Red-detuned speckle por the redEdetuned spekle potentilD we proeed similrlyF rere howeverD we do not re muh out the potentil wells whih hve energy minim typilly of the order of -V 0 D tht isD in rnge where the lssil pproximtion works well @see (gure QFRAF sndeedD we re interested here in the energy rnge round = 0, so tht it is the potentil maxima whih re relevntF e thus mke use of n inverted hrmoniEosilltor pproximtionF sn this se howeverD the representtion @QFSRA of the spetrl funtion is not onvenient due to the ontinuous nture of the spetrum of the inverted hrmoni osilltor4 F e therefore prefer to work in the time dominD using formultion @QFQRA for the spetrl funtion @written for Ih spekleAX

A k ( ) ρ ∞ -∞ dt 2π e i t/ k| e -iH IHO t/ |k , @QFSUA
where

H IHO = p 2 /(2m) -V i -mω 2 (x -x i ) 2 /2F he Ih inverted hrmoniEosilltor propgtor in k spe is given y IPT k| e -iH IHO t/ |k = 2πe iV t/ i 2πmω sinh(ωt) exp - i k 2 mω coth(ωt) - 1 sinh(ωt)
. @QFSVA he disorder verge is then rried out s in eqution @QFSTAD y verging over V nd ω with the help of the joint distriution P (V, ω)F fy returning the potentil V (x) → -V (x), we re k to the lueEdetuned potentil so the joint distriution of the mxim P red (V, ω) is nothing ut the joint distriution P blue (V, ω) for the minim of lueEdetuned spekleF his symmetry lso implies tht the density of mxim for redEdetuned spekle is equl to the density of minim ρ for lueEdetuned spekleF

QS

xote thtD in prinipleD one ould use the propgtor of the hrmoni osilltor in the time domin for lulting the spetrl funtion of the lueEdetuned spekle potentil s wellF his pproh turns however indequte due to the presene of n in(nite numer of singulrities ! when ωt is n integer multiple of π ! rising in the time integrl over the propgtorF sn prinipleD the present method pplies to ny dimensionD with the ounterprt of the joint distriution P (V, ω) eoming priori inresingly di0ult to lulte s the dimension inresesF e hve suessfully pplied this method to the Ih nd Ph ses in pulished pperD reprodued in setion QFSF e here detil only the Ih seF he Ph se is quite similrD we merely omment on it in susetion QFQFUF

Spectral function and density of states in one dimension: results

lugging the joint distriution of minim nd urvture round minim @PFRSAD s well s the density of minim @PFRWAD in the formuls of susetion QFQFQD it is not di0ult to otin the speE trl funtion nd the hoF he resulting preditions re however it umersome nd not very enlighteningD we thus hoose not to reprodue them hereF he interested reder n (nd them in our pulished pperD reprodued in setion QFSF st is more ppeling to ompre them to numerE il simultionsF his is done in (gure QFT where we show systemtilly the @invertedA hrmoni osilltor predition s red urve nd the results from numeril simultions s lue pointsF por omprisonD we lso show the lssil limits equtions @QFQWA nd @QFRHA s solid green urvesF es visile in (gures QFT nd QFTD in the lueEdetuned seD the hrmoni osilltor pproximE tion desries very well the spetrl funtion @(gure QFTA nd the ho @(gure QFTA t low energiesF e n thus interpret the lowEenergy fetures of the spetrl funtion nd the ho from this perE spetiveF he pek of the spetrl funtion in the lueEdetuned se origintes from the ground stte of the hrmoni osilltorsF sts reltively nrrow hrter omes from the ω distriution in eqution @PFRRA rther well peked round ω = ω 0 F he ensuing ump uilds up upon dding exE ited sttes 5 F he nrrow pek in turn results in ump in the ho t low energyD see (gure QFTF sndeedD upon inresing k the pek of the spetrl funtion eomes less nd less pronouned ut remins t the sme energyD whih results in smooth ump fter summtion over kF et lrger energies > V 0 @not shown for the ho in (gure QFTAD the hrmoniEosilltor pproximtion reks down nd the purely lssil limit tkes overD eventully leding to

A k=0 ( ) = exp -/V 0 /V 0 nd ν( ) ν 0 ( ) = m/(2 )/(π ) for → ∞ IITF
sn the redEdetuned spetrl funtion seD shown in (gure QFTD the inverted hrmoniEosilltor predition is in good greement with the numeril results for energies ner HF et smller energies @ -V 0 AD the desription of the spekle potentil in terms of inverse hrmoni osilltors eomes poorD while the lssil limit provides n exellent pproximtionF et vrine with the lueEdetuned ho @(gure QFTAD the redEdetuned ho @(gure QFTdA is well desried y the lssil limit t ll energiesF his n e understood qulittively from the qutzwiller tre formul IPIDIPUD whih expresses the ho s the sum of the lssil ontriutionD eqution @QFRHAD nd of osilltory ontriutions oming from periodi oritsF eround = 0, the periodi orits in redEdetuned spekle re long ones with hrteristi properties @tionD periodFFFA whih strongly depend on the disorder reliztionD so tht ll osilltory ontriutions nel outF his is indeed in strk ontrst with the lueEdetuned spekle where periodi orits round = 0 re short orits trpped in the deep potentil minim nd olletively ontriute to umps in the hoF he sene of invertedEhrmoni osilltor predition for the redEdetuned ho omes from n ultrviolet divergeneF his divergene lredy ppers in the ho of the inverted hrmoni osilltorD for whih it origintes of the ontinuous nture of the spetrumF st thus ppers tht for the ho of redEdetuned speklesD the desription of singulr quntum orretions requires to go pigure QFT ! petrl funtion A k=0 ( ) nd density of sttes ν( ) s funtion of energy in Ih lue@redEAEdetuned spekle potentil with qussin orreltion funtionD for η = 128F he @invertedA hrmoniEosilltor pproximtion of susetion QFQFQ is systemtilly shown s solid red urveD nd the lssil limitD equtions @QFQWA nd @QFRHA for the spetrl funtion nd the density of sttes respetivelyD s solid green urveF flue dots re the result of ext numeril simultionsF he prmeters used in the numeril simultions n e found in our pulished pperD reprodued in setion QFSF eyond the inverted hrmoniEosilltor pproximtionF sn ny seD these orretions re extremely smllD s visile in (gure QFTdF xote tht very reent experimentl mesurements of the spetrl funtion IIH orroorte these results t qulittive levelF sn prtiulrD the predited dissymmetry etween redE nd lueEdetuned speklesD s well s the peked struture of the lueEdetuned spetrl funtionD were unmiguously oservedF e quntittive omprison is impossileD s the experiment ws performed with Qh speklesF 3.3.5 Validity of the harmonic-oscillator approximation e simple rgument n e used to estimte the energy rnge where the hrmoniEosilltor pproximtion is vlidF eording to the viril theoremD equiprtition etween kineti nd potentil energy imposes tht n = mω 2 x 2 n for the energy of n eigenstteF sn order for the spekle potentil to e orretly desried y n hrmoniEosilltor pproximtionD ll sttes suh tht n = in eqution @QFSRA should hve n extension

x 2 n muh smller thn the orreltion length σD whih imposes n upper limit for the energyX mω 2 σ 2 @in se this ondition is not ful(lledD nhrmoni terms would lso ome into plyAF es seen in susetion PFQFID the most likely vlue of ω is ω 0 D so the ondition eomes V 0 . @QFSWA yn the other hndD the lssil pproximtion is expeted to desrie well the spetrl funtion down to energies of order V 0 / √ η IHSF hereforeD in the region V 0 / √ η V 0 oth the hrmoniE osilltor nd the lssil pproximtion provide good desription of the spetrl funtion nd of the hoF iqution @QFSWA provides restrition on the highEenergy til of the spetrl funtion A k ( ) for the ltter to e orretly desried y our hrmoniEosilltor pproximtionF e similr rgument imposes n dditionl restrition for the momentum kF sndeedD equiprtition etween kineti nd potentil energy for the hrmoni osilltor lso implies

2 k 2 n 2m = 1 2 mω 2 x 2 n , @QFTHA where x 2
n should e gin muh smller thn σ for the hrmoniEosilltor pproximtion to holdF ith ω ∼ ω 0 D ondition @QFTHA reds

2 k 2 n m V 0 . @QFTIA
he ontriution of eh eigenstte to the sum in eqution @QFSRA eing proportionl to |ψ n (k)| 2 D the sum is dominted y eigensttes hving k 2 n of the order of kD suh tht riterion @QFTIA leds to k V 0 . @QFTPA sn ny seD the hrmoni osilltor pproximtion is good one in the region , k ∼ ω 0 where the quntum orretions re importntD while the purely lssil result @QFQWA tkes over t higher energy , k ∼ V 0 .

Validity of the inverted harmonic-oscillator approximation he rekdown of the inverted hrmoniEosilltor pproximtion t energies

-V 0 n e understood from resoning on the lssil tion tht ppers in eqution @QFSPAF sndeedD for the sttionry phse pproximtion to e vlidD the time spn t ssoited with lssil trjetory should e suh tht the lssil tion V 0 t/ is lrgeD imposing t /V 0 F inergies orresponding to suh long times ful(ll

| | V 0 . @QFTQA
xote tht this ondition is fully similr to tht for lueEdetuned speklesD eqution @QFSWAD though it is here dedued from slightly di'erent rgumentF henD the motion of lssil tom of energy = kmωx 2 /2 desries well the dynmis in redEdetuned spekle s long s the exursion x 2 remins muh smller thn σD nmely s long

s k + | | mωσ 2 /2F ine ω ∼ ω 0 nd | | V 0 D this leds to k V 0 , @QFTRA
whih is the sme vlidity ondition s for lueEdetuned speklesF 3.3.7 Higher dimensions rving settled the Ih situtionD we now rie)y omment on the se of higher dimensionsF he shrp ehvior of the lueEdetuned spetrl funtion t low energy oserved in one dimension is rooted in the presene of most minim t very low V V 0 @see susetion PFQFIAF sn this respetD one expets shrper spetrl funtion in two dimensionsD reminisent of the minim sitting exactly t V = 0 @see susetion PFQFQAF his is indeed wht is numerilly oserved in (gure QFU where the spetrl funtion for Ih nd Ph lueEdetuned spekle potentils re shown respetively s lue nd green squresF e note tht the Qh spetrl funtionD shown s red squresD is even shrperF yn the other hndD t high energies the IhD Ph nd Qh spetrl funtions ll ollpse on the lssil limit eqution @QFQWA shown s solid lk urveF

0.0 0.2 0.4 0.6 0.8 1.0 /V 0 0.0 0.5 1.0 1.5 2.0 A k = 0 ( )V 0 Blue detuned η = 128 1D 2D 3D Classical limit
pigure QFU ! petrl funtion A k=0 ( ) s funtion of energy in lueEdetuned spekle potentils with qussin orreltion funtionD for η = 128F he result for the oneEdimensionl se is shown s lue squres nd is ompred with the results otined in two nd three dimensionsD respetively shown s green nd red squresF he lssil limitD eqution @QFQWAD is the sme in ll dimensions nd is shown s solid lk urveF he prmeters used in the Ih nd Ph numeril simultions n e found in our pulished pperD reprodued in setion QFSF sn three dimensionsD we hve used disrete grid of size L 3 = (6πσ) 3 with 120 3 grid pointsD nd verged the results over UPH disorder reliztionsF eside from the minim exactly t V = 0D the Ph se is quite similr to the Ih oneF esults similr to those presented in susetion QFQFR n e found in our pulished pperD reprodued in setion QFSF he Qh se onstitutes tehnil hllengeD euse the points of vnishing potentil responsile for the shrp Ph lueEdetuned spetrl funtion turn to urvesF he pproximtion of isolted hrmoni osilltors used throughout this setion thus reks downD one hs to ope with vlleys insted of isolted minimF e leve this hllenging tsk for lter workF

Conclusion

sn this hpterD we hve disussed mny properties of the spetrl funtionD fundmentl uilding lok in the desription of mtter wve pketsF e hve introdued this onept y strting from the wekEdisorder limitD nd we hve then disussed its importne in the hrteriztion of the energy distriutionD nd thus the dynmis of wve pketsF hiretly onneted with the spetrl funtionD the density of sttes me longF elthough we hve put less emphsis on itD it is lso n importnt quntityD prtiulrly in the disussion of phses of dirty interting osons THD UID IHRF sn the ore of this hpterD we hve introdued two semilssil methods iming to desrie the spetrl funtion nd the density of sttes in spekle potentilsF e hve seen how the orretions to the lssil limit n e otined through n expnsion in powers of of the spetrl funtion or the density of sttes in the lrge energy setorF his expnsion is however not su0ient t low energiesD due to the disontinuity of the onEsite potentil distriutionF sn order to overome this di0ultyD we hve developed novel nlytil method sed on semilssil desription of the dynmis omined with the sttistil properties of potentil extremF epplying this pproh to Ih lueE nd redEdetuned speklesD we hve rried out the lultion of the spetrl funtion nd the hoF he Ph se is very similr to the Ih one with some dditionl tehnil di0ultiesD we hve preferred to left it side here nd refer the interested reder to our pulished pperD reprodued in setion QFSF sn oth the Ih nd Ph sesD y onneting the two methodsD rther omplete desription of the whole energy spetrum is possileF yur semilssil desription dditionlly provides simple interprettion of intriguing fetures of the spetrl funtion nd hoF sn prtiulrD for lueEdetuned potentils we hve shown tht the lowEenergy pek of spetrl funtions is essentilly ssoited with the ground stte of n tom in potentil well of the spekleD while the seondry ump is ssoited with exited sttesF e hve lso emphsized tht in spite of their simple symmetryD redE nd lueEdetuned spekles exhiit remrkly di'erent fetures in the semilssil regimeD oming from the fundmentl di'erent nture of the lssil trjetories involved ner zero energyX for lueEdetuned speklesD these lssil trjetories lie in deep potentil wellsD while for redEdetuned spekles they lie in the viinity of the top of inverted wellsF es logil ontinution of this workD it would e of gret interest to ddress the se of threeE dimensionl spekle potentilsD involved in importnt questions relted to enderson loliztion QVD RHD RRD IHWD IISD IPVF his tsk ppers hllenging thoughD due to the existene of urves long whih the potentil is vnishingD mking the pplition of hrmoni osilltor pproximtion less oviousF 3.5 We present an analytical method for calculating the spectral function and the density of states in speckle potentials, valid in the semiclassical regime. Our approach relies on stationary phase approximations, allowing us to describe the singular quantum corrections at low energies. We apply it to the calculation of the spectral function and the density of states in one-and two-dimensional speckle potentials. By connecting our results with those of previous work valid in the high-energy sector, we end up with a consistent description of the whole energy spectrum, in good agreement with numerical simulations. DOI: 10.1103/PhysRevA.94.022114

I. INTRODUCTION

Anderson localization, the absence of wave diffusion due to destructive interference between partial waves multiply scattered by a disordered potential [1], has been observed in a number of experiments involving atomic matter waves quasiperiodically kicked by laser pulses [2,3] or subjected to one-dimensional (1D) [4] and three-dimensional (3D) [5,6] quenched speckle potentials, as well as ultrasound waves in 3D disordered dielectric media [7]. In cold-atom setups, the control of atom-atom interactions (through, e.g., Feshbach resonances) together with a weak coupling to the environment constitute precious assets for the observation of interference effects in disorder. Furthermore, atom-optics experiments offer the possibility to directly probe localization phenomena inside the atomic system, as well as to follow their evolution in the course of time [8,9].

If atoms are injected into a disordered potential with an initial momentum k, they no longer have a well defined energy but rather an energy distribution called the spectral function, denoted by A k ( ). The spectral function thus defines a quasiparticle, and generally speaking can provide important physical insights to the complex problem of disorder scattering even without the knowledge of the system's eigenstates [10]. Even more, it turns out that to achieve a quantitative understanding of cold-atom experiments in speckle potentials and in particular to properly characterize Anderson localization, a good knowledge of the spectral function is crucial. Indeed, when disorder is strong enough the spectral function is broad, which can have important consequences for the global motion of an atomic cloud. For instance, a cloud of atoms that are individually diffusive may exhibit a global subdiffusive behavior as a result of the superposition of the various energydependent atomic diffusion coefficients, thus mimicking the onset of localization [11,12]. Furthermore, even if the cloud contains localized atoms, usually a finite part of it remains diffusive and a precise characterization of the spectral function is then required in order to pinpoint the location of the mobility edge [6,13]. Related to the spectral function, the density of states (DOS) in strong speckle potentials is also poorly understood. This question is however essential as the DOS plays a central role in atomic physics, in particular in the discussion of phases of interacting bosons [14][15][16].

Despite its importance, the calculation of the spectral function of speckle potentials in the strong disorder regime has been little addressed, the main difficulty stemming from the inapplicability of weak-disorder approximations in this regime. Recently however, a systematic semiclassical expansion of the spectral function around the classical solution has been proposed [17]. Although successful in the large-energy sector, the approach of [17] fails at capturing the singular quantum corrections at low energies. As far as the DOS is concerned, important progress has been recently accomplished by Falco et al. [18], who used a classical approximation for describing high energies in speckle potentials. Again however, this approach remains inaccurate to capture the low-energy sector. As a matter of fact, the difficulty of treating low energies in speckle potentials lies in the singular nature of quantum corrections in this region of the spectrum. Such singular corrections are absent for Gaussian random potentials [17] frequently used in condensed-matter physics [19]. To our knowledge, they have not been described yet.

In this paper, we calculate the spectral function and the density of states in one-(1D) and two-dimensional (2D) speckle potentials, making use of a semiclassical approach based on stationary phase approximations, thereby allowing for a nonanalytic perturbation expansion in . Our theoretical predictions are in good agreement with exact numerical simulations in the low-energy sector where quantum corrections are singular. By connecting our results with those of [17], we eventually end up with a consistent description of the whole energy spectrum. Section II is devoted to the definition of the relevant quantities and to a discussion of the results previously obtained in [17]. Our semiclassical approach is also introduced and discussed. In Sec. III, we derive important statistical properties of 1D speckles needed to implement our semiclassical theory. Results for the 1D spectral function and DOS are presented in Sec. IV. The approach is then extended to the 2D case in Secs. V and VI. In Sec. VII, we finally summarize our findings and discuss some open questions.

II. DEFINITIONS AND METHODS

A. Framework

We consider a cloud of noninteracting atoms of mass m, subjected to a random potential V (r). Its dynamics is governed by the Hamiltonian

H = p 2 2m + V (r), (1)
where p = -i ∇. The coordinate vector r ∈ [0,L] d lies in a d-dimensional cubic volume of linear size L that we will eventually make tend to infinity. In the following, averaging over the random potential will be indicated by an overline: (. . . ). In practice, speckle potentials are obtained by transmission or reflection of a laser through a rough plate. The resulting potential V (r) felt by atoms subjected to this light is proportional to the square of a complex Gaussian field [20], with a sign that depends on the laser detuning with respect to the considered two-level transition. This potential has the following on-site distribution:

P [V (r)] = 1 V 0 θ [±V (r)]exp ∓ V (r) V 0 , ( 2
)
where θ (. . . ) is the Heaviside θ function. The disorder strength V 0 > 0 enters both the average V (r) = ±V 0 and the variance

V (r) 2 -V (r) 2 = V 2 0 .
In Eq. ( 2), the upper sign refers to a blue-detuned speckle potential, bounded by zero from below, and the lower sign to a red-detuned speckle potential, bounded by zero from above. Another quantity that we will frequently encounter in the following is the two-point correlation function V (r)V (r ) -V (r) 2 . For the isotropic speckles considered in this paper, the two-point correlation function depends only on |rr |. It decays over a typical distance σ , referred to as the correlation length [20]. σ defines an important characteristic energy scale, the so-called correlation energy [21]:

E σ = 2 mσ 2 . ( 3
)
The two-point correlation function can take various forms depending on the experimental setup [20]. The approach developed in this paper in principle applies to any shape of the correlation function, but the results for the spectral function and the DOS turn out to very weakly depend on it, provided the proper value of σ is chosen. Consequently, for definiteness we will only consider the Gaussian case in the following:

V (r)V (r ) -V (r) 2 = V 2 0 exp - |r -r | 2 2σ 2 . ( 4
)
As an example, we show in Fig. 1 FIG. 1. Numerical realizations of a red-(left) and a blue-detuned (right) 1D speckle potential. The on-site distribution is given by Eq. ( 2) and the two-point correlation function by Eq. ( 4). The procedure used to numerically generate the speckle is explained in the main text. potential. To generate these realizations, we use a numerical procedure that precisely describes the experimental scenario: we first generate a spatially uncorrelated complex random Gaussian field in Fourier space, simulating the transmission through the rough plate. This field is then multiplied by a proper cutoff function-that physically describes the shape of the plate-which we take Gaussian to reproduce the two-point correlation function (4). Finally, (the opposite of) the modulus square of the field in coordinate space gives the blue-(red-)detuned speckle potential visible in the observation plane [20].

B. Definitions, semiclassical regime

The figure of merit of this paper is the spectral function, defined as

A k ( ) = k| δ( -H ) |k . ( 5
)
Physically, the spectral function is the probability density for a plane wave |k to have energy in the potential V (r). At vanishing disorder, the spectral function is a Dirac δ function centered at energy 2 k 2 /2m. Upon increasing the disorder, this peak acquires a finite width and, at strong disorder, starts to develop intriguing structures that we wish to explore.

Introducing the Fourier representation of the Dirac δ function in Eq. ( 5), it follows that

A k ( ) = ∞ -∞ dt 2π e i t/ k| e -iH t/ |k , (6)
which establishes the connection with the evolution operator e -iH t/ . The spectral function is related to the DOS per unit volume ν( ) through the relation

ν( ) = 1 L d Tr δ( -H ) = d d k (2π ) d A k ( ). (7)
There are several energy scales in the problem: E, E σ , V 0 , and only their ratio matter. Of special importance is the parameter

η = V 0 E σ = mσ 2 V 0 2 . ( 8
)
In this paper, we focus on the so-called semiclassical regime characterized by the condition [17,18]

η 1. (9)
This inequality has a simple interpretation: √ η is the ratio of the disorder correlation length σ to the de Broglie wavelength of a particle with energy V 0 , so that, in the semiclassical regime, the quantum particle can resolve all the potential fluctuations. Alternatively, a quantum particle with energy V 0 encountering a potential barrier of height V 0 and thickness σ will have a vanishingly small probability exp(-√ η) to tunnel through it, making the dynamics almost classical.

In the deep semiclassical limit η → ∞, the noncommutation between position and momentum can be neglected, so that k| e -iH t/ |k ≈ e -i k 2 t/2m e -iV (r)t/ and Eq. ( 6) yields

A cl k ( ) = ∞ -∞ dt 2π e i( -k )t/ 1 ± itV 0 / = P ( -k ), ( 10
) 022114-2
where k = 2 k 2 /(2m) and P ( ) is the on-site potential distribution [Eq. ( 2)]. In the classical limit, the spectral function thus mimics the on-site distribution (2) [17]. With this result in hand, the classical DOS then follows from Eq. ( 7):

ν cl ( ) = ∞ 0 d k ν 0 ( k )P ( -k ), (11)
where ν 0 is the free-space DOS [17,18].

C. Smooth quantum corrections

For both the spectral function and the density of states, it is possible to calculate the smooth quantum corrections to the classical limits ( 10) and ( 11) from an analytic expansion in . The calculation of the first quantum correction has been recently carried out in [17] in the energy domain from Wigner-Weyl formalism [22]. The calculation is also possible in the time domain from an expansion of the evolution operator, as we show in Appendix A. In any dimension d, either of the two approaches leads to

A k ( ) = ∞ -∞ dt 2π e i( -k )t/ 1 ± itV 0 / × 1 + dit 3 V 2 0 E σ / 3 12(1 ± itV 0 / ) + t 4 V 2 0 E σ / 4 12(1 ± itV 0 / ) k , (12)
with again the + (-) sign for the blue-(red-)detuned speckle.

As was noticed in [17], Eq. ( 12) is correct only at large energies. For k = 0, this can be readily seen from the observation that the first quantum correction term should remain small for the perturbation theory to be valid. This term is of the order of t 2 V 0 E σ / 2 ∼ (t/ ) 2 V 2 0 /η. It is useful to define the natural frequency unit in this context:

ω 0 = V 0 mσ 2 = V 0 √ η , ( 13
)
which is the typical oscillation frequency in a potential well of height V 0 and size σ. The condition of validity of Eq. ( 12) simply reads ω 0 t 1. The Fourier integral over time is then well approximated if

ω 0 = V 0 √ η . ( 14
)
If one performs the Fourier integral in Eq. ( 12) at energies smaller than V 0 / √ η, the failure of the perturbation expansion manifests itself as unphysical singularities (δ functions and derivatives). One should then resort to another approach, which is the object of the next section. In fact, as noted in [17], for speckle potentials the low-energy region is nontrivial. While the classical spectral function, Eq. ( 10), has a discontinuity at k , the exact spectral function is widely different: for a blue-detuned speckle, it rigorously vanishes below = 0 and, for k = 0, it rapidly increases between = 0 and ∼ V 0 / √ η. These difficulties are absent for Gaussian potentials [17].

D. Treatment of low energies

Harmonic-oscillator approximation

We now would like to describe the quantum corrections to the classical limit in the low-energy region ∼ V 0 / √ η for speckle potentials. For this purpose, we propose an approach inspired of Gutwiller theory [START_REF] Gutzwiller | Chaos in Classical and Quantum Mechanics[END_REF], for which we here sketch the essential ideas. The starting point is the Van Vleck form of the propagator, valid in the semiclassical regime [24,25]:

r| e -iH t/ |r α (. . . )e iS α (r,r ,t)/ , ( 15
)
where the sum runs over classical trajectories leading from r to r during the time span t. S α (r,r ,t) is the classical action associated with the classical trajectory α. We do not give the expression of the prefactors here. Their exact value is not important for the present preliminary discussion, where we remain at a qualitative level and want only to discuss which classical trajectories give the most important contributions.

The sum over all classical trajectories is a very complicated one, obviously different for each disorder realization, making the averaging a priori rather complex. One can nevertheless envision that the statistical properties of the potential may have a strong influence. For a blue-detuned speckle at low energy, there will be essentially short trajectories trapped in the potential wells, so that it is easy to understand that the peculiar distribution of energy minima will play a crucial role. The spectral function is related to the propagator (15) through the relation

A k ( ) = dt 2π d d r L d e i t/ -ik• r r| e -iH t/ |r , ( 16
)
where r = rr . The integral over time can be performed by a stationary phase approximation, which restricts the contributing classical trajectories to those with energy [25].

At the low energies < ω 0 we are targeting, such classical trajectories lie in potential wells (respectively inverted potential wells) for blue-detuned (respectively red-detuned) speckles. We propose to approximate these wells by independent harmonic oscillators [26]. Under this approximation, the stationary phase approximation becomes exact so one can simply replace the propagator (15) by the known propagator of a harmonic oscillator (respectively inverted harmonic oscillator) [29].

Blue-detuned speckle

Within the harmonic oscillator approximation described above, Eq. ( 16) simply reduces to a sum of spectral functions of infinitely many random harmonic oscillators i whose minima V i are centered at r i . For the case of a 1D, blue-detuned speckle potential, this reads

A k ( ) 1 L x i ∞ n=0 ψ i n (k) 2 δ -i n , ( 17
) 022114-3
where ψ i n (k) is the eigenfunction of the 1D ith oscillator in k space:

ψ i n (k) = (2π ) 1/2 √ 2 n n! πmω i 1/4 e -k 2 /2mω i +ikx i H n ⎛ ⎝ k 2 mω i ⎞ ⎠ ,
normalized according to dk/(2π )|ψ i n (k)| 2 = 1 and with associated eigenenergy i n = V i + ω i (n + 1/2). We now make use of the assumption that the harmonic wells are statistically independent, which allows us to take the sum over x i out of the disorder average. The latter is then over the random frequency ω i and the potential minimum V i of a single oscillator only. By introducing the joint distribution P (V i ,ω i ) of these two random variables, we rewrite Eq. ( 17) as

A k ( ) = ρ ∞ n=0 dV i dω i P (V i ,ω i ) ψ i n (k) 2 δ -i n , ( 18
)
where ρ is the average density of potential minima. Calculation of the distribution P (V i ,ω i ) will be the object of Sec. III. Note that Eq. ( 18) is only justified if a typical harmonic well accommodates many states. In the semiclassical regime (9), this is indeed the case: as the typical frequency of the oscillator will be ω 0 and the typical depth of a potential well V 0 , the number of states contained in the well is ∼ V 0 / ω 0 = √ η 1.

Red-detuned speckle

For the red-detuned speckle potential, we proceed similarly. The potential wells which can accommodate a harmonic series of bound states have energy minima typically of the order of -V 0 , that is in a range where the classical approximation works well (see below). In contrast, we are interested in the energy range around E = 0, near the maximum allowed potential, and it is the potential maxima which are relevant. We thus make use of an inverted harmonic-oscillator approximation. In this case however, the representation (17) of the spectral function is not convenient due to the continuous nature of the spectrum of the inverted harmonic oscillator [30]. We therefore prefer to work in the time domain, using formulation (6) for the spectral function (written for a 1D speckle):

A k ( ) ρ ∞ -∞ dt 2π e i t/ k| e -iH IHO t/ |k , ( 19
)
where

H IHO = p 2 /(2m) -V i -mω 2 i (x -x i ) 2 /2.
The 1D inverted harmonic-oscillator propagator in k space is given by [29] k|e

-iH IHO t/ |k = 2πe iV i t/ i 2πmω i sh(ω i t) × exp - i k 2 mω i coth(ω i t) - 1 sh(ω i t) . ( 20
)
The disorder average is then carried out as in Eq. ( 18), by averaging over V i and ω i with the help of the joint distribution P (V i ,ω i ). By "returning" the potential V (x) → -V (x), we are back to the blue-detuned potential so the joint distribution of the maxima P red (V i ,ω i ) is nothing but the joint distribution P blue (V i ,ω i ) for the minima of a blue-detuned speckle. This symmetry also implies that the density of maxima for a reddetuned speckle is equal to the density of minima ρ for a blue-detuned speckle. Note that, in principle, one could use the propagator of the harmonic oscillator in the time domain for calculating the spectral function of the blue-detuned speckle potential as well. This approach turns however inadequate due to the presence of an infinite number of singularities-when ω i t is an integer multiple of π -arising in the time integral over the propagator.

III. STATISTICS OF 1D SPECKLE POTENTIALS

A. Joint distribution P(V i ,ω i )

In this section, we calculate the joint probability distribution P (V i ,ω i ) discussed above. From here on we drop the subscript i and merely write P (V ,ω) to lighten the notations. We derive it for the blue-detuned speckle potential, for which it corresponds to the joint probability of minima and potential curvature around minima.

The distribution P (V ,ω) is closely related to the joint, conditional probability distribution P (V (x), V (x)|V (x) = 0, V (x) > 0) of V (x) and its second derivative V (x) given that V (x) = 0 and V (x) > 0, that we propose to calculate first. From here on we use the following abbreviated notation for the potential and its derivatives at point x:

V ≡ V (x), V x ≡ V (x), V xx ≡ V (x). (21)
The above distribution follows from

P (V ,V xx |V x = 0,V xx > 0) = N × lim V x →0 P (V ,V x ,V xx ) P (V x ) . ( 22
)
The numerical constant N that appears in Eq. ( 22) stems from the fact that only positive curvatures are selected on the left-hand side, whereas on the right-hand side all possible values are understood. It will be later determined from the normalization condition. In order to compute the joint distribution P (V ,V x ,V xx ), we follow Goodman [20] and write the potential as

V = Re(x) 2 + Im(x) 2 . ( 23
)
Up to a constant multiplicative factor, Re(x) and Im(x) respectively describe the real and imaginary parts of the laser electric field at point x, from which the speckle potential V is built. As for the potential, we introduce the following short-hand notations:

≡ Re(x), x ≡ d dx Re(x), xx ≡ d 2 dx 2 Re(x) (24)
≡ Im(x), x ≡ d dx Im(x), xx ≡ d 2 dx 2 Im(x). The motivation for introducing the fields and is that they are independent Gaussian variables with zero mean and equal variance [20]. Their derivatives are likewise Gaussian, since any linear transformation of a Gaussian retains Gaussian statistics. They also have a zero mean. As a consequence, the six random variables of interest obey the multidimensional Gaussian distribution

P ( , , x , x , xx , xx ) = e -u t C -1 u/2 8π 3 √ det(C) , ( 25
)
where u t is a row vector with entries ( , , x , x , xx , xx ), and C is the covariance matrix

C = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ x x xx xx x x xx xx x x x x x x x xx x xx x x x x x x x xx x xx xx xx xx x xx x xx xx xx xx xx xx xx x xx x xx xx xx xx ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
.

The entries of this matrix can be explicitly calculated for a blue-detuned speckle potential. This yields

C = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ F (0) 0 0 0 F (0) 0 0 F (0) 0 0 0 F (0) 0 0 -F (0) 0 0 0 0 0 0 -F (0) 0 0 F (0) 0 0 0 F (4) (0) 0 0 F (0) 0 0 0 F (4) (0) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
, where F (x) is related to the two-point correlation function of the potential V through

F (x -x ) = 1 2 V (x)V (x ) -V (x) 2 . ( 26
)
We then introduce in Eq. ( 25) the change of variables

= √ V cos θ, = √ V sin θ, ( 27
)
from which we calculate the distribution P (V ,θ,V x ,θ x , V xx ,θ xx ), with a corresponding Jacobian equal to 1/8. By explicitly evaluating the entries of the C matrix for the Gaussian correlation function (4) and calculating the remaining integrals over θ , θ x , and θ xx with Mathematica [START_REF]Mathematica[END_REF], we find

P (V ,V x ,V xx ) = σ 4 4 √ 2πV 3 0 V e -[24V +16V xx σ 2 +(V 2 x -2V V xx ) 2 σ 4 /V 3 ]/16V 0 × I -1/4 V 2 x -2V V xx 2 σ 4 16V 3 V 0 + I 1/4 V 2 x -2V V xx 2 σ 4 16V 3 V 0 -V 2 x + 2V V xx V 0 V , ( 28
)
where I 1/4 and I -1/4 are the modified Bessel functions of the first kind. Note that this expression is valid only when V 2 x -2V V xx < 0, a condition fulfilled since only minima of the potential are considered [START_REF]The maxima of blue-detuned speckle-or equivalently the minima of a red-detuned one-could be studied along the same lines, but they are not relevant for the behavior of the spectral function[END_REF]. The distribution P (V ,V x ,V xx ) is regular with respect to the limit V x → 0. In Eq. ( 22), we can thus take this limit separately in the numerator and denominator, reducing the latter to a numerical constant which can be absorbed in the normalization prefactor N.

From the joint distribution (28), we are now in position to access the probability P (V ,V xx |V x = 0,V xx > 0) using Eq. ( 22). The result is

P (V ,V xx |V x = 0,V xx > 0) = N √ V xx V e -(6V 2 +4V V xx σ 2 +V 2 xx σ 4 )/4V 0 V × I -1/4 V 2 xx σ 4 4V V 0 + I 1/4 V 2 xx σ 4 4V V 0 . ( 29
)
By imposing that the distribution is normalized, we find N = σ 5 /(2cV

5/2 0 ), where c = [ √ 3 (1/4) (5/4) -(-1/4) (7/4)]/(3 3/4 √ 2π ) 1.
006 85, which will be replaced by 1 in the following.

The last stage of the calculation consists of connecting P (V ,V xx |V x = 0,V xx > 0) to the sought for distribution P (V ,ω). This amounts to changing the variables from V x = 0 to x such that V x (x) = 0, and from V xx to ω such that mω 2 = V xx . The associated Jacobian is |dV x /dx × dV xx /dω| = 2m 5/2 ω 3 . We finally infer

P (V ,ω) = 1 V ω 0 ω ω 0 4 e -3/2(V /V 0 ) 2 -(ω/ω 0 ) 2 -V 0 /4V (ω/ω 0 ) 4 × I -1/4 V 0 4V ω ω 0 4 + I 1/4 V 0 4V ω ω 0 4 . ( 30
)
The joint distribution is shown in Fig. 2. At a given potential minimum V , we observe that it is maximum for ω ∼ ω 0 . At smaller ω, the distribution rapidly falls to zero, which supports our description of the speckle potential landscape in terms of purely harmonic wells at low energies.

As we are primarily interested in low-energy minima V V 0 , it is instructive to express the distribution P (V ,ω) in the FIG. 2. Joint distribution P (V ,ω) of minima and potential curvature around minima, for a 1D, blue-detuned speckle potential with Gaussian correlation function [Eq. ( 30)]. 36), is shown as a solid red curve, and the classical limit, Eq. ( 10), as a solid green curve. Blue dots are the result of exact numerical simulations. limit V → 0 [START_REF] Watson | A Treatise on the Theory of Bessel Functions[END_REF]:

P (V ,ω)V 0 ω 0 ∼ V →0 2 π V 0 V ω ω 0 2 e -(ω/ω 0 ) 2 . ( 31
)
This asymptotic expression shows that most minima lie at very low V V 0 . This phenomenon is ultimately responsible for the sharp behavior of the spectral function at low energy. A broader distribution of energy minima would smooth out all peaks and oscillations in the spectral function and DOS, as visible in Figs. 3 and4. The harmonic oscillator-approximation, Eq. (37), is shown as a solid red curve, and the classical limit, Eq. ( 11), as a solid green curve. Blue dots are the result of exact numerical simulations.

B. Density of minima

The last unknown quantity is the density of minima ρ. To evaluate it, we follow [START_REF] Weinrib | [END_REF]35] and consider the general identity

dxδ[V (x)]f (x) = n 1 |V (x n )| f (x n ), (32)
valid for any function f . The sum is over all points x n where V (x) ≡ V x vanishes. If we choose f (x) to be |V (x)| ≡ |V xx |, then the integral is equal to the number of points at which V x vanishes. This defines the density of extrema per unit length as

δ(V x )|V xx |. ( 33
)
The corresponding density restricted to minima of the potential is

δ(V x )V xx θ (V xx ), (34)
with θ the Heaviside function. The disorder-averaged density of minima then reads

ρ = dV x dV xx P (V x ,V xx )δ(V x )V xx θ (V xx ). ( 35
)
Using Eq. ( 28), we obtain ρ = c /σ , where c 0.284 026.

IV. 1D SPECTRAL FUNCTION AND DOS: RESULTS

A. Spectral function for 1D blue-detuned speckles

We now evaluate the theoretical prediction (18) of the spectral function for 1D, blue-detuned speckle potentials, using Eq. ( 30) for the joint distribution of minima and curvature around minima. By carrying out the integral over ω i that ranges from 0 to ∞, we find

A k ( ) = c σ n 0 dV |ψ n (k)| 2 (n + 1/2) P V , -V (n + 1/2) θ ( ).
(36) This prediction is shown in Fig. 3 as a function of energy, for k = 0 and η = 128 (solid red curve). As discussed in Sec. II D 1, we expect it to describe low energies. At large energies, the classical limit (10) (solid green curve in Fig. 3)-and its smooth quantum corrections (12)-is on the other hand a very good approximation. In order to assess the accuracy of these two limits, we have performed numerical simulations of the spectral function. For these simulations we use a discrete grid of size L = 200σ with 4000 grid points and periodic boundary conditions, and compute the spectral function from definition (6), using the same approach as described in [17] to carry out the time evolution. The results are averaged over 50 000 disorder realizations, and are shown in Fig. 3 as blue dots. We see that the harmonic-oscillator prediction is in excellent agreement with the numerics at low energies. In particular, the high and narrow peak near /V 0 ∼ 0.05 and the secondary "bump" near /V 0 ∼ 0.25 are very well described. The peak originates from the ground state of the harmonic oscillator [term n = 0 in the sum (36)]; its relatively narrow character originates from the ω distribution in Eq. ( 30) rather well peaked around ω = ω 0 . The bump comes from the excited states.

B. Density of states for 1D blue-detuned speckles

From definition (7) and Eq. ( 36), we can compute the DOS for 1D blue-detuned speckles. Carrying out the integral over k, we find

ν( ) = c σ n 0 dV 1 (n + 1/2) P V , -V (n + 1/2) θ ( ).
(37) This prediction is shown in Fig. 4 as a function of energy, for η = 128 (solid red curve), together with the classical limit, Eq. ( 11) (solid green curve). We have also performed numerical simulations of the DOS, by first computing many spectral functions for k ranging from 0 to 13σ -1 and then summing over k, using a number of grid points between 4000 (at small k) and 40 000 (for the largest k). These results are shown in Fig. 4 as blue dots. The DOS displays a bump at low energies, which is reminiscent of the narrow peak that shows up in the profiles of the spectral function; see Fig. 3. Indeed, upon increasing k the peak of the spectral function becomes less and less pronounced but remains at the same energy, which results in a smooth bump after summation over k. As seen in Fig. 4, at low energies numerical results are very well captured by the harmonic-oscillator prediction. At larger energies > V 0 (not shown in Fig. 4), the harmonic-oscillator approximation breaks down and the purely classical limit takes over, eventually leading to ν( ) ν 0 ( ) = √ m/(2 )/(π ) for → ∞ [18].

C. Validity of the harmonic-oscillator approximation

A simple argument can be used to estimate the energy range where the harmonic-oscillator approximation is valid. According to the Virial theorem, equipartition between kinetic and potential energy imposes that n = mω 2 x 2 n for the mean energy of an eigenstate. In order for the speckle potential to be correctly described by a harmonic-oscillator approximation, all states such that n = in Eq. ( 17) should have an extension

x 2 n much smaller than the correlation length σ , which imposes an upper limit for the energy: mω 2 σ 2 (in case this condition is not fulfilled, anharmonic terms would also come into play). As seen in Sec. III A, the most likely value of ω is ω 0 , so the condition becomes

V 0 . ( 38
)
On the other hand, the classical approximation is expected to describe well the spectral function down to energies of order V 0 / √ η [17]. Therefore, in the region V 0 / √ η V 0 both the harmonic-oscillator and the classical approximation provide a good description of the spectral function and of the DOS.

Equation (38) provides a restriction on the high-energy tail of the spectral function A k ( ) for the latter to be correctly described by our harmonic-oscillator approximation. A similar argument imposes an additional restriction for the momentum k. Indeed, equipartition between kinetic and potential energy for the harmonic oscillator also implies

2 k 2 n 2m = 1 2 mω 2 x 2 n , (39)
where x 2 n should be again much smaller than σ for the harmonic-oscillator approximation to hold. With ω ∼ ω 0 , condition (39) reads

2 k 2 n m V 0 . ( 40
)
The contribution of each eigenstate to the sum in Eq. ( 17) being proportional to |ψ n (k)| 2 , the sum is dominated by eigenstates having k 2 n of the order of k, such that criterion (40)

leads to k V 0 . ( 41
)
In any case, the harmonic oscillator approximation is a good one in the region , k ∼ ω 0 where the quantum corrections are important, while the purely classical result (10) takes over at higher energy , k ∼ V 0 .

D. Spectral function for 1D red-detuned speckles

For 1D, red-detuned speckle potentials, we make use of the approach explained in Sec. II D 3 to calculate the spectral function. Using Eq. ( 19) and (20) together with the joint distribution (30) and carrying out the integral over V , we find

A k ( ) = (2π ) 2 c σ 4 m 5/2 √ 2V 5/2 0 ∞ -∞ dt 2π e i t/ ∞ 0 dω ω 4 I 2 -1/4 mω 2 σ 2 2V 0 3 -2itV 0 / -I 2 1/4 mω 2 σ 2 2V 0 3 -2itV 0 / × i 2πmωsh(ωt) exp - i k 2 mω coth(ωt) - 1 sh(ωt) - mω 2 σ 2 V 0 . ( 42
)
This prediction is shown in Fig. 5 for k = 0 (solid red curve), together with the classical limit, Eq. ( 10) (solid green curve). Both limits are compared with the result of numerical simulations (blue dots) that use a discrete grid of size L = 200σ with 4000 grid points, periodic boundary conditions, and 50 000 disorder realizations.

As seen in Fig. 5, the harmonic-oscillator prediction is in good agreement with the numerical results for energies near 0. At smaller energies ( -V 0 ), the description of the speckle potential in terms of inverse harmonic oscillators becomes poor, while the classical limit provides an excellent approximation.

E. Validity of the inverted harmonic-oscillator approximation

The breakdown of the inverted harmonic-oscillator approximation at energies -V 0 can be understood from a 022114-7 numerics inverted harmonic oscillator classical limit red-detuned FIG. 5. Spectral function A k=0 ( ) as a function of energy in a 1D, red-detuned speckle potential with Gaussian correlation function, for η = 128. The inverted harmonic-oscillator approximation, Eq. ( 42), is shown as a solid red curve, and the classical limit, Eq. ( 10), as a solid green curve. Blue dots are the result of exact numerical simulations.

reasoning on the classical action that appears in Eq. ( 15). Indeed, for the stationary phase approximation to be valid, the time span t associated with a classical trajectory should be such that the classical action V 0 t/ is large, imposing t /V 0 . Energies corresponding to such long times fulfill

| | V 0 . ( 43
)
Note that this condition is fully similar to that for blue-detuned speckles, Eq. ( 38), though it is here deduced from a slightly different argument. Then, the motion of a classical atom of energy = kmωx 2 /2 describes well the dynamics in a red-detuned speckle as long as the excursion x 2 remains much smaller than σ , namely as long as

k + | | mωσ 2 /2. Since ω ∼ ω 0 and | | V 0 , this leads to k V 0 , (44)
which is the same validity condition as for blue-detuned speckles.

F. Density of states for 1D red-detuned speckles

We show in Fig. 6 as blue dots the DOS in a 1D, red-detuned speckle potential, computed from numerical simulations where we have summed over 208 spectral functions with k ranging from 0 to 13σ -1 , varying the number of grid points from 4000 (for small k) to 40 000 (for the largest k). We also show as the solid green curve the classical prediction (11). As seen in the figure, the latter already provides an excellent description of the exact results. This can be understood qualitatively from the Gutzwiller trace formula [START_REF] Gutzwiller | Chaos in Classical and Quantum Mechanics[END_REF]36] which expresses the density of states as the sum of the classical contribution, Eq. ( 11), and of oscillatory contributions coming from periodic orbits. Around E = 0, the periodic orbits in a red-detuned speckle are long ones with characteristic properties (action, period...) which strongly depend on the disorder realization, so that all oscillatory contributions cancel out. This is in stark contrast with the blue-detuned speckle where periodic orbits around red-detuned numerics classical limit FIG. 6. Density of states ν( ) as a function of energy in a 1D, red-detuned speckle potential with Gaussian correlation function. The classical limit, Eq. ( 11), is shown as a solid green curve. Blue dots are the result of exact numerical simulations. E = 0 are short orbits trapped in the deep potential minima and collectively contribute to "bumps" in the DoS.

In principle, quantum corrections to the DOS can be obtained from Eq. ( 42) by evaluating the Fresnel integral over k; see Eq. (7). The latter can be performed, but the remaining integral over t displays an ultraviolet divergence. This divergence already appears in the DOS of the inverted harmonic oscillator, for which it originates of the continuous nature of the spectrum. It thus appears that for the DOS of red-detuned speckles, the description of singular quantum corrections requires to go beyond the inverted harmonicoscillator approximation, a task that we leave for later work.

V. STATISTICS OF 2D SPECKLE POTENTIALS

We now turn to the study of 2D speckle potentials which we aim to describe, at low energies, by a 2D harmonic-oscillator approximation. By analogy with the 1D case, we propose to model the speckle potential around an extremum V (x i ,y i ) by a 2D harmonic oscillator (respectively inverted harmonic oscillator) of the form ±V ± mω 2

x (xx i ) 2 /2 ± mω 2 y (yy i ) 2 /2 with again the + (respectively -) sign for blue (respectively red)-detuned speckles, with random frequencies ω x and ω y . Such a description requires the preliminary knowledge of the joint probability distribution P (V ,ω x ,ω y ) of extrema and potential curvature around extrema. Study of this quantity is the object of the present section. We here focus on bluedetuned speckle potentials, and then infer the corresponding distribution for red-detuned speckles by the same symmetry argument as in one dimension.

A. Density of minima at V = 0 2D speckle potentials have a important difference with 1D potentials: they present a finite density of points exactly at V = 0 [START_REF] Weinrib | [END_REF]. In writing the blue-detuned speckle potential as

V (x,y) = Re(x,y) 2 + Im(x,y) 2 , ( 45
)
022114-8

(minima at (minima at )

) FIG. 7. Blue curve: integrated density of minima (density of minima whose depth is greater than V ) for a 2D, blue-detuned speckle potential with Gaussian correlation function. The results have been obtained numerically on a discrete grid of size L × L = 400σ × 400σ.

these points are minima that correspond to the intersections of the curves Re(x,y) = 0 and Im(x,y) = 0. Before considering the distribution P (V ,ω x ,ω y ), let us first examine the proportion of minima at V = 0 and at V = 0. To this end, we have numerically computed the integrated density of minima, i.e., the density of minima whose depth is greater than V . To distinguish between minima at V = 0 and minima at V = 0, we have exploited the sensitivity (respectively insensitivity) of the minima at V = 0 (respectively V = 0) with respect to a change in the spatial discretization (number of grid points). The results of these simulations are shown in Fig. 7. They have been obtained on a discrete grid of size L × L = 400σ × 400σ , by varying the number of grid points between 18 000 and 26 000 along x and y. The discontinuity of the integrated density of minima at V = 0 visible in Fig. 7 defines ρ 0 , the density of minima at V = 0. We find that approximately ρ 0 /ρ ∼ 65% of all minima lie at V = 0 [37]. Note that this result is confirmed by an analytical prediction derived in [START_REF] Weinrib | [END_REF]:

ρ 0 = -4πF (0) ∇ 2 r F (r)| r=0 -1 , ( 46
)
where

F (r -r ) = V (r)V (r ) -V (r) 2 /2.
For the Gaussian correlation function (4), this explicitly gives ρ 0 = 1/(4πσ 2 ) 0.08/σ 2 .

In two dimensions, the majority of minima thus lies at V = 0. To keep the discussion and the calculation as simple as possible, as a first approximation, we keep only the minima at V = 0 in the 2D semiclassical description. We will discuss the validity of this approximation in Sec. VI A. The joint distribution of interest P (V ,ω x ,ω y ) reduces to

P (V ,ω x ,ω y ) P (ω x ,ω y )δ(V ), (47)
where P (ω x ,ω y ) is the 2D joint distribution of potential curvatures around a minimum (x i ,y i ) where V (x i ,y i ) = 0.

B. Joint distribution P(ω x ,ω y )

The distribution P (ω x ,ω y ) is closely related to the joint, conditional probability distribution P (ω x ,ω y |V (x i ,y i ) = 0) of potential curvatures given that V (x i ,y i ) = 0. To calculate this distribution, we first expand V (x,y) up to second order in the vicinity of (x i ,y i ) as

V (x,y) 1 2 XAX t , ( 48
)
where X = (xx i ,yy i ) and

A = ∂ 2 x V (x,y) ∂ x ∂ y V (x,y) ∂ y ∂ x V (x,y) ∂ 2 y V (x,y) . ( 49
)
By diagonalizing the quadratic form (48) (which is possible since the matrix A is symmetric), we can describe a well of the speckle potential in terms of two independent 1D harmonic oscillators, whose curvatures ω x and ω y are related to the eigenvalues λ 1 and λ 2 of A through ω x = √ λ 1 /m and ω y = √ λ 2 /m. The calculation of the joint distribution of the eigenvalues (λ 1 ,λ 2 ) is done in Appendix B for clarity. The corresponding result for P (ω x ,ω y |V = 0) is

P (ω x ,ω y |V = 0) = 2 ω 4 0 ω 2 y -ω 2 x e -(ω 2 x +ω 2 y )/ω 2 0 . ( 50
)
The sought for distribution P (ω x ,ω y ) then follows from the change of variables from V (x i ,y i ) = 0 to (x i ,y i ) such that V (x i ,y i ) = 0: P (ω x ,ω y ) = (d 2 V /dxdy)P (ω x ,ω y |V = 0), where d 2 V is the change in the surface element defined by the 2D curve V (x,y) when x varies from x i to x i + dx and y varies from y i to y i + dy. Since V (x,y) mω 2

x (xx i ) 2 /2 + mω 2 y (yy i ) 2 /2 in the vicinity of a minimum, we expect this change to be proportional to ω x ω y dxdy, such that

P (ω x ,ω y ) ∝ ω x ω y P (ω x ,ω y |V = 0). ( 51
)
The unknown prefactor is determined from normalization, which eventually leads to

P (ω x ,ω y ) = 4 ω 6 0 ω x ω y ω 2 y -ω 2 x e -(ω 2 x +ω 2 y )/ω 2 0 . ( 52
)
A density plot of P (ω x ,ω y ) is shown in Fig. 8. As in one dimension, the distribution rapidly falls to zero at small frequencies, which again supports our description of the speckle potential landscape in terms of purely harmonic wells at low energies. We have confirmed Eq. ( 52) by numerical simulations of the distribution P (ω x ,ω y ), deduced from numerically generated speckle potentials. We show in Fig. 9 the numerical cut P (ω x ,ω y = 1.25ω 0 ) as a function of ω x (blue dots), together with Eq. (52) (red curve), and find a very good agreement.

VI. 2D SPECTRAL FUNCTION AND DOS: RESULTS

A. Spectral function for 2D blue-detuned speckles

We are now in position to compute the spectral function for 2D, blue-detuned speckle potentials. The 2D counterpart 022114-9 of Eq. ( 18) reads

A k ( ) = ρ 0 ∞ n x ,n y =0 dω x dω y P (ω x ,ω y ) × |ψ n x (k x )| 2 |ψ n y (k y )| 2 δ( -n x ,n y ), ( 53
)
where P (ω x ,ω y ) is the joint distribution of curvatures around minima at V = 0 given by Eq. (52), n x ,n y = ω x (n x + 1/2) + ω y (n y + 1/2) and the eigenfunctions ψ n x (k x ) are given by Eq. (17) with n replaced by n x and k replaced by k x , and similarly for ψ n y (k y ). By performing the integral over ω y and FIG. 9. Cut P (ω x ,ω y = 1.25ω 0 ) of the joint distribution of curvatures around a minima at V = 0 for a 2D, blue-detuned speckle potential. Blue dots are the results of numerical simulations and the red curve is Eq. ( 52). 54), is shown as a solid red curve, and the classical limit, Eq. ( 10), as a solid green curve. The corrected harmonic-oscillator approximation, Eq. ( 56), is shown as a solid black curve. Blue dots are the results of exact numerical simulations.

using that ρ 0 = 1/(4πσ 2 ), we find

A k ( ) = 1 4πσ 2 n x ,n y / (n x +1/2) 0 dω x |ψ n x (k x )| 2 (n y + 1/2) θ ( )
×|ψ n y (k y )| 2 P (ω x ,ω y )| ω y =( -ω x (n x +1/2))/ (n y +1/2) . ( 54
)
This prediction is shown in Fig. 10 as a function of energy, for k = 0 and η = 128 (solid red curve). The classical limit (10), expected to describe large energies, is also shown as a solid green curve. These results are compared to numerical simulations of the spectral function (blue dots) which use a system size L × L = (20πσ ) 2 with 600 grid points along x and y, periodic boundary conditions and 40 000 disorder realizations. Several observations can de made. Like in one dimension, the harmonic approximation quantitatively describes the spectral function for energies ∼ V 0 / √ η = ω 0 . The large peak is at an energy about twice larger than in one dimension-compare with Fig. 3-because it is the ground-state energy of a 2D (instead of 1D) harmonic oscillator. It is also slightly higher and the minimum around /V 0 = 0.2 as well as the second bump above are slightly more visible than in one dimension. This is because most potential minima are exactly at V = 0 in two dimensions, while this is not true in one dimension, so that an additional smoothing takes place in the latter case. This must however be taken with a grain of salt: the 2D low-energy peak of the spectral function is not entirely controlled by the ground state of the harmonic oscillator: excited states also contribute for roughly 25% of the peak height. As seen in Fig. 10, deviations of the harmonic-oscillator prediction from the numerical result occur at smaller energy than in one dimension. This phenomenon can be understood from the expression of the spectral function in terms of the propagator of the 2D harmonic oscillator:

A k ( ) = ρ 0 ∞ -∞ dt 2π e i t/ k| e -iH HO t/ |k , ( 55
)
where H HO = p 2 /(2m) + mω 2 x x 2 /2 + mω 2 y y 2 /2. In two dimensions, the propagator k| e -iH HO t/ |k ∝ 1/t at short times [29]. This singularity is more pronounced than in one dimension where the propagator diverges as 1/ √ t. In two dimensions there is thus more weight on short times, which are by construction not well captured by the harmonic-oscillator approximation. On the other hand, we know that short times are fairly well described by the classical limit, Eq. (10). To improve on the quality of the harmonic-oscillator description, we thus propose to replace the contribution from the pole at t = 0 by the classical contribution. The contribution from this pole is simple to calculate from Eq. (55): we find θ ( )/V 0 . The classical contribution is given in Eq. (10). The above prescription thus leads to

A corr k ( ) A k ( ) - θ ( ) V 0 + θ ( ) V 0 exp - -k V 0 , ( 56
)
where A k ( ) is the prediction of the harmonic-oscillator description, Eq. (54). Equation ( 56) is shown in Fig. 10 as a solid black curve, and is in very good agreement with the numerical simulations.

The excellent agreement with the numerical calculations justifies a posteriori the approximation of keeping only the minima at V = 0. Such an agreement may surprise the attentive reader as approximately 35% of the minima have been left aside. The reason for it lies in two mechanisms reducing the contribution to the spectral function of minima at V = 0 as compared to minima at V = 0. First, among the 35% of minima at V = 0, only a fraction contributes to the spectral function: as we are interested in very low energies ( V 0 ), we should keep only the harmonic wells with associated minimum smaller than . Second, the smoothing due to the dispersion in V -compare the 1D oscillations in Fig. 3 with such a dispersion and the 2D oscillations in Fig. 10 where the dispersion is absent-makes the contribution of minima at V = 0 negligible after application of the corrected harmonic-oscillator prescription [Eq. ( 56)].

B. Density of states for 2D blue-detuned speckles

From definition (7) and Eq. (54), we can compute the DOS for 2D blue-detuned speckles. Carrying out the integral over k, we readily find

ν( ) = 1 4πσ 2 n x ,n y / (n x +1/2) 0 dω x θ ( ) (n y + 1/2) ×P ω x , -ω x (n x + 1/2) (n y + 1/2) . ( 57
)
This prediction is shown in Fig. 11 as a function of energy, for η = 128 (solid red curve), together with the classical limit, Eq. ( 11) (solid green curve). We have also performed numerical simulations of the DOS. In two dimensions however, the strategy of numerically computing first spectral functions at different k and then summing of k is numerically demanding. We have thus used a different scheme that consists of blue-detuned numerics harmonic oscillator classical limit corrected harm. osc. FIG. 11. Density of states ν( ) as a function of energy in a 2D, blue-detuned speckle potential with Gaussian correlation function. The harmonic-oscillator approximation, Eq. (57), is shown as a solid red curve, and the classical limit, Eq. ( 11), as a solid green curve. The corrected harmonic-oscillator description, Eq. ( 60), is shown as a solid black curve. Blue dots are the result of exact numerical simulations.

expressing the trace in Eq. ( 7) in real space rather than in momentum space:

ν( ) = 1 L 2 Tr δ( -H ) = 1 L 2 d 2 r r| δ( -H )|r . ( 58
)
The system being translation invariant on average, the integrand is in fact independent of r so

ν( ) = r = 0| δ( -H ) |r = 0 = ∞ -∞ dt 2π e i t/ 0| e -iH t/ |0 . ( 59
)
From Eq. (59), it thus appears that the DOS can be obtained by numerically propagating a particle initially located at the origin, then recording the value of the wave function at the origin for many different times t, and finally taking the Fourier transform with respect to time and averaging over disorder. We have applied this strategy for a system size L × L = (10πσ ) 2 with 400 grid points along x and y and 40 000 disorder realizations. Results are shown in Fig. 11 as blue dots. As for the 2D spectral function, we observe deviations of the theoretical prediction (59) from the numerical results at relatively small energies due to a pole ∝ 1/t 2 in the propagator in (59). We again correct them by replacing the contribution of this pole by the classical result (11). This gives

ν cor ( ) = ν( ) - m θ ( ) 2π 2 V 0 + mθ ( ) 2π 2 1 -e -/V 0 . ( 60
)
This prediction is plotted in Fig. 11 (solid black curve), and describes very well the exact numerical results.

022114-11 numerics inverted harmonic oscillator classical limit red-detuned corrected inverted harm. osc.

FIG. 12. Spectral function A k=0 ( ) as a function of energy in a 2D, red-detuned speckle potential with Gaussian correlation function, for η = 128. The inverted harmonic-oscillator approximation, Eq. ( 62), is shown as a solid red curve, and the classical limit, Eq. ( 10), as a solid green curve. The corrected inverted harmonic-oscillator approximation, Eq. ( 63), is shown as a solid black curve. Blue dots are the results of exact numerical simulations.

C. Spectral function for 2D red-detuned speckles

To evaluate the spectral function for 2D, red-detuned speckle potentials, we proceed as in one dimension and write

A k ( ) ρ 0 ∞ -∞ dt 2π e i t/ k| e -iH IHO t/ |k , ( 61
)
where H IHO = p 2 /(2m)mω 2 x x 2 /2mω 2 y y 2 /2. Making the average over disorder explicit, we have

A k ( ) = 1 4πσ 2 ∞ -∞ dt 2π e i t/ ∞ 0 dω x dω y P (ω x ,ω y ) × k x | e -i[p 2 x /(2m)-mω 2 x x 2 /2]t/ |k x × k y | e -i p 2 y /(2m)-mω 2 y y 2 /2 t/ |k y , ( 62
)
where the 1D inverted harmonic-oscillator propagator is given by Eq. ( 20). Equation ( 62) is shown in Fig. 12 (solid red curve), together with the classical limit, Eq. ( 10) (solid green curve). Results of numerical simulations that use a system size L × L = (20πσ ) 2 with 600 grid points along x and y and 40 000 disorder realizations are also shown (blue dots). As for the blue-detuned speckle, the pole 1/t in the propagator gives rise to deviations of the oscillator description from the exact numerical results that are more significant than in one dimension. We again cure them by replacing the contribution of the pole by the classical limit:

A cor k ( ) = A k ( ) - θ (-) V 0 + θ (-) V 0 e ( -k )/V 0 . ( 63
)
This prediction is plotted in Fig. 12 (solid black curve), and describes very well the exact numerical results.

numerics classical limit

red-detuned FIG. 13. Density of states ν( ) as a function of energy in a 2D, red-detuned speckle potential with Gaussian correlation function. The classical limit, Eq. ( 11), is shown as a solid green curve. Blue dots are the results from exact numerical simulations based on Eq. (59).

D. Density of states for 2D red-detuned speckles

We show in Fig. 13 the DOS in a 2D, red-detuned speckle potential computed from numerical simulations using a system size L × L = (10πσ ) 2 with 2000 grid points in each direction and 8000 disorder realizations, based on Eq. (59) (blue dots). As for 1D red-detuned speckles, the oscillator correction to the DOS diverges due to an ultraviolet divergence in the propagator; see Sec. IV F. Nevertheless, as seen in Fig. 13, the classical prediction (11) (solid green curve) already constitutes an excellent approximation of the exact result.

VII. CONCLUSION

In this paper, we have pointed out that an expansion in powers of of the spectral function or the density of states in speckle potentials is not sufficient at low energies, due to the discontinuity of the potential distribution. In order to overcome this difficulty, we have developed an analytical method based on a semiclassical description of the dynamics combined with the statistical properties of potential extrema. Applying this approach to 1D and 2D blue-and red-detuned speckles, we have carried out the calculation of the spectral function and the DOS. By connecting our results with those of previous works valid at high energies [17,18], we have been able to describe the whole energy spectrum, and have found a good agreement with exact numerical simulations.

Our semiclassical description additionally provides a simple interpretation of intriguing features of the spectral function and DOS. In particular, for blue-detuned potentials we have shown that the low-energy peak of spectral functions is essentially associated with the ground state of an atom in a potential well of the speckle, while the secondary bump is associated with excited states. We have also emphasized that in spite of their simple symmetry, red-and blue-detuned speckles exhibit remarkably different features in the semiclassical regime, coming from the fundamental different nature of the classical trajectories involved near zero energy: for blue-detuned speckles, these classical trajectories lie in deep potential wells, while for red-detuned speckles they lie in the vicinity of the top of inverted wells.

As a logical continuation of this work, it would be of great interest to address the case of three-dimensional speckle potentials, involved in important questions related to Anderson localization [5,6,38,39]. This task appears challenging though, as the isolated points of zero potential in two dimensions become curves in three dimensions, making the application of a harmonic-oscillator approximation less obvious.

where E 1 and E 2 are independent Gaussian variables with zero mean and equal variance [20]. Defining X = -iV (x)t and denoting by Y i the corrections appearing in Eq. (A2), we obtain for the nth cumulant:

κ n X + m i=1 Y i = κ n (X) + m i=1 nκ(X, . . . ,X n-1 terms ,Y i ) + 2 j =1 n 2 κ ⎛ ⎝ X, . . . ,X n-2 terms , - t 2 2m ∂ x E 2 j (x) i k, - t 2 2m ∂ x E 2 j (x) i k + O 3 . (A6)
We now need to calculate the various cumulants entering this equation. For this purpose, we use a theorem due to Leonov and Shiryaev [START_REF] Leonov | [END_REF][START_REF]Here we use a somewhat simpler and less general version of Leonov and Shiryaev theorem[END_REF]. Before discussing the theorem itself, it is useful to introduce some terminology. Consider the matrix ⎛

⎜ ⎜ ⎜ ⎝ X 11 . . . X 1J . . . . . . X J 1 . . . X J J ⎞ ⎟ ⎟ ⎟ ⎠ , ( A7
)
and a partition P 1 ∪ P 2 ∪ • • • ∪ P M of its entries. We choose this matrix square for simplicity, but the formalism is straightforwardly generalizable to rectangular matrices. If the rows are denoted by R 1 , . . . ,R J , then a partition is said to be indecomposable if and only if there exist no sets P m 1 , . . . ,P m N , (N < M), and rows R i 1 , . . . ,R i P , (P < J), with

P m 1 ∪ • • • ∪ P m N = R i 1 ∪ • • • ∪ R i P . (A8)
The theorem then goes as follows [START_REF] Leonov | [END_REF]. Consider a matrix of random entries X ij (i,j = 1, . . . ,J ) and the J random variables hence defining

Y i = J j =1 X ij , i = 1, . . . ,J. ( A9
E j (x) = dp i 2π e ipx E j (p). (A11)
The cumulant of interest then reads

κ E 2 1 , . . . ,E 2 1 ,∂ 2 x E 2 1 = - 2n i=1 dp i 2π (p 2n-1 + p 2n ) 2 κ[E 1 (p 1 )E 1 (p 2 ), . . . ,E 1 (p 2n-3 )E 1 (p 2n-2 ),E 1 (p 2n-1 )E 1 (p 2n )]. (A12)
The corresponding matrix (A7) is

⎛ ⎜ ⎜ ⎜ ⎝ E 1 (p 1 ) E 1 (p 2 ) . . . . . . E 1 (p 2n-1 ) E 1 (p 2n ) ⎞ ⎟ ⎟ ⎟ ⎠ . (A13)
As E 1 is Gaussian distributed, only joint cumulants involving two fields should be kept in the right-hand side of Eq. (A10). Our indecomposable partitions are then made of pairs of E 1 and all give the same contribution. Let us now count them, taking into account the two following constraints for making pairs so to obtain an indecomposable partition: (i) A pair cannot be formed out of two fields lying on the same line, i.e., the choice

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ E 1 (p 1 ) E 1 (p 2 ) . . . . . . E 1 (p 2n-1 ) E 1 (p 2n ) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (A14)
is forbidden.

(ii) Two pairs right nearby cannot be formed, i.e., the choice

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ E 1 (p 1 ) E 1 (p 2 ) E 1 (p 1 ) E 1 (p 2 ) . . . . . . E 1 (p 2n-1 ) E 1 (p 2n ) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (A15) is forbidden.
Therefore, to form the first pair, we have (2n) choices for the first field and (2n -2) choices for the second, and similarly for the next pairs. This leaves us with 2n(2n -2) 2 (2n -4) 2 . . . = 2 2n n! 2 /(2n) choices of pairing. There is however a redundancy in this counting, due to the invariance of the partition with respect to both swapping of the two fields inside one pair (2 n possibilities) and swapping of different pairs (n! possibilities). This leaves us with only 2 n n! 2 /(2n2 n n!) = 2 n n!/(2n) choices of pairing. Calculating the contribution from one of them, we obtain

κ E 2 1 , . . . ,E 2 1 n-1 terms ,∂ 2 x E 2 1 = 2 n n! 2n F n-2 (0)2 ×[F (0)F (0) + F 2 (0)], (A16)
where F (x) is defined by Eq. (26). A similar derivation is then performed for all terms in Eq. (A6) that are not found to vanish on the basis of the property (3) above [START_REF] Brillinger | Time Series Data Analysis and Theory[END_REF]. Upon summing various geometric series and recognizing the expansion of a logarithm, we find

k| e -i[p 2 /2m+V ](t/ ) |k = e -i k t/ 1 ± itV 0 / 1 + it 3 V 2 0 E σ / 3 12(1 + itV 0 / ) + t 4 V 2 0 E σ / 4 12(1 + itV 0 / ) k . ( A17
)
This result is not difficult to generalize to dimension d, where Eq. (A2) becomes

k| e -i[ p 2 /2m+V ](t/ ) |k = e -iV (r)t e (i 2 t 3 /3m) d i=1 [∂x i V (r)] 2 ×e (-2 t 2 /4m)[2ik•∇V (r)+∇ 2 V (r)] ×e -(i 2 t 3 /3m) d i,j =1 [∂ x i ∂ x j V (r)]( 2 k i k j /2m) e O( 3 ) . (A18) In the sum d i,j =1 [∂ x i ∂ x j V (r)] 2 k i k j /2m
, to leading order in the crossed terms (i = j ) do not contribute to the disorderaveraged propagator as their contributions are proportional to first-order derivatives of the field correlation function (26) evaluated at 0, which vanish. Also, derivatives of the potential with respect to different directions are independent. Therefore, the propagator in dimension d is simply the product of d 1D propagators. Finally, the result for the red-detuned speckle is deduced by changing m to -m and t to -t (which amounts to changing the sign of V ). The general result then reads

k| e -i[p 2 /2m+V ](t/ ) |k = e -i k t/ 1 ± itV 0 / 1+ dit 3 V 2 0 E σ / 3 12(1 ± itV 0 / ) + t 4 V 2 0 E σ / 4 12(1 ± itV 0 / ) k , ( A19
)
with the + (respectively -) sign for blue-(respectively red-)detuned speckles. This immediately leads to Eq. ( 12) of the main text.

APPENDIX B

In this Appendix, we derive the joint probability distribution P (λ 1 ,λ 2 ) of the eigenvalues λ 1 and λ 2 of the matrix

A = ∂ 2 x V (x,y) ∂ x ∂ y V (x,y) ∂ y ∂ x V (x,y) ∂ 2 y V (x,y) , (B1)
in the vicinity of a minimum V (x,y) = 0. As in Sec. III A we write the potential as

V (x,y) = Re(x,y) 2 + Im(x,y) 2 , ( B2
)
where Re(x,y) and Im(x,y) are independent Gaussian variables with zero mean and equal variance σ 2 c = V 0 /(4σ 2 ) [20]. Making use of the shorthand notation x ≡ ∂ x Re(x,y),

x ≡ ∂ x Im(x,y), (B3) y ≡ ∂ y Re(x,y), y ≡ ∂ y Im(x,y), we rewrite the matrix A as

A = 2 2 x + 2 x x y + x y x y + x y 2 y + 2 y .
(B4)

x , y , x , and y are independent, Gaussian distributed random variables with zero mean and variance σ c . The distribution P (u,v) can then be expressed as

P (u,v) = d x d y d x d y P ( x )P ( y )P ( x )P ( y ) × δ[u -λ 1 ( x , y , x , y )]δ[v -λ 2 ( x , y , x , y )].
To tackle this integral, we first change variables to "intensity" and "phase":

x = I 1 cos θ 1 , x = I 1 sin θ 1 , (B5) y = I 2 cos θ 2 , y = I 2 sin θ 2 .
The Jacobian of the transformation is 1/4, and

I 1 ,I 2 ∈ [0, + ∞[ and θ 1 ,θ 2 ∈ [-π,π].
The integral reduces to

P (λ 1 ,λ 2 ) = 1 32π 2 σ 4 c +∞ 0 dI 1 dI 2 π -π dθ 1 dθ 2 e -(I 1 +I 2 )/2σ 2 c ×δ λ 1 -I 1 + I 2 -I 2 1 + I 2 2 + 2I 1 I 2 cos 2(θ 1 -θ 2 ) ×δ λ 2 -I 1 + I 2 + I 2 1 + I 2 2 + 2I 1 I 2 cos 2(θ 1 -θ 2 ) ,
where we have assumed λ 2 > λ 1 without loss of generality, and added a corresponding renormalization prefactor 1/2. We then introduce

ϕ = θ 1 + θ 2 , φ = 2(θ 1 -θ 2 ), ( B6
)
and carry out the integral over ϕ. This eventually yields

P (λ 1 ,λ 2 ) = 1 8πσ 4 c +∞ 0 dI 1 dI 2 π 0 dφe -(I 1 +I 2 )/2σ 2 c ×δ λ 1 -I 1 + I 2 -I 2 1 + I 2 2 + 2I 1 I 2 cos φ ×δ λ 2 -I 1 + I 2 + I 2 1 + I 2 2 + 2I 1 I 2 cos φ .

This expression can be further simplified by writing

∞ 0 dI 1 dI 2 = ∞ 0 dI 1 I 1 0 dI 2 + ∞ 0 dI 1 ∞ I 1 dI 2
and noticing the equality of these two integrals due to the symmetric role played by I 1 and I 2 :

P (λ 1 ,λ 2 ) = 1 4πσ 4 c +∞ 0 dI 1 I 1 0 dI 2 π 0 dφe -(I 1 +I 2 )/2σ 2 c × δ λ 1 -I 1 + I 2 -I 2 1 + I 2 2 + 2I 1 I 2 cos φ × δ λ 2 -I 1 + I 2 + I 2 1 + I 2 2 + 2I 1 I 2 cos φ .
We then change the variable φ to z so that

z = I 1 + I 2 + I 2 1 + I 2 2 + 2I 1 I 2 cos φ, ( B7
)
where z spans the interval [0,2I 2 ]. The corresponding Jacobian is

∂φ ∂z = 2|I 1 + I 2 -z| √ z(2I 1 -z)(2I 2 -z)(2I 1 + 2I 2 + z) . ( B8
)
Performing the integrals over I 2 and z, we straightforwardly find

P (λ 1 ,λ 2 ) = 1 8πσ 4 c λ 2 /2 (λ 1 +λ 2 )/4 dI 1 e -(λ 1 +λ 2 )/4σ 2 c × (λ 2 -λ 1 )θ (λ 1 ) √ λ 1 λ 2 (λ 2 -2I 1 )(2I 1 -λ 1 ) . ( B9
)
The remaining integral can be done analytically, yielding

P (λ 1 ,λ 2 ) = (λ 2 -λ 1 )e -(λ 1 +λ 2 )/4σ 2 c 32σ 4 c √ λ 1 λ 2 θ (λ 1 ) (λ 2 > λ 1 ). (B10)
This relation has been obtained assuming λ 2 > λ 1 . The opposite case λ 1 < λ 2 is fully symmetric:

P (λ 1 ,λ 2 ) = (λ 1 -λ 2 )e -(λ 1 +λ 2 )/4σ 2 c 32σ 4 c √ λ 1 λ 2 θ (λ 2 ) (λ 2 < λ 1 ). (B11)
Using Eqs. (B10) and (B11) together with the relations λ 1 = mω 2 x , λ 2 = mω 2 x , we finally obtain Eq. (50) of the main text.

Chapter 4

Quantum boomerang eect in one-dimensional random potentials rving lri(ed how rndom potentils re generted in oldEtom experiments in hpter PD nd disussed the generl frmework for desriing the dynmis of wve pkets in setion QFPD we now onsider onrete dynmil senrioF he ltter resemles the experiment performed in liseu in PHHV QUD where n initilly nrrow wve pket spreds in oneEdimensionl rndom potentilD nd the density pro(le is reorded in the ourse of timeF rere howeverD we propose to dditionlly give the toms (nite verge veloity nd trk the wve pket enterEofEmss motionF he hpter strts with rief sttement of the prolem in setion RFIF sn setion RFPD we develop n intuitive derivtion of the pket enterEofEmssD sed on lssil rgumentsF his somewht nive pproh is ompred to numeril simultions in setion RFQF he numeril simultions revel n unexpeted phenomenonX fter n initil llisti motionD the pket enterE ofEmss experienes retrore)etion nd ppers to slowly return to its initil positionF e dued this phenomenon the quntum oomerng e'etF e show in setion RFR tht it is triggered y enderson loliztionF etion RFS is devoted to n nlytil tretment of the enter of mssD sed on ferezinskii digrmmti tehnique IQF his derivtion is supplemented y ppendies RFe nd RFfF sn pplying ferezinskii digrmmti tehniqueD we mke some ssumptions on the rndom potentil nd on the initil wve pketD whih re disussed in setion RFTF sn setion RFU (nllyD we derive surprising reltion etween enter of mss nd men squre displementF etion RFV onludes the hpter nd gives some perspetives for future workF 4.1 Initial condition e onsider the evolution of n initil qussin wve pket with phseD

ψ(x, t = 0) = Ψ k 0 (x) = exp -x 2 /2σ 2 + ik 0 x π 1/4 σ 1/2
, @RFIA whih we will refer to s kiked qussin wve pket in the followingF et vrine with the initil wve pket used in liseu in PHHV QUD @RFIA is imprinted with men veloity k 0 /mF he evolution is governed y the rmiltonin

H = -2 ∆/2m + V, @RFPA SU SV
ghpter RF untum oomerng e'et in oneEdimensionl rndom potentils where V is qussin1 D unorrelted rndom potentilX

V (x) = 0 nd V (x)V (x ) = γδ(x -x ), @RFQA
the overr denoting verging over disorder reliztionsF trting from qussinEorrelted poE tentilD eqution @PFQIAD the unorrelted potentil @RFQA is otined y tking the limit σ → 0 nd V 0 → ∞ with γ = V 2 0 σ √ 2πF yf ourse the dynmis of the wve pket for positive nd negtive k 0 is perfetly symmetriF ithout loss of generlityD from here on we onsider only k 0 > 0F por the ske of simpliityD we ssume in the following tht the wve pket n e pproximted y qusiEmonohromti wve pket @iFeF its dynmis is supported y only one energy ompoE nentAF his pproximtion is justi(ed in the limit of n initilly nrrow momentum distriutionD k 0 σ 1D nd of wek energy rodening y the disorder @k 0 1AF o mke it lerD we dopt the lnguge of setion QFP nd write the energy distriution P ( ) s

P ( ) = W (x, p)A p ( )dxdp, @RFRA
where A p ( ) is the spetrl funtion nd W (x, p) is the igner distriution of the initil wve pket @RFIAX

W (x, p) = 1 π exp -x 2 /σ 2 exp -(p -k 0 ) 2 σ 2 . @RFSA
nder the ssumption k 0 σ 1D we n simplify the initil igner distriution to

W (x, p) 1 √ πσ exp -x 2 /σ 2 δ(p -k 0 ), @RFTA
suh tht eqution @RFRA redues to P ( ) = A k 0 ( )F fy further ssuming wek disorder @k 0 1AD the spetrl funtion n e pproximted y its free form A k 0 ( ) δ( -2 k 2 0 /2m)D suh tht P ( ) δ( -2 k 2 0 /2m)F ynly one energy omponent @ 0 = 2 k 2 0 /2mA is indeed t plyF sn setion RFSD we will ome k more rigorously on this nd rely diretly on the ssumptions k 0 σ 1 nd k 0 1F

he disorder prmeter k 0 introdued ove involves the sttering men free pth = k 0 τ /mD relted to the sttering men free time τ D given y @susetion QFIFQA τ = k 0 3 2mγ @RFUA t the forn pproximtionF e fous primrily on the visul se of wve pket whih is initilly muh smller thn the men free pthF es it turns outD wht we disuss in the following holds s well for wve pket roder thn the men free pthD s shown in susetion RFTFQF 4.2 Classical approach o get (rst insight on the dynmis of the kiked wve pketD let us strt with lssil onsidertionsF o tht endD we (rst relte the enter of mss to the verge momentumD through ihrenfest theoremX

∂ t x = p m . @RFVA
sn this nottionD the symol . refers to oth the quntum expettion vlue nd the disorder vergeF iqution @RFVA provides good strting point in tht the evolution of the momentum distriution tkes rther simple form in the lssil frmeworkF sndeedD under the onditions introdued in setion RFID toms hve either positive momentum k 0 or negtive onek 0 D nd they re sttered from one momentum stte to the other t rte 1/2τ @see susetion QFIFQ for more detilsAD leding to two oupled foltzmnn equtions for the evolution of the verge popultion in

k 0 @n + A nd -k 0 @n -AX dn + dt = n - 2τ -n + 2τ , dn - dt = n + 2τ -n - 2τ ,
@RFWA with of ourse n + + n -= 1F hese two equtions re strightforwrd to solveF ith the initil ondition n + = 1D we (nd

n + = 1 + e -t/τ 2 nd n -= 1 -e -t/τ 2 . @RFIHA lugging this result k in eqution @RFVAD ∂ t x class = k 0 m (n + -n -) , @RFIIA
nd using x(t = 0) = 0D we (nd the following lssil solution for the enter of mssX

x class = 1e -t/τ . @RFIPA ithin lssil pprohD the initil llisti motion of the enter of mss thus quikly stuE rtes t sttering men free pthF ss this the full storyD or does enderson loliztion modify this intuitive ehviorc o nswer this questionD we propose to perform numeril simultionsF fefore doing soD we note tht eqution @RFIPA pplies in ny dimensionF 4.3 Numerical solution his setion eing devoted to numeril tehniqueD we set to I to lighten the nottionsF o propgte numerilly the initil wve pketD eqution @RFIAD we strt y disretizing the spe on gridF he vplin in the hrödinger eqution thus eomes disrete oneF sn order to orretly desrie the ontinuous limitD the disretiztion should e (ne enough to pture the sptil vritions of the wve funtionF hen using orrelted potentilsD one should lso mke sure tht the disretiztion llows to resolve the orreltion lengthF he spe eing of (nite sizeD we hve to impose oundry onditionsF hroughout this thesisD we use periodi oundry onditionsF xote tht we use lrge enough systems for the wve pket not to reh the oundriesF he wy rndom potentils re numerilly generted ws disussed in susetion PFPFRF st remins to perform the evolutionD for tht we follow IPWD IQHF he ide is to tke dvntge of the forml solution of the hrödinger eqution s

|ψ(t) = Û (t) |ψ(t = 0) = e -i
dx T n (x)T m (x) √ 1 -x 2 = π 2 δ n,m (2 -δ n,0 ), @RFIRA
nd they n e determined itertively using IQI

T n+1 (x) = 2xT n (x) -T n-1 (x), T 0 (x) = 1D T 1 (x) = x. @RFISA
smportntlyD the gheyshev polynomils form omplete sisD nd the gheyshev series of given funtion f X

f (x) = n c n T n (x), @RFITA onverges to f D for x ∈ [-1, 1]D if f is
R = E max -E min 2 , @RFIUA G = E min , @RFIVA Ĥnorm = Ĥ -(R + G) R , @RFIWA
where E min nd E max re respetively lower nd n upper ound of the rmiltonin spetrum @suh ounds lwys exist in the disrete (nite systems used in numeril simultionsAF ith these de(nitionsD the eigenvlues of Ĥnorm re in [-1, 1] nd Û (t) = e -i Ĥt = e -i(R+G)t e -iR Ĥnormt . @RFPHA e n now write e -iR Ĥnormt s sum of gheyshev polynomilsX

e -iR Ĥnormt = ∞ k=0 a k T k ( Ĥnorm ).
@RFPIA he a k re determined from the gheyshev polynomils orthogonlity reltions @RFIRAD

a k π 2 (2 -δ k,0 ) = 1 -1 dx T k (x)e -iRxt √ 1 -x 2 = π(-i) k J k (Rt), @RFPPA
where J k denotes the k th order fessel funtion of the (rst kindF

Practical implementation

xumerillyD we nnot sum the in(nite series in @RFPIAD we hve to stop t some pointF his is where the hoie of gheyshev polynomils proves usefulD sine for lrge enough kD a k deys exponentilly with k IHIF et this stgeD two importnt omments re in orderF pirstD the e0ieny of the proedure strongly depends on the ounds on the rmiltonin spetrum tht one is le to provide E max nd E min in eqution @RFIWAF esolute ounds re not di0ult to otin y minimizingGmximizing seprtely the kineti nd potentil energiesX

E abs min = V min nd E abs max = 1 2m π ∆x 2 + V max @RFPQA
where V min @V max A is the smllest @lrgestA vlue tken y the given reliztion of the rndom potentil nd ∆x is the grid stepF roweverD it is generlly more e'etive to use tighter ounds E min nd E max D whih n e estimted from

( Ĥ -E abs max ) n |ψ 2 ∼ n→∞ E min -E abs max 2n , @RFPRA ( Ĥ + E abs min ) n |ψ 2 ∼ n→∞ E max + E abs min 2n , @RFPSA
with |ψ n ritrry stte with non vnishing projetion on the eigenstte with lowest @respF highestA eigenvlue of ĤF sn prtieD t wek disorder we use the eigenstte minimizing or mximizing the kineti energyD while t strong disorderD the eigensttes minimizing nd mximizing the potentil energy re usedF e ssume tht the symptoti ehviors @RFPRA nd @RFPSA re rehed when we oserve n essentilly geometril progression @iFeF when the rtio of two onseutive vlues is roughly onstntAF o e on the sfe sideD E min @E max A is eventully redued @inresedA y few perentsF e seond importnt point is the question of numeril stilityF sndeedD omputing the gheyE shev polynomils in @RFPIA itertively using eqution @RFISA up to high order my led to numeril instilitiesF o void suh troulesD it is generlly preferle to divide the full evolution into smll time stepsF sndeedD the mximum order in the gheyshev expnsion dereses when the time step deresesF he mximum order in the gheyshev expnsion eing inversely proportionl to the time step J k (Rt) in eqution @RFPPA tkes nonEnegligile vlues for k RtD the division of the full evolution in smll time steps does not impt muh the performnesF e hve lso notied tht rerrngement of the series @RFPIA tends to impt the numeril stility s well s the performnesF rtillyD we hve oserved tht rerrnging the series @RFPIA under the form

kmax k=0 a k T k ( Ĥnorm ) = kmax k=0 b k Ĥk norm , @RFPTA
with time steps llowing for k max to e round PHD is oth numerilly stle nd e0ientF iquE tion @RFPTA simply rerrnges the gheyshev polynomils @eFgF T

1 = xD T 2 = -1 + 2x 2 A in monomils @eFgF -1D xD 2x 2 AF

Results for x

e re now in position to perform the numeril simultions for x F o meet the onditions of setion RFID we disretize the rmiltonin on Ih grid of size 16000π/k 0 D divided into PSIQSP grid points @roughly (ve grid points per k -1 0 AF he initil wveEpket width is set to σ = 10/k 0 D nd γ = 5.8 10 -3 4 k 3 0 /m 2 @k 0 86.5AF he resulting enter of mssD verged over RSHHH disorder reliztionsD is shown in (gure RFI s lue pointsD with sttistil error rsF es visile in (gure RFID while t short times the initil llisti motion is well desried y the lssil predition @RFIPA @in greenAD the enter of mss experienes retrore)etion insted of sturting t men free pth @quntum oomerng e'etAF his surprising ehvior is due to enderson loliztionD s demonstrted in the next setionF e note nevertheless tht the lssil predition is n exellent pproximtion t short timesF sndeedD ntiipting on the results presented in setion RFSD we hve ompred the lssil prediE tion @RFIPA to the ext shortEtime expnsion eqution @RFTWA in (gure RFIF yne needs to keep the (rst six terms of the shortEtime expnsion @mgent urveA to improve upon the lssil preditionF

< x(t) > /

Classical approach Numerics Short-time series up to t 5 Short-time series up to t 6 @A pigure RFI ! genter of mss x s funtion of timeD respetively in units of the sttering men free pth nd timeD of wve pket initilly lunhed with (nite veloity in rndom potentilF he lssil preditionD eqution @RFIPAD shown s green urveD is ompred with numeril simE ultions shown s lue pointsF pigure AX fter n initil llisti motionD the pket enterEofEmss experienes retrore)etionD in shrp ontrst with the lssil expettionF pigure AX fous on short timesD with omprison etween the lssil predition nd the ext shortEtime expnsion eqution @RFTWAD the lssil predition is n exellent pproximtion in this limitD six terms of the shortEtime expnsion @mgent urveA re needed to improve upon itF he prmeters used in the numeril simultions re given in the min textD in susetion RFQFRF 4.4 Convergence of the density to its innite-time limit he reson why quntum wve pkets ehve so di'erently n e understood y the following rgumentF et ny timeD the density distriution n e expnded over the eigensis

{ n , |φ n } of H s |Ψ(x, t)| 2 = n,m φ n |Ψ k 0 Ψ k 0 |φ m φ n (x)φ * m (x)e -i( n-m)t/ .
@RFPUA ine eigensttes re lolizedD the system is onstrined to volume set y the loliztion length ξ = 2 IQPF his de(nes typil men level sping ∆ = 1/(ρξ) @ρ is the density of sttes per unit volumeAD with orresponding reisenerg time τ H = 2π /∆ = 4τ eyond whih the o'Edigonl osilltory terms n = m in eqution @RFPUA vnishD levingX

|Ψ(x, ∞)| 2 = n | φ n |Ψ k 0 | 2 |φ n (x)| 2 , @RFPVA
where the overr (. . . ) denotes verging over the rndom potentilF hue to timeEreversl inE vrineD the lolized eigensttes φ n n e hosen relF purthermoreD we hve

| φ n |Ψ k 0 | 2 = | φ * n |Ψ * k 0 | 2 = | φ n |Ψ -k 0 | 2 X
eqution @RFPVA is independent of the sign of k 0 F iqution @RFPVA thus oinides with the longEtimeD sptilly symmetriD verged density distriution tht would hve een otined with n initil wve pket hving symmetri veloity distriutionF his shows tht the enter of mss must return to its initil position t long timesD s result of enderson loliztionF sn order to lrify whih spei( ehvior of the sptil distriution |Ψ(x, t)| 2 tully gives rise to the quntum oomerng e'etD we show in (gure RFP the x > 0 @lue urveA nd x < 0 @red urveA omponents of the sptil pro(le |Ψ(x, t)| 2 D otined numerilly t three suessive timesF es shown oveD this distriution is expeted to onverge towrd symmetri oneD eqution @RFPVAD whih oinides with the (nl distriution ssoited to n initil wve pket of the form Ψ k 0 (x) ∝ exp(-x 2 /2σ 2 ) cos(k 0 x) @iFeF stte hving momentum distriution symmetri with respet to k = 0AF es it turns outD this (nl distriution is the soElled qogolin density pro(le IRX

|Ψ(x, ∞)| 2 = ∞ 0 dηπ 2 32 η 1 + η 2 2 sinh(πη)e -(1+η 2 )|x|/8 [1 + cosh(πη)] 2 . @RFPWA
iqution @RFPWA is derived in IR using ferezinskii digrmmti tehnique IQF he ltter is disussed in detils in setion RFS nd pplies in the onditions detiled in setion RFI @qusiE monohromti nrrow initil wve pket nd wek disorderAF iqution @RFPWA is lso shown in (gure RFP for omprisonF et short timesD the initil llisti motion is visile s pek moving with veloity v 0 = k 0 /mF efter this pek hs een ttenutedD one sees tht the retrore)etion does not stem from rigidD forth nd k motion of the wve pketD ut rther from proess of reEsymmetriztion of the shpe of the pro(le round x = 0F pigure RFP ! everge density pro(le otined numerilly t three di'erent timesF he solid upper lue nd lower red urves re the x > 0 nd x < 0 omponents of the pro(leD respetivelyF he longEtime limit of the pro(leD eqution @RFPWAD is shown s dshed lk urveF rving lri(ed the origin of the quntum oomerng e'etD we now would like to nlyze the enter of mss t (nite times from n nlytil perspetiveF 4.5 Center-of-mass motion from Berezinskii diagrammatic technique o lighten the nottionsD we use = 1 throughout this setionF 4.5.1 Center of mass in terms of Green functions o egin withD we express the quntity of interestD here x D in terms of qreen funtions @the reder not fmilir with qreen funtions my (nd susetion QFIFI usefulAF his is most onveniently done y (rst mking use of the irenhfest theoremD ∂ t x = p /mD whih leds to

p = dp 2π |ψ(p, t)| 2 p = i 4π dx a dx b dp ∂ xa -∂ x b e -ip(xa-x b ) ψ(x a , t)ψ * (x b , t). @RFQHA henD we express ψ(x a , t) respF ψ * (x b , t) in terms of G R (x a , x c , t) respF G A (x b , x d , t)X ψ(x a , t) = dx c G R (x a , x c , t)Ψ k 0 (x c ) respF ψ * (x b , t) = dx d G A (x b , x d , t)Ψ * k 0 (x d ) .
@RFQIA e n perform the integrl over pD equl to 2πδ(x ax b )D to (nd

p = i 2 dxdx c dx d ∂ xa -∂ x b G R (x a , x c , t)G A (x b , x d , t) xa=x b =x Ψ k 0 (x c )Ψ * k 0 (x d ). @RFQPA
st is esier to work in the frequeny domin thn in time dominD we thus introdue the pourier trnsforms

G R (x a , x c , ω 1 ) = dte iω 1 t G R (x a , x c , t) nd G A (x b , x d , ω 2 ) = dte -iω 2 t G A (x b , x d , t). @RFQQA
ghnging vriles to nd ω through ω 1 = nd ω 2 = -ωD we otin

p(ω) = i 4π dxdx c dx d d ∂ xa -∂ x b G R (x a , x c , )G A (x b , x d , -ω) xa=x b =x Ψ k 0 (x c )Ψ * k 0 (x d ),
@RFQRA where p(ω) = dte iωt p(t) F he pourier trnsform of the enter of mssD x(ω) = dte iωt x(t) D immeditely follows from

∂ t x(t) = p(t) /m ⇔ x(ω) = - 1 imω p(ω) . @RFQSA
iqution @RFQRA is fully generl nd does not rely on ny ssumptionF 4.5.2 Diagrammatics es we hve seen in susetion QFIFID qreen funtions re very onvenient for perturtive lE ultionsF e perturtive tretment is however insu0ient hereD euse the quntum oomerng e'et is due to enderson loliztionD whih is in essene nonEperturtive phenomenonF he ferezinskii digrmmti tehnique IQ llows for systemti resummtion of the perturtive seE riesD thus providing n nlytil desription of enderson loliztion in one dimensionF ferezinskii digrmmti tehnique opertes in frequeny rnge where ω D hene a priori desriing only times longer thn 1/ F xote thtD ntiipting on the results presented elowD we n reple y 0 D suh tht the wek disorder limit k 0 1 n equivlently e written 0 τ 1D nd the ondition t > 1/ 0 does not impose nything on t/τ F Diagrammatic representation of G R (x a , x c , )G A (x b , x d ,ω) e strt y expnding the qreen funtionD under the form of forn series eqution @QFTAD

G A/R (x, x , ) =G A/R 0 (x -x , ) + dx 1 G A/R 0 (x -x 1 , )V (x 1 )G A/R 0 (x 1 -x , )+ dx 1 dx 2 G A/R 0 (x -x 1 , )V (x 1 )G A/R 0 (x 1 -x 2 , )V (x 2 )G A/R 0 (x 2 -x , ) + . . . , @RFQTA where G A/R 0 (x -x , )
is the qreen funtion of the freeEprtile rmiltoninD given y

G A/R 0 (x -x , ) = ±i m k e ∓ik|x-x | , @RFQUA
with k = √ 2m F he following formul will e useful in wht followsX

G A 0 (x -x , -ω) = i m k e -ik(1-ω 2 )|x-x | when |ω| . @RFQVA
he di'erent terms ppering in the rightEhnd side of eqution @RFQTA re represented s diE grmsX the unverged digrms onsist of two prtile lines going respetively from x c to x a nd x d to x b @respetively for G R nd G A AF ih line onsists of segments (x , x 1 ), . . . , (x i , x i+1 ), . . . , (x n , x) Y the segments re the free qreen funtions while the points x i refer to the ftors V (x i )F efter vE ergingD these ftors group together in pirs @y pplition of ik theoremAD eh pir eing ssoited with the potentil orreltion funtion @RFQAF sn the digrmsD the potentil orreltion funtions re represented y wvy lines nd free qreen funtions y ordinry @for the retrded qreen funtionD G R A nd dshed lines @for the dvned qreen funtion G A AF en exmple of suh digrm is shown in (gure RFQ @for the ske of lrity with

x a = x b = x nd x c = x d = x AF x x x 1 x 2 x 3 x 4 x 5 x 6 x 7 L Z R
pigure RFQ ! ixmple of digrm ontriuting to G R (x, x , )G A (x, x ,ω)F he wvy lines repreE sent the potentil orreltion funtionD the ordinry @dshedA lines represent the free qreen funtions for the retrded @dvnedA qreen funtionF he vertil unfolding is neessry to represent the diE grmD the vertil diretion hs no sustneF Initial vertices sn prinipleD to sum ll the digrmsD we should onsider ll possile initil vertiesF hey re depited in (gure RFRF nder the wekEdisorder ondition introdued in setion RFI @k 0 1AD only two of them mtterF o exhiit these relevnt initil vertiesD the (rst step is to order the sttering points x i @over whih the integrtion is performed to lulte digrmA s

-∞ < x 1 ≤ • • • ≤ x i ≤ x ≤ x i+1 ≤ • • • ≤ x j ≤ x ≤ x j+1 ≤ • • • ≤ x n < ∞.
@RFQWA ith this orderingD the free qreen funtions re now ftorle thnks to the (xed sign of x ix j over the region of integrtion given y eqution @RFQWAF por exmpleD if

x i > x j D we hve G R 0 (x i -x j , ) = -i m k e ik|x i -x j | = -i m k e ikx i -i m k e -ikx j . @RFRHA
his llows us to formlly ssoite the (rst ftor to the vertex x i nd the seond one to the vertex x j F e generlize this proedure to ll G R 0 nd G A 0 lines of the digrmsD therey trnsferring the dependene on the x i 9s from the lines to the vertiesF sn eh digrmD we n now onsider the initil verties seprtely from the restF he di'erent initil verties re shown in (gure RFRF erties RFR nd RFRd rry rpidly osillting ftor e ±ik(xc+x d ) D integrting it mkes negligile ontriution to G R G A F e thus keep only initil verties RFR nd RFRF D llowing us to perform the integrl over x cx d in eqution @RFQRAF o this endD we hnge vrile to r = x cx d nd x = (x c + x d )/2 in eqution @RFQRA @we lso use eqution @RFQSA to express the enter of mss x in terms of the momentum expettion vlue p AX

x c x d @A x c x d @A x c x d @A x c x d
x(ω) = - 1 4πmω dxdx drd ∂ xa -∂ x b G R (x a , x + r/2, )G A (x b , x -r/2, -ω) xa=x b =x × Ψ k 0 (x + r/2)Ψ * k 0 (x -r/2).
@RFRIA he rnge of Ψ k 0 eing muh smller thn the sttering men free pthD we n ssume tht no sttering event tkes ple etween

x c nd x d D nd ftorize e ±ikr in the produt G R G A s illustrted in (gure RFSX G R (x a , x + r/2, )G A (x b , x -r/2, -ω) e ∓ikr G R (x a , x , )G A (x b , x , -ω), @RFRPA
with the upper sign for the vertex RFR nd the lower sign for the vertex RFRF e re now in position to perform the integrtion over rF por verties RFR nd RFR we otinX

dre ∓ikr Ψ k 0 (x + r/2)Ψ * k 0 (x -r/2) = 2πk m ∞ 0 dpW (x , ±p)δ( -p 2 /2m), @RFRQA x c x d x e -ikr @A x c x d x e ikr @A
pigure RFS ! elevnt verties t the strting pointX opertion of hnging vriles from (x c , x d ) to (r, x ) through r = x cx d nd x = (x c + x d )/2D under the ondition |r| @no sttering event etween x c nd x d AF where we hve introdued @we remind tht k

= √ 2m A 1 = k m ∞ 0 dpδ( -p 2 /2m), @RFRRA
nd the igner distriution of the initil wve pket W (r, p) eqution @RFSAF e identify δ(p 2 /2m) s the free spetrl funtion A 0 p ( ) @see susetion QFIFR for more detilsAF lugging equtions @RFRPA nd @RFRQA in eqution @RFRIAD we otin

x(ω) = -dxdx dpd k 2m 2 ω ∂ xa -∂ x b G R (x a , x , )G A (x b , x , -ω) xa=x b =x ×   W (x , p) vertex 4.4a + W (x , -p) vertex 4.4b    A 0 p ( ).
@RFRSA xote tht the free spetrl funtion A 0 p ( ) ppers here insted of the verge spetrl funtion A p ( ) in the presene of the rndom potentilF his is due to our pproximtion of no sttering event etween x c nd x d IIIF enywyD the free spetrl funtion A 0 p ( ) is n exellent pproximtion of A p ( ) in the wek disorder limit we re onsideringF nder the onditions of setion RFI A

p ( ) = A 0 p ( ) = δ( -p 2 /2m) nd W (x , p) = δ(x )δ(p -k 0 )D eqution @RFRSA simpli(es to x(ω) = - k 0 2m 2 ω dx ∂ xa -∂ x b G R (x a , 0, 0 )G A (x b , 0, 0 -ω) xa=x b =x
, @RFRTA with vertex RFR s only possile initil vertex in the digrmmti representtion of G R G A F iquE tion @RFRTA onstitutes our strting point for the lultion of x with ferezinskii digrmmti tehniqueF o summrizeD the onditions llowing us to simplify equtions @RFQRA nd @RFQSA to equE tion @RFRTA re iA the qusiEmonohromtiityD whih mkes the integrl over dominted y 0 D iiA the initilly nrrow wve pketD llowing for Ψ k 0 (x) δ(x)D iiiA the kiking termD exp(ik 0 x)D together with the ssumption k 0 σ 1D whih impose n initil motion to the right @iFeF initil vertex RFR onlyAF Selection of relevant diagrams por the ske of generlityD we momentrily forget the restrition to the initil vertex RFR in the digrmmti representtion of G R G A D nd inlude the initil vertex RFR s wellF sn order to evlute G R G A extlyD one should in priniple sum ll possile digrmsD whih is formidle tskF es for initil vertiesD t wek disorder @k 0 1A only sulss of digrms mttersF o exhiit these digrmsD we onsider the intervls delimited y the suessive sttering points of eqution @RFQWA nd to eh ssoite numer pir (g, g )D where g nd g re the numer of G R 0 nd G A 0 lines in the intervlF por exmpleD to the digrm in (gure RFQ orresponds the sequene @from left to rightA @PDPAD@QDQAD@QDIAD@IDIAD@IDIAD@PDPAD@RDRAD@PDPAF hen going from one intervl to neighoring oneD the numers g nd g undergo de(nite hnges ∆g nd ∆g whih re uniquely determined y the type of vertex seprting the two intervlsF hi'erent types of verties re displyed in (gure RFTF es n exmpleD the digrm f in (gure RFT orresponds to ∆g = ∆g = 2F uh vertex ppers in (gure RFQ etween the intervls t the left nd t the right of x 1 F e re now in position to distinguish two fmilies of vertiesF yn the one hndD the (rst fmily hs the property tht ∆g = ∆g F es n e seen from eqution @RFQVAD in the limit ω/ 0 → 0D the verties of this fmily re phselessF yn the other hndD the remining verties rry phse e i(∆g -∆g)k 0 x F sntegrting this rpidly osillting ftor mkes negligile ontriution to G R G A F e thus keep only digrms mde of verties hving the property ∆g = ∆g F uh verties re represented in (gure RFT @for the inner onesA nd RFU @for the externl onesAF

a(a') b(b') c(c') d e f
pigure RFT ! hi'erent inner verties tht form the relevnt digrms when ω 0 F he verties 9D9 nd 9D not shown on the (gureD di'er from the verties D nd y hving dshed rown lines in ple of ordinry linesF he vertex @9A orresponds to term of the form

G 0 (x i-1 -x i )V (x i )G 0 (x i - x i )V (x i )G 0 (x i -x i+1 )F o those verties orrespond the following ftorsX -γ(m/k 0 ) 2 D9DD9DD9D γ(m/k 0 ) 2 dD γ(m/k 0 ) 2 exp(iωk 0 x i / 0 ) eD γ(m/k 0 ) 2 exp(-iωk 0 x i / 0 ) fF x a Γ +,. x b Γ -,. x c Γ .,+ x d Γ .,-
pigure RFU ! hi'erent verties t the strting point x nd the ending point x entering the relevnt digrms when ω 0 F e denote suh digrms y Γ ±,± D the left susript ± indites the type of x vertex @ or AD while the right susript indites the type of x vertex @ or dA @eFgF digrm ontining nd will e denoted Γ -,+ AF o those verties orrespond the following ftorsX

(m/k 0 ) exp(-iωk 0 x /(2 0 )) D (m/k 0 ) exp(iωk 0 x /(2 0 )) D (m/k 0 ) exp(iωk 0 x/(2 0 )) D (m/k 0 ) exp(-iωk 0 x/(2 0 )) dF
x in terms of diagrams goming k to our initil prolemD the lultion of x through eqution @RFRTAD we see tht two importnt points should e tken into ountX iA we should only keep digrms with initil verties of type RFR nd iiA we should ount for the derivtives with respet to the initil pointF iA mounts to keeping only digrms of type Γ +,. in (gure RFU nd iiA simply introdues preftors @from the ption of (gure RFRD ∓2ik 0 for Γ .,± AF king iA nd iiA into ountD we otin

x(ω) = - 1 iω d(x -x ) Γ +,+ (x -x ) -Γ +,-(x -x ) , @RFRUA
we derive the eqution @RFSHA king the limit δx → 0D we otin

Rm (x -δx) = Rm (x) + γ 4   -4mδx
- d Rm (x) dx = γ 4 -2m 2 Rm (x) + m 2 Rm-1 (x)e iωx + m 2 Rm+1 (x)e -iωx .
@RFSIA his eqution n e solved under the form Rm (x) = e iωmx R m D with R m solution of

iνR m + m (R m+1 + R m-1 -2R m ) = 0 for m > 0 nd R 0 = 1, @RFSPA
where ν = 4ω/γF por Rm (x )D the proedure is identil with x + δx insted of xδx nd e ↔ f (gure RFTF yne (nds

d Rm (x ) dx = γ 4 -2m 2 Rm (x ) + m 2 Rm -1 (x )e -iωx + m 2 Rm +1 (x )e iωx , @RFSQA iFeF Rm (x) = Rm (-x)F por Z .,m (
dx = ± iω 2 Z .,m (x) + γ 4 m 2 e -iωx Z .,m-1 + (m + 1) 2 e iωx Z .,m+1 -(m 2 + (m + 1) 2 )Z .,m ,
@RFSSA ± depending on the kind of ending vertexD + for Γ .,+ ndfor Γ .,-F Z m,. (x ) is in turn found equl to Z .,m (-x)F prom the initil nd (nl verties presented in (gure RFUD we see tht the Γ introdued in equtions @RFRWA n e expressed s

Γ ++ Γ +- Γ -+ Γ -- = ∞ m =0 ∞ m=0 Rm Z m ,m Rm Rm Z m ,m Rm+1 Rm +1 Z m ,m Rm Rm +1 Z m ,m Rm+1
. @RFSTA Some simplications vet us ome k to eqution @RFRWAF e (rst onsider the ontriution of Γ +,+ (x < x)F e express it through3

x <x d(x -x)e ik(x -x) Γ +,+ (x -x) = 4γ ∞ m =0 R m (ω) Q++ m (ω, k), @RFSUA x x -δx 1 2 . . . 2m -1 2m 1' 2' . . . 2m -1 2m x x -δx 1 2 . . . 2m -1 2m 1' 2' . . . 2m -1 2m x x -δx 1 2 . . . 2m -1 2m 1' 2' . . . 2m -1 2m x x -δx 1 2 . . . 2m -1 2m 1' 2' . . . 2m -1 2m x x -δx 1 2 . . . 2m -1 2m 1' 2' . . . 2m -1 2m x x -δx 1 2 . . . 2m -1 2m 1' 2' . . . 2m -1 2m x x -δx 1 2 . . . 2m -1 2m 1' 2' . . . 2m -1 2m x x -δx 1 2 . . . 2m -1 2m 1' 2' . . . 2m -1 2m = + + + + + + + . . . Rm (x -δx) Rm (x) a(a') b(b') c(c') d e f
pigure RFV ! hemti representtion of the proess of onstruting the right hnd prts nd equE tion @RFSHAF he rightEhnd prts re represented with retngles Y the urves re the prtile lines inside the rightEhnd prt nd the entrl prtF he digrms 9D9 nd 9 re the sme s D nd with the vertex elow the xExisF ynly one exmple for eh kind of vertex is representedD summtion over ll possile hoie for inoming nd outgoing legs is impliedF

whereD ording to @RFSTAD R m Q++ m (ω, k) = γ 4 ∞ m=0 ∞ 0 dre -ikr Rm (x -r)Z m ,m (x -r, x) Rm (x). @RFSVA Q++ m is then de(ned s Q++ m (ω, k) = γ 4 ∞ m=0 x -∞ dx e ik(x -x) e -iωm x Z m ,m (x , x)e iωmx R m .
@RFSWA e rell tht Rm (r) = e -iωm r R m D with R m independent of rF o derive n eqution for Q++ m (ω, k)D we strt y the following integrtion y prtX

x -∞ dx e ik(x -x) e -iωm x Z m ,m (x , x)e iωmx R m = e ik(x -x) e -iωm x ik -iωm Z m ,m (x , x)e iωmx R m x -∞ - x -∞ dx e ik(x -x) e -iωm x ik -iωm dZ m ,m (x , x) dx e iωmx R m .
@RFTHA UP ghpter RF untum oomerng e'et in oneEdimensionl rndom potentils e hve the ondition Z m ,m (x , x ) = δ m ,m D nd the eqution for Z m ,m @from eqution @RFSSA nd

Z m,. (x ) = Z .,m (-x)A - dZ m ,. dx = iω 2 Z m ,. + γ 4 m 2 e iωx Z m -1,. + (m + 1) 2 e -iωx Z m +1,. -(m 2 + (m + 1) 2 )Z m ,.
. @RFTIA lugging it in eqution @RFTHAD we (nd the sought for eqution

R m - 4ik γ Q++ m + 4iω γ m + 1 2 Q++ m + (m + 1) 2 ( Q++ m +1 -Q++ m ) -m 2 ( Q++ m -Q++ m -1 ) = 0. @RFTPA he sme proedure is then rried out for Γ +,-(x < x)F he ontriutions from Γ +,+ (x < x) nd Γ +,-(x < x) re gthered through the introdution of Q 1 m (ω, k) = Q++ m -Q+- m F Γ +,+ (x > x) nd Γ +,-(x > x)
give the sme ontriution with n opposite kD Q 1 m (ω, -k)F qthering the ontriutions from the Γ +,. we otin

x(ω) Γ +,. = - 2 iωγ ∞ m=0 R m Q 1 m (ω, k = 0) + Q 1 m (ω, -k = 0) , @RFTQA with P 1 m -iκQ 1 m + iν m + 1 2 Q 1 m + (m + 1) 2 (Q 1 m+1 -Q 1 m ) -m 2 (Q 1 m -Q 1 m-1 ) = 0, @RFTRA
where we hve introdued ν = 4ω/γD κ = 4k/γ nd P 1 m = R m -R m+1 F edding the ontriution from Γ -,. D we eventully otin

x(ω) = - 2 iωγ ∞ m=0 P 1 m Q 1 m (ω, k = 0) + Q 1 m (ω, -k = 0) . @RFTSA
he prolem is redued to solving equtions @RFSPA for R m D @RFTRA for Q 1 m nd summing them s presried y eqution @RFTSAF he solution of eqution @RFSPA hs n expliit form IQQX R m (s) = sΓ(m + 1)Ψ(m + 1, 2; -s), @RFTTA where Ψ is the on)uent hypergeometri funtion of the seond kindD nd s = iνF st is sometimes more onvenient to work with the integrl representtion of R m D whih reds

R m = -iν ∞ 0 dse iνs s s + 1 m = -iν ∞ 0 dse iνs 1 + s -1 -m
. @RFTUA iqution @RFTUA n e diretly heked from eqution @RFSPA through integrtion y prtF yn the other hndD to the est of our knowledge no losed form solution for Q 1 m existsF xote tht the derivtion of qogolin density pro(le eqution @RFPWA presented in IR involves n eqution kin to @RFTSAF wore preiselyD the in(niteEtime density pro(le is similrly written s sum over

P 0 m nd Q 0 m F P 0 m = (R m + R m+1 )/2
is the ounterprt of P 1 m D nd the eqution for Q 0 m di'ers from @RFTRA only through the upper indexF he in(niteEtime limit then llows to pproximte m y ontinuous vrileD mking the lultion trtleF

Results

he prolem is now only of mthemtil ntureF o extrt the long time symptoti ehvior of x D one n use the generting funtions of R m nd Q 1 m to derive n integroEdi'erentil eqution for x F his integroEdi'erentil eqution nnot e solved in full generlity ut n e used to extrt the limit of long timesF his progrm is rried out in ppendix RFeD the result writes

x = 64ln(t/4τ )τ 2 t 2 + O 1 t 2 . @RFTVA
iqution @RFTVA is shown in (gure RFWD s solid red urveD nd is in exellent greement with the numeril simultions t long timesF e n lso solve eqution @RFTRA for Q 1 m y mking use of n expnsion in powers of tF he method is presented in ppendix RFf nd llows to extrt the shortEtime expnsion of x up to ritrry orderF ueeping only the (rst few termsD we otin the shortEtime series

x = t τ - t 2 2τ 2 + t 3 6τ 3 - 3t 4 64τ 4 + 7t 5 576τ 5 - 629t 6 207360τ 6 + O t 7 .
@RFTWA he series hs (nite onvergene rdius @estimted t 4τ from the (rst IHH termsAF iqution @RFTWA is ompred with numeril simultions t shortEtimes in (gure RFID it desries very well the numeril results up to t ≈ τ F o desrie the enter of mss eyond the onvergene rdius of the shortEtime series @RFTWAD we propose dé resummtionF he knowledge of the longEtime limit @RFTVA suggests to express x t ny time under the form

x = ln(1 + t/4τ )τ 2 t 2 lim N →∞ N n=0 a n t/τ n N n=0 b n t/τ n Padé approximant
, @RFUHA the oe0ients a n nd b n re dedued from the shortEtime series @RFTWAF sn prtieD @RFUHA onverges quiklyD nd n exellent pproximtion of x for times up to 120τ is otined with N = 7F his is demonstrted y the solid green urve in (gure RFWD whih perfetly oinides with the numeril resultsF 4.6 General case o frD we hve only onsidered the se of qussin unorrelted potentilF es disussed in setion PFPD this model for the rndom potentil is not relisti for oldEtom experimentsF st ws hosen here to llow for the full nlytil tretment of setion RFSF xeverthelessD in the wek disorder limitD kind of universlity with respet to the potentil onEsite distriution nd orreltion funtion is expeted RPF 4.6.1 Case of a non-Gaussian potential he onEsite distriutionD if resonleD is not expeted to hve ny qulittive e'et on the dynmisF his is indeed wht we oserve in (gure RFIHD where the numeril simultions using lueEdetuned spekle unorrelted potentil 4 re in exellent greement with our predition for qussin potentilF his is in greement with the oneEprmeter sling theory setion IFQD whih sttes tht the dynmis depends on the mirosopi detils only through the nonEinterferentil @lssilA limit nd τ hereD see eqution @RFIPAF

Analytics -long times Numerics

@A pigure RFW ! pigure AX enter of mss x of wve pket initilly lunhed with (nite veloity in rndom potentilD s funtion of timeF he nlytil resultsD eqution @RFTVA for the long time symptoti nd the dé pproximnt @RFUHAD respetively shown s red nd green urvesD re ompred with numeril simultions shown s lue pointsD with sttistil error rsF snset of (gure AX di'erene etween the dé pproximnt of x nd the numeril result @∆x = x greenx blue A s funtion of timeF pigure AX enter of mss multiplied y (t/τ ) 2 s funtion of timeF he symptoti result @RFTVA red urve is ompred to the numeril preditionF he prmeters used in the numeril simultions re given in susetion RFQFRF o llow for more urte omprisonD we hve used more preise estimtes @only y roughly perentA of the sttering men free pth nd timeD = 4 (1cos 2 (k 0 a))/2m 2 a 2 γ nd τ = ma/ sin(k 0 a) respetively IQRD where a = 0.2/k 0 is the lttie sping used in the numeril simultionsF tust like @RFUAD these forms re vlid to lowest order in the potentilD the only di'erene is tht they tke the sptil disretiztion into ount @RFUA orresponds to the limit a → 0F 4.6.2 Case of a correlated potential he introdution of orreltions requires some reF sndeedD the sttering on orrelted poE tentils is desried y two time slesD the sttering @τ A nd trnsport men free time @τ b AF he ltter is de(ned s the verge time it tkes for n tom to experiene signi(tive hnge in its diretion of propgtionF sn the se of n unorrelted potentilD the sttering is isotropi nd the trnsport men free time is equl to the sttering men free timeF sn strongly orrelted potentils in ontrstD sttering is strongly nisotropi nd the trnsport men free time well exeeds the sttering men free time IQSF gorrespondinglyD the trnsport men free pth @ b = k 0 τ b /mA is the verge time trveled y the toms efore experiening signi(tive hnge in their diretion of propgtionF es it turns outD ferezinskii equtions n e derived for orrelted potentilsD the net result is tht the replement (τ, ) → (τ b , b ) fully ounts for the orreltions IRF his is gin fully onsistent with the oneEprmeter sling theoryD s here the lssil limit is set y b nd τ b @in setion RFPD desriing the lssil limitD the replement (τ, ) → (τ b , b ) is immediteAF hen performing the simple replement (τ, ) → (τ b , b ) to desrie the orrelted seD one should ensure tht the vritions of τ b nd b with energy re not too importnt in the energy rnge overed y the wve pketF sndeedD s disussed in setion QFPD not ful(lling this ondition n led to surprisesF es n exmpleD one n imgine hving energy omponents with shrply distint loliztion lengthsD s n e the se in spekle potentils RUF sf this hppensD it ould led to n pprent hlt of the quntum oomerng e'etD witing for the retrore)etion of slow energy

< x > /

Analytics -Padé approximant Numerics

@A norrelted spekle potentil pigure RFIH ! genter of mss s funtion of time for two types of rndom potentilsF he numeril results for qussin orrelted potentil @A nd n unorrelted potentil with onEsite lueE detuned spekle distriution @A re shown in lueD with their error rsD nd re ompred to the nlytil predition otined in setion RFSF por AD the potentil orreltion funtion is given y @RFUIAF he spekle distriution @A orresponds to n exponentil distriutionD de(ned only for positive V F por A respF AD we hve used disorder strength V 0 = 0.1 2 k 2 0 /m respF γ = 0.0058 4 k 3 0 /m 2 F sn oth sesD we hve used n initil wve pket of size σ = 10/k 0 F he results re verged over PSHHH respF ISHHH disorder reliztions for A respF AF sn the orrelted se @AD the orreltion length σ c is equl to 0.5/k 0 F o gurntee tht the orreltions re orretly resolvedD we hve used thinner disretiztion of roughly IH points per k -1 0 in the orrelted se @A thn in the unorrelted se @A for whih roughly S points per k -1 0 were su0ientF rnsport @A nd sttering @A men free time nd pth re otined y (tting the short times with the lssil predition @RFIPA @with nd τ repled y τ b nd b for AD see min text for more detilsAF he (tted vlues 87 nd b 96 re lose to the preditions of perturtion theory 86 from eqution @RFUA nd b 100 from eqution @RFUPAF xote tht in the orrelted se @AD b 1.5 F omponentsF o void the pperne of suh e'etsD we hve hosen to use orrelted potentil tht hs smooth vritions of the trnsport men free pth nd timeF his potentil is desried y n exponentil orreltion funtionX

V (x)V (x ) = V 2 0 exp - |x -x | σ c , @RFUIA
from whih we infer the sttering nd trnsport men free pth t the forn pproximtion QX

= 4 k 2 0 m 2 V 2 0 2σ c + 2σ c 1 + 4k 2 0 σ 2 c -1 nd b = 4 k 2 0 2m 2 V 2 0 1 + 4k 2 0 σ 2 c 2σ c . @RFUPA
pigure RFIH shows tht in this seD the simple replements (τ, ) → (τ b , b ) in the results of setion RFS is indeed su0ient to desrie the orrelted seF 4.6.3 Case of a broad initial wave packet e hve so fr restrited ourselves to the se of wve pket initilly muh smller thn the men free pthF ht hppens to the quntum oomerng e'et if we relx this ssumptionc o UT ghpter RF untum oomerng e'et in oneEdimensionl rndom potentils nswer this questionD we elorte on the resoning leding to eqution @QFQPAF e ssume tht the initil wve pket plys only role through its onvolution with propgtorX

|ψ(x, t)| 2 = dx P (x -x, t)|Ψ k 0 (x )| 2 ,
@RFUQA with P (x -x, t) the proility for prtile initilly t x to reh x in time spn t nd |Ψ k 0 (x )| 2 the initil wve pketF he enter of mss follows from integrtion over x weighted y xF he hnge of vrile x = xx then llows to immeditely onlude tht the width of the initil wve pket plys no roleF o vlidte the ove resoningD we hve performed numeril simultions strting from wve pkets muh roder thn the men free pthF he results re reported in (gure RFII nd on(rm the minor role @if nyA plyed y the initil wve pket width on the quntum oomerng e'etF

< x > /

Analytics -Padé approximant Numerics pigure RFII ! genter of mss s funtion of time in the se of n initilly rod wve pket @σ AF he enter of mss otined numerillyD strting from rod wve pketD is shown in lueD with its error rs nd is ompred to the nlytil predition otined in setion RFSF he numeril prmeters mth the ones used for the numeril results of (gure RFWD exept for the initil width of the wve pketD equl to σ = 750/k 0 9 F he sttering men free pth @ A nd time @τ A re lulted using the formul given in the ption of (gure RFWF sn onlusionD lthough rod wve pket ppers essentilly motionless t the sle of its sizeD (ne nlysis revels tht it experienes the sme quntum oomerng e'et thn nrrow wve pketF 4.7 A simple relation between x and x 2 snterestinglyD the similrity of the lultion of x nd of the urrentEurrent orreltion funE tionD oserved in susetion RFSFPD is rooted in simple reltion etween x nd x 2 @note tht x 2 is diretly relted to the di'usion oe0ient nd thus to the ondutivityD whih in turn is relted to the urrentEurrent orreltion funtionAF o exhiit this onnetionD we strt y pplying ihrenfest theorem to the menEsqure displementD ∂ t x 2 = x 2 , p 2 /(2i m)D nd split the prtile disE triution into two lsses of positive nd negtive veloitiesX |Ψ(x, t)

| 2 = n + (x, t) + n -(x, t)F his RFVF gonlusion UU leds to 5 ∂ t x 2 = 2v 0 x + -2v 0 x -, @RFURA where v 0 = k 0 /mF rere x ± = ∞ -∞ x n ± (x, t) dxD with oviously x = x + + x -F [...] [...] [...] [...] a) b) c) d)
pigure RFIP ! A eritrry multiple sttering pth going from x = 0 to x, ontriuting to x - @the pth is unfolded to the top for lrityAF fy timeEreversing nd trnslting this pth y -xD we otin pth AD whih gives n opposite ontriution to x -D ensuring tht x -vnishesF A th ontriuting to x + F sts timeEreversed nd trnslted ounterprt dA strts with momentum -k 0 nd is thus not populted t t = 0D so tht x + n e nonEvnishingF e now onsider n ritrry pth ontriuting to x -(gure RFIPF he pth strts t x = 0 with momentum k 0 nd rehes x with momentumk 0 t time tF fy timeEreversing nd trnslting this pth of distne -xD one n lwys (nd omplementry pth strting with momentum k 0 t x = 0 nd rehing -x t time t (gure RFIPF hue to timeEreversl nd trnsltionl invrineD these two pths ontriute with the sme weight to n -(x, t)D whih is thus n even funtion of xD yielding x -a HF his resoning does not pply to x + sine the timeE reversedGtrnslted ounterprt of n ritrry pth ontriuting to x + strts y onstrution with momentumk 0 whih is not initilly populted @see (gures RFIPEdAF e hve thus shown

∂ t x 2 = 2v 0 x . @RFUSA

Conclusion

sn this hpterD we hve onsidered the evolution in Ih disordered potentil of n initil wve pket rrying (nite veloityF e hve (rst oserved numerilly thtD unexpetedlyD fter n initil llisti motionD the pket enterEofEmss experienes retrore)etion nd slowly returns to its initil position @quntum oomerng e'etAF e hve ttriuted this phenomenon to enderson loliztionF he ore of this hpter onsisted in pplying ferezinskii digrmmti theory IQ to this senrioF e hve een le to derive the longEtime symptoti return of the pket enterEofEmss to its initil positionF purtherD we hve presented new method to solve ferezinskii equtions under the form of dé pproximntD llowing for desription of the enterEofEmss motion t ll timesF e hve lso derived n intriguing reltion etween the enter of mss nd the men squre displementD whih turns out to e t the root of our lultionsF e hve then tested the roustness our results with respet to hnge in the sttistis of the rndom potentil nd of the size of the initil wve pketF e hve found thtD t lest for wek disorderD the sttistis of the potentil @inluding onEsite distriution nd possile orreltionsA does not visily 'et the enterEofEmss motionF imilrlyD the enterEofEmss motion ppers independent of the width of the initil wve pketF he resoning presented in setion RFRD nd explining the quntum oomerng e'et in terms of rel lolized modesD pplies in ny dimensionF he quntum oomerng e'et is thus expeted to tke ple in higher dimensions s wellD provided the dynmis is enderson lolizedF his o'ers interesting perspetives for future workF por exmpleD long with reent works on the oherent kE wrd nd forwrd sttering peks ross enderson trnsition RRD IPVD it would e very interesting to perform (ne numeril nlysis of the quntum oomerng e'et in three dimensionsF enother interesting diretion of reserh would e to look for similr phenomen in other symmetry lsses @eFgF in the unitry lss where the qulittive understnding of retrore)etion in terms of real We unveil a novel and unexpected manifestation of Anderson localization of matter wave packets that carry a finite average velocity: after an initial ballistic motion, the packet center-of-mass experiences a retroreflection and slowly returns to its initial position. We describe this effect both numerically and analytically in dimension 1, and show that it is destroyed by weak particle interactions which act as a decoherence process. The retroreflection is also present in higher dimensions, provided the dynamics is Anderson localized. Anderson localization (AL), the absence of wave diffusion due to destructive interference in disordered potentials [1], is ubiquitous in condensed-matter systems, wave physics or atom optics. This offers many experimental platforms for its characterization, as was recently demonstrated experimentally with light [2,3] (see however [4,5]) or ultrasound waves [6]. Very recently, AL of atomic matter waves has also been observed [7][8][9][10][11], as well as its many-body counterpart [12,13]. A precious asset of atom optics experiments is to allow for direct tests of fundamental manifestations of AL, such as the time evolution of wave packets. In this context, a common experimental scenario for probing localization consists in preparing a spatially narrow atomic wave packet in a trap, then opening the trap and recording the time evolution of the gas [14,15]. After it has been released, the wave packet spreads symmetrically around its initial position and, after a transient ballistic expansion, quickly becomes localized in space. What happens, now, if a nonzero average velocity is additionally imprinted to the gas? In a naive picture, one expects the randomization of velocities due to scattering on the random potential to stop the initial ballistic motion of the wave packet centerof-mass (CoM) at roughly a mean free path , and then a symmetric localization of the packet around this new central position due to AL. We show in this Letter that the evolution is in fact very different. Quite unexpectedly, after an initial ballistic motion where the CoM indeed increases up to , the wave packet slowly returns to its initial position, recovering a symmetric shape at long times. The final state of the system is in turn identical to the one that would have been reached if no velocity had been transferred to the gas.

In this Letter, we thoroughly study this phenomenon both numerically and analytically. In dimension 1, we give an exact solution to this problem. We then study the CoM retroreflection in the presence of a weak nonlinearity describing particle interactions in the mean field approximation. The CoM freezes at a position that depends on the strength of interactions, very much like a decoherence process.

Let us consider a one-dimensional (1D) system described by the Hamiltonian H = -2 ∆/(2m) + V (x), where V (x) is a Gaussian, uncorrelated random potential: V (x) = 0 and V (x)V (x ) = γδ(xx ), where the overbar denotes averaging over disorder realizations. We wish to study the time evolution of a normalized Gaussian wave packet, Ψ k0 (x) ∝ exp -x 2 /(2σ 2 ) + ik 0 x , to which a finite momentum k 0 is imprinted. We choose k 0 > 0 without loss of generality. To simplify the discussion, we assume throughout this Letter a sharp initial velocity distribution, k 0 σ 1, and weak disorder, k 0 1, thereby allowing for a simple description of the wave packet in terms of two velocity components 6), is shown as a solid red curve, and the re-summation of the short-time series, Eq. ( 10), as a solid green curve. The latter perfectly overlaps with the numerical result (blue dots). The dashed curve is the classical result, Eq. (1). Inset: center of mass multiplied by (t/τ ) 2 as a function of time. The asymptotic result (6) (red curve) is compared to the numerical prediction, displayed with its statistical error bars. The parameters used in the simulations are given in the main text.

± k 0 /m, with energy E 0 = 2 k 2 0 /(2m).
The average evolution in the random potential is governed by two microscopic scales, the scattering mean free time τ and the scattering mean free path = v 0 τ , where v 0 = k 0 /m. Throughout this Letter, τ and are calculated to the leading order in 1/(k 0 ) 1, using the Born approximation at energy E 0 [16]. The assumption of uncorrelated random potential is not crucial for our discussion: all the results that follow hold as well for short-range correlated potentials, provided that and τ are replaced by the transport mean free path and time, respectively [17,18].

By numerically propagating Ψ k0 (x), we obtain the disorder-averaged density profile |Ψ(x, t)| 2 , from which we infer the CoM x ≡ x|Ψ(x, t)| 2 dx. The result is shown in Fig. 1 : x first increases rapidly, reaches a maximum at t ∼ τ and then slowly decreases to zero. In other words, after a transient motion rightward, the center of mass of the wave packet slowly returns to its initial position x = 0. For these simulations we discretize the Hamiltonian on a 1D grid of size 16000π/k 0 , divided into 251352 grid points. The initial wave-packet width is set to σ = 10/k 0 , and γ = 0.0058 4 k 3 0 /m 2 (k 0 = 4 k 3 0 /(2m 2 γ) 86.5). The results are averaged over 45000 disorder realizations. In the simulations, the evolution operator is expanded in a series of Chebyshev polynomials, as explained in [19,20]. The surprising behavior observed in Fig. 1 is dramatically different from the classical expectation, which can be simply deduced from Ehrenfest theorem:

∂ t x class = p /m = k 0 (n + -n -)/m
where n ± is the population of particles with momentum ± k 0 (n + +n -= 1). Using the classical Boltzmann equations ∂ t n ± = (n ∓n ± )/(2τ ) with the initial condition n + = 0, we find

x class = 1 -e -t/τ , (1)
which is shown in Fig. 1 as a dashed curve. Within the classical picture, the CoM thus quickly saturates to the mean free path , but never experiences retroreflection. The reason why quantum wave packets behave so differently can be understood by the following argument. At any time, the density distribution can be expanded over the eigenbasis

{ n , |φ n } of H as |Ψ(x, t)| 2 = n,m φ n |Ψ k0 Ψ k0 |φ m × φ n (x)φ * m (x)e -i( n -m)t . (2)
Since eigenstates are localized, the system is constrained to a volume set by the localization length ξ = 2 . This defines a typical mean level spacing ∆ = 1/(ρξ) (ρ is the density of states per unit volume), with a corresponding localization time τ loc = 2π /∆ = 4τ beyond which the off-diagonal oscillatory terms n = m in Eq. ( 2) vanish, leaving:

|Ψ(x, ∞)| 2 = n | φ n |Ψ k0 | 2 |φ n (x)| 2 . (3)
Due to time-reversal invariance, the φ n (x) are real so that φ n |Ψ k0 = φ n |Ψ -k0 * : Eq. ( 3) is independent of FIG. 2. a) A typical multiple scattering path going from x = 0 to x, contributing to x -(the path is unfolded to the top for clarity). The momentum reverses at each scattering event. By time-reversing and translating this path by -x, we obtain path b), which gives an opposite contribution to x -, ensuring that x -vanishes. c) Path contributing to x + . Its time-reversed and translated counterpart d) starts with momentum -k0 and is thus not populated at t = 0, so that x + = 0.

the sign of k 0 , and thus coincides with the long-time, spatially symmetric density distribution that would have been obtained with an initial wave packet having a symmetric velocity distribution. This shows that the CoM must return to its initial position at long times, as a result of AL.

Let us now be more quantitative and analyze the CoM at finite times. For this purpose, we start by applying Ehrenfest theorem to the mean-square displacement, ∂ t x 2 = x 2 , p 2 /(2i m), and split the particle distribution into two classes of positive and negative velocities:

|Ψ(x, t)| 2 = n + (x, t) + n -(x, t). This leads to [21] ∂ t x 2 = 2v 0 x + -2v 0 x -. (4)
Here x ± = ∞ -∞ x n ± (x, t) dx, with obviously x = x + + x -. We now consider an arbitrary path contributing to x -[Fig. 2(a)]. The path starts at x = 0 with momentum k 0 and reaches x with momentum k 0 at time t. By time-reversing and translating this path of a distance -x, one can always find a complementary path starting with momentum k 0 at x = 0 and reaching -x at time t (Fig. 2b). Due to time-reversal and translational invariance, these two paths contribute with the same weight to n -(x, t), which is thus an even function of x, yielding x -= 0. This reasoning does not apply to x + since the time-reversed/translated counterpart of an arbitrary path contributing to x + starts by construction with a momentumk 0 which is not initially populated (see Figs. 2c-d). We have thus

∂ t x 2 = 2v 0 x , (5)
a property that we can use to infer the long-time limit of x from x 2 , which was previously studied in [22]. This yields [START_REF] Gutzwiller | Chaos in Classical and Quantum Mechanics[END_REF] x

= 64ln(t/4τ )τ 2 t 2 + O 1 t 2 . (6)
Eq. ( 6) is shown in Fig. 1 and matches well the exact numerical prediction at long times. The inset of Fig. 1 also confirms the presence of the logarithmic term in Eq. ( 6).

In fact, one can go one step further and exploit Eq. ( 5) to compute x at any time. For this purpose, we use Berezinskii diagrammatic technique [24] which, combined with Eq. ( 5), gives [START_REF] Gutzwiller | Chaos in Classical and Quantum Mechanics[END_REF] x

= dω 2π e -iωt   - 2 iω ∞ m=0 P 1 m (ω)Q 1 m (ω)   , (7)
where P 1 m (ω) = sΓ(m+1)[Ψ(m+1, 2; -s)-(m+1)Ψ(m+ 2, 2; -s)], with s = 4iωτ , Γ the Gamma function and Ψ the confluent hypergeometric function of the second kind.

The Q 1 m (ω) are solutions of [4iτ (m + 1/2)ω -(m + 1) 2 -m 2 ]Q 1 m (ω) +(m + 1) 2 Q 1 m+1 (ω) + m 2 Q 1 m-1 (ω) + P 1 m (ω) = 0. (8)
At short times, one can solve these equations with the expansion Q 1 m (ω) = +∞ n=0 q m,n /(iω) n . To compute the q m,n , we first notice that q m,i = 0 if i ≤ m, which follows from the large-frequency expansion of P 1 m (ω) (which has no terms 1/ω i with i < m). We use this result to expand Eqs. (8) order by order in 1/ω and reduce them to a triangular system. This method provides us with the coefficients χ n of the expansion x = n χ n (t/τ ) n at arbitrary order [18]. We find for instance

x = t τ - t 2 2τ 2 + t 3 6τ 3 - 3t 4 64τ 4 + O t 5 . (9)
The method cannot be directly used to estimate x at any time because the series has a finite convergence radius, estimated at 4τ from the first 100 terms. Nevertheless, the observed exponential decay of the χ n makes this series a good candidate for a Padé resummation. The knowledge of the long-time limit (6) suggests to express the CoM at any time under the form

x = ln(1 + t/4τ )τ 2 t 2 lim n→∞ R n (t), (10)
where R n (t) is a diagonal Padé approximant of order n, deduced from the χ n coefficients [25]. In practice, R n (t) converges quickly, and an excellent approximation of x for times up to 120τ is obtained with n = 7. This is demonstrated by the solid green curve in Fig. 1, which perfectly coincides with the numerical results. In order to clarify which specific behavior of the spatial distribution |Ψ(x, t)| 2 actually gives rise to the phenomenon of retroreflection, we show in Fig. 3 the x > 0 (blue curve) and x < 0 (red curve) components of the spatial profile |Ψ(x, t)| 2 , obtained numerically at three successive times. The profiles display a ballistic peak responsible for the increase of x at short times. After this peak has been attenuated, the profile re-symmetrizes itself around x = 0, which gives rise to the retroreflection phenonemon. As discussed above, this distribution is expected to converge toward a symmetric one, Eq. ( 3), which coincides with the so-called Gogolin density profile [17,26]:

|Ψ(x, ∞)| 2 = ∞ 0 dηπ 2 32 η 1 + η 2 2 sinh(πη)e -(1+η 2 )|x|/8 [1 + cosh(πη)] 2 , (11)
which is shown in Fig. 3 for comparison. Note that although we start from a rather narrow wave packet with σ < in our simulations, the retroreflection phenomenon is present as well when σ > .

We finally discuss the effect of particle interactions on x (t), by considering a weakly interacting, condensed bosonic gas.

Its dynamics is governed, at the mean field level, by the Gross-Pitaevskii equation

i ∂ t Ψ = [-2 ∆/2m + V (x) + g|Ψ| 2 ]Ψ.
For wave packets with k 0 = 0, it was shown that the interaction term g|Ψ| 2 leads to a destruction of AL at very long times, in favor of a regime of subdiffusion where x 2 ∼ t α with α < 1 [27,28]. Here we take a different perspective and investigate numerically how the nonlinearity affects x . For these simulations, we write the evolution operator over a small time step δt as

U (δt) = exp[-ig Ψ(δt) 2 δt/2]exp[-i(-2 ∆/2m + V )δt]exp[-ig Ψ(0) 2 δt/2
] and treat the linear part as before, through an expansion in a series of Chebyshev polynomials. We use a system of size 7500π/k 0 discretized into 23562 grid points, and propagate a wave packet of width σ = 10/k 0 in a random potential of strength γ = 0.0196 4 k 3 0 /m 2 (k 0 18.1). The results are averaged over 600000 (8.85 millions) disorder realizations when g = 0 (respectively g = 0). We show in Fig. 4 the CoM x g as a function of time obtained with this procedure, for two values of g = 0. We observe that x g decreases more slowly than in the non-interacting limit and saturates at a finite value at long times. The CoM retroreflection is thus interrupted by the nonlinearity.

To better understand the role of the nonlinearity, we have also studied how the CoM is affected by decoherence, modeled by the Hamiltonian H = -2 ∆/(2m) + V (x) + V φ (x, t). Here V (x) is the same random potential as above and V φ (x, t) = h(x)f (t), where h(x) has the same statistical properties as V (x) and f (t) is a random, Gaussian distributed function of time with zero average. f (t) is fully characterized by its time-time correlation function which we choose Gaussian,

f (t)f (t ) = A 2 f exp[-(t -t ) 2 /2σ 2 t ],
with A f 1 to ensure that the fluctuating potential is weaker than the static one. We have checked that the potential V φ (x, t) does induce decoherence: at long times, it restores classical diffusion with x 2 = 2D φ t, where D φ = ξ 2 /τ φ = 4 2 /τ φ is the diffusion coefficient and τ φ the decoherence time [17]. Because V φ (x, t) preserves time-reversal and translational invari-ance after a disorder average, Eq. ( 5) still holds for this model, demonstrating that the CoM, x φ , converges to a finite value 4 τ /τ φ at long times. x φ is displayed in Fig. 4 (dashed curves), for two values of A f . The two values of g chosen in the Gross-Pitaevskii model were adjusted so that x g coincides with these two curves x φ in the long-time limit. Surprisingly, the obtained curves x φ match extremely well the nonlinear curves x g in the whole time window. This suggests that at least regarding x and for times short enough for subdiffusion not to play a major role, the nonlinearity acts similarly to a decoherence process. From this observation, we associate to the nonlinearity an effective decoherence time τ φ (g). To find this quantity, we first determine D φ from the evolution of x 2 with time in the model of decoherence, and then find the associated g by matching the curves x φ and x g at long time. The results, shown in the inset of Fig. 4, demonstrate that /τ φ (g) ∝ g/ξ, which can be interpreted as the average interaction energy within a localization volume ξ = 2 . A similar time scale for the dynamical alteration of localization by interactions was found in [29,30].

We expect the retroreflection phenomenon to be a rather general property of systems displaying Anderson localization. In particular, it is not restricted to 1D systems, as can be inferred from a straightforward extension to any dimension of the reasoning leading to Eq. ( 3). We have also numerically checked that the CoM indeed goes back to its initial position in two-dimensional (2D) random potentials, see the dotted curve in Fig. 4, obtained using the 2D version of the potential V (x) (with k 0 2.5). At weak disorder, the decay of x (t) is however much slower in 2D because the localization time is much longer than in 1D. The fact that the retroreflection is significantly affected by weak interactions at short times suggests that it could be advantageously used as a sensitive probe of Anderson localization in interacting disordered systems. Appendix 4.A Solution of Berezinskii equations at long times sn this ppendixD we use the ferezinskii onvention = 2m = k 0 = 1F o otin the longEtime @smll frequenyA symptoti ehvior of x from ferezinskii equtions equtions @RFTRAD @RFTSA nd @RFTTAD it is onvenient to introdue the generting funtions

R(ζ) = ∞ m=0 R m ζ m nd Q(ζ) = ∞ m=0 Q 1 m ζ m . @RFUTA R(ζ
) n e dedued from the losedEform solution of R m eqution @RFTUAD the result writes

R(ζ) = 1 1 -ζ 1 -ζF iν 1 -ζ , F (z) = ze -z ii(z), @RFUUA
where ii(z) is the exponentil integrl funtionF por Q(ζ) we n only infer di'erentil eqution from eqution @RFTRAX

iν ζ dQ dζ + 1 2 Q + (1 -ζ) d dζ ζ d dζ (1 -ζ)Q + P (ζ) = 0, @RFUVA
where

P (ζ) = ∞ m=0 P 1 m ζ m is strightforwrd to dedue from R(ζ)F goming k to eqution @RFTSAD we introdue the solution for R m eqution @RFTUA nd otin x = - 4ν ωγ ∞ 0 dse iνs ∞ m=0 1 s + 1 Q m (ω, k = 0) s s + 1 m = - 4ν ωγ ∞ 0 dse iνs B(s), @RFUWA
where B(s) is then given y

B(s) = 1 s + 1 Q s s + 1 = 1 s + 1 ∞ m=0 Q m (ω, k = 0) s s + 1 m .
@RFVHA prom eqution @RFUVAD we dedue the eqution oeyed y B(s)X

d ds s(s + 1) dB(s) ds + iν s d ds ((s + 1B(s)) + B(s) 2 + iνe -iν(s+1) ii(iν(s + 1)) = 0. @RFVIA
st is onvenient to use the lterntive form

x = - 4ν iω ∞ 0
ds(e iνs -1) dB(s) ds , @RFVPA otined through integrtion y prtF e re now in position to ddress the smll frequeny @νA expnsionF e strt y hnging vrile to u = -iνsD

x = - ν 2 iωγ -i∞ 0 (e -u -1) dB(u) du du. @RFVQA iqution @RFVIA eomes d du u 2 dB(u) du -u d du uB(u) + iν - d du uB(u) + u dB(u) du + B(u) 2 + iνe -iν e u ii(-u + iν) = 0.
@RFVRA RFeF olution of ferezinskii equtions t long times VS e re interested in the few (rst orders in νD we n therefore redily neglet the term enlosed in the resD to simplify eqution @RFVRA to

d du u 2 dB(u) du -u d du uB(u) + iνe -iν e u ii(-u + iν) = 0, @RFVSA
yet it is di0ult mtter to solve this equtionF he wy round this di0ulty is to (rst onsider eqution @RFVIA in the limit |νs| = |u| 1F sndeedD in this limit the terms enlosed in the squre rkets n e negletedD nd using the expnsion ii(z) ≈ ln(-z) + C @where C is the iuler onstntAD we diretly otin

dB(s) ds ≈ -iν C -1 s + 1 + ln (-iν)(s + 1) s + 1 + ln(s + 1) s(s + 1) for |νs| 1.
@RFVTA en expnsion of eqution @RFVTA for lrge s leds to

1 -iν dB(s) ds ≈ C -1 + ln(s) s + d ds ln(s) + C -1 s + ln(s) s 2 + . . . for |νs| 1. @RFVUA qoing k to u = -iνsD we (nd dB(u) du = -iν ln(u) + C -1 u + (iν) 2 d du ln(u) + C -1 u + 1 u 2 ln u -iν + . . . @RFVVA
prom this lst equtionD it is ler tht B(u) s ν expnsion of the form

B(u) = iνB 1 (u) + (iν) 2 ln(-iν) B(u) + (iν) 2 B 2 (u) + . . . , @RFVWA
with the symptoti forms s u → 0

dB 1 (u) du → - ln(u) + C -1 u , d B(u) du → - 1 u 2 , dB 2 (u) du → 2 -C u 2 . @RFWHA st is di0ult to fully determine B 1 (u) nd B 2 (u)F B(u)
is however quite simple to otinF prom the expnsion of the nonhomogeneous term of eqution @RFVRAX d du

u 2 dB(u) du -u d du uB(u) + iν - d du uB(u) + u dB(u) du + B(u) 2 iνe u ii(-u) -(iν) 2 e u ii(-u) + 1 u + O ν 3 = 0, @RFWIA
we identify the eqution oeyed y B(u)D

d du u 2 d B(u) du -u d du u B(u) = 0, @RFWPA
whose solutionD onsistent with eqution @RFWHAD is simply given y B(u) = 1/uF sn the integrl of eqution @RFVQAD the ontriution of the smll u prt is not very ler nd requires some re @divergenes pper due to the ove expnsion for lrge sD whih lerly does not extend to u = 0AF o hndle itD we split the integrl over two regionsX [0, s 0 ] nd [s 0 , ∞]D where s 0 1D ut |ν|s 0 1F e strt with the region s < s 0 D where we n pproximte e iνs -1 y iνs nd use eqution @RFVTAD the integrl is then not di0ult to evlute expliitlyX

1 -iν s 0 0 s dB(s) ds ds ≈ s 0 0 s C -1 s + 1 + ln (-iν)(s + 1) s + 1 + ln(s + 1) s(s + 1) ds = (C -2)s 0 + (s 0 + 2 -C)ln(s 0 + 1) + ln(-iν)(s 0 -ln(s 0 + 1)).
@RFWQA e lrge s 0 expnsion provides

1 -iν s 0 0 s dB(s) ds ds ≈ (C -2)s 0 + (s 0 + 2 -C)ln(s 0 ) -ln(-iν)ln(s 0 ) + . . . . @RFWRA
ixpressing this lst result in terms of u 0 = -iνs 0 D we (nd the following ontriution to x X

4 iωγ iνu 0 (ln(u 0 ) + C -2) -(iν) 2 ln(-iν)(C -2 -ln(u 0 )) + (iν) 2 (C -2)ln(u 0 ) + ν 2 ln 2 (-iν) + . . . .
@RFWSA es for the remining ontriutionD from the region s > s 0 D we go from s to u = -iνs nd use eqution @RFVWAD nd onsider only the ontriution from BD relevnt for the enter of mssF B eing equl to 1/uD in the limit u 0 → 0D we re free to move the integrtion k to the relsD suh tht its ontriution to x is

- 4 iωγ (iν) 2 ln(-iν) ∞ u 0 e -u -1 u 2 = - 4 iωγ (iν) 2 ln(-iν)(ln(u 0 ) + C -1) + O(u 0 ). @RFWTA
he limit u 0 → 0 is now well de(ned @with the ontriutions from B 1 nd B 2 t smll uD the ln(u 0 A nel eh otherAF umming equtions @RFWSA nd @RFWTAD we (nlly otin

x = πα + 16iν γ 2 ln 2 (-iν) + (2C -3)ln(-iν)) + O(ν), @RFWUA
where α is numeril onstnt whih is di0ult to determine using the ove method @it omes from B 1 A nd irrelevnt for the enter of mssF st is given in IQ s α = 32(π 2 -C 3 )/γ 2 F B 2 only ontriutes to O(ν)F he result for the enter of mss otined y pourier trnsforming @RFWUA is

x = 64 ln(t/4τ )τ 2 t 2 + τ 2 2t 2 + . . . @RFWVA
Appendix 4.B Solution of Berezinskii equations from short times o e onsistent with the nottions used in our sumitted pper reprodued in setion RFWD we reintrodue D k 0 nd m in this ppendixF o go eyond the longEtime limit treted in ppendix RFeD we propose to solve the equtions for the enter of mss t ny timeF he solution tkes the form of short time expnsionD

x = ∞ n=0 χ n t τ n , @RFWWA
rerrnged under the form of dé pproximntD

x = ln(1 + t/4τ )τ 2 t 2 lim N →∞ N n=0 a n t/τ n N n=0 b n t/τ n . @RFIHHA
sndeedD it is possile to work out the χ n of eqution @RFWWA up to ritrry orderD ut the resulting series hs (nite onvergene rdiusD estimted t 4τ from the (rst IHH terms6 F yne rerrnged under the form of dé pproximntD s done in eqution @RFIHHAD we oserve numeril onvergene nd the resulting x is in exellent greement with our numeril simultions @(gure RFWAF st is the knowledge of the longEtime limit eqution @RFWVA tht suggests the form @RFIHHAF sn prtieD @RFIHHA onverges quite rpidlyD the results for N = 7 demonstrte no visile di'erene with the results for N = 14 for times up to 120τ F vet us (nd the χ n thenF e rell the expression of the enter of mss eqution @RFTSAX

x = - 2 iω   ∞ m=0 P 1 m Q 1 m   , @RFIHIA
where P 1 m = sΓ(m + 1)[Ψ(m + 1, 2; -s) -(m + 1)Ψ(m + 2, 2; -s)]D with s = 4iωτ D Γ the qmm funtion nd Ψ the on)uent hypergeometri funtion of the seond kindF he Q 1 m re solutions of

[4iωτ (m + 1/2) -(m + 1) 2 -m 2 ]Q 1 m + (m + 1) 2 Q 1 m+1 + m 2 Q 1 m-1 + P 1 m = 0.
@RFIHPA he di0ulty lies in the lultion of the Q 1 m F pollowing the route outlined oveD we write them under the form of lrge frequeny expnsion

Q 1 m (ν) +∞ n=0 q n,m (iν) n .
@RFIHQA end introdue this deomposition in eqution @RFIHPAD with P 1 m redued to its symptoti formD

O 1 (iν) m + m + 1 2 +∞ n=0 q n,m (iν) n-1 + (m + 1) 2   +∞ n=0 q n,m+1 (iν) n - +∞ n=0 q n,m (iν) n   -m 2   +∞ n=0 q n,m (iν) n - +∞ n=0 q n,m-1 (iν) n   = 0.
@RFIHRA por m = 0D we see tht q 0,0 = 0F henD for m = 1D we see tht q 0,1 nd q 1,1 re vnishingF his pttern leds to

q n,m = 0 if n ≤ m ⇔ Q 1 m (ν) = n>m q n,m (iν) n .
@RFIHSA e n now turn to the nonEvnishing termsF o tht endD let us move to n lgorithmi point of viewD nd in this respet rest eqution @RFIHPA to the generl form

(α m ν + β m )Q 1 m + γ m Q 1 m+1 + m 2 Q 1 m-1 + δ 0 m (ν) = 0, @RFIHTA
where α m , β m , γ m nd δ 0 m (ν) re knownF lugging in eqution @RFIHSAD we (nd our si eqution

(α m ν + β m )   n>m q n,m (iν) n   + γ m   n>m+1 q n,m+1 (iν)
n   + m 2   n>m-1 q n,m-1 (iν) n   + δ 0 m (ν) = 0. @RFIHUA
o strt withD one n sn m nd onsider the resulting equtions from @RFIHUA t lowest order in 1/νF ynly α m νQ 1 m nd m 2 Q 1 m-1 ontriuteD they re thus strightforwrd to solveD to (nd ll the q n=m+1,m F yne n then sor the known q n=m+1,m nd δ 0 m (ν) in new quntityD δ 1 m (ν)D to (nd n eqution similr to @RFIHUAX

(α m ν + β m )   n>m+1 q n,m (iν) n   + γ m   n>m+2 q n,m+1 (iν) n   + m 2   n>m q n,m-1 (iν) n   + δ 1 m (ν) = 0. @RFIHVA
e hve highlighted in red the importnt di'erenes etween equtions @RFIHUA nd @RFIHVAF eE peting the ove proedureD one (nds the q n=m+2,m D in turn sored in new δ * m (ν)D nd so on nd so forthF st is then immedite to dedue the χ n of eqution @RFWWAF por prtil purposesD we hve inluded opy of the wthemti IHH noteook used to generte the results presented in (gure RFW in wthemti IHH noteook RFIF xote tht n lterntive numeril solution of ferezinskii equtions exists IQTF nterms a ISY @B xumer of hin omputedF hould e greter thn I BA tmx a IPHY @B wximum time for whih `xb is omputedD in units of the sttering men free time BA tstep a HFHIY @B mple the time with time step tstepF hould e smller thn tmx BA ftr mD s Xa sBGammam C IBHypergeometricUm C ID PD -s ftp mD s Xa ftrmD sftr m C ID s @B mI BA SeriesCoecientftpmD sD {sD InnityD n}Y p a Table7D {mD HD nterms -I}D {nD HD nterms -I}Y @B tore mI lrge lrge nu expnsion for ll m in mtrix BA svp a pY q a Table HD {xD nterms}D {yD nterms}Y @B tore mI lrge lrge nu expnsion for ll m in mtrix BA SumpmD iBqmD jBKroneckerDeltanD i C jD {mD nterms}D {iD nterms}D {jD nterms}D {nD QD nterms C P}Y @B um over m mI mID gives the series in frequeny BA xtime a Table IGPBxomegnB@-IA@nAG@Gamman C IBR@n -IAAD {nD nterms}Y @B a hin BA xt Xa SumxtimenBtnD {nD nterms -I}Y xpde a deepproximnt tPGLogI C tGRBxtD {tD HD Floor@nterms -IAGP}Y txpde a Table{tD xpdeBLogI C tGRGtP}D {tD tstepD tmxD tstep}Y (lenme a 4menxpdeorder4 `b ToStringFloor@nterms -IAGP `b 4Fdt4Y Export(lenmeD txpdeY wthemti IHH noteook RFI ! gomputes x under the form of dé pproximntF he sript outputs two olumn (leD the (rst olumn is the time in units of τ D nd the seond olumn is the orresponding x in units of F he prmeter nterms (xes the numer of χ n omputedF he mximum time n e (xed with the prmeter tmxD nd the time smpling with the prmeter tstepF Chapter 5 Weakly interacting wave packets o frD we hve negleted tomEtom intertions nd disussed only singleEprtile physisF sntertions reD howeverD often present in experimentsD it is thus importnt to hrterize their e'etsF sn the present thesisD we restrit ourselves to dilute ondensed osoni gsesD whih re used in the liseu QV nd plorene RH groupsF he limittion to wekly interting ondensed osoni gses llows us to pproximte the full quntum mnyEody dynmis y nonliner eqution for lssil (eldF etion SFI rells some detils on this pproximtionF e present method to propgte numerilly wve pkets in the presene of nonlinerity in setion SFPF es n illustrtionD we pply this method to n extensively studied senrioX the spreding of n initilly nrrow wve pket in disordered potentilF st is the opportunity to disuss the interply etween enderson loliztion nd intertionsF etion SFQ is devoted to the e'et of the nonlinerity on the quntum oomerng e'et of hpter RF sn setions SFR nd SFSD we trde the Ih initilly nrrow wve pket for Qh plne wvesD with setion SFR fousing on the dynmis of the energy distriution nd setion SFS exmining the e'et of intertions on the oherent k sttering pekF he sujet of wekly interting wve pkets evolving in rndom potentils is vstD the present hpter is only onerned with some spets nd is not intended to over it exhustivelyF usetion SFPFI nd setions SFQD SFR nd SFS present originl results otined during the present thesisF etions SFR nd SFS present onEgoing workF 5.1 Bose-Einstein condensates in random potentials 5.1.1 Many-body Hamiltonian e ommon experimentl senrio for proing loliztion onsists in prepring sptilly nrrow tomi wve pket in trpD then opening the trp to relese the toms in disordered potentil nd reording the time evolution of the gs RID RPF efter it hs een relesedD its dynmis is enoded in the mnyEody rmiltoninD written in seond quntiztionD

Do qID I C i a -PBpID iY DoqmD m C i a @-IG@m -IGPAAB@@m -IAPBqm -ID m -I C i C p mD m C i -IAD {mD PD nterms -iD I} GY @nterms -iA b IY DopmD m C i Ea @mP C @m -IAPAB qmD m C iD {mD nterms -i} GY @nterms -iA b HY DopmD m C i C I Ca mPB qm C ID m C I C iD {mD nterms -i -I} GY @nterms -i -IA b HY D {i D nterms -I} p a svpY xomeg a Table
H = dr Ψ † (r) - 2 ∆ 2m + V ext (r) Ψ(r) + 1 2 drdr V int (r -r ) Ψ † (r) Ψ † (r ) Ψ(r ) Ψ(r), @SFIA
where V ext is the externl @here disorderedA potentil nd V int is the twoEody intertion potentilF he (eld opertor Ψ oeys the nonEtrivil ommuttion reltion @ssuming osoni tomi loudA

Ψ(r), Ψ † (r ) = δ(r -r ). @SFPA WI WP
ghpter SF ekly interting wve pkets sn the reisenerg pitureD the (eld opertor evolves ording to

i ∂ t Ψ(r, t) = Ψ(r, t), H = - 2 ∆ 2m + V ext (r) + dr V int (r -r ) Ψ † (r , t) Ψ(r , t) Ψ(r, t). @SFQA

Bogoliubov approximation

iqution @SFQA is fully generlD ut rther omplitedD it is nonliner eqution for quntum (eldF sn the limit of wekly interting foseEiinstein ondenste @figAD we n proeed with the fogoliuov pproximtionF he (rst step onsists in singling out the ondenste mode in the (eld ΨX Ψ(r, t) = φ(r, t) a 0 + δ Ψ(r, t). @SFRA rere a 0 nnihiltes prtile in the ondenste modeD desried y the wve funtion φ(r, t)F δ Ψ(r, t) ounts for nonEondensed tomsF henD sine the initil stte is figD we expet Ψ(r, t) to e dominted y φ(r, t) a 0 nd we thus neglet δ Ψ(r, t)F pinllyD the fig ontining mny tomsD the opertor a 0 is well pproximted y slr @ a 0 nd a † 0 re pproximtely equl to the squre root of the numer of toms in the ondensteD N AF epling Ψ(r, t) y φ(r, t) √ N in eqution @SFQAD we (nd

i ∂ t φ(r, t) = - 2 ∆ 2m + V ext (r) + N dr V int (r -r )|φ(r , t)| 2 φ(r, t).
@SFSA he dynmis of the fig is desried y the lssil (eld φ(r, t)D the wve funtion of the onE densteD oeying the nonliner eqution @SFSAF xote tht φ(r, t) is normlized to unityF

Two-body low energy collisions and scattering potential

o simplify the prolem even furtherD we n use the ft thtD t the very low energies we re onsideringD the sttering properties of the toms through the full intertion potentil V int (rr ) re enoded in single length sleD the sttering length a IQUF sn the dilute limitD it is thus su0ient to model the intertion potentil y potentil reproduing the pproprite sttering lengthF he simplest hoie is

V int (r -r ) = 4πa 2 m δ(r -r ) = g N δ(r -r ).
@SFTA ustituting eqution @SFTA into eqution @SFSAD we otin the fundmentl eqution of the present hpterD the qrossEitevskii equtionD desriing the evolution of the ondenste wve funtionX

i ∂ t φ(r, t) = - 2 ∆ 2m + V ext (r) + g|φ(r, t)| 2 φ(r, t).
@SFUA he mnyEody evolution eqution @SFQA is thus rest under the form of singleEprtile hrödinger eqution eqution @SFUA with n dditionl nonliner potentilD g|φ(r, t)| 2 D desriing kind of self intertionF he ltter pproximtes the intertion of n tom with the other toms in the ondenste in menE(eld wyF es we will see lter onD the nonliner nture of the qrossE itevskii eqution mkes the ssoited physis very rihD s the nonlinerity n hve widely di'erent e'ets depending on the initil onditionsF 5.1.4 Comments on the Gross-Pitaevskii equation fefore exploring the onsequenes of the nonlinerityD two omments re in orderF pirstD even though we gve here somewht simplisti derivtion of the qrossEitevskii eqution @SFUAD one should not onlude tht this hpter is on thin ieF sndeedD the qrossEitevskii eqution n e derived rigorously IQV nd turns out to explin very well mny experiments involving wekly interting old osons @see IQW for reviewAF xeverthelessD the qrossEitevskii eqution uE rtely desries the evolution of the ondenste wve funtion onlyD nd in prtie some toms re inevitly not ondensedF st is possile to tke them into ount pproximtelyD for instne within fogoliuov theory UH!UTD IRHF e will however sty t the qrossEitevskii level in this thesisF xote tht the nonEondensed frtion my inrese in time through the depletion of the ondenE steF o the est of our knowledgeD this issue ws only ddressed in ontexts tht di'er slightly from the ones onsidered in this hpterF o e more spei(D s opposed to USD UTD IRH where the toms re ondensed in the lowest energy stteD we onsider the evolution of toms ondensed in ritrry wve funtionsF enother exmple is the study of trnsport of ondensed toms through disordered wveguidesD in tomElserElike on(gurtions UPF sn these ontextsD for wek enough potentils nd intertionsD the depletion of the ondenste remins smllF tillD even though the qrossEitevskii eqution @SFUA represents formidle simpli(tion of the full mnyEody dynmis eqution @SFQAD relile results remin generlly di0ult to otin nlytillyF xumeril simultions of the qrossEitevskii eqution in the wekly interting limit therefore turn out quite vluleF 5.2 Numerical integration of the Gross-Pitaevskii equation his setion eing devoted to numeril tehniqueD we set to I to lighten the nottionsF e hve seen in setion RFQ tht the hrödinger eqution n e0iently e solved numerilly y mens of gheyshev polynomil expnsionF he qrossEitevskii eqution @SFUA is a priori not well dpted to this method euse of the timeEdependent term g|φ(r, t)| 2 F sn this thesisD howeverD intertions re lwys onsidered s wek perturtion of the disordered rmiltoninD suh tht it is tempting to integrte eqution @SFUA similrly to the usul hrödinger equtionD with only smll modi(tion ounting for the nonlinerityF 5.2.1 Numerical scheme and error estimate o integrte the qrossEitevskii eqution numerillyD we show in the present susetion tht the evolution opertor over smll time step n e onveniently seprted into liner opertor nd two simple nonliner termsF o this endD we strt from the forml solution

|φ(t) = U (t) |φ(t = 0) = T exp -i t 0 dt H GP (t ) |φ(t = 0) , @SFVA
where

H GP (t) = p 2 2m + V ext (r) + g|φ(r, t)| 2 = H g=0 + g|φ(r, t)| 2 , @SFWA
nd T is the timeEordering opertorF e proeed with ylor expnsion of the timeEdependent term

g|φ(r, t )| 2 = g |φ(r, t = 0)| 2 + ∂ t |φ(r, t )| 2 t =0 t + O(t 2 ). @SFIHA
he key point is to notie tht the term liner in t in eqution @SFIHA n e tken out of the exponentil in eqution @SFVAD with only n O(t 3 ) errorX

T exp -i t 0 dt H GP (t ) = exp -ig ∂ t |φ(r, t)| 2 t=0 t 2 2
exp -i H GP (t = 0)t + O(t 3 ). @SFIIA sndeedD terms rising from the nonEommuttivity of ∂ t |φ(r, t)| 2 t=0 t nd the kineti term re O(t 3 ) @ t 2 omes from the two integrls over time nd t from the time dependene of the nonliner prtAF o further simplify @SFIIAD we use the symmetrized rotter formul IRIX τ dis τ g where the time sles re respetively ssoited to the kinetiD disorder nd intertion termsF rtillyD we wnt to reh long times @t τ g A to see the e'ets of the nonlinerityD while the numeril simultions re onstrined to desrie orretly the kineti dynmis @t τ k AF o mke the prolem more trtleD we hve hosen to relese the onstrint of desriing the ontinuous sitution s fithfully s possile nd simply onsider the lttie sitution in susetion SFPFQ nd setion SFQF he orresponding Ih rmiltonin writes

exp -i H g=0 + g|φ(r, t = 0)| 2 t = exp -i g|φ(r, t = 0)| 2 2 t
H a = i - φ i+1 + φ i-1 -2φ i 2ma 2 + V i + g|φ i | 2 , @SFIUA
where a is the lttie spingD φ i = φ(ai)D nd the sum runs over the lttie sitesF xote tht throughout the thesisD lttie is used in the numeril simultionsF roweverD in the rest of the thesisD we lwys hoose lttie sping muh smller thn the smllest length sle of the prolemF st is this ondition tht is relxed in susetion SFPFQ nd setion SFQD we expet the results presented there to desrie the ontinuous sitution t qulittive level onlyF xote tht in the nonEinterting limitD there is little di'erene etween ontinuous nd disrete systemD s fr s loliztion on lrge sptil sles is onerned IHVF hroughout this hpterD we use n unorrelted qussin potentil of strength set y γ @in dimension dAX V i V j = γδ i,j /a d , @SFIVA with δ i,j the uroneker deltF he rest of this hpter is devoted to the pplition of the present numeril pproh to the qrossEitevskii eqution in vrious situtionsF henever possileD we will underline the qulittive physil piture t plyF etions SFPFQD SFQ nd SFRFI disuss the interply etween loliztion nd nonlinerity in the Ih seF prom susetion SFRFP onwrdD on the other hndD we fous on the Qh geometry in the di'usive regimeF 5.2.3 Nonlinearity and spreading o illustrte the ove numeril methodD we propose to onsider the ommon experimentl senrio for proing loliztion in oldEtom experimentsF st onsists in relesing in rndom potentil n initilly nrrow tomi wve pketF enderson loliztion is trked in the evolution of the men squre displement x 2 of the wve pketF sndeedD s opposed to di'usionD for whih x 2 grows linerly in timeD enderson loliztion ounds x 2 to roughly the squre of the loliztion lengthF sn this setionD we ddress the question of how the nonlinerity modi(es this ehviorF xote tht this question hs lredy een extensively studied numerilly nd theoretilly IRP!IRVF Numerical experiment pollowing IRPD we propgte numerilly wve pket initilly pled t x = 0 @φ 2 (x, t = 0) = δ x,0 /aD with δ x,0 the uroneker deltA in strong rndom potentilF he resulting men squre displement x 2 s funtion of time is shown in (gure SFQD nd unmiguously displys sudi'usion t long timesX

x 2 ∝ t α with 0 < α < 1. @SFIWA he sudi'usive exponent @the slope of the dshed line in the (gureA is pproximtely equl to HFPWD in greement with IRPF his sudi'usive ehvior ws lso oserved numerilly in IRQ!IRV nd experimentlly in SHF udi'usion is however predited to rekdown t extremely long timesD repled y slower spreding @if nyA @see SID SP for rigorous proofs nd SQD SR for possile physil mehnismsAF e however never reh this regime in the present thesis @with possily the exeption of the results presented in (gure SFQAF

Numerics

pigure SFI ! wen squre displement s funtion of timeF sn logElog sleD the long time limit is well (tted y stright lineD hene demonstrting sudi'usionF he ssoited sudi'usive exponent @α in @SFIWAD iFeF the slope of the dshed line in the (gureA is pproximtely equl to HFPWF por these simultionsD we hve used nonliner strength g = 0.5 2 /ma nd disorder strength γ = 0.5625 4 /m 2 aF e time step δt = 0.04 ma 2 / ws found su0ientF he results re verged over ISHHH disorder reliztionsF Discussion he nonlinerity thus 'ets enderson loliztionD without destroying it ompletely @whih would result in the restortion of pure di'usive spredingAF he reson for this mixed ehvior lies in trdeEo' etween destrution of loliztion y the timeEdependent nonlinerity on one hndD nd dilution of the nonlinerity itself y di'usive ehvior on the other hndF sn the litertureD one (nds two pprohes ttempting t quntifying this trdeEo'F yne trend is to tret the nonlinerity s mixing the liner lolized modes nd hene restoring trnsport y llowing the toms to jump from one lolized mode to nother @see RVD RW for reent reviewsAF he mixing depending on the strength of the nonlinerityD it would dispper t long times if di'usion previledF enother pproh tht hs een proposed onsists in generlizing the selfEonsistent theory of loliztion @see setion IFI for more detilsA to the nonliner se IRWF rere s well loliztion would not e destroyed y the nonlinerity if di'usive trnsport were restoredF foth pprohes predit tht sudi'usion is the ondition to lne destrution of loliztion nd dilution of the nonlinerityF elthough the numeril vlue of the sudi'usive exponent α in @SFIWA is still detedF yn the one hndD the generlized selfEonsistent theory of loliztion predits α = 1/2F yn the other hndD the mixing of liner lolized modes pper to e dependent on the hoti properties of the qrossEitevskii hin @SFIUA IRRF hese hoti properties re only prtilly understood RWDSUDISHD resulting in di'erent preditions for the sudi'usive exponentD eFgF α = 1/2 IRRD ISID α = 2/5 IRSD IRU or α = 1/3 IRQD IRRD ISPD ISQF @SFPHA e show in (gure SFP the resulting enter of mss x g s funtion of timeD for two vlues of g = 0F e oserve tht x g dereses more slowly thn in the nonEinterting limit nd seems to sturte t (nite vlue t long timesF he enterEofEmss retrore)etion is thus interrupted y the nonlinerityD the nonliner quntum oomerng does not ome k to the originF por these simultionsD we used system of size 7500π/k 0 disretized into PQSTP grid pointsD nd propgte wve pket of width σ = 10/k 0 in rndom potentil of strength γ = 0.0196 4 k 3 0 /m 2 @k 0 18.1AF he results re verged over THHHHH @VFVS millionsA disorder reliztions when g = 0 @respetively g = 0AF e time step etween 1.47 m/( k 2 0 ) @for the lrgest gA nd 2.27 m/( k 2 0 ) @for the smllest gA ws found su0ientF 0. 028gm/ 2 k 0 @A pigure SFP ! pigure AX enter of mss x g s funtion of timeD for g = 0 @solid lower lue urveAD g = 0.09 2 k 0 /m @solid middle ornge urveA nd g = 0.15 2 k 0 /m @solid upper mgent urveAF wo dshed urves show the enter of mss x φ otined with the model of deohereneD for A f = 0.02 nd 0.03 @from ottom to topAF he sttering men free pth @ A nd time @τ A re lulted t the forn pproximtionD using the formul given in the ption of (gure RFWF pigure AX τ /τ φ (g)D well (tted y liner regressionF he e'etive deoherene time τ φ (g) ssoited with the nonlinerity thus ehves like /τ φ (g) ∼ g/ F he prmeters used in the numeril simultions re given in the min textF

Comparison with decoherence

o etter understnd the role of the nonlinerityD we hve lso studied how the enter of mss is 'eted y decoherenceD modeled y the rmiltonin

H = - 2 ∆ 2m + V (x) + V φ (x, t). @SFPIA rere V (x) is the usul rndom potentil nd V φ (x, t) = h(x)f (t), @SFPPA
where h(x) hs the sme sttistil properties s V (x) nd f (t) is rndomD qussin distriuted funtion of time with zero vergeF f (t) is fully hrterized y its timeEtime orreltion funtion whih we hoose qussinD

f (t)f (t ) = A 2 f exp[-(t -t ) 2 /2σ 2 t ], @SFPQA
with A f 1 to ensure tht the temporlly )ututing potentil is weker thn the stti oneF por this modelD we use the sme numeril sheme s for the qrossEitevskii eqution with g|φ(x, t)| 2 repled y V φ (x, t) nd tke σ t = 4 m/( k 2 0 ) nd time step δt = 0.735 m/( k 2 0 )F e hve heked tht the potentil V φ (x, t) does indue deohereneX t long timesD it restores lssil di'usion with x 2 = 2D φ tD where D φ = ξ 2 /τ φ = 4 2 /τ φ is the di'usion oe0ient nd τ φ the deoherene time IRF feuse V φ (x, t) preserves timeEreversl nd trnsltionl invrine fter disorder vergingD eqution @RFUSA still holds for this modelX x = ∂ t x 2 2v 0 . @SFPRA ine x 2 = 2D φ t t long timesD x φ onverges to the (nite vlue 4 τ /τ φ @we hve used = v 0 τ AF x φ is displyed in (gure SFP @dshed urvesAD for two vlues of A f F he two vlues of g hosen in the qrossEitevskii model were djusted so tht x g oinides with these two urves x φ in the longEtime limit1 F urprisinglyD the otined urves x φ mth extremely well the nonliner urves x g in the whole time windowF his suggests tht t lest regrding x D the nonlinerity ts similrly to deoherene proessF prom this oservtionD we ssoite to the nonlinerity n e'etive deoherene time τ φ (g)F o (nd this quntityD we (rst determine D φ from the evolution of x 2 with time in the model of deohereneD nd then (nd the ssoited g y mthing the urves x φ nd x g t long timeF he resultsD shown in (gure SFPD demonstrte tht /τ φ (g) ∝ g/ξD whih n e interpreted s the verge intertion energy within loliztion volume ξ = 2 F e similr time sle for the dynmil ltertion of loliztion y intertions ws found in IRUD IRWF xote tht the sudi'usion regime ensuing the dynmil ltertion of loliztion @see susetion SFPFQA is only visile for times muh longer thn τ φ (g) IRUD IRWD wheres (gure SFP present times of order τ φ (g) @eFgF for g = 0.15 2 k 0 m ! mgent urve in (gure SFP ! τ φ (g) ≈ 250τ AF 5.4 Nonlinearity and energy distribution e hve seen in setion QFP tht in the liner seD it is quite nturl to express the dynmis of given wve pket s sum over its energy omponentsF por exmpleD the density t time t is onveniently written s |ψ(r, t)| 2 = d P ( )Φ (r, t), @SFPSA where P ( ) is the verge wve pket energy distriution nd Φ (r, t) desries the dynmis of the energy omponent F sn the nonliner seD the energy distriution my vry in timeD due to the timeEdependene of the nonliner termF his setion ims t studying the evolution of its vergeF e de(ne the verge energy distriution t time t s

P ( , t) = φ(t)|δ( -H)|φ(t) , @SFPTA
where H is the qrossEitevskii rmiltonin t time t nd φ(t) the wve funtion t time tF fefore studying the evolution of the energy distriutionD let us provide some motivtion through seond look t the spreding senrio of susetion SFPFQF 5.4.1 Motivations: a second look at spreading Numerical experiment sn susetion SFPFQD we hve used n initil wve pket of the form φ 2 (x) = δ x,0 /a nd strong rndom potentilF e propose to onsider now wve pket of the form φ(x) ∝ cos(k 0 x)e -x 2 /2σ 2 , @SFPUA evolving in wek rndom potentilF hese onditions o'er more ontrol over the initil energy distriutionD sine t wek disorder the kineti energy domintes nd the momentum distriution of @SFPUA n e tunedF sn prtiulrD in the limit k 0 σ 1D the momentum distriution nd thus the energy distriution ! see susetion QFIFR for more detils of @SFPUA is shrply peked round k 0 2 k 2 0 /2mF he resulting men squre displement x 2 is shown in (gure SFQ s funtion of time for two vlues of k 0 F yn the one hndD the results for k 0 = π/4a @(gure SFQA unmiguously disply sudi'usion t long timesD x 2 ∝ t α D with sudi'usive exponent @the slope of the dshed line in the (gureA α 0.3F yn the other hndD the se k 0 = π/2a @(gure SFQA ppers lerly di'erentD with muh smller sudi'usive exponent @α 0.19A nd possile devition from sudi'usion t long timesF wore work is however needed to fully hrterize this possile devition from sudi'usionF Numerics @A k 0 = π/4a pigure SFQ ! wen squre displement s funtion of timeD for two vlues of k 0 aF sn logElog sleD the long time limit is well (tted y stright lineD hene demonstrting sudi'usionF xote however the possile devition from sudi'usion t long times in (gure AD lthough more work is needed to e onlusiveF he ssoited sudi'usive exponent @α in @SFIWA nd slope of the line hereA di'ers for the two vlues of k 0 F por A @respetively AD we hve used disorder strength γ = 0.0225 4 /m 2 a 3 @respetively γ = 0.01 4 /m 2 a 3 AD so tht = 25a for oth k 0 t the forn pproximtion @formul in ption of (gure RFWAF sn oth sesD we hve used nonliner strength g = 0.5 2 /ma nd n initil wve pket of size σ = 10aF e time step δt = 0.05 ma 2 / ws found su0ient for oth (guresF he results re verged over QHHH @ISHHA disorder reliztions for (gure A @respetively AF Possible interpretation e propose to interpret the di'erene etween the two results of (gures SFQ nd SFQ from the perspetive of wve pket exploring the energy lndspeF et wek disorderD the loliztion length is mximl t energy 2 k 2 m /2m with k m = π/2a IQRD thus wve pket strting with wve vetor k 0 = π/2a = k m n only explore energy regions where the loliztion length is smller thn the one t = 2 k 2 0 /2m while the wve pket strting with wve vetor k 0 = π/4a = k m will expnd to energy region of lrge loliztion length ompred to the one t = 2 k 2 0 /2mF es resultD the wve pket strting with wve vetor k 0 = π/4a spreds more rpidly thn the one strting with k 0 = π/2aF xote tht in the ontext of mnyEody loliztionD somewht similr resoning led qornyiD wirlinD wüeller nd olykov to question the very possiility of mnyEody loliztion in the IHH ghpter SF ekly interting wve pkets ontinuum limit ISRD for whih the loliztion grows without ounds s the energy inresesF 5.4.2 Numerical study of the energy distribution dynamics e now ome k to the primry ojet of this setionX the dynmis of the energy distriutionF Denition o simplify the disussionD we onsider the evolution of n initil plneEwveD φ(x, y, z, t = 0) ∝ e ik 0 x , @SFPVA evolving in wek Qh qussin deltEorrelted potentilD V r V r = γδ r,r /a 3 , @SFPWA with δ r,r the uroneker deltF he motivtion for onsidering the wekly disordered Qh se is twofoldF yn the one hndD we im t desriing the e'et of the nonlinerity on the oherent k sttering pek in setion SFSD the ltter is di0ult to see in IhD s it does not stnd out from the kground @see susetion SFSFI for more detilsAF yn the other hndD it o'ers oneptul simpli(tion y pling us on the di'usive side of enderson trnsitionF

Results at intermediate times sn the numeril simultionsD we ompute the verge energy distriution using

P ( , t) = φ(t)|δ( -H)|φ(t) = ∞ -∞ dt 2π φ(t)|e i( -H)t / |φ(t) , @SFQHA
where |φ(t) is the wve funtion t time t nd H is the full rmiltonin @inluding the nonlinerityA t time tF he evolution is performed with the usul gheyshev method @presented in setion RFQAF he resulting verge energy distriution t di'erent times is shown in (gure SFR s lue urvesF he energy distriution visily explores the energy lndspeF xote tht = φ(t)|H|φ(t) is not onserved during the nonliner evolutionF he onserved quntity is

E = φ(t)| p 2 /2 + V + g|φ(t)| 2 /2 |φ(t) = (t) - g 2 dr|φ(r, t)| 4 . @SFQIA
roweverD the nonlinerity eing very smllD the distriutions of nd E re not expeted to di'er muhF yur gol of desriing the e'et of the nonlinerity on the oherent k sttering pek @see susetion SFSFI for more detils out itA led us to hoose over EF e re however not quite ertin tht E is not more suited to the tskF Results at short times (t ≈ τ )

xivelyD one expets the initil energy distriution to e set y the spetrl funtion A k 0 ( ) with given selfEenergy 2 X P ( , t = 0 pigure SFR ! inergy distriution t di'erent timesF he solution of the kineti eqution @SFQUAD is shown s mgent urves nd is ompred to numeril simultions shown s lue urvesF he initil energy distriution @omputed t time t = 36 ma 2 / 2τ s presried y susetion SFRFRA is shown for omprison s lk dshed lineF he prmeters used in the numeril simultions n e found in the ption of tle SFIF nd to then slowly evolve through inelsti ollisions on the nonliner term of the qrossEitevskii equtionD not muh hppening t short timesF urprisinglyD the numeril results present very shortEtime dynmis @t ≈ τ A of the verge energy distriutionF e hve hrterized this shortE time dynmis y (tting the verge energy distriution with timeEdependent selfEenergy Σ t X

+ ) = A k 0 ( ) = (Σ) π 1 -(k 0 ) -(Σ) 2 + (Σ) 2 , @SFQPA
P ( , t) = (Σ t ) π 1 -(k 0 ) -(Σ t ) 2 + (Σ t ) 2 .
@SFQQA he results re reported in tle SFIF

= (k 0 ) + (Σ t ) - g 2 dr|φ(r, t)| 4 = E.
@SFQSA enother e'et tkes ple t short timesD the sreening of the rndom potentil y the nonlinE erity ensuing from the toms preferring low potentil regions to high potentil onesF i'etivelyD the sreening of the rndom potentil redues the selfEenergy @in solute vlueAF snitilly sentD oth these e'ets uild up rpidlyD over roughly one sttering men free timeD in greement with the results reported in tle SFIF o e it more quntittiveD we propose to estimte the shift of the rel prt of the selfEenergy (Σ) ssoited to the rndomiztion of the wve pketF essuming tht φ(r, t > τ ) is omplex qussin rndom vrileD vr |φ| 2 = |φ| 2 D suh tht the rndomiztion of the wve pket shifts (Σ) y pproximtely 0.5g/ol 4×10 -3 2 /ma 2 D iFeF roughly hlf of the oserved shift of (Σ)F sn onlusionD oth e'ets @the sreening nd the rndomiztionA hve omprle e'et on (Σ)F 5.4.4 Intermediate-time picture: kinetic equation e now turn to times longer thn the sttering men free timeD t whih the energy distriution evolves through inelsti ollisions on the nonliner potentil @g|ψ| 2 AF sn the di'usive regimeD it is possile to ount for these inelsti ollisions within kineti equtionD vi ollision integrlF he ltter n e derived using digrmmti theory STDSVF sn this ontextD it is more onvenient to use vrint of the energy distriutionD f D relted to P ( ) eqution @SFPTAD through f = P ( )/ν( ) with ν( ) the density of sttes per unit volume @hoAF he orresponding energy distriution is then normlized ording to f ν( )d = 1. @SFQTA urprisinglyD in the se of n initil plne wveD the evolution eqution for the energy distriuE tionD f D is losed SWD ISSX

∂ t f = ∞ -∞ d 1 ∞ -∞ d 2 W ( ; 1 , 2 ) (f + f 1 + 2 -)f 1 f 2 -f f 1 + 2 -(f 1 + f 2 ) , @SFQUA
where

W ( ; 1 , 2 ) = m 3 g 2 8π 3 7 ν 2ν 1 + |ν 1 + ν 2 + ν 3 -ν 4 | -|ν 1 -ν 2 -ν 3 + ν 4 | , @SFQVA
with (ν 1 , ν 2 , ν 3 , ν 4 ) eing permuttion of (ν 1 , ν 2 , ν , ν 1 + 2 -) suh tht ν 1 ≤ ν 2 ≤ ν 3 ≤ ν 4 F sn ftD the energy distriution ontins ll the neessry informtion to desrie the inelsti ollisionsF his is somewht reminisent of the spetrl funtionD the energy distriution of plne wveD ontining ll informtion out the elsti ollisions @men free pth nd timeAD s we hve seen in susetion QFIFRF he disorder enters in eqution @SFQUA through the hoD ndD more importntlyD in the initil ondition whih is set y the spetrl funtion eqution @SFQPAF he shortEtime e'ets disussed in susetion SFRFQ re not inluded in eqution @SFQUAD whih desries the evolution of the energy distriution t times t τ F sn ftD they only 'et the initil energy distriution to e fed into eqution @SFQUAF he initil energy distriution is set y the selfEenergyD through the spetrl funtion eqution @SFQPAF e thus tke the shortEtime e'ets into ount y simply using orreted selfEenergy in @SFQPAD tht orresponds to the selfEenergy t t 2τ F 5.4.5 Density of states o integrte eqution @SFQUA nd ompute the verge energy distriution t time tD one needs the hoF o ompute itD we ssume tht is not 'eted y the nonlinerityD whih should e the se s long s the ltter is smll ISTF he potentil eing deltEorrelted eqution @SFPWAD vrious pproximte strtegies n e employedD rnging from the forn pproximtion introdued in setion QFI to more sophistited methodsD the selfEonsistent forn pproximtion @gfeA nd the oherent potentil pproximtion @geA IHWD ISUD ISVF he orresponding formuls re given in ppendix SFeF o ttest the ury of these pproximte methodsD we ompre them to the numerilly otined ho in (gure SFSD we lso show in green the ho of the disorderEfree lttie for omprisonF he numeril ho follows from ISW

ν( ) = r = 0| δ ( -H) |r = 0 = ∞ -∞
dt 2π e i t/ r = 0| e -iHt/ |r = 0 , @SFQWA with the onvention |r = 0 = δ r,0 a 3/2 , @SFRHA where a is the lttie sping nd δ r,0 is the uroneker deltF he evolution is performed with the usul gheyshev method @see setion RFQ for more detilsAF es visile in (gure SFSD the ge pproximtion desries very well the numeril hoF st will thus e our hoie for the integrtion of eqution @SFQUAF 5.4.6 Comparison with numerics o estimte the ury of eqution @SFQUAD we solve it numerillyD using the ge ho nd tking into ount the shortEtime e'ets s presried in susetion SFRFRD nd ompre the solution to numeril simultions in (gure SFRF et this stgeD we hd to divide W eqution @SFQVA y P for the solution of @SFQUA to mth our numeril simultionsF e presume tht ftor P is missing in @A pigure SFS ! hensity of sttes s funtion of energyF hi'erent pproximte methods for the density sttes re ompred to numeril resultsF pigure A presents the full energy rngeD while (gure A fous is on the smll energy prtF sn (gure @AD the initil energy distriution is lso shownD with n ritrry sle long the yExisF he numeril simultions were performed on Qh lttie mde of 120 3 sitesD with disorder strength γ = 0.18 4 /m 2 a nd verged over IHHH disorder reliztionsF eqution @SFQVAF pigure SFR shows wht the evolution of the energy distriution is well desried y the ollision integrl eqution @SFQUAF he energy distriution tends to explore the energy lndspe on time sle τ g = 3 /mg 2 F his time sle di'ers from the time sle found in setion SFQ for the ltertion of the quntum oomerng e'etD whih is proportionl to 1/gF he quntum oomerng e'et ppers more sensitive to intertions thn the energy distriutionD we postulte tht it is 'eted y proesses leving the energy distriution unhngedF his di'erene n e understood in the frmework of deoherene nd dissiption @see ITH nd referenes thereinAD deoherene proesses 'et only the quntum oomerng e'etD nd dissiption is responsile for the slower dynmis in energyF 5.4.7 Innite-time equilibrium et in(nite timeD we expet the system to reh n equilirium stteD with therml yleighE tens distriution mximizing entropy ITID ITPX

0 1 2 3 4 5
f eq = 1 Ω T -µ , @SFRIA
where Ω is the volume of the systemD T nd µ(≤ 0) re nlogues of the usul temperture nd hemil potentil respetivelyF lugging @SFRIA into the ollision integrl @SFRFRAD one indeed (nds ∂ t f eq = 0 THF histriution @SFRIA is generlly ssumed to hold up to some uto'D eFgF max = T + µ THD ITQ @eyond whih @SFRIA strts to devite from the foseEiinstein distriution UVAF nfortuntelyD this long time regime is eyond the reh of our numeril simultionsF xote tht the entropy whose mximiztion leds to @SFRIA is de(ned with plneEwve sttes ITID this is good pproximtion sine we re on the di'usive side of enderson trnsitionF his hoie is onsistent with the ollision integrl @SFQUAD whih is derived negleting loliztion e'etsF he se of strongly lolized Ih hins ws onsidered in SUD ITRD where they ssume foltzmnn distriutionF SFSF xonlinerity nd the oherent k sttering pek IHS 5.5 Nonlinearity and the coherent back scattering peak 5.5.1 Coherent back scattering peak sn setion SFRD we hve studied how the energy distriution of n initil plne wve evolves in Qh rndom potentil through the nonliner term of the qrossEitevskii eqution eqution @SFRAF he energy distriution is not esily mesurle in oldEtom experimentsD one often prefers to mesure the relted momentum distriution RHF et times longer thn the sttering men free timeD one nively expets no preferred diretion for the momentumD nd thus n isotropi momentum distriution given y the projetion of the energy distriution on the momentum speX

|φ(p, t)| 2 = d A p ( )f (t), @SFRPA
where the spetrl funtion A p ( ) onnets energy nd momentum in the rndom potentil @see susetion QFIFR for more detilsAF iqution @SFRPA relies on lssil point of view nd neglets n importnt ontriution to the momentum distriutionD the oherent k sttering @gfA pekF o exhiit itD let us expnd on the disussion of the wek loliztion phenomenon in susetion IFIFIF st ws rgued there tht two lsses of pths ontriuting to the rel spe density survive the disorder vergingF he momentum distriution n e similrly deomposed s sum of pthsD with however di'erent initil nd (nl points for the wve funtion nd its onjugteD ording to |φ(p, t)| 2 = 1 2 dr 3 dr 4 e -ip.(r 3 -r 4 )/ φ(r 3 , t)φ * (r 4 , t). @SFRQA pigure SFT dpts (gure IFI to the present se y showing the two min pths tht ontriute to φ(r 3 , t)φ * (r 4 , t)F essuming n initil plne wve with wve vetor k 0 nd performing the pourier trnsformD the hi'uson rings n isotropi ontriutionD desried y eqution @SFRPA wheres the gooperon ontriution is strongly nisotropi nd peked roundk 0 ITSD s visile in (gure SFUF sf phse oherene is fully preservedD the pek rehes twie the kground height ITSF @A gooperon pigure SFT ! wo kinds of pir of sttering pths ontriuting to φ(r 3 , t)φ * (r 4 , t)F fetween the (rst nd lst sttering eventsD oth pths follow the sme sequene of sttering events y either propgting @A in the sme diretion or @A in opposite diretionsF he present setion is devoted to the e'et of intertions on the gf pekF pigure SFU ! pigure from U showing the experimentlly mesured momentum distriution fter di'erent propgtion times in Ph disordered potentilF he initil momentum distriution @(rst imgeA is shrply peked round k 0 F efter few sttering men free times @lst imgeAD the moE mentum distriution presents n isotropi prt @ssoited the hi'usonA nd the oherent k sttering @gfA pek roundk 0 @ssoited to the gooperonAF st should e mentioned tht nE other pekD the oherent forwrd sttering pekD is expeted t longer timesD when the loliztion regime is rehed RQD ITTF xote tht the sitution is similr in three dimensionsD with spheril isotropi shell nd pek tk 0 F

Numerics

o study the e'ets of intertions on the gf pekD we propgte n initil plne wve extly s we did in setion SFRD ut keep trk of the momentum distriution |φ(p, t)| 2 = 1 dre -ip.r/ φ(r, t)

2

, @SFRRA insted of the energy distriutionF e de(ne the gf pek ontrst s the height of the distriution t the top of the pek @loted t p =k 0 A divided y the kground mplitudeD mesured t momentum k 0 in diretion orthogonl to k 0 F e show in (gure SFV the numerilly otined gf pek ontrstF sn the non interting limit @g = 0AD the gf pek uilds up rpidly @over few sttering men free timesA nd sturtes t twie the kground s expetedF sn the nonliner se @g = 0AD the gf does not uild up ompletely t short times nd sturtes well elow twie the kground efore deresing on muh longer time sleF 5.5.3 Physical picture he gooperon digrms of (gure SFT re uilt y letting the dshed pth follow the sme trjetory s the ontinuous pth ut in the opposite diretionD in order to gurntee tht the phse umulted long the ontinuous pth is neled y the one umulted long the dshed pthF sn doing soD we impliitly ssumed tht on eh segment the qusiEprtiles re the sme for the ontinuous nd dshed pths @iFeF sme energy nd selfEenergyD see setion QFI for more detilsAF his ssumption my e hllenged y the nonliner timeEdependent term of the qrossEitevskii eqution @SFRAF o mke it more lerD we reprodue the digrms in (gure SFTD displying the time s olor @going from lue t short times to red t long timesA in (gure SFWF st ppers tht the phse nelltion on eh segment is hmpered y the mehnisms introdued in susetions SFRFR pigure SFV ! goherent k sttering @gfA pek ontrst s funtion of timeD with sttistil error rsF e de(ne the gf pek ontrst s the height of the distriution t the top of the pek @|φ(-

k 0 )| 2 A divided y the kground mplitude @|φ( k ⊥ 0 )| 2 )D where k ⊥ 0 .k 0 = 0 nd |k ⊥ 0 | = k 0 F he nonliner se @g = 0AD
shown in greenD is ompred to the liner one @g = 0AD shown in lueF he simultions were performed on Qh lttie mde of 600 × 96 2 sites @the long diretion is the one ssoited to k 0 AD with disorder strength γ = 0.054 /m 2 a @τ ≈ 71 ma 2 / A nd n initil wve vetor k 0 = π/4aF he results re verged over SHHH disorder reliztionsF sn the nonliner seD we hve used nonlinerity g/ol = 0.004 2 /ma 2 D nd time step δt = 1.6 ma 2 / F nd SFRFQF yn the one hndD s result of the shortEtime e'ets of susetion SFRFQD di'erent qusiE prtiles re ssoited to the ontinuous nd dshed pths in the viinities of the strting nd ending pointsF his 'ets ll ontriutions to the gooperon nd results in the prtil rise of the gf pek t short times 4 F yn the other hndD the energy redistriution of susetion SFRFR my hnge the energy fter some timeF his e'et tkes ple on muh longer time sleX in the wek nonlinerity limit we re onsideringD mny sttering events on the disordered potentil re needed efore the energy redistriution eomes e'etiveF e expet this e'et to eventully destroy the gf pek s inelsti ollisions dd upF o summrizeD the shortEtime e'ets re responsile for the gf pek not rising to twie the kground t short times @t ≈ τ AD nd the energy redistriution to the slow redution of the gf ontrst on muh longer time sleF

Conclusion

sn this hpterD we hve touhed upon the e'et of wek intertions on the evolution of onE densed osoni gses in disordered potentilsF his study onsisted of three prtsF e hve (rst shown how the omplex quntum mnyEody dynmis pproximtely redues to nonliner equE tion on lssil (eldF e hve then developed numeril sheme to integrte sid equtionF sn third stgeD we hve pplied the numeril method to vrious physil situtionsF he min onlusions tht we hve drwn from this study followsF pirstD ting similrly to deoherene proess @with deoherene time proportionl to ξ/gAD the nonlinerity seems to interrupt the quntum oomerng e'et nd prevent the full retrore)etion of wve pkets lunhed pigure SFW ! gouple of sttering pths of gooperon typeD with time shown s olor @going from lue t short times to red t long timesAF with (nite veloityF eondD we hve shown tht the nonlinerity llows the wve pket to explore the energy lndspeF qiven n initil plne wveD the dynmis of the energy distriution oeys losed eqution nd inelsti ollisions hppen on time sle τ g = 3 /mg 2 F hird these inelsti ollisions hit the gf pek nd re expeted to eventully led to its destrutionF pourthD the gf pek is lso 'eted y shortEtime e'etsD the sreening of the disordered potentil y the nonlinerity s well s the shift of the rel prt of the selfEenergy ssoited to the rndomiztion of the wve pket 5 F es resultD the gf pek does not rise to twie the kground height t short timesF hile we hve underlined the physil piture ehind these phenomenD our work ws essentilly numerilF purther nlytil hrteriztion of the e'ets introdued in this hpter should provide deeper insights nd o'er interesting perspetives for future workF Appendix 5.A Approximations of the density of states sn this ppendixD we give the formuls for the density of sttes @hoA used in susetions SFRFS nd SFRFTF pour pproximtions of the ho re presentedD a priori in inresing order of ury @disorderEfreeD fornD gfe nd geAF hese formuls re tken from IHW nd pply to deltE orrelted potentilsD for whih the selfEenergy n e ssumed to depend only on energy Σ(k, ) = Σ( )F

5.A.1 Density of states in term of self-energy he ho is omputed through the verge qreen funtion

G( ) setion QFIX ν( ) = - 1 π G( ) , @SFRSA whih is in turn expressed in terms of the selfEenergy Σ( ) setion QFIX G( ) = π -π d 3 k (2π) 3 1 -(k) -Σ( ) , @SFRTA where (k) is the lttie dispersion reltion (k) = (3 -cos(k x a) -cos(k y a) -cos(k z a)) 2 /ma 2 F he integrl over k n e performed to express G( ) s ITU G( ) = P (6/E) E , where E = 2ma 2 2 -Σ( ) -6 nd P (z) = 1 -9ξ 4 (1 -ξ) 3 (1 + 3ξ) 2 π K(k 1 ) 2
. @SFRUA 5 The eect of the randomization of the wave packet on the CBS peak should be taken with a grain of salt, see footnote 4.

rere ξ nd k 1 re funtions of z de(ned s

ξ(z) = 1 -1 -z 2 /9 1 + √ 1 -z 2 1/2 , k 1 (z) 2 = 16ξ 3 (1 -ξ) 3 (1 + 3ξ) , @SFRVA
nd K is the omplete ellipti integrl of the (rst kindF 5.A.2 Approximations of the self-energy he simplest pproximtion is to ignore the disordered potentil nd set Σ to HD hene reovering the disorderEfree hoX

ν free ( ) = - 1 π G 0 ( ) , @SFRWA
where G 0 ( ) is the disorderEfree qreen funtion @whih n e dedued from G( ) y setting Σ to HAF elterntivelyD one n ompute the selfEenergy perturtively in the disorder @see setion QFI for more detilsAF et lowest order @forn pproximtionAD the selfEenergy writes

Σ( ) = V 2 0 G 0 ( ).
@SFSHA e slightD ut simpleD improvement is the selfEonsistent forn pproximtion @gfeA where the disorderEfree qreen9s funtion in eqution @SFSHA is modi(ed selfEonsistentlyD leding to

Σ( ) = V 2 0 G 0 ( -Σ( )).
@SFSIA Σ( ) n then e omputed numerillyD eFgF with rootEserhing lgorithmF e supposly etter pproximtion sheme is the oherent potentil pproximtion @geA ISUD ISVD whih leds to n eqution involving the potentil onEsite distriution P (V ) here P

(V ) = exp(-V 2 /2V 2 0 )/V 0 √ 2πX dV V -Σ( ) 1 -(V -Σ( ))G 0 ( -Σ( ))
P (V ) = 0. @SFSPA eginD rootEserhing lgorithm llows to ompute Σ( ) numerillyF Chapter 6

The kicked rotor, a paradigmatic simulator for Anderson localization e hve so fr disussed enderson loliztion nd its mnifesttions in the dynmis of prtiles evolving in rndom potentilsF uh systems do not hve the exlusivity thoughD enderson lolizE tion lso ppers in deterministi modelsF sn this hpterD we onsider n exmple of deterministi model displying enderson loliztionD the kiked rotorF his model hs een of gret experimentl importne ITVD in prtiulr in the ontext of the erly oservtions with oldEtoms of enderson loliztion in one dimension ITWD of the enderson trnsition in three dimensions QT nd more reently of enderson loliztion in two dimensions QWF fesides these experimentl suessesD the kiked rotor hs lso onstituted theoretil hllengesD strting from the numeril oservtion of dynmil loliztion IUH nd its onnetion to enderson loliztion through mpping onto disorder model IUIF vter the onnetion ws lso estlished through supersymmetri (eld theory IUPF wore reentlyD the emergene of topologil fetures in kikedErotor models for prtile with hlfEinteger spins ws reported IUQ!IUSF he im of this hpter is to show tht mny res re still to e exploredF e strt y rie)y introduing the kiked rotor nd its onnetion with enderson loliztion in setion TFIF etion TFP introdues the three simplest universlity lsses nd some of their universl propertiesF prom these notionsD setion TFQ hllenges ommon wisdom y exploring the possiility of spinless kiked rotor in the sympleti lssF sn setion TFRD we strt y introduing n importnt possiility o'ered y the kiked rotorD the simultion of high dimensions with spei( Ih qusiEperiodilly modulted kiked rotorsF his possiility is then pplied to kiked rotor elonging to the sympleti universlity lssF e onlude setion TFR y investigting the properties of new type of qusiEperiodilly modulted kiked rotorF xote tht this hpter presents works in progressF 6.1 From chaos to disorder he kikedErotor model desries the motion of prtile living on irle of length 2π nd periodilly kikedF he orresponding rmiltonin tkes the form

H = p 2 2 + V (θ) n δ(t -n), @TFIA
III with p = -i ∂ θ F he singleEvluedness of the wve funtion quntizes the momentum in units of @p = l with l ∈ ZAF e ommonly studied kiked rotor hs the simple form

H = p 2 2 + K cos (θ) n δ(t -n), @TFPA iFeF V (θ) = K cos (θ)F
sn oldEtomi reliztions of model @TFIA ITVD the prtile does not stritly live on irleD ut rther in virtully in(nite Ih spe prmetrized y xD nd is kiked with periodi optil potentil @of period 2π/kAF he rmiltonin tkes the form

H = p 2 2m + V (kx) n δ(t -nτ ), @TFQA
with V periodi funtion of period 2πF he periodiity of the potentil llows one to write the dynmis on irle y pplition of the floh theoremF his proedure shifts the momentumD quntized in integer units of k on the irle @θ ∈

[-π/k, π/k[AD y qusiEmomentum β k @β ∈ [0, 1[AX H = (k ) 2 (l + β) 2 2m + V (kθ) n δ(t -nτ ).
@TFRA he qusiEmomentum β is onversed during the evolutionF he theoretiin model @TFIA orresponds to using the units m = τ = k = 1F sn these unitsD is proportionl to m/k 2 τ F sn prtieD it n thus e'etively e experimentlly tuned vi τ F hroughout out this hpterD we work in units m = τ = k = 1F

Classical chaotic dynamics

vet us strt y relling wht is the lssil dynmis ssoited to @TFPAF prom rmilton equtionsD one n express the lssil dynmis s ghirikov9s stndrd mp IUTX p t+1 = p t + K sin(θ t ), @TFSA θ t+1 = θ t + p t+1 , @TFTA where θ n @p t A is the position @momentumA of the lssil prtile t time t @right efore kikAF es it turns outD ghirikov9s stndrd mp is hoti for lrge enough K IUTF he ssoited extreme sensitivity on the initil onditions results in deterministi di'usion proess in momentum speX for lrge enough K nd t long enough timesD the verge of p 2 over the initil ngle θ grows linerly in time IUTF ith respet to enderson loliztionD this di'usion proess in momentum spe plys the role of the lssil rel spe di'usion of disorder modelsF 6.1.2 Quantum evolution: Floquet operator et the quntum levelD the dynmis is similrly otined y pplying repetedly the ploquet opertor F = U kinetic U kick , @TFUA whih governs the evolution over one periodF rereD U kick desries the kik it is the quntum ounE terprt of @TFSA nd U kinetic desries the free propgtion etween kiks it orresponds to @TFTAF por the kiked rotor @TFPAD they re respetively given y U kick = exp -iK cos(θ)/ nd U kinetic = exp -ip 2 /2 . @TFVA 6.1.3 Quantum evolution: pseudo-random kicked rotor e now turn to the kiked rotor @TFRAD relevnt for oldEtom experimentsF st is useful to express U kick nd U kinetic in the momentum sisF he momentum eing quntized in units of D we prmetrize the momentum sis y integers lX p |l = l |l F yn the one hndD the kik hnges the momentumD nd is thus non digonlX

l|U kick |l = π -π dθe -iK cos(θ)/ e -i(l-l )θ = i l -l J l -l (K/ ), @TFWA
where J l -l denotes the (ll) th fessel funtion of the (rst kindF yn the other hndD U kinetic is digonlX l|U kinetic |l = exp -i (l + β) 2 /2 δ l ,l , @TFIHA with δ l ,l the uroneker deltF vet us now onsider the di'erene etween two onseutive phses of @TFIHAX

2 (l + 1 + β) 2 -(l + β) 2 = l + β + /2.
@TFIIA sn the limit K/ 1D @TFWA popultes lrge momentD suh tht two onseutive phses typilly di'er y muh more thn 2πF elterntivelyD the sme holds true when 1F sn either one of the two limitsD (l + β) 2 /2 in @TFIHA is essentilly equivlent to pseudoErndom momentumEdependent phseX l|U kinetic |l ≈ exp (-iφ l ) , φ l ∈ [0, 2π[, @TFIPA with di'erent reliztions of the phses depending on β IUUD IUVF sn experimentsD mny β ompoE nents re initilly populted suh tht the mesurements n e viewed s verged over β IUWF sn the rest of the hpterD we pproximte the φ l of eqution @TFIPA y uniformly distriuted rndom phsesF ithin this pproximtionD only ppers in the kik term nd n thus e sored in the kik strengthX we set it to I to lighten the nottionsF he reson for using this pproximtion is essentilly oneptulD it is used to ring out the e'ets disussed nd filitte their understndingF he onsisteny of the results for pseudoErndom sequene is systemtilly heked numerillyD with the exeption of the results presented in (gure TFTF xote tht this pproximtion is only vlid for vlues inommensurte with π @eFgF for = 4πD two onseutive phses di'er only y β [2π]D whih n e very smllA IVHD IVIF 6.1.4 From classical diusion to localization sn the spirit of wht we hve done in setion IFI for disordered systemsD we write the quntum proility to go from momentum p 1 to momentum p 2 in time spn t s sum over lssil pths weighted y omplex numers A i @verging over the qusiEmomentum β is impliedAX

P (p 1 → p 2 , t) = path i A i 2 = paths i A i A * i classical + paths i =j A i A * j quantum .
@TFIQA ith only the soElled lssil ontriutionD the momentum spe dynmis would e di'usiveF pollowing the resoning of setion IFID to this ontriution we should dd the phseless quntum ontriutionsD mong whih the importnt gooperon ontriutionD uilt from pth i nd its timeE reversed ounterprt jF sn wyD the hoti properties of the lssil dynmis selet the lssil pthsD nd thus ply role similr to disorderF sn lose nlogy with disorder models @see setion IFI for more detilsAD the umultion of gooperon loops1 leds to loliztionX the lssil momentum spe di'usion is stopped t long times nd repled y loliztion IVPF hese perturtive similrities etween the kiked rotor nd disordered systems n e formlized through mpping of the kiked rotor onto Ih lttie with pseudoErndom onEsite potentil IUIF he equivlene etween the kiked rotor nd disorder models n lso e shown through suE persymmetri (eld theory IUPF he ltter method onnets the kiked rotor to qusiEIh disorder modelsD with thus n importnt di'erene with respet to the Ih disorder models of hpter RD the presene of di'usive regime t short timesF 6.2 Universality classes and random matrix theory 6.2.1 From a perturbative perspective es disussed in susetion TFIFRD the gooperonD formed y oupling pth nd its timeEreversed ounterprtD plys n importnt role in the trnsport propertiesF st is sed on timeEreversl symE metryD thus suggesting tht this symmetry plys entrl roleF sn the prtiulr se of the kiked rotorD for whih the gooperon is uilt in momentum speD the relevnt timeEreversl symmetries re omintions of sptil prities nd onventionl timeEreversl symmetries IVQX

t → t 0 -t, p → p, θ → θ 0 -θ, @TFIRA
with ritrry t 0 nd θ 0 F he kiked rotor @TFPA lerly hs this symmetry @eFgF with t 0 = θ 0 = 0AD it thus elongs to the soElled orthogonl lss nd the gooperon is nonzeroF hen ll timeEreversl symmetries of type @TFIRA @iFeF for ll t 0 nd θ 0 A re rokenD the system elongs to the unitry lss nd the gooperon is zeroF sn the se of prtiles with hlfEinteger spinD the sitution is somewht more sutleD with the timeEreversl symmetry hving di'erent e'ets depending on the presene of dditionl geometril symmetriesF e ome k to these sutleties in susetion TFPFPF por nowD we just note tht when the only symmetry is of the form

t → t 0 -t, p → p, θ → θ 0 -θ, S → -S, @TFISA
where S is the spinD the ssoited kiked rotor elongs to the sympleti lss IVR nd the gooperon is presentD lthough it mnifests itself s n antilocalization ontriution @we ome k to this point in susetion TFQFIAF fsed on the interply etween gooperon nd timeEreversl symmetriesD we distinguish three lssesF pirst the orthogonl lssD for spinless kiked rotors invrint under timeEreversl symmeE tryD feturing gooperon inhiiting trnsportF eond the unitry lssD for spinless kiked rotors without ny time reversl invrineD for whih the gooperon is irrelevntF hird the sympleti lssD for spinEIGP prtile with timeEreversl symmetry of type @TFISA s only symmetryD feturing gooperon enhning trnsportF 6.2.2 Universal properties and random matrix theory Generalities he present susetion ims t pointing out universl properties ssoited with eh lssD sed on the frmework of rndom mtrix theoryF he ide ehind rndom mtrix theory is tht lol spetrl properties of disordered or quntum hoti rmiltonins2 re universl nd depend only on spei( symmetries of the rmiltoninF es n exmpleD given the ordered eigenvlues @λ 1 , λ 2 , . . . , λ n A of n rmiltoninD the distriution of the level spings @s 1 = λ 2 -λ 1 Ds 2 = λ 3 -λ 2 Ds n-1 = λ n -λ n-1 A is universlD provided the level spings re expressed in units of the lol men level sping @ s i AF o (nd the orresponding universl propertiesD one uses ensemles of rndom mtriesF ndom mtries were introdued y igner to model the nulei of hevy toms IVSF his fruitful ide then spred to mny (eldsD from disordered nd quntum hoti systems to twoE dimensionl grvity nd string theory IVTF erhps more surprisinglyD it ws lso found useful in numer theoryD to study the distriution of zeros of the iemnn zet funtion IVUF sn this thesisD we onsider only prtiulr lss of rndom mtrix ensemlesD the soElled hyson ensemlesD whih re ensemles of rermitin mtries feturing sttistilly independent entriesD with only one possile symmetryX time reverslF xote tht in the ontext of disordered nd quntum hoti systemsD other rndom mtrix ensemles ply n importnt roleD they inlude the possiility of other symmetries IVVD IVWF sn the following two susetionsD we disuss the di'erent rndom mtrix ensemles desriing systems with nd without time reversl symmetryF he disussion follows the ook y rke IPQF por the ske of revityD we only gther here some importnt results for the upoming setion TFQD nd refer the reder to IPQ for more thorough disussionF Hamiltonians not invariant under time reversal fefore exploring the onsequenes of time reversl symmetry on the lol spetrl properties of disordered or quntum hoti rmiltoninsD let us onsider the nonEsymmetri seF gonretely spekingD we wnt to de(ne the rndom mtrix ensemle tht desries given disordered or hoti rmiltonin feturing no time reversl symmetryF por simpliityD we ssume tht the rndom mtrix ensemle hs sttistilly independent entriesF o hrterize this rndom mtrix ensemleD we need to identify under whih trnsformE tions it should e sttistilly invrintF e generi trnsformtion hnges the rmiltonin H to H = AHA -1 F ine no time reversl symmetry pplies hereD the only requirement is tht the trnsformtion should retin rermitiityX

H † = (AHA -1 ) † = AHA -1 ⇔ H, AA † = 0.
@TFITA he generl trnsformtions for nonEtimeEreversl invrint systems re thus unitryD up to n irrelevnt glol ftorF por N × N mtriesD they form the U (N ) groupF ynly one rndom rermitin mtrix ensemle hs sttistilly independent entries nd is stE tistilly independent under U (N ) trnsformtionsD the qussin unitry ensemle IWHF his ensemle of rermitin mtries hs rel @respF omplexA sttistilly independent qussin @respF nonEAdigonl entries 3 F Time reversal operator o explore the onsequenes of timeEreversl symmetriesD we need to represent them y opertors ting on wve funtionsF es it turns outD timeEreversl opertors T re ntiunitry IPQX T ψ| T φ = φ|ψ , @TFIUA nd n lwys e written IPQ T = U K, @TFIVA where U is n ritrry unitry mtrix nd K is the ntiunitry omplex onjugtion opertor in position representtionF he se U = 1 is ssoited to onventionl timeEreversl symmetryX

t → -t, p → -p, θ → θ.
@TFIWA st is resonle to require tht every timeEreversl opertors T ting twie on wve funtion does not hnge it @up to phse ftorD iFeF T 2 ψ = e iα ψAF st n e shown tht T 2 = 1 @α = 0A nd T 2 = -1 @α = πA re the only possiilities onsistent with the ntiunitrity of T IPQF vet us (rst onsider the se T 2 = 1F essuming n rmiltonin invrint under T @[H, T ] = 0AD its lol spetrl properties should e desried y rndom mtrix ensemle similrly invrint under T F o exhiit the onstrint imposed y this invrineD we (rst show tht T Einvrint sis existsF o tht endD given n ritrry vetor φ 1 nd omplex numer a 1 D we uild the vetor

ψ 1 = a 1 φ 1 + T a 1 φ 1 , @TFPHA
whih is mnifestly T EinvrintF epeting the sme proedure with vetor φ 2 orthogonl to ψ 1 D we get the T Einvrint vetor ψ 2 = a 2 φ 2 + T a 2 φ 2 , @TFPIA whih turns out to e orthogonl to ψ 1 X

ψ 2 |ψ 1 = a * 2 φ 2 |ψ 1 + a 2 T φ 2 |ψ 1 = a 2 T 2 φ 2 |ψ 1 * = a 2 φ 2 |ψ 1 * = 0. @TFPPA
prom this proedureD one n uild omplete sis of orthogonl vetorsF ith T Einvrint sis t hndD we onsider n ritrry mtrix

O invrint under T @[O, T ] = 0AF he mtrix O is rel in the T Einvrint sisX O µν = ψ µ |Oψ ν = T ψ µ |T Oψ ν * = ψ µ |T OT 2 ψ ν * = ψ µ |T OT ψ ν * = O * µν .
@TFPQA he entries of the rndom mtries invrint under T n thus e desried y rel distriutionsF he ensemle of mtries tht leve N × N rel mtrix rel is the O(N ) groupD sugroup of the U (N ) groupF he rndom rermitin mtrix ensemle tht hs sttistilly independent entries nd is sttistilly independent under O(N ) trnsformtions is the qussin orthogonl ensemle IWHF his ensemle of symmetri mtries hs sttistilly independent rel qussin entriesF he se T 2 = -1 is somewht riherD with di'erent distriutions depending on the presene of geometril symmetriesF sndependently of the presene of geometril symmetriesD the T 2 = -1 se hs distinguishle fetureD the presene of urmers9 degeneryF sndeedD if ψ is n eigenstte of the rmiltonin with energy ED so is T ψ sine [H, T ] = 0D moreover ψ nd T ψ re orthogonlX ψ|T ψ = T ψ|T 2 ψ * = -T ψ|ψ * = -ψ|T ψ = 0, @TFPRA IIU ll eigenvlues re thus douly degenerteF en importnt onsequene of this doule degenery is tht the rilert spe dimensionD if (niteD must e even @we denote it 2N AF his result is onsistent with the ommon wisdom tht T 2 = -1 is only possile for hlfEinteger spin prtilesF sn the sene of geometril symmetriesD one n use sis orgnized in pirs @|i DT |i AF sn this sisD mtrix invrint under T n e deomposed in 2 × 2 loks formed y the pirs @|i DT |i AF imilrly to the rel T Einvrint mtries when T 2 = 1D when T 2 = -1 eh lok of T Einvrint mtrix n e desried y four rel numers giving the weights ssoited to the mtries @ID-iσA with σ the uli mtriesF purther rermitiity onnets the loks on eh side of the digonl nd redues the desription of digonl loks to single rel numer IPQF hese mtries re thus formlly equivlent to N × N rermitin quternioni mtries4 F yn the one hndD for o'Edigonl elementsD there is oneEtoEone orrespondene etween the four rel numers omposing quternion nd the four rel numers desriing lok5 F yn the other hndD the N rel digonl elements of the rermitin quternioni mtrix desries the digonl loksF he ensemle of trnsformtions leving this struture unhnged is the sympleti Sp(N ) groupF he rndom rermitin mtrix ensemle tht hs sttistilly independent entries nd is sttistilly independent under Sp(N ) trnsformtions forms the qussin sympleti ensemle IWHF his ensemle of 2N × 2N rermitin mtries n e desried y N × N rermitin mtries with rel @respF quternioniA sttistilly independent qussin @respF nonEAdigonl entriesF he presene of n dditionl geometril symmetry @eFgF prityA splits the mtries in two loks trnsposed of one notherX

H = H + 0 0 H T + .
@TFPSA urmers9 degenery is diret onsequene of this strutureF he ssoite rndom rermitin mtrix ensemle is then tht of H + D whih is only onstrined y rermitiityD it is thus the qussin unitry ensemleF he presene of n dditionl geometri symmetry imposes rel )ututions for H + nd the orresponding ensemle is thus the qussin orthogonl ensemleF Level spacing distribution in the three Dyson random matrix ensembles e hve de(ned three ensemles of rermitin rndom mtriesD the qussin orthogonl ensemE le @qyiAD the qussin unitry ensemle @qiAD nd the qussin sympleti ensemle @qiAF sn eh ensemleD one n work out lol spetrl properties tht re expeted to e universlF e ommonly used quntity is the distriution of level spingsF qiven the ordered eigenvlues @λ 1 , λ 2 , . . . , λ N A of given mtrixD it is the distriution of s

i @s 1 = λ 2 -λ 1 D s 2 = λ 3 -λ 2 D . . . D s N -1 = λ N -λ N -1 AF
st is expressed in units of the lol men level sping @ s i AF por su0iently lrge mtriesD the pproximte results re respetively for the orthogonlD unitry nd sympleti ensemles IWHX

P GOE (s) = π 2 se -π 4 s 2 , @TFPTA P GUE (s) = 32 π 2 s 2 e -4 π s 2 , @TFPUA P GSE (s) = 2 18 3 6 π 3 s 4 e -64 9π s 2 .
@TFPVA e omprison etween these three results nd the oisson distriution P Poisson (s) = e -s , @TFPWA whih would previl if the eigenvlues were just uniformly nd independently distriuted over some intervlD is shown in (gure TFIF he rndom mtrix ensemles show di'erent degrees of levelElevel repulsionD with level sping distriution strting s s for the orthogonl ensemleD s 2 for the unitry ensemle nd s 4 for the sympleti ensemleF 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.

P(s)

Poisson GOE GUE GSE pigure TFI ! iigenvlue sping distriutions for di'erent rndom mtrix ensemlesF he pproxiE mte results for the three hyson ensemles re shownX eqution @TFPTA for the orthogonl ensemle in greenD eqution @TFPUA for the unitry ensemle in red nd eqution @TFPVA for the sympleti ensemle in light lueF por omprisonD the oisson distriution @TFPWA @whih would previl if the eigenvlues were just uniformly nd independently distriuted over some intervlA is shown in drk lueF 6.2.3 Time reversal symmetry for Floquet models qoing k to our initil modelD eqution @TFIAD we see tht it does not (t in the ove frmeworkF sndeedD @TFIA eing timeEdependentD it does not mke muh sense to look t its @timeEdependentA eigenvluesF sn ftD it is the ploquet opertor @TFUAD desriing the evolution over periodD tht plys the role of the rmiltonin in this seF he universl lol spetrl properties re found in the eigenvlues of the ploquet opertorF he ploquet opertor eing unitryD it is not desried y the ove qussin ensemlesD ut y very similr ensemlesD the irulr ensemlesF he eigenvlues of unitry opertors eing omplex numers of modulus ID the interesting spetrl properties lie in their phseF sn ftD for lrge enough mtriesD de(ning the level sping s the distne etween two onseutive eigenphsesD one (nds the sme distriutions of level spings thn for the qussin ensemlesF he interply with the timeEreversl symmetry is slightly hngedF sndeedD sine the ploquet opertor propgtes given stte over periodD its timeEreversed ounterprt should propgte stte kwrd in time over periodD suh tht the timeEreversed ploquet opertor should e the inverse of the ploquet opertorX T F T -1 = F -1 = F † . @TFQHA iqution @TFQHA reples [H, T ] = 0 for ploquet opertorsF e ploquet opertor without property @TFQHA hs the universl properties of the irulr unitry ensemle @giAF e ploquet opertor with property @TFQHA nd T 2 = 1 respetively T 2 = -1 hs the universl properties of the irulr orthogonl @gyiA respetively sympleti @giA ensemleF IIW o good pproximtionD their level sping distriution re respetively given y @TFPUAD @TFPTA nd @TFPVAF 6.2.4 Critical exponents he ove lssi(tion of rmiltonins in terms of their lol spetrl properties my seem it unusul to the sttistil physiistF sndeedD in sttistil physis it is more ommon to group models in universlity lsses hrterized y some ritil exponents IWPF sn the se of enderson trnsitionsD the sme hrteriztion pplies nd di'erent universlity lss fetures di'erent ritil exponentsF st is now ler tht the ove three lsses @orthogonlD unitry nd sympletiA de(ne three di'erent universlity lssesD eh displying di'erent universl divergene of the loliztion length ner the ritil point IVWF gonerning the ritil propertiesD to the est of our knowledgeD nothing is known for kiked rotors esides spinless timeEreversl invrint ones tht re known to elong to the orthogonl lssD together with disorder models QTD IUWF 6.3 A spinless kicked rotor in the symplectic class? st is ommonly epted tht the timeEreversl opertor n only squre to EI for prtiles with hlfEinteger spin IPQF es orollryD it is elieved tht the sympleti enderson universlity lss n only e explored with hlfEinteger spin prtilesF his setion ims t hllenging this restritionF

Weak antilocalization Model

o strt withD we look for kikedErotor model exhiiting wek ntiloliztionF his hrE teristi of the sympleti lss orresponds to gooperon with sign opposite to the hi'uson QF he gooperon is uilt from pth nd its timeEreversed ounterprtF e pth itself is de(ned s sequene of free propgtions with moment p t D seprted y kiksF he intuitive ide is to use kiking potentil V (θ) with the property

p 1 |e -iV (θ) |p 2 = -p 2 |e -iV (θ) |p 1 for ll (p 1 , p 2 ),
@TFQIA suh thtD for eh kikD the gooperon umultes minus sign s ompred to the usul timeE reversl invrint situtionD for whih

p 1 |e -iV (θ) |p 2 = p 2 |e -iV (θ) |p 1 .
@TFQPA e need to go one step further nd impose tht the gooperon only hosts n odd numer of kiksD otherwise the umultion of positive nd negtive signs would simply kill itF his n e esily hieved with n lterntion of two di'erent kiksF pigure TFP presents pitoril view of these idesF o hieve property @TFQIAD we need kiking term e -iV (θ) whose pourier trnsform f (p) = dθe -ipθ e -iV (θ) , @TFQQA is odd f (p) = -f (-p)F prom the properties of pourier trnsformsD it is su0ient nd neessry for e -iV (θ) to e odd s wellF en exmple of potentil with property @TFQIA is pigure TFP ! gooperon with lterntion of two di'erent kiksD respetively shown s lue squres nd red strsD lines re free propgtionsF yn the one hndD for odd numer of kiksD we n use property @TFQIA to express the gooperon s minus its hi'uson ounterprtF yn the other hndD the lterntion of two kiks mke it impossile for gooperon to hve n even numer of kiksF he potentil @TFQRA hs however serious issue with respet to enderson loliztionX it is disonE tinuousF sndeedD one kik with potentil @TFQRA is su0ient to populte ritrry high momentD with only n lgeri dey of the momentum densityD whih is inonsistent with exponentil enderson loliztion IWQF sn order to hieve the desired properties without destroying enderson loliztionD we introdue n dditionl dimensionD mde of only two sitesD nd ple the disontinuity long itF e end up with the following rmiltonin @with only two sites long yD y = ±1AX

V (θ) = K cos(θ) + π 2
H CSE = p 2 θx 2 + t y + g CSE (θ x , y, α) n δ(t -2n) + g CSE (θ x , y, -α) n δ(t -2n + 1), @TFQSA
where g CSE (θ x , y, α) = K(1 + α) cos(θ x ) + y sin(2θ x ) + y π 2 . @TFQTA t y ounts for the tunneling etween the two sites long yF sn prtieD we tret it like p 2 θx /2X in momentum speD we pproximte it y p y Edependent rndom numer uniformly distriuted in [0, 2π[ @see susetion TFIFQ for more detilsAF rereD the term in y sin(2θ x ) is introdued to ouple the θ x nd y diretionsY if they were unoupled the dynmis long the two diretions would e seprtedF st is hosen invrint under (θ x , y) → (-θ x , -y)D suh tht @TFQIA holdsF e will lso onsider n orthogonl vrint of @TFQSA for omprisonX

H COE = p 2 θx 2 + t y + g COE (θ x , y, α) n δ(t -2n) + g COE (θ x , y, -α) n δ(t -2n + 1), @TFQUA
where g COE (θ x , y, α) = K(1 + α) cos(θ x ) + y sin(2θ x ) . @TFQVA Signatures sn order to hek tht model @TFQSA indeed fetures wek ntiloliztionD we perform two types of numeril simultionsF e (rst signture of wek ntiloliztion is visile in rel speF sn usul timeEreversl invrint kiked rotorsD strting from wve pket pled t (θ x , y) = (θ 0 , 1)D the gooperon of susetion TFP mnifests itself s pek in the density t (θ x , y) = (-θ 0 , -1) @the gooperon is uilt in momentum speD we re thus looking t n nlogue of the oherent k sttering pekAD on top of the uniform kground ssoited to the hi'uson IWRF hould its sign e reversedD the gooperon mnifests itself s dip in the density t (θ x , y) = (-θ 0 , -1)F his is indeed wht we oserve numerillyD s visile in (gure TFQF

Orthogonal Symplectic

pigure TFQ ! everge densities s funtion of θ x D long y = -1D otined y kiking PHHH times wve pket initilly pled t (θ x = 1, y = 1)F he numeril prmeters re K = 7.5D = 0.2 nd α = 1/3F he spe @[-π, π[A is disretized in IPSVR pointsF he results re verged over ISHHH rndom reliztions of the kineti opertorF he results were respetively otined with rmiltonin @TFQSA @lueA nd rmiltonin @TFQUA @greenAF xote tht the sme wve funtion normliztion is used in oth sesY for some reson the density otined with rmiltonin @TFQSA is slightly imlned etween y = -1 nd y = 1F e seond signture is visile in momentum speF sn usul timeEreversl invrint kiked rotorsD strting from plne wveD the wek loliztion orretion slows down the shortEtime di'usionF hould the sign of the gooperon e reversedD wek ntiloliztion orretion ppers nd enhnes the shortEtime di'usionF hese e'ets re most onveniently oserved in vrint of the β funtionD de(ned s PI

β(g) = d ln(g) d ln(L) , @TFQWA with L = p 2 nd g = p 2
/tF ith these de(nitionsD the β funtion writes

β(g) = 1 - 2 d ln( p 2 ) d ln(t)
.

@TFRHA e lulte this funtion numerilly y propgting plne wve nd reording p 2 versus timeF hi'usion t short times mnifests itself s β funtion going to EI t lrge gF ek @ntiA loliztion orretion rings @positiveA negtive orretion to the β funtion t lrge gF his is lerly visile in (gure TFRF e hve thus found spinless kiked rotor exhiiting fetures typil of the sympleti lssF xote tht in the numeril simultionsD we use (nite grid hosen with reF sndeed for reltion @TFQIA to hold on (nite gridD eh point θ x ∈ [-π, π[ must hve prtner -θ x D in prtiulr the grid should not inlude the point θ x = 0F ypillyD 2N grid points re usedD with uniform smplingX {±2πn/N ± π/N / n ∈ N}F eltion @TFQIA is true in the ontinuous se sine the point θ x = 0 is of mesure zeroF 0.0 0.5 1.0 1.5 2.0 2.5 3.0 1/g pigure TFR ! β(g) s funtion of 1/gF he numeril prmeters re K = 10D α = 1/3 nd = 0.2F he spe @[-π, π[A is disretized in TPWP pointsF he results re verged over SHHH rndom reliztions of the kineti opertorF he results were respetively otined with rmiltonin @TFQSA @lueA nd rmiltonin @TFQUA @greenAF es there re strong )ututions from one kik to the otherD due to the struture of the rmiltoninD we hve only kept mesurements done fter n even numer of kiks to ompute βF 6.3.2 Eigenvalue statistics o see how fr the nlogy with spinEIGP prtiles goesD we now onsider the eigenvlue sttistisF sn the limit of n in(nite systemD lolized eigensttes do not overlpD suh tht their ssoited eigenvlue re independentF he oisson distriution is thus expetedD independently of the rmilE tonin symmetriesF yf ourseD we n only simulte numerilly (nite systemsF por kiked rotorsD eigensttes re lolized in momentum speD the system size is thus set y the sptil disretiztionX ssuming tht the sptil grid vries y step of ∆ x D moment tke vlues in [-π/∆ x D π/∆ x [F por the eigenvlues to oey the rndom mtrix preditions of susetion TFPFPD we must ensure tht the loliztion length is muh lrger thn π/∆ x @this is generlly hieved y using huge kikE ing strengthsAF eprt from thtD the loliztion properties of the rmiltonin re not espeilly importnt for the eigenvlue sttistisF e thus onsider diretly the Ih model6 @TFRIA es disussed in susetion TFPFQD the relevnt opertor to e digonlized is the ssoited ploquet opertorF he ploquet opertor eing unitryD its eigenvlues re omplex numers of modulus IF he sttistil study is performed on the phses of the eigenvluesF o e more spei(D given the ordered eigenphses of the eigenvlues @φ 1 , φ 2 , . . . , φ N AD we onsider the distriution of the level spings s i @s 1 = φ 2φ 1 D s 2 = φ 3φ 2 D . . . D s N -1 = φ Nφ N -1 AF vevel sping distriutions omputed numerilly for the ploquet opertor of @TFRIA with di'erent prmeters re shown in (gure TFSF 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3. pigure TFS ! histriutions of level sping for the ploquet opertor of @TFRIAF por the three (guresD we hve used K = 100 nd omputed ll the eigenvlues for IHHHH rndom reliztions of the ploquet opertorF pigure @A ws otined for α = 0.3 nd sptil disretiztion mde of QP points in ]0, π[ nd their prtner otined y prity @the grid ws of the form {±2πn/N ± π/N / n ∈ N}AF pigure @A ws otined for α = 0 nd the sme sptil grid s (gure @AF pigure @A ws otined for α = 0.3 nd sptil grid inluding the point θ = 0 @the grid ws of the form {±2πn/N / n ∈ N}AF ndom mtrix theory preditionsD eqution @TFPTA @gyiAD eqution @TFPUA @giA nd eqution @TFPVA @giAD s well s the oisson distriutionD eqution @TFPWAD re shown for omprisonF sn dditionD given the level spingsD one n de(ne the quntity IWS ri = min(s i , s i-1 ) mx(s i , s i-1 ) . @TFRPA sts verge over oth the spetrum nd di'erent reliztions of the ploquet opertor is denoted r F e hve ompred the r found for the ploquet opertor of @TFRIA to the rndom mtrix results of IWS in tle TFIF le TFI lso reports the presene of urmers9 degeneryF rmeters urmers9 degenery r r from IWS me s (gure TFS resent 0.6751 ± 0.0004 0.6744 ± 0.0001 @qiA me s (gure TFS resent 0.6012 ± 0.0004 0.5996 ± 0.0001 @qiA me s (gure TFS esent 0.5313 ± 0.0004 0.5307 ± 0.0001 @qyiA le TFI ! r s otined for the ploquet opertor of @TFRIA nd di'erent prmetersF he quntity r is de(ned in the min textF por omprisonD we hve reported the vlues found in IWS for rndom mtries in the qussin orthogonl @qyiAD unitry @qiA nd sympleti @qiA ensemlesF e hve lso reported the presene of urmers9 degeneryF es disussed in susetion TFQFQD only the square of the ploquet opertor ssoited to (gure TFS exhiits urmers9 degeneryF he ploquet opertor itself exhiits pirs of opposite eigenvluesF he results for the eigenvlue sttistis reported in this setion on(rm the nlogy with spinE IGP prtilesF he results re indeed fully onsistent with the sympleti lss when the onditions of susetion TFQFI re met @(gure TFSAX iA α = 0 @iFeF lterntion of two di'erent kiksAD nd iiA sptil grid uilt in suh wy tht eh point θ s prtner -θ so s to gurntee @TFQIAF snterestinglyD the se α = 0 @(gure TFSA seems to orrespond to the se of spinEIGP prtile in the presene of geometril symmetry @urmers9 degenery nd unitry )ututionsAF por the ske of ompletenessD we hve lso inluded the results otined with grid inluding θ = 0D even though it is not relevnt to the ontinuous se sine the point θ = 0 should e of mesure zeroF uite surprisinglyD dding the point θ = 0 @(gure TFSA leds to eigenvlue sttistis onsistent with the orthogonl lssF 6.3.3 Time-reversal invariance he ove results suggest to look for timeEreversl symmetry of the ploquet opertor of @TFRIA squring to -1F o this endD we write the ploquet opertor s

F = U 1 U 2 ,
@TFRQA with U 1 = e -ip 2 θ /2 e -i[K(1+α) cos(θ)+ π 2 sgn(θ)] , @TFRRA U 2 = e -ip 2 θ /2 e -i[K(1-α) cos(θ)+ π 2 sgn(θ)] . @TFRSA es disussed in susetion TFPFID in the se of the kiked rotorD one usully onsiders the produt of the prity @P A nd the onventionl timeEreversl @T A opertors7 @T K = P T AF es suhD it squres to I @T 2 K = 1AF sts ssoited symmetry writes

       t → -t, θ → -θ, p → p.
@TFRTA Case of even number of sites, α = 0 (gure 6.5a)

glerly the ploquet opertor of @TFRIA is not invrint under T K X

T -1 K F T K = U † 1 U † 2 = F † .
@TFRUA e n nevertheless onsider the timeEreversl opertor T = U 1 T K . @TFRVA yne then (nds @it is esy to hek tht

T -1 = T K U † 1 A T -1 F T = T K U † 1 F U 1 T K = T -1 K U 2 U 1 T K = U † 2 U † 1 = F † . @TFRWA
por U 1 given y eqution @TFRRAD one n use the importnt reltion

T -1 K U 1 T K = -U † 1 , @TFSHA to ompute T 2 X T 2 = U 1 T K U 1 T K = U 1 T -1 K U 1 T K = U 1 (-U † 1 ) = -1.
@TFSIA sn onlusionD the ploquet opertor is invrint in the sense of eqution @TFQHA under the tion of timeEreversl opertor squring to -1F his explins the sympleti sttistis of eigenvlues oserved in (gure TFS nd tle TFID s well s the presene of urmers9 degeneryF Case of even number of sites, α = 0 (gure 6.5b) sn the se α = 0D the ploquet opertor redues to F α=0 = U 1 nd we still hve eqution @TFSHA

T -1 K U 1 T K = -U † 1 ⇔ T -1 K F α=0 T K = -F † α=0 .
@TFSPA st hs n importnt onsequene on the spetrumF o exhiit itD we strt from the eigenvlue eqution

F α=0 |φ v = e -iφv |φ v ⇔ F † α=0 |φ v = e iφv |φ v .
@TFSQA e then pply T K on oth sides of the right equlity nd insert T 2 K = 1 to (nd

T K F † α=0 T K T K |φ v = e -iφv T K |φ v ⇔ F α=0 T K |φ v = -e -iφv T K |φ v . @TFSRA
T K |φ v is n eigenstte of F α=0 with eigenvlue -e -iφv F essuming (nite dimension 2N D we (nlly use the sis {φ 1 , . . . , φ N }D {T K φ 1 , . . . , T K φ N } to write F α=0 s

F α=0 = F + 0 0 -F + .
@TFSSA pigure TFS nd tle TFI show the eigenvlue sttistis of

F 2 α=0 = F 2 + 0 0 F 2 + .
@TFSTA he struture of F 2 α=0 explins the results of (gure TFS nd tle TFID in the spirit of urmers9 degenery with one geometri symmetry eqution @TFPSAF Case of odd number of sites (gure 6.5c) sn the se of n odd numer of sitesD eqution @TFSHA does not hold due to the presene of the site θ = 0 on whih T K hs no e'etF iqution @TFRWA however does holdF he question is thus whether T squres to plus or minus IF he se -1 implying urmers9 degeneryD it is possile only if the rilert spe dimension is evenF he numer of sites is oddD so is the rilert spe dimensionD suh tht ll timeEreversl symmetries must squre to IF e onlude tht the ploquet opertor is invrint in the sense of eqution @TFQHA under the tion of timeEreversl opertor squring to ID explining the orthogonl sttistis of eigenvlues oserved in (gure TFS nd tle TFIF

Conclusion

e hve presented in this setion spinless kiked rotor onsistent with the sympleti lssF st shows ler signtures of wek ntiloliztion nd the sttistis of its eigenvlues unmiguously ple it in the sympleti lssF st is indeed invrint under the e'et of timeEreversl opertor squring to EIF etD it is desirle to on(rm these oservtions y n nlysis of its ritil properties in higher dimensionsF vet us tke tht rodF 6.4 From incommensurate frequencies to higher dimensions n δ(tn), @TFSUA with K(t) = K 1 + η cos(ω 1 t + ϕ 1 ) cos(ω 2 t + ϕ 2 ) , @TFSVA to simulte Qh disorder prolemF @TFSUA ws shown to e equivlent to Qh disorder modelsD provided ω 1 nd ω 2 re hosen inommensurte with eh other nd 2π IUWD IWTD IWU @we use ω 1 = 2π √ 5 nd ω 2 = 2π √ 13 in the followingAF he urrent understnding of this orrespondene etween Ih modulted kiked rotors nd Qh prolems is sed on the equivlene etween the dynmis generted y @TFSUA nd the dynmis generted y the Qh pseudoErotor (θ) = (1 + α) cos(θ) + sin(2θ)σ z nd g o 1 2

(θ) = (1α) cos(θ) + µ sin(2θ)σ x . @TFTIA rere σ x nd σ z re uli mtriesF K(t) is given y eqution @TFSVAF @TFTHA is invrint under @TFISA @with θ 0 = 0 nd hlfEintegers t 0 A nd does not hve ny geometril symmetry @this is ensured y the sin(2θ) termsAD it thus elongs to the sympleti lss IVQF Finite time scaling he kiked rotor @TFTHA presents phse trnsitionF o extrt the ssoited ritil exponentD whih governs the divergene of the loliztion length ner the trnsitionD we follow IUW nd perform (nite time sling nlysis of p 2 F rtillyD p 2 is otined y propgting n initil plne wve with the rmiltonin @TFTHA nd verging the result over rndom reliztions of the kineti term nd the phses ϕ 1 nd ϕ 2 F xer the phse trnsition nd t long enough timesD p 2 follows the universl sling lw p 2 = t 2/3 F (K -K c )t 1/3ν , @TFTPA where K c is the ritil point nd ν is the ritil exponent hrterizing the trnsitionF o extrt the ritil exponentD we de(ne the quntity ln Λ = ln( p 2 /t 2/3 ) nd (t it with ylor expnsion of ln(F )X ln(Λ) = ln(Λ c ) + (K -K c )t , @TFTRA where σ(K, t) is the numeril unertinty of ln Λ data (K, t) F he numeril dt ln Λ data (K, t) for di'erent times t re orreltedF por the (tting proedure to e meningfulD we resmple them with ootstrp methodF he error estimtes on the position of the ritil point K c nd the vlue of the ritil exponent ν given elow orrespond to their respetive stndrd devition during the ootstrp proessF

Results

e hve performed numeril simultions of @TFTHA with the numeril prmeters α = 0.5D = 0.5D η = 0.15D µ = 0.5 nd for vrious vlues of KF he spe @[-π, π[A ws disretized in TPVR pointsF o void devitions from @TFTPA due to (nite time e'etsD we hve rried out the (nite time nlysis with vlues of p 2 mesured etween PHHHHH nd IHHHHHH kiksF qenerillyD the results re verged over IPH rndom reliztionsF e hve used more reliztions ner the ritil pointD with RVH reliztions for K ∈ [1.68, 1.695[ nd ISTH reliztions for K ∈ [1.695, 1.75]F he results re reported in (gure TFT nd exlude the possiility of @TFTHA eing in the sme universlity lss s spinEIGP disorder modelsD with ritil exponent more thn 4σ wy from the one found in spinEIGP disorder modelsF yur numeril simultions strongly suggests tht the spinEIGP qusiEperiodilly modulted kiked rotor is not in the sympleti universlity lssF his result is onsonnt with similr oserE vtions for unitry kiked rotors PHHF his issue fores us to postpone the hrteriztion of the ritil properties of the spinless kiked rotor presumly in the sympleti lssF 6.4.3 Beyond the incommensurate frequencies st is presently not ler why qusiEperiodilly modulted kiked rotors pper not to fll in the expeted unitry nd sympleti universlity lssesF he reson my e tht the qusiEperiodi trik is limited to the orthogonl universlity lss nd is not suited to the unitry nd sympleti universlity lssesF yne my envision tht the ommonly used osine qusiEperiodi sequenes reprodue the )ututions of the orthogonl lssD nd should not e used in n ttempt t exploring the unitry nd sympleti universlity lssesF sn this line of thoughtsD we propose to onsider the possiility of other qusiEperiodi sequenesF fefore trgeting spei(lly the unitry nd sympleti universlity lssesD we propose to explore the possiility of simulting enderson loliztion on frtls with qusiEperiodilly modulted kiked rotorsF uh kiked rotors my e useful to study the sympleti lssD nd in prtiulr to identify the lower ritil dimensionD whih is elieved to e somewhere etween I nd P PHID with intriguing onsequenes PHPF yther possile pplitions inlude the vlidtion of d = 2 + expnsions PPD PHQF st is indeed di0ult tsk to simulte numerilly @not to mention experimentllyA suh prolems on tul frtls PHI!PHSF Casati et al. quasi-periodic sequences fefore exploring other qusiEperiodi sequenesD we need to develop some understnding of the ommonly used qusiEperiodi sequenes IWTF hese sequenes re uilt y smpling trigonometri funtionsF wore preiselyD one studies Ph enderson loliztion with kik sequene K 2 (t) = K 1 + η cos(ω 1 t + ϕ 1 ) for t ∈ N, @TFTSA pigure TFT ! win plotX sling quntity ln(Λ) s funtion of ln(ξ/t 1/3 )F ξ is proportionl to |K -K c | -ν F he numeril dt in lueD with their sttistil error rsD re well (tted y funtion of the form @TFTQAD with the expnsion stopped t F 1 F he χ 2 per degree of freedom ssoited to the (tD equl to IFID is indeed stisftory @lose to IAF he (t provides the following estimtes for the ritil point nd ritil exponentX K c = 1.695 ± 0.001 nd ν = 1.68 ± 0.07F he vlue of the ritil exponent is more thn 4σ wy from the one found in spinEIGP disorder modelsD whih lies in the rnge [1.35, 1.39] IWWF snsetX ln(Λ) s funtion of K t vrious times etween PHHHHH nd IHHHHHH kiksF por K < K c @on the lolized side of the trnsitionAD ln(Λ) dereses in timeD while for K > K c @on the di'usive side of the trnsitionAD ln(Λ) inreses in timeF fetween these two limitsD ln(Λ) is onstnt t the ritil point K = K c F e ritil point K c ∈ [1.694, 1696] @represented s lue retngleA is fully onsistent with the resultsF he numeril prmeters used re given in the min textD in susetion TFRFP esultsF

where ω 1 is inommensurte with 2πD nd studies higher dimensions y dding inommensurte frequeniesD eFgF three dimensions with K 3 (t) = K 1 + η cos(ω 1 t + ϕ 1 ) cos(ω 2 t + ϕ 2 ) for t ∈ N. @TFTTA o (nd other relevnt qusiEperiodi sequenesD our ngle is to retin two importnt properties of these sequenesD nmely the qusiEtimeEreverslEinvrine nd the qusiEperiodiityF sndeedD it is with these properties tht the digrms responsile for enderson loliztion n e uilt PPD IVPD PHTF usiEtimeEreverslEinvrine nd qusiEperiodiity orrespond to pproximtions of ω 1 /2π @or ω 1 /2π nd ω 2 /2πA y rtionlsF sn this veinD the importnt di'erene etween K 2 nd K 3 is the sling of how di0ult it gets to hieve qusiEtimeEreverslEinvrine @qusiEperiodiityA s one eomes more nd more demndE ing on the preision of the timeEreversl @qusiEperiodAF sn mthemtil termsD this sttement tkes the following formX lling α the numer of eptle rtionl pproximtions to ω 1 /2π @or ω 1 /2π nd ω 2 /2πA nd β the mximl distne etween ω 1 /2π @or ω 1 /2π nd ω 2 /2πA nd its rtionl pproximtionsD we re interested in how α hnges with βF por exmpleD for K 2 one (nds α ∝ βD while for K 3 one otins α ∝ β 2 F wore generllyD for K d one gets α ∝ β d-1 F his resoning neglets the properties of the rtionl pproximtions tht my hnge with βF IPW Two theorems from number theory wo theorems from numer theory re of interest for the mtter t hndF he (rst one is rurwitz9s theorem PHUD it sttes tht for every irrtionl numer ξ there re in(nitely mny oprime integers mD n suh tht ξ -m n < 1 √ 5 n 2 . @TFTUA he seond one is oth9s theorem PHVD whih sttes tht every irrtionl lgeri numer ξ hs pproximtion exponent equl to PD iFeFD for ny ε > 0D the inequlity ξ -m n < 1 n 2+ε @TFTVA n hve only (nitely mny solutions in oprime integers m nd nF prom these two theoremsD we onlude tht the error in pproximting ω/2π y rtionl is inversely proportionl to the squre of the ssoited periodF Quasi-periodic sequences from series prom the ove disussionD we wnt funtion f (t) tht hs pproximte time reversls @periodsAD with the numer of time reversls @periodsA sling s d-1 D where is the preision of the timeE reversl @periodAF sn dditionD the lengths of the timeEreversls @periodsA should sle s √ F e propose the following funtion f (t

Results

e hve tested the sequene @TFTWA numerilly with the rmiltonin

H = p 2 2 + K 1 + ηf (t) cos(θ) n δ(t -n), @TFUIA
the results re reported in (gure TFUF he results show tht loliztion previls t smll K nd turns into sudi'usion s K is inresedF urprisinglyD sudi'usion does not onern only one vlue of K ut full rnge of vlues of K @η eing (xedAD with sudi'usive exponent inresing with KF o go furtherD we hve ompred the results displying sudi'usive exponent of 2/3 @K = 1.64 in (gure TFUA with the known ritil pro(le of the usul qusiEperiodi kiked rotor eqution @TFSUA PHWX |ψ(p)| 2 = 3α 2 ei(α|p|), @TFUPA where ei is the eiry funtionF es visile in the min plot of (gure TFVD the numeril results re very well (tted y @TFUPA with α 0.0056D thus suggesting tht it orresponds to n endersonE Qh ritil pointF xote tht eqution @TFUPA orresponds to Qh di'usion proess with frequeny dependent di'usion oe0ient D(ω) = (-iω) 1/3 D integrted over two diretions PHWDPIHF hevitions from @TFUPA re expeted t smll pD due to the multifrtl struture of the ritil eigenfuntions PHW!PIQD they re lerly visile in the inset of (gure TFVF hese devitions onstitute more spei( signture of ritil pointD lthough more work is needed to desrie them in terms of the multifrtl exponents of the ritil eigenfuntions PIRD whih would provide n unmiguous orrespondene with n endersonEQh ritil pointF edmittedlyD these results re not onsistent with @TFUIA eing equivlent to dEdimensionl disorder model @for whih one expets sudi'usion for only one vlue of K ! t (xed ηAD ut we elieve tht they open the door to exiting perspetivesF e theoretil frmework is lking to put these results into perspetiveF e n only notie similrity with the powerElw rndom nded mtrix @fwA ensemle IWQD whih fetures set of ritil pointsF en importnt di'erene with this model resides in the lolized regime @smll K in the present ontextAD where we oserve exponentil loliztion @(gures TFV nd TFVA s opposed to lgeri loliztion in the fw ensemleF pigure TFV ! vogrithms of the verge momentumEdensity pro(les for n initil plne wve evolving ording to @TFUIAD with η = 0.45F sn (gures A nd AD the dshed lines re (ts of the form a|x| + bF por (gures A nd AD the spe @[-π, π[A is disretized in IPTH points nd the results re verged over IHHHH reliztions of the kineti termF por (gure AD the spe is disretized in TPWP points nd the results re verged over PHHH reliztions of the kineti termF pigure A min plotX t lrge sles the numeril results in green re well (tted y @TFUPA with α 0.0056F snsetX verge momentumEdensity pro(leD devitions from @TFUPA with α 0.0056 re lerly visile t smll pD they ould e mnifesttion of the multifrtl struture of the ritil eigenfuntions PHW!PIQF he study of the sympleti lss led us to seond diretionD the pprent filure of the qusiE periodi modultion trik IWT in the unitry nd sympleti lssesF o egin ddressing this issueD one ould perform numeril simultions with tul Qh kiked rotorsD s done in PIS for the orthogonl lssD rther thn through the introdution of inommensurte frequeniesF uh study would either point to the qusiEperiodi modultionD or to more fundmentl di'erene etween kiked rotors nd disorder models in the unitry nd sympleti lssesF e third spet onerns the possiility of using new types of qusiEperiodi kiking sequenesD with di'erent possile intentionsF elthough our ttempt t exploring enderson loliztion on frtls hs not yet gone throughD we elieve tht our preliminry study shows promising results nd open exiting perspetives for future workF Chapter 7 Conclusion sn this thesisD we theoretilly investigted severl interesting e'ets relted to enderson loE liztionD fousing on the ontext of oldEtomi systemsF e nturlly strted y hrterizing the spekle potentils typilly used in oldEtom experimentsF sn prtiulrD we lulted sttisE til properties of intensity minimF sn shortD these minim typilly lie t low energies nd the distriution of their urvture is peked round the verge vlueF ith these results t hndD we emrked into the desription of the spetrl properties of toms in spekle potentilsF wore preiselyD we onsidered the spetrl funtion nd the density of sttesD in the strong disorder limitF emilssil methods turned out very powerful nd llowed for good desription of the whole energy spetrumF fesidesD our lultions provide simple interprettion of intriguing fetures of the spetrl funtion nd density of sttes nd of the di'erene etween redE nd lueEdetuned speklesF he study of spetrl properties ws lso the opportunity to set the generl frmework for desriing the dynmis of wve pkets in rndom potentilsF e pplied it to prtiulr senrioD di'ering from the experiment performed in liseu in PHHV QU only in tht toms re given (nite veloity t t = 0F emrklyD we found tht fter n initil llisti motionD the pket enterEofEmss experienes retrore)etion nd slowly returns to its initil positionD mimiking oomerngF e ssoited this quntum oomerng e'et to enderson loliztionD nd desried it thoroughly in one dimensionD oth nlytilly nd numerillyF e then onsidered the e'ets of wek intertions on the evolution in disordered potentils of ondensed osoni gsesF o tht endD we hve developed numeril integrte sheme of the qrossEitevskii equtionD whih desries the evolution of n interting gs t the menE(eld levelF his method ws pplied to vrious situtionsD with three min onlusionsF pirstD nonliner intertions tend to inhiit the quntum oomerng e'et of wve pkets lunhed with (nite veloityF eondD euse of inelsti ollisions on the nonliner potentilD the wve pket explores the energy lndspeF hirdD the oherent k sttering @gfA pek of mtter wves prepred s plne wve sttes is strongly 'eted y the nonlinerityF e identi(ed two kind of proesses reduing the gf ontrstX shortEtime e'ets tht rek timeEreversl invrine on the one hndD nd inelsti ollisions eventully leding to its destrution on muh longer time sle on the other hndF hese results point to interesting perspetives for future workF gonerning the spetrl propE ertiesD we hve not treted the threeEdimensionl seD involved in importnt questions relted to enderson loliztion QVD RHD RRD IHWD IISD IPVF sts desription would e of gret interestF por exE mpleD the spetrl funtion ould e used to estimte the position of the moility edgeD through generlized so'eEegel riterion IHWD PIT!PIVF es fr s the quntum oomerng e'et is onernedD while we hve given thorough desription of the oneEdimensionl seD more work is needed in higher dimensionsF por exmpleD in light of reent IQQ IQR ghpter UF gonlusion work on the oherent kwrd nd forwrd sttering peks ross enderson trnsition RRD IPVD it would e very interesting to perform (ne numeril nlysis of the quntum oomerng e'et in three dimensionsF enother interesting possiility is to look for similr phenomen in other symmetry lsses @eFgF in the unitry lss where the qulittive understnding of retrore)etion in terms of lolized real modes does not pplyAF yur study of the e'ets of wek intertions on the evolution of ondensed osoni gses ws essentilly numerilF hile we underlined the physil piture explining our oservtionsD muh remins to e done to desrie them nlytillyF por exmpleD we pointed out strong similritiesD regrding the quntum oomerng e'etD etween the nonlinerity nd simple deoherene proessF ixhiiting the mirosopi origin of these similrities would proly e very enlighteningF yn more fundmentl levelD we treted the intertions through menE(eld pproximtionD thus possiility missing oherent mnyEody e'etsF heir inlusions my led to qulittively di'erent onlusions @eFgF mnyEody oherent ksttering pek VTD PIWD PPHAF pinllyD we hve shown tht the kiked rotor onstitutes good pltform to hrterize vrious spets of enderson loliztionD nd hve identi(ed three interesting diretions of reserh for future workF pirstD going ginst ommon wisdomD we hve presented strong evidene tht the sympleti lss n e explored with spinless prtilesF eondD we hve oserved tht the qusiEperiodi modultion trik of gsti et al. IWT seems to fil in the unitry nd sympleti lssesF hirdD we hve explored the possiility of using new types of qusiEperiodi kiking sequenesF e elieve tht our preliminry study in this diretion shows promising results nd opens exiting perspetives for future workF Bibliography I F F endersonD hysF evF 109D IRWP @IWSVAF P qF tolzD gontemporry wthemtis 552D UI @PHIIAF Q iF ekkermns nd qF wontmuxD Mesoscopic Physics of Electrons and Photons @gmridge niversity ressD PHHUAF R qF fergmnnD hysF epF 107D I @IWVRAF S pF ierreD eF fF qougmD eF enthoreD rF othierD hF isteveD nd xF yF firgeD hysF evF f 68D HVSRIQ @PHHQAF T gF rinutD sF wniD F ghiirenuD tF pF glémentD F emmouriD tF gF qrreuD F zriftgiserD qF vemriéD xF gherroretD nd hF helndeD hysF evF vettF 118D IVRIHI @PHIUAF U pF tendrzejewskiD uF wüllerD tF ihrdD eF hteD F lissonD F fouyerD eF espetD nd F tosseD hysF evF vettF 109D IWSQHP @PHIPAF V qF veyrieD F urpiukD tF pF h'D fF qrémudD gF winiturD nd hF helndeD iuro hysF vettF 100D TTHHI @PHIPAF W hF ollhrdt nd F öl)eD hysF evF f 22D RTTT @IWVHAF IH F öl)e nd hF ollhrdtD sntF tF wodF hysF f 24D ISPT @PHIHAF II pF ivers nd eF hF wirlinD evF wodF hysF 80D IQSS @PHHVAF IP yF sF vokis nd F vF everD hysF evF i 71D HIIIIP @PHHSAF IQ F vF ferezinskiiD hF ikspF eorF pizF 65D IPSI @IWUQA ovF hysF ti 38D TPH @IWURAF IR eF eF qogolinD hysis eports 86D I @IWVPAF IS gF F tF feenkkerD evF wodF hysF 69D UQI @IWWUAF IT F vF ever nd tF furkhrdtD tF eoustF oF emF 96D QIVT @IWWRAF IU F vF ever nd yF sF vokisD hysF evF vettF 84D RWRP @PHHHAF IV iF erhmsD F F endersonD hF gF viirdelloD nd F F mkrishnnD hysF evF vettF 42D TUQ @IWUWAF IW rF qF husterD F hysF f 31D WW @IWUVAF PH hF ollhrdt nd F öl)eD hysF evF vettF 48D TWW @IWVPAF PI F ystrovskyD F xkymD uF eF wuttliD nd F öl)eD xew tF hys 15D HSSHIH @PHIQAF PP uF ifetovD Supersymmetry in Disorder and Chaos @gmridge niversity ressD IWWWAF PQ qF peherD hysF evF 114D IPIW @IWSWAF PR qF peher nd iF eF qereD hysF evF 114D IPRS @IWSWAF PS fF urmer nd eF wuinnonD epF rogF hysF 56D IRTW @IWWQAF PT F eF vee nd F F mkrishnnD evF wodF hysF 57D PVU @IWVSAF IQS IQT filiogrphy PU uF fyzukD F rofstetterD nd hF ollhrdtD ghpter PH in 50 Years of Anderson LocalizationD edF iF erhms @orld ienti(D ingporeD PHIHAY reprinted in sntF tF wodF hysF f 24D IUPU @PHIHAF PV fF eF vn iggelenD ghpter I in Diuse Waves in Complex MediaD edF tF F pouque @pringer xetherlndsD IWWWAF PW F hlihouhD tF F ermstrongD F hultzD F wF ltzmnD nd F vF wgllD xture 354D SQ @IWWIAF QH vF eD qF godyD wF houD F hengD nd eF xF xorrisD hysF evF vettF 69D QHVH @IWWPAF QI eF eF ghnovD wF toythevD nd eF F qenkD xture 404D VSH @PHHHAF QP F hwrtzD qF frtlD F pishmnD nd wF egevD xture 446D SP @PHHUAF QQ wF törzerD F qrossD gF wF eegerterD nd qF wretD hysF evF vettF 96D HTQWHR @PHHTAF QR F perlingD vF hertelD wF ekermnnD qF tF euryD gF wF eegerterD nd qF wretD xew tF hysF 18D HIQHQW @PHITAF QS rF ruD eF tryulevyhD tF rF geD F iF kipetrovD nd fF eF vn iggelenD xture hysF 4D WRS @PHHVAF QT tF ghéD qF vemriéD fF qrémudD hF helndeD F zriftgiserD nd tF gF qrreuD hysF evF vettF 101D PSSUHP @PHHVAF QU tF fillyD F tosseD F uoD eF fernrdD fF rmrehtD F vugnD hF glémentD vF nhezE leniD F fouyerD nd eF espetD xture 453D VWI @PHHVAF QV pF tendrzejewskiD eF fernrdD uF wüllerD F gheinetD F tosseD wF irudD vF ezzéD vF nhezE leniD eF espetD nd F fouyerD xture hysF 8D QWV @PHIPAF QW sF wniD tF pF glémentD F ghiirenuD gF rinutD tF gF qrreuD F zriftgiserD nd hF helndeD hysF evF vettF 115D PRHTHQ @PHISAF RH qF emeghiniD wF vndiniD F gstilhoD F oyD qF pgnolliD eF renkwlderD wF pttoriD wF sngusioD nd qF wodugnoD xture hysF 11D SSR @PHISAF RI qF wodugnoD epF rogF hysF 73D IHPRHI @PHIHAF RP fF hpiroD tF hysF eX wthF heorF 45D IRQHHI @PHIPAF RQ F urpiukD xF gherroretD uF vF veeD fF qrémudD gF eF wüllerD nd gF winiturD hysF evF vettF 109D IWHTHI @PHIPAF RR F qhoshD gF winiturD xF gherroretD nd hF helndeD hysF evF e 95D HRITHP@A @PHIUAF RS F F uondovD F F wqeheeD tF tF irelD nd fF hewroD iene 334D TT @PHIIAF RT F F wqeheeD F F uondovD F uD tF tF irelD nd fF hewroD hysF evF vettF 111D IRSQHQ @PHIQAF RU F vugnD eF espetD vF nhezEleniD hF helndeD fF qrémudD gF eF wüllerD nd gF winiturD hysF evF e 80D HPQTHS @PHHWAY irrtum hysF evF e 84D HIWWHP @PHIIAF RV F plhD hpter I in Nonlinear Optical and Atomic Systems: At the Interface of Physics and MathematicsD edF gF fesse nd tF gF qrreu @pringerD ghm PHISAF RW eF sominD gompF wthF epplF 73D WIR @PHIUAF SH iF vuioniD fF heisslerD vF nziD qF otiD wF ntiD wF wodugnoD wF vrherD pF hlfovoD wF sngusioD nd qF wodugnoD hysF evF vettF IHTD PQHRHQ @PHIIAF SI qF fenettinD tF pröhlihD nd eF qiorgilliD gommunF wthF hysF 119D WS @IWVVAF SP F wF ng nd F hngD tF ttF hysF 134D WSQ @PHHWAF SQ iF wihely nd F pishmnD hysF evF i 85D HRTPIV @PHIPAF

K = 2 K = 3 K = 4 K = 5 @A d = 2F

  UU RFW ertileX When Anderson localization makes quantum particles move backward F F F F UV eppendix RFe olution of ferezinskii equtions t long times F F F F F F F F F F F F F F F F VR eppendix RFf olution of ferezinskii equtions from short times F F F F F F F F F F F F F F VT 5 Weakly interacting wave packets 91 SFI foseEiinstein ondenstes in rndom potentils F

  pitureD A i is the prtil omplex mplitude ssoited with the sttering pth iF st hs phse whih depends on the pth iF sn eqution @IFIAD we hve seprted two ontriutions to I P ghpter IF sntrodution

  yn the other hndD perturtion theory round the yhm lw provides the lrge g symptoti IVX β(g) = d -2a/g + . . . @IFWA with a > 0 for spinless timeEreversl invrint systemsF fy interpolting etween these two limitsD erhmsD endersonD viirdello nd mkrishnn skethed the β funtion in ll dimensionsD s reprodued in (gure IFQF pigure IFQ ! keth @reprodued from IVA of the β funtion versus ln(g) otined y interpolting smoothly etween the symptoti limits @IFVA nd @IFWAF es disussed in the min textD in d > 2 the β funtion rosses HD thus exhiiting phse trnsitionF sts slope ross HD shown s solidEirled lineD enodes the ritil exponent ssoited to this phse trnsition IVF e jump of ondutivity in d = 2D deted t the time IWD leds to the dshed lineD inonsistent with the hypothesis of regulrityD whih insted rules in fvor of loliztion of ll sttes in d = 2F

  σ = D/kR σ = D/kR σ = D/ √ 2kR

  -detuned spectral function b) Blue-detuned density of states c) Red-detuned spectral function d) Red-detuned density of states

  a numerical disorder realization of both a blue and a red-detuned 1D speckle

FIG. 3 .

 3 FIG.3. Spectral function A k=0 ( ) as a function of energy in a 1D, blue-detuned speckle potential with Gaussian correlation function, for η = 128. The harmonic-oscillator approximation, Eq. (36), is shown as a solid red curve, and the classical limit, Eq. (10), as a solid green curve. Blue dots are the result of exact numerical simulations.

FIG. 4 .

 4 FIG.4. Density of states ν( ) as a function of energy in a 1D, blue-detuned speckle potential with Gaussian correlation function. The harmonic oscillator-approximation, Eq. (37), is shown as a solid red curve, and the classical limit, Eq. (11), as a solid green curve. Blue dots are the result of exact numerical simulations.

FIG. 8 .

 8 FIG. 8. Density plot of the joint distribution P (ω x ,ω y ) at a point where V = 0 for a 2D, blue-detuned speckle potential with Gaussian correlation function.

  FIG.10. Spectral function A k=0 ( ) as a function of energy in a 2D, blue-detuned speckle potential with Gaussian correlation function, for η = 128. The harmonic-oscillator approximation, Eq. (54), is shown as a solid red curve, and the classical limit, Eq. (10), as a solid green curve. The corrected harmonic-oscillator approximation, Eq. (56), is shown as a solid black curve. Blue dots are the results of exact numerical simulations.

  hi'erent verties t the strting pointF o those verties orrespond the following ftorsX (m/k) exp(ik(x dx c )) exp(-iωkx d /(2 )) D (m/k) exp(ik(x cx d )) exp(iωkx d /(2 )) D (m/k) exp(ik(x c + x d )) exp(-iωkx d /(2 )) D (m/k) exp(-ik(x c + x d )) exp(iωkx d /(2 )) dF es spei(ed in setion RFID the initil wve pket is ssumed muh smller thn the sttering men free pthX |x cx d |

a and a' -2m 2

 2 δx -2m(m -

  PACS numbers: 72.15.Rn, 67.85.-d, 05.60.Gg

FIG. 1 .

 1 FIG. 1. (Color online) Main plot: center of mass as a function of time. Its long-time asymptotics, Eq. (6), is shown as a solid red curve, and the re-summation of the short-time series, Eq. (10), as a solid green curve. The latter perfectly overlaps with the numerical result (blue dots). The dashed curve is the classical result, Eq. (1). Inset: center of mass multiplied by (t/τ ) 2 as a function of time. The asymptotic result (6) (red curve) is compared to the numerical prediction, displayed with its statistical error bars. The parameters used in the simulations are given in the main text.

FIG. 3 .

 3 FIG. 3. (Color online)Average density profile obtained numerically at three different times. The solid upper blue and lower red curves are the x > 0 and x < 0 components of the profile, respectively. The long-time limit of the profile, Eq. (11), is shown as a dashed black curve.

FIG. 4 .

 4 FIG. 4. (Color online)Main plot: center of mass x g as a function of time, for g = 0 (solid lower blue curve), g = 0.09 2 k0/m (solid middle orange curve) and g = 0.15 2 k0/m (solid upper magenta curve). Two dashed curves show the center of mass x φ obtained with the model of decoherence, for A f = 0.02, 0.03. Dotted curve: CoM in 2D. Inset: τ /τ φ (g), well fitted by a linear regression. The effective decoherence time τ φ (g) associated with the nonlinearity thus behaves like /τ φ (g) ∼ g/ .

5. 3

 3 Nonlinear quantum boomerang 5.3.1 Numerical experiment he nonlinerity 'eting enderson loliztionD nturl question to sk is whether it prevents the quntum oomerng e'et of hpter RF vet us perform the numeril simultion thenD nd propgte the initil wve funtion φ(x) ∝ e -x 2 /2σ e ik 0 x .

  grid sizeD α = 0F

6. 4 . 1 = p 2 θ 2 +

 412 Generalitieshe kiked rotor o'ers very powerfulD yet intriguingD wy to explore high dimensions experE imentlly or numerillyF st onsists in modulting qusiEperiodilly the kik sequene of Ih kiked rotorF por exmpleD one n use the kiked rotor H K(t) cos (θ)

H = p 2 θ 2 + ω 1 p 1 + ω 2 p 2 +

 212 K 1 + η cos(θ 1 ) cos(θ 2 ) cos (θ) n δ(tn), @TFSWA with initil onditions θ 1 = ϕ 1 nd θ 2 = ϕ 2 F he equivlene ppers diretly when writing down the evolution of given wve pket with the two models IWTF xote tht similrly to the Ih usul kiked rotorD the pseudoErotor @TFSWA n e mpped onto Qh disorder model IUWD IWT nd n e shown to e equivlent to disorder models through supersymmetri (eld theory IWUF 6.4.2 Incommensurate frequencies in the symplectic classo the est of our knowledgeD the ide of using qusiEperiodi modultion to study higher dimensions hs only een pplied to kiked rotors in the orthogonl lssF fefore pplying it to our spinless kiked rotor presumly elonging to the sympleti lssD we propose to hek the vlidity of the pproh in the more fmilir sitution of spinEIGP kiked rotor IVRD IWVF Model e use the following kiked rotor

  round m/2X f m (m/2 + t) = f m (m/2t)D nd period mF d supposly enodes the sptil dimension of the equivlent disorder modelF sn the followingD we verge over the origin of time to ensure sttistil timeEtrnsltionl invrineF

2 0

 2 . 8 x + 3 0 . 8 9 x + 3 . 1

3 @ 3 @

 33 A d = 3Fpigure TFU ! p 2 s funtion of time for n initil plne wve evolving ording to @TFUIAD with η = 0.45F he spe @[-π, π[A is disretized in PSITV points nd the results re verged over WT reliztions of the kineti termF he dshed lines re liner (tsF6.5 Conclusionsn this hpterD we hve disussed vrious spets of enderson loliztion from the perspetive of kikedErotor modelsF he min onlusions of these studies is tht mny spets of enderson loliztion re still to e explored with the kiked rotorF e hve put forwrd three min diretionsF he (rst one onerns the possiility of exploring the sympleti lss without the need of spinF his hs oth theoretil nd experimentl motivtionsF yn the theoretil sideD it goes ginst ommon wisdom tht spinless model elongs to the sympleti lssF prom n experimentl point of viewD while it would open exiting possiilities IURD the introdution of spin neessittes quite some e'orts ITVD voiding it is desirleF e hve presented strong evidenes in fvor of this possiilityF A d = 3D K = 1.64F

  Weak localizationsn this setionD we present n intuitive piture of enderson loliztionF o tht endD we ssume tht prtile is initilly pled t some point r in disordered mediumD nd formlly express its density distriution t nother point r t time t s sum over ll possile sttering pthsX

	n(r, t) =
	paths i

1.1.1

spectral function and density of states in speckle potentials

  Article: Semiclassical spectral function and density of states in speckle potentials [Phys. Rev. A 94, 022114 (2016)] Tony Prat, Nicolas Cherroret, and Dominique Delande Laboratoire Kastler Brossel, UPMC-Sorbonne Universités, CNRS, ENS-PSL Research University, Collège de France; 4 Place Jussieu, 75005 Paris, France (Received 8 June 2016; published 19 August 2016)

	Semiclassical

  When Anderson localization makes quantum particles move backwardTony Prat 1 , Dominique Delande 1 , Nicolas Cherroret 1 1 Laboratoire Kastler Brossel, UPMC-Sorbonne Universités, CNRS, ENS-PSL Research University, Collège de France; 4 Place Jussieu, 75005 Paris, France (Dated: 19/04/2017)

	lolized modes does not pplyAF
	4.9 Article: When Anderson localization makes quantum particles move backward [Submitted to Phys. Rev. Lett. (arXiv:1704.05241)]

2

  At short times, the momentum and energy distributions are sharply peaked, the self-energy Σ corresponds to Σ( = 2 k 2 0 /2m, k = k0). The reader not familiar with the concepts of spectral function or self-energy can nd some

					IHI
		12	t = 400. 0 ma 2 /		12	t = 800. 0 ma 2 /
	6 8 10 × 2 /ma 2	Numerics t 2τ Numerics Collision integral	6 8 10 × 2 /ma 2	Numerics t 2τ Numerics Collision integral
	P( )	4		P( )	4
		2			2
		0 12	0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 × ma 2 / 2 t = 1200. 0 ma 2 /		0 12	0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 × ma 2 / 2 t = 1600. 0 ma 2 /
	6 8 10 × 2 /ma 2	Numerics t 2τ Numerics Collision integral	6 8 10 × 2 /ma 2	Numerics t 2τ Numerics Collision integral
	P( )	4		P( )	4
		2			2
		0	0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 × ma 2 / 2		0	× ma 2 / 2 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2
	details in section 3.1.

  5.4.3 Short-time (t ≈ τ ) picture: shifting and screening sn ftD t erly times one needs to ount for the rpid rndomiztion of φF sndeedD initilly φ is plne wveD thus |φ| 2 does not )utute nd |φ| 4 = |φ| 2 2 D ut s φ gets rndomized y the disorderD we expet |φ| 4 to inlude )ututing prtD in ddition to |φ| 2 2 X |φ| 4 = |φ| 2 2 + |φ| 4 -|φ| 2 2 = |φ| 2 2 + vr |φ| 2 .le SFI ! ttering men free time τ = -/2 (Σ) nd rel prt of the selfEenergy otined numerilly for vrious timesF he simultions were performed on Qh lttie mde of 600 × 96 2 sites @the long diretion is the one ssoited to k 0 AD with disorder strength γ = 0.18 4 /m 2 aD nonlinerity g/ol = 0.0075 2 /ma 2 nd n initil wve vetor k 0 = π/4aF he results re verged over IS disorder reliztionsF e time step δt = 0.4 ma 2 / ws found su0ientF he error estimtes ome from the (tting proedureF he e'et of this rndomiztion orresponds to shift of the rel prt of the selfEenergy ounting for the inrese of |φ| 4 nd the onserved quntity E eqution @SFQIA 3 X

	ime @ma 2 / A	τ (ma 2 / )	(Σ) ( 2 /ma 2 )
	H T IP IV PR QH QT			17.44 ± 0.04 (-8.863 ± 0.005) × 10 -2 18.28 ± 0.04 (-8.475 ± 0.005) × 10 -2 18.56 ± 0.04 (-8.280 ± 0.005) × 10 -2 18.68 ± 0.04 (-8.135 ± 0.005) × 10 -2 18.76 ± 0.04 (-8.040 ± 0.005) × 10 -2 18.76 ± 0.04 (-7.975 ± 0.008) × 10 -2 18.76 ± 0.08 (-7.938 ± 0.008) × 10 -2
	d P ( , t) -	g 2	dr|φ(r, t)| 4
				@SFQRA

  6

	0.30		0.14
	0.25		0.12	Initial P( )
	0.00 0.05 0.10 0.15 0.20 × 2 ν( ) a/m	× ma 2 / 2 Numerics Disorder-free Born SCBA CPA	0.5 0.00 0.02 0.04 0.06 0.10 0.08 × 2 a/m ν( )	0.0	0.5 × ma 2 / 2	1.0 Numerics Disorder-free Born SCBA CPA
		@A

  1/3ν F 1 + . . . @TFTQA TFRF prom inommensurte frequenies to higher dimensions IPU he est (t ln Λ t (K, t) is otined y minimizing χ 2 = K,t ln Λ data (K, t)ln Λ t (K, t) σ(K, t)

	2

Conclusion

For a derivation of equation (1.3), see e.g.[3].

The correction brought by each loop is an only an average correction, possible uctuations from one loop to the other are not accounted for.

We will come back on the conditions under which equation (1.6) is valid in section

4.4.

Note that these correlations are important in experiments to keep a small diraction angle, so as to concentrate the speckle pattern in a limited region of space. Taking them into account only changes the large scale structure of the speckle pattern.

To simplify the discussion, we assume that the complex electric eld phase is uniformly distributed in [0, 2π[, such that E(x, y) = 0. Experimentally, this corresponds to using a plate rough at the scale of λ.

The maxima of the potential could be studied along the same lines, but they are not relevant for our calculations of spectral properties in chapter 3.

To evaluate equation(3.16) at negative times, we use complex analysis and close the contour in the upper-half complex plane: the analyticity of G(k, ) guarantees that G(k, t < 0) = 0 (causality). If the imaginary part of the self-energy were positive, a pole in the upper-half complex plane would contribute to G(k, t < 0), thus violating causality.

See section 2.2 for more details about this decomposition.

Note that for blue-detuned speckle potentials, the spectral function and the density of states exhibit a Lifshitz tail at very small energies[124, 125]. This tail is intrinsically quantum and cannot be described by semiclassical approximations.

It could be possible to perform a Wick rotation and express everything as a sum of imaginary energies. This treatment is not very enlightening.

Note that |ψn(k = 0)| 2 vanishes for odd n, such that only even n contribute to A k=0 ( ) in equation (3.56).

As discussed in chapter

2, this model of random potential is not realistic for cold-atom experiments. It is chosen here because it allows for much simpler perturbative developments thanks to Wick theorem (see subsection 3.1.2 for more details), and eventually to fully solve the problem in section 4.5. In the weak disorder limit, the choice of potential on-site distribution, if reasonable, is anyhow not expected to change the physics discussed in this chapter[START_REF] Leonov | [END_REF].This issue is further discussed in section 4.6.1.

Note that using directly the results of[13] to calculate x , we nd a result o by a factor of 2.

k is absent in the equation for x [equation (4.49)], and thus set to 0 in the nal expressions [equations (4.63) and (4.65)].It is here introduced for the sake of generality, to conform with Berezinskii notations[13].

This potential has an on-site exponential distribution and is dened only for positive V , see section 2.2 for more details.

This can be justied more rigorously by using the phase space Wigner representation W (x, p, t) of the wave packet (see section 3.2 for more details). Under the conditions of section 4.1, W (x, p, t) is non vanishing only near p = ± k0, so that it can be split in two parts W (x, p, t) ≈ n+(x, t)δ(p -k0) + n-(x, t)δ(p + k0).

More work is needed to identify the origin of this nite convergence radius.

Practically, a least-square t method has been used for center-of-mass data corresponding to t > 150τ .

Note that we dene the real part of the self-energy with respect to the energy = φ(t)|H|φ(t) . One could work with E [equation (5.31)] instead, and the self-energy would then not be shifted by the randomization of φ. At this stage, this choice is arbitrary.

Continuing on the remark of footnote 3, we would like to stress that at the moment, it is not clear to us which denition of the couple energy/real part of the self-energy is relevant for the CBS peak. Nevertheless, both denitions lead to the same qualitative conclusion of a shift of the self-energy at short times, only the amplitude of the shift is in question.

Note that other more complicated interferential contributions also play an important role for localization[22].They also have analogues in disorder models[22].

By quantum chaotic Hamiltonian, we mean a quantum Hamiltonian whose classical counterpart displays a chaotic dynamics.

Of course, strictly speaking, an Hermitian matrix cannot have statistically independent entries, since the upper and lower triangular part of the matrix are related by complex conjugation. Only the upper (lower) triangular part of the matrix has literally statistically independent entries.

An Hermitian quaternionic matrix is composed of real entries on the diagonal and o-diagonal quaternionic entries.

In fact, the analogy between the structure of the 2 × 2 blocks and quaternions goes much deeper, see e.g.[191].

While the conventional time-reversal symmetry (t → -t, p θ → -p θ , θ → θ) plays no role for the kicked rotor, it could disturb the spectral analysis[183]. For the random kicked rotor, using independent random phases (as opposed to phases with parity properties, e.g. φ l = φ -l ) in equation (6.12) allows to stay clear of this parasitic eect.

For the sake of clarity, we omit the factors e ±ip 2 θ /2 that should enter the denitions of the time-reversal operators[123], they do not change anything to the reasoning. This remark applies to the whole subsection.

As the functions fm 1(t) are slowly varying and positive, we have added the (-1) n+1 to ensure some timetranslational-invariance. The factor 8 is simply here for the sum not to be negligibly small.

Remerciements

ACKNOWLEDGMENTS

The authors would like to thank T. Thiery and C. Müller for useful discussions. This work was granted access to the HPC resources of TGCC under the allocation 2016-057083 made by GENCI (Grand Equipement National de Calcul Intensif). We thank Christian Miniatura for discussions at the early stages of this work.

IQU

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HTQTQR @PHITAF IIP wF irudD F vugnD F fouyerD eF espetD nd vF nhezEleniD hysF evF e 83D HQITHQ@A @PHIIAF IIQ gF eF wüller nd fF hpiroD hysF evF vettF 113D HWWTHI @PHIRAF IIR gF eF wüllerD hF helndeD nd fF hpiroD hysF evF e 94D HQQTIS @PHITAF IIS wF sekD qF yrsoD nd hF helndeD hysF evF vettF 118D IUHRHQ @PHIUAF IIT qF wF ploD eF eF pedorenkoD tF qiomelliD nd wF wodugnoD hysF evF e 82D HSQRHS @PHIHAF IIU fF qrmmtios nd eF orosD ennF hysF 123D QSW @IWUWAF IIV F wgnusD gommF ure epplF wthF 7D TRW @IWSRAF IIW hF F frillingerD Time series data analysis and theory @imD PHHIAF IPH F F veonov nd eF xF hiryevD heorF proF pplF 4D QIW @IWSWAF xote tht semiEinvrint is nother word for joint umulntF filiogrphy IQW IPI wF gF qutzwillerD Chaos in classical and Quantum mechanics @pringerD IWWHAF IPP wF orD hysi 6DD IWS @IWVQAF IPQ pF rkeD Quantum Signatures of Chaos @pringerD PHIHAF IPR sF wF vifshitzD edvF hysF 13D RVQ @IWTRAF IPS qF wF plo nd eF eF pedorenkoD hysF evF e 92D HPQRIP @PHISAF IPT eF 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IQS F lissonD F fourdelD nd gF eF wüllerD iurF hysF tF peF opF 217D UW @PHIQAF IQT eF eF qogolin nd F sF wel9nikovD hysF ttF olF @A 88D QUU @IWUVAF IQU vF hF vndu nd iF wF vifshitzD Quantum Mechanics (Non-relativistic theory) @ergmonD IWUUAF IQV vF ird®sD fF hleinD nd rF F uD hysF evF vettF 98D HRHRHR @PHHUAF IQW pF hlfovoD F qiorginiD vF F itevskiiD nd F tringriD evF wodF hysF 71D RTQ @IWWWAF IRH uF qF ingh nd hF F okhsrD hysF evF f 49D WHIQ @IWWRAF IRI wF uzukiD tF wthF hysF 26D THI @IWVSAF IRP eF F ikovsky nd hF vF hepelynskyD hysF evF vettF 100D HWRIHI @PHHVAF IRQ F plhD hF yF urimerD nd gF kokosD hysF evF vettF 102D HPRIHI @PHHWAY irrtum hysF evF vettF 102D PHWWHQ @PHHWAF IRR F F vptyevD tF hF fodyfeltD hF yF urimerD gF kokos nd F plhD iurophysF vettF 91D QHHHI @PHIHAF IRS hF vF hepelynskyD hysF evF vettF 70D IUVU @IWWQAF IRT qF uopidkisD F uominesD F plhD nd F euryD hysF evF vettF 100D HVRIHQ @PHHVAF IRU eF sominD hysF evF i 81D HIUTHI @PHIHAF IRV fF winD F viD wF osenkrnzD nd F 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@PHIRAF IUS gF inD F ghenD nd tF ngD hysF evF f 93D HUSRHQ @PHITAF IUT fF F ghirikovD hysF epF 52D PTQ @IWUWAF IUU wF qrinisty nd F pishmnD hysF evF vettF 60D IQQR @IWVVAF IUV xF frenner nd F pishmnD xonlinerity 4D PII @IWWPAF IUW qF vemriéD tF ghéD F zriftgiserD tF gF qrreuD fF qrémudD nd hF helndeD hysF evF e 80D HRQTPT @PHHWAF IVH pF wF szrilevD hysF epF 196D PWW @IWWHAF IVI gF in nd eF eltlndD xew tF hys 12D HRQHRQ @PHIHAF IVP eF eltlndD hysF evF vettF 71D TW @IWWQAF IVQ F flümel nd F milnskyD hysF evF vettF 69D PIU @IWWPAF IVR wF hhD F flümelD nd F milnskyD hysF evF i 48D IUTR @IWWQAF IVS iF F ignerD ennF wthF 62D SRV @IWSSAF filiogrphy IRI IVT Applications of Random Matrices in PhysicsD edF iF frezinDF uzkovD hF ernD F iegE mnnD nd eF rodin @pringer xetherlndsD PHHTAF IVU tF uetingD The Riemann Zeta-Function and Quantum Chaology @inrio permi snterntionl hool of hysisD IWWQAF IVV eF eltlnd nd wF F irnuerD hysF evF f 55D IIRP @IWWUAF IVW pF ivers nd eF hF wirlinD evF wodF hysF 80D IQSS @PHHVAF IWH wF vF wehtD Random Matrices @edemi ressD PHHRAF IWI tF rF gonwy nd hF mithD On Quaternions and Octonions @e u etersGgg ressD PHHQAF IWP tF grdyD Scaling and Renormalization in Statistical Physics @gmridge niversity ressD IWWTAF IWQ eF hF wirlinD F F pyodorovD pF wF hittesD tF uezdD nd F rF eligmnD hysF evF i 54D QPPI @IWWTAF IWR qF vemriéD gF eF wüllerD hF quéryEydelinD nd gF winiturD hysF evF e 95D HRQTPT @PHIUAF IWS F F etsD iF fogomolnyD yF qirudD nd qF ouxD hysF evF vettF 110D HVRIHI @PHIQAF IWT qF gstiD sF qurneriD nd hF vF hepelynskyD hysF evF vettF 62D QRS @IWVWAF IWU gF inD eF eltlndD nd wF qrstD hysF evF vettF 107D HURIHI @PHIIAF IWV F hrfD tF hysF e 22D RPPQ @IWVWAF IWW F esdD uF levinD nd F yhtsukiD tF hysF oF tpnF 74D PQV @PHHSAF PHH hF helndeD privte ommunitionF PHI F esdD uF levinD nd F yhtsukiD hysF evF f 73D HRIIHP@A @PHHTAF PHP hF tilet nd eF ekhmerovD hysF evF f 94D ITIIIS@A @PHITAF PHQ wF hreier nd rF qrusshD hysF evF vettF 76D ITVU @IWWTAF PHR sF rv¥ne nd F wrko²D hysF evF f 65D IIQIHW @PHHPAF PHS eF uosior nd uF hD hysF evF f 95D IHRPHT @PHIUAF PHT F wiklitzD gF eF wüllerD nd eF eltlndD hysF evF f 91D HTRPHQ @PHISAF PHU eF rurwitzD wthemtishe ennlen @in germnA 39D PUW @IVWIAD see lso @in englishA qF rF rrdyD iF wF rightD F rethEfrownD tF ilvermn nd eF ilesD An introduction to the Theory of Numbers @yxford niversity ressD PHHVAF PHV uF pF othD wthemtik 2D I @IWSSAF PHW qF vemriéD rF vignierD hF helndeD F zriftgiserD nd tF gF qrreuD hysF evF vettF 105D HWHTHI @PHIHAF PIH qF vemriéD hFhF thesisD niversité F et wF gurieD ris @PHHWAF PII tF F ghlker nd qF tF hniellD hysF evF vettF 61D SWQ @IWVVAF PIP tF F ghlkerD hysi e 167D PSQ @IWWHAF PIQ F uetzmerikD uF uruseD F urutD nd F qeiselD hysF evF vettF 79D IWSW @IWWUAF PIR F ekrids nd hF helndeD to e pulishedF PIS tF ng nd eF wF qríEqríD hysF evF i 79D HQTPHT @PHHWAF PIT iF xF ionomouD gF wF oukoulisD nd eF hF detsisD hysF evF f 30D ITVT @IWVRAF PIU eF edjour nd fF eF n iggelenD iurF hysF tF h 59D PRW @PHIHAF PIV wF irudD vF ezzéD nd vF nhezEleniD xew tF hysF 15D HUSHHU @PHIQAF PIW F inglD tF hF rinD nd uF ihterD hilF rnsF F oF e 374D PHISHISW @PHITAF PPH F hlghek nd tF hujrdinD ennF hysF @ferlinA 529D ITHHQII @PHIUAF

APPENDIX A

In this Appendix, we calculate the leading-order smooth quantum corrections to the classical limit of the spectral function, Eq. ( 12), using an alternative approach to the one used in [17]. The calculation is first carried out for 1D, blue-detuned speckles, then generalized to any dimension, and finally to red-detuned speckles by a simple symmetry argument.

The first stage of our approach is a commutator expansion of the evolution operator based on the Zassenhaus formula [40]:

The second stage consists of carrying out the disorder average. This can be done by mean of the following cumulant expansion:

where κ n denotes the nth cumulant. To evaluate the cumulants of sums of random variables that appear in Eq. (A2), we make use of the expansion

where we have introduced the joint cumulants κ, defined as [START_REF] Brillinger | Time Series Data Analysis and Theory[END_REF] κ

B∈π i∈B X i .

(A4) Here π runs through the list of all partitions of {1, . . . ,n}, B runs through the list of all blocks of the partition π , and |π | is the number of parts in the partition. Joint cumulants have the following important properties [START_REF] Brillinger | Time Series Data Analysis and Theory[END_REF]:

(1) they are linear in all variables;

(2) κ(X, . . . ,X) = κ n (X);

(3) κ(X 1 , . . . ,X n ) = 0 if any set of the X i 's are independent of the remaining X j =i 's.

After these premises, let us now write the random potential as

TW where we hve used the ft tht G R G A depends only on |xx |D due to trnsltionl invrine fter verging over the rndom potentilF por ltter onvenieneD it is useful to symmetrize equE tion @RFRUAD whih presently desries the enter of mss of wve pket initilly kiked to the rightD y dding the orresponding eqution for wve pket initilly kiked to the left

sndeedD under this form x(ω) n e treted on the sme footing s the urrentEurrent orreE ltion funtion with the ferezinskii pproh 2 IQF e will ome k to the onnetion etween enter of mss nd the urrentEurrent orreltion funtion in setion RFUF

Equations for the center of mass

Setting up the equations prom this point onD we use ferezinskii onvention = 2m = k 0 = 1F D m nd k 0 will e reintrodued in the (nl resultsF o lulte the digrms involved in eqution @RFRWAD we strt y distinguishing the leftEhndD rightEhnd nd entrl prts of the digrms s the prt of the digrms lying to the left of x D to the right of xD nd etween x nd x respetively @we ssume x > x for the momentAF sn (gure RFQD these respetively orrespond to LD R nd ZF ine for ll the seleted verties @see (gure RFTA ∆g = ∆g = 0, ±2D in eh intervl elonging to the leftEhnd nd rightEhnd prt we hve g = g = 2nD while in eh intervl elonging to the entrl prt we hve g = g = 2n + 1D where n is n integer whih my depend on the intervlF he ontriution of eh digrm is n integrl over the region @RFQWAF his integrl reks up into produt of three integrls over @x 1 , . . . , x i AD @x i+1 , . . . , x j A nd @x j+1 , . . . , x n AD whih we ll respetively the ontriutions of the leftEhndD entrl nd rightEhnd prtF vet us denote y Rm (x) the sum of the ontriutions of ll the rightEhnd prts tht hve t the oundry with the entrl prt @iFeF immeditely to the right of point xA the stte g = g = 2mD y Rm (x ) the nlogous sum of the ontriutions of the leftEhnd prtsD nd y Z m ,m (x , x) the sum of the ontriutions mde y the entrl prts with left nd right oundry sttes g = g = 2m + 1 nd g = g = 2m + 1F o lulte Rm (x)D the ide is to onsider how it vries s x is inresed of n in(nitesiml mount δxF sn prtieD one n relte Rm (xδx) to Rm (x) y dding ll the possile verties in (gure RFT etween x-δx nd xF roweverD in doing thisD one should e reful not to otin digrms with prtile loops or digrms tht er no reltion with the originl digrmsF sn prtiulrD one must only keep digrms for whih the lines G R 0 nd G A 0 re ontinuous from x to xF por this purposeD we n numer the lines on the oundry of the rightEhnd prt y ssigning 5I to the segment orresponding to the (rst entry of the prtile line into the rightEhnd prtD 5P to the (rst emergene from itD 5Q to the seond entryD nd so forth from I to 2mF henD in onstruting the digrmsD we should keep in mind tht the ngles t the verties in (gure RFT n only e formed y the segments of the G 0 Elines with onseutive numersF epplying this proedure to Rm in (gure RFVD Abstract his thesis theoretilly investigtes severl e'ets relted to enderson loliztionD fousing on the ontext of disordered nd hoti oldEtomi systemsF sn oldEtomi systemsD optil spekle ptterns re often used to rete the disorderF he resulting potentils felt y the toms di'er from qussin rndom potentilsD ommonly ssumed in the desription of ondensedEmtter systemsF sn the (rst prt of the thesisD we disuss their spei(itiesD with n emphsis on the spetrl properties of toms in suh potentilsF sn prtiulrD we derive severl pproximtions for the spetrl funtionF etomEoptis experiments o'er interesting possiilitiesD suh s the possiility to diretly proe the toms inside the disordered potentilF sn view of these possiilitiesD we onsider in the seond prt of the thesis the spreding of mtter wve pkets initilly lunhed with nonEzero veloityF e (nd tht fter n initil llisti motionD the pket enterEofEmss experienes retrore)etion nd slowly returns to its initil positionD mimiking oomerngF e show tht this unexpeted quntum oomerng e'et is onsequene of enderson loliztionD nd desrie it oth numerilly nd nlytilly in dimension IF etomEtom intertions re then introdued in third prtF e onsider dilute ondensed osoni gsesD nd tret the intertions t the menE(eld @qrossEitevskiiA levelF rious situtions re studied numerillyD in prtiulr the quntum oomerng senrioD nd the dynmil spreding ! oth in momentum nd energy ! of mtter wves prepred s plne wvesF sn the lst prtD we show tht hoti models o'er interesting prospets for the study of enderson loliztionF yn the one hndD going ginst ommon wisdomD we present strong evidenes in fvor of spinless kiked rotor in the sympleti ensemleF yn the other hndD seond look t ommonly studied qusiEperiodilly modulted kiked rotor revels intriguing resultsF