Localization and spreading of matter waves in disordered potentials

Dipartimento di Fisica

Loalization and spreading

of matter waves

in disordered potentials

Ph.D. thesis by

Maro Larher

1 Introdution 1

1.1 Outline ofthethesis . . . 2

2 Loalization properties in disordered quantum systems 5 2.1 Disorder induedloalization . . . 5

2.2 Onedimensional disorderedsystems . . . 14

2.3 Experimental observations ofAnderson loalization . . . 16

3 Noninterating partiles in quasiperiodi potentials 21 3.1 From bihromatioptial lattiesto theAubry-André model. . . 22

3.2 Loalization propertiesof theAubry-André model . . . 26

3.3 Spreading ofwavepakets intheAubry-André model . . . 29

3.3.1 Inommensurate vs. ommensurate ase . . . 31

3.4 Loalization ofultraold atoms inmomentumspae. . . 34

3.4.1 Aubry-André modelinmomentumspae . . . 35

3.4.2 Periodiosillations intheAubry-André model . . . 37

3.4.3 Deteting the Aubry-André transitioninmomentumspae . . 41

4 Weakly interating bosons in quasiperiodi potentials 45 4.1 DisretenonlinearShrödinger equation . . . 46

4.2 Eetsof the interation . . . 48

4.2.1 Self-Trapping . . . 48

4.2.2 Destrutionof Anderson loalization . . . 52

4.3 Experimental observationof subdiusion . . . 54

4.3.1 Experimental setup. . . 54

4.3.2 Comparisonwiththedisrete nonlinear Shrödinger equation 58 5 Subdiusion of nonlinear waves in quasiperiodi potentials 61 5.1 DNLSinnormalmode spae . . . 62

5.2 Relevant energy sales . . . 64

5.3 Expetedspreading regimes . . . 66

5.3.1 Spreading laws . . . 68

5.3.2 Resonane probability . . . 70

5.4 Numerialobservations . . . 72

5.4.1 Resultsfor squarewavepakets . . . 73

5.4.2 Roleof the shape oftheinitial wavepaket . . . 76

5.5 Appliation to oldatoms . . . 78

6 Deloalization phenomena in 1D models with orrelated disorder 83

6.1 Dipolarinteration . . . 85

6.2 The physial model . . . 87

6.3 Statistial properties oftherandom potential . . . 90

6.4 Nature ofthe spetrum . . . 92

6.5 Role oforrelations . . . 94

6.5.1 Short rangeorrelations . . . 94

6.5.2 Long rangeorrelations . . . 97

Introdution

Disorderisubiquitousinnatureandaetsthepropertiesofmanyphysialsystems. A deep understanding of its eets is therefore of fundamental importane. This ispartiularly truefor quantumsystems,wheredisorderan inuenedramatially thetransportpropertiesof eletrons leading to aphenomenon thattodayis known as Anderson loalization [1℄. Thistype of loalization is a subtle eet that arises from aninterferene proess due to oherent multiple sattering from thedisorder. It leads to a omplete absene of diusion and to exponentially loalized single-partile wavefuntions [2℄. Anderson's disovery representeda breakthrough inthe studyoftransportpropertiesofquantumpartiles,sineitintroduedaompletely new approah to the problem. Previous theories of transport onsidered disorder justasaweakperturbation,prediting adiusivemotiondetermined byinoherent sattering [3℄.

Shortly after the disovery of Anderson loalization it has been shown that, in onedimensional systems,loalizationtakesplae forallquantumstates[4,5℄. This is ounterintuitive espeially when the energy of a given state is muh larger than thetypial energy utuations assoiated to thedisorder. A properunderstanding of loalization in higher dimensions required more time, but nowadays it is nally aeptedthatalsointwodimensionalsystemsallstatesareloalized,whileinthree dimensionsametal-insulatortransitionanour,asalreadysuggestedbyAnderson inhisoriginal paper[6,7℄.

The phenomenonofAnderson loalization hasbeen originallyintrodued inthe ontextofeletronspropagatingindisorderedsolidstatematerials. Onlylaterithas beenrealizedthatthesametypeofloalizationalsoourswithlassialwaves,suh as light or sound [8, 9℄. This ledto the rst observations of Anderson loalization inoptis[10,11℄and aoustis [12℄.

[16℄ and the formation of quantized vorties [17℄. More reently a great interest is foused on properties of many-body systems, where new quantum phases an emerge. Examples are the observation of a Tonks-Girardeau gasin one dimension [18,19℄or the transitionfrom superuidto Mottinsulator [20℄.

Thegreatsuess ofultraoldatoms ismostlydue tothehighdegreeof ontrol that an be reahed in experiments [21℄. Both bosoni and fermioni atoms an be ooled down to degeneray and external trapping potentials an be used to ontrol the dimensionalityof thesystem. Feshbahresonanes areusedtotunethe interation between atoms andmany dierent observables an bedeteted ranging from the atomi density to the momentum distribution. Laser light an be used to design dierent kind ofpotentialsfor theatoms, suhasperfet periodioptial latties indierent dimensionalities[22℄.

Reently optial potentials have been usedalso for thegeneration of ontrolled random potentials [23℄ and a new eld of study started with the rst diret ob-servation ofAnderson loalizationof matterwavesin twodierent researh groups [24, 25℄. This observation represented a remarkable result and showed that ultra-old atoms an represent a powerful experimental tool for exploring a number of problems relatedto thetheoryof loalization [26,27,28℄.

In this thesis we will onentrate on two main issues,namely the interplay be-tween loalization and interation indisordered systems and theproblem of loal-ization in orrelated random potentials. The former is a long standing problem that has been raised shortly after the disovery of Anderson loalization [29℄, be-auseofitsfundamentalimportaneineletrontransportindisorderedsolids,where Coulomb interation between eletrons an not be negleted. One naively expets that interation ats against loalization, but a detailed study of this interplay is highly nontrivial. The study of the role played by orrelations in the loalization proess isalso of great interest, sine, stritly speaking, in real world unorrelated potentials do not exist. It is known that orrelations an lead to deloalization ef-fets, however,a fullunderstanding isstill missingand one ofthemost hallenging questionsis whetheror not they an introdue a metal-insulator transitionalready inonedimension,where theeetofdisorderisknowntobethestrongest[30℄. The possibilitytodesign dierent kindof disorderedpotentialsand toontrol the inter-atomiinterationsarethetwokeyfeaturesthatmakesultraoldatomspartiularly suitable to takle these twoproblems.

1.1 Outline of the thesis

The outlineof the thesisisthefollowing :

◦

In hapter 2 we present the basi onepts of the theory of loalization of quantum partiles in disordered systems. We introdue the phenomenon of Anderson loalizationand theonept of mobility edgein three dimensional systems. We disuss the role played bythe dimensionality in relation to the problemofloalizationwithspeialfousto theonedimensional ase, thatis themostrelevantforthisthesis. Weintroduedierentmodelsofdisorderand disuss their implementations withultraold atomi gases. Finally we review the experimental observations of Andersonloalization.

◦

Inhapter 3theproblemofloalization inquasiperiodisystemsisdisussed. After introduing the Aubry-André model and explaining in detail its onnetion with ultraold atoms in bihromati optial latties, we review its loalization properties. We then onsider the spreading of an initially loalized wavepaket, both inreal and momentum spae, asa possible toolto studytheloalizationpropertiesoftheAubry-Andrémodel. Speialattention isgiventoproperties whih areobservableinexperiments. Partof theresults presented inthis hapter arepublishedin:

M. Larher, F. Dalfovo, and M. Modugno, Eets of interation on the diusion of atomimatter waves inone-dimensional quasiperiodi poten-tials,Physial Review A,80, 053606(2009)[31℄.

M. Larher, M. Modugno, and F.Dalfovo, Loalizationin momentumspae ofultraoldatomsininommensuratelatties,PhysialReviewA,83,013624 (2011) [32℄

◦

Chapters 4 is devoted to the study of the eets of interation on the spreading of an ultraold atomi gasina bihromati optial lattie. This is done by onsidering a disretized mean-eld equation, whih generalizes the Aubry-André modelby addinganonlinear term thatinludestheinteration between atoms. This model is also known as disrete nonlinear Shrödinger equation (DNLS). We solve this equation numerially and analyze the interplay between two ompeting eets of the interation, namely, self-trapping and destrution of Anderson loalization. Finally we ompare the numerial results that an be extrated from this model with experimental measurements performed in Florene. The results of this hapter have been publishedintherst papermentionedabove aswell asin:

E. Luioni, B. Deissler, L. Tanzi, G. Roati, M. Zaanti, M. Modugno, M. Larher, F. Dalfovo, M. Ingusio, and G. Modugno, Observation of subdiusion in a disordered interating system, Physial Review Letters, 106,230403(2011) [33℄.

more attention is devoted to the investigation of the spreading behaviour of wavepakets, that were loalized inthe noninterating ase. We haraterize in detail the phenomenon of destrution of Anderson loalization and the resulting subdiusive expansion indued by the interation, identifying dierent spreading regimes and prediting the assoiated spreading laws. Finally an extensive numerial analysis is performed and the results are ompared with thetheoretial expetations. The ontent of this hapter has been publishedin:

M Larher, T. Laptyeva, J. Bodyfelt, F. Dalfovo, M. Modugno and S. Flah, Subdiusion of nonlinear waves in quasiperiodi potentials, New Journal of Physis, 14, 103036(2012)[34℄.

◦

In hapter 6 we propose a model of disorder that an be realized ex-perimentally using dipolar ultraold gases and that presents orrelation properties that leads to interesting deloalization eets. The model is rst introdued and its statistial properties are haraterized. In partiular we show that both short and long orrelations are naturally present in the disordered systemthat we propose. We thenstudy itsloalization properties alulating the loalization length of the eigenstates by means of an exat renormalization-deimation tehnique. Using these results, we disuss the role of short and longrange orrelations and their interplay. The materialin this hapteris thebasisfor apaperwhihwill besoon submitted:

Loalization properties in

disordered quantum systems

It was rst realized by Anderson that disorder an have a dramati impat on the transport properties of a quantum partile. More preisely, by studying the on-dutane of eletrons in solids, he disovered that disorder an lead to a omplete absene ofdiusionand aonsequent metal-insulator transition. Thisphenomenon is known as Anderson loalization [1℄. Sine the revolutionary disovery by An-derson a huge ativity on the physis of quantum disordered system has started and nowadays it isstill an ative researh eldthat involves manyareas of physis [36,26℄.

This hapter is devoted to the introdution of the loalization problem in dis-ordered quantum systems. We will review some basi onepts that will form the bakground for the understanding of the results presented in the others hapters of this thesis. In setion 2.1 we outline the main ahievements of the theory of loalizationfornoninterating quantumpartilesand lassialwaves. Insetion2.2 we speializeto theloalizationpropertiesof onedimensionalsystems,whihisthe dimensionality that we will onsider for the rest of this thesis. In setion 2.3 we reviewthe experimental observations of Anderson loalizationwith a speialfous on theloalizationof matterwavesand ultraoldatomi systems.

2.1 Disorder indued loalization

Inthissetionwepresentsomeofthe keyresultsofthequantumtheoryof loaliza-tion. Thereareanumberofinterestingintrodutionstothiseldthatanbefound inthesienti literature(see forinstane [37,3,38,36,39℄).

In partiularwe rstintrodue theproblemofquantumtransportindisordered systems from a omparison with the lassial ase. We then present the onepts of Anderson loalization and of mobility edge and disuss the role played by the dimensionality of the system starting from the results of the saling theory of loalization. Finally we introdue some models of disorder in onnetion with the eldof ultraold atomigases.

Classial vs. Quantum

Figure 2.1: Loalization properties of a lassial partile in a disordered potential depending on the value of its energy. If the energy of the partile is smaller than the highest barriers of thepotential(

E

1

< E

)themotion of thepartile isrestrited to a niteregionof spae. Conversely ifthe energy of thepartile islargerthan the highest barriers of the potential (

E

2

> E

0

) the motion of the partile is unbounded and it will propagate through the disordered potential. Figure takenfrom Ref. [2℄.

give anintuition ofwhythestudy ofAnderson loalization is highlynontrivial and introdues the two main eets that play a key role for the determination of the transportbehaviourofa quantum partile.

Letusonsideralassialpartileinadisorderedpotential

V

(

x

)

(weonsidered the one dimensional ase for simpliity) like the one that is depited in Fig. 2.1 and let us onsider the situation where the potential is upperbounded by a max-imum value that we all

E

0

. The behaviour of the lassial partile an be easily determined byasimpleomparison oftheenergy ofthepartilewiththemaximum valueassumedbythepotential. Inpartiulariftheenergyofthepartileissmaller than

E

0

the motionwillbeboundedina niteregionofspaebetweentwo barriers of the potential, transport over long distanes is suppressed and loalization takes plae. Ontheontraryiftheenergy islargerthan

E

0

thepartilewillyabovethe barriers of the disorderedpotential and onaverage themotionwill be ballisti.

isAnderson loalization [1,2℄. Anderson loalization

The eets of disorder on the propagation of quantum partiles has been ini-tiallystudied inthe ontext of ondensed matter physis for the desriptionof the propagation of eletrons in solids. The natural starting point for the analysis of thisproblemis aperfet rystal,whose propertiesarewellknownandaregoverned by the Bloh theorem. In partiular the eigenstates of the system are extended Bloh wavesthatan propagate throughthe rystal[40℄.

The traditional view, before thedisovery of Anderson loalization, onsidered as a starting point for the study of theeets of disorder the extendedwaves of a perfetrystal[36℄. Asaonsequeneinthesemilassialtheoryofeletroni trans-port, eletrons are still onsidered as waves whose wavefuntion remains extended throughout the sample but the propagation is modied by inoherent sattering due to the preseneof disorder inthe system. The result of these ollisions auses a lossof the phase oherene of the waves on the length of the mean free path

ℓ

. Thisleads toadiusive motionofeletronsthroughthedisorderedpotential,whih allows eletrons to propagate to innity and results in a nite ondutane of the sample. Aninrease ofthe strengthof thedisorderleads to adereaseof themean free path

ℓ

and to a onsequent derease of the diusion onstant and of the on-dutane of the sample. This turnsout to be true wheninterferene eets an be negleted.

The rst to understand that interferene eets play a key role for the deter-mination of the transportbehaviour ofa quantum partile was Philip Anderson in 1958. He showed that the onsequene of these destrutive interferene proesses between dierent sattered wavesis not asimple redution oftheondutivitybut aomplete absene of diusion[1℄.

In his seminal paper he onsidered the transport of a partile (spin) in a dis-retizedlattiethatantunnelviaquantumjumpsbetweendierentsitesand disor-derisintroduedbyrequiringthattheon-site energiesassoiated withthedierent lattie sites varies randomly. More preisely Anderson introdued the following modelfor thedesription ofthepropagationof thepartile (spin)

i

∂ψ

j

∂t

=

ε

j

ψ

j

−

X

k

6

=

j

J

j,k

ψ

k

(2.1)

where

ψ

j

isthe probabilityamplitude thata partileis onsite

j

,

J

j,k

desribesthe hopping amplitude between dierent sites and

ε

j

are the random on-site energies haraterized by aprobabilitydistribution

P

(

ε

)

.

than

1

/

|

j

−

k

|

3

andifthe disorder

W

isstrongenough ifomparedwiththeaverage valueofthe hoppingamplitude

J

thentherewillbeaompleteabseneofdiusion. The initial amplitude

|

ψ

j

(0)

|

staysloalized around theinitiallyoupiedsites and fallsoexponentiallywiththe distane.

Thisabseneofdiusionisassoiatedwiththefatthatthesingle-partile eigen-statesofadisorderedsystemareexponentiallyloalizedifdisorderisstrongenough [3℄. Morepreisely, this means that, on the average, the envelopes of their ampli-tudes areexponentiallydeaying inspae at innity

|

φ

(

~r

)

| ∼

e

−|

~

r

−

~

r

o

|

/L

loc

(2.2) where

~r

0

is the loalization enter and

L

loc

is the loalization length. Partile desribed by these kindof states annot ontribute to transport sine they oupy a niteregion ofspaeinopposition topartilesinextendedstatesthatanesape to innity. Therefore themain two manifestationsof Anderson loalization, whih are losely onneted, are the absene of diusion and the fat that the single partile eigenstates areexponentiallyloalized.

Mobility edge

Anderson alreadyunderstoodthatomplete loalization takesplaeonly ifthe dis-order is strong enough [1℄. In this situation all the single partile eigenstates are loalized. Below a ertain disorder strength, instead, loalization takes plae only for a frationofstates whiletheremaining statesare extended.

Ten years after thepubliation of thepaperby Anderson,Mott introdued the onept of mobility edge [6℄ whih represents an energy whih separates loalized and extendedstates. He understood thatno loalizedstatesan exist inanenergy regionofextendedstateswiththefollowingargument. Assumethatitispossibleto

Figure 2.2: Shemati representation of the onept of mobility edge. The states are loalized in energy regions where the density of states is small

E < E

c

and

E > E

′

c

. Conversely they are extended in energy regions where the density of states is large. The two energies

E

c

and

E

′

Figure2.3: PhasediagramforthethreedimensionalAndersonmodelasafuntion of the disorder stength

W

and of the energy of the states

E

. The states in the spetrum of the system are divided in two regions and an be extended or loalized. Thepointsandthethiksolidlinerepresentsthemobilityedge,i.e. theritialenergy that separates these two regions of the spetrum. More preisely the points are the result of an exat numerial alulation, whilethe thik solid line isthe outome of the self onsistent theory of loalization. The thin line indiates the position of the upper edge of the spetrum, only the region on the left of this line belongs to the spetrum. Figure taken fromRef. [41℄.

ndaloalizedstateand anextendedstatewithinnitelylose energiesfora given onguration of disorder, then an innitesimal hange of the onguration would hybridize them,formingtwo extended states. Hene, for a given energy, almostall statesshould be eitherloalized or extended.

In Fig. 2.2 we show a pitorialview of the onept of mobility edgefor a tight binding modelsimilarto Eq.(2.1). Thevertialdashedlinesrepresent theposition of the two mobility edges

E

c

and

E

′

c

while the solid line represents the density of states of the system. The regions with the loalized states are those where the density of states is small. As thedisorder strength is inreased the mobilityedges move towards the band enter and eventually, for a ritial value of the disorder strength, they meetat the enterof the band. Abovethis ritial value of disorder there arenomore extendedstatesinthesystem.

indiates instead the upper bound of the spetrum. This quantitative alulation onrms the qualitative piture that we have just desribed. One an see that as the disorder strength is inreased the loalized states appear at the edge of the spetrum and then move gradually towards the enter of the band. Above

W/J

≈

16

,whihrepresentstheritialdisorderstrengthfor thethreedimensional Andersonmodelunderonsideration,thereareonlyloalizedstatesinthespetrum. Role of dimensionality: saling theory

The dimensionality of the system,

d

, plays a rather important role for the determination of the loalization properties of a quantum disordered system. In partiular one ofthemainresultsofthetheoryofloalization isthatinone dimen-sional(1D)and twodimensional(2D) systemsallthe singlepartileeigenstatesare exponentiallyloalizedwhileinthree dimensions(3D)bothextendedand loalized states an exist.

This result has been rst suggested by Abrahams, Anderson, Liiardello and Ramakrishnan who gave a rst formulation of the so alled one-parameter saling theory of loalization [7℄. A saling theory desribes the relevant properties of a physial systemunderahange of size

L

→

bL

(

b >

1)

.

In partiular Abrahams et al. introdued a dimensionless ondutane

g

=

G

~

/e

2

by noting that the ondutane

G

of a sample is dimensionless one is ex-pressed in units of

e

2

/

~

. They desribed the behaviour of the dimensionless

on-Figure 2.4: The saling funtion

β

(

g

)

in dimensions

d

= 1

,

2

,

3

. The dimension-less ondutane

g

grows with the size of the system

L

if

β >

0

but dereases for

dutaneofahyperubeofsize

L

d

asafuntionofthesystemsize

L

bydeningits logarithmi derivative

β

=

d

ln

g

d

ln

L

.

(2.3)

and assumingthatitdependsonly onthe dimensionlessondutane itself andnot on theother mirosopi propertiesof the sample. Thisis themain assumption at the basis of the theory and it is known as the one parameter saling hypothesis. The behaviour of

β

(

g

)

is qualitatively obtained by Abrahams et al. starting from the two limiting behaviours for strong and weak disorder. In partiular for weak disorder the lassial (i.e. without interferene) behaviourof theondutane

g

is assumed. This orrespondsto theOhm's law namely theondutane dependson thesurfae

A

=

L

d

−

1

of the sampleandon its length

L

aordingto

g

∼

σ

A

L

=

σL

d

−

2

(2.4) where

σ

istheondutivityofthesample,whihisanintensivequantityindependent onthesystemsize. Intheoppositelimitofstrongdisorder, exponential loalization is assumed in all dimensions and therefore ondutivity is also expeted to deay exponentiallywiththesystemsize

g

∼

e

−

L/L

loc

.

(2.5) From Eqs.(2.4) and(2.5) one obtains

β

(

g

)

∼

(

d

−

2

weak disorder

ln

g

+

const.

strong disorder

(2.6)

Interpolating between the two limiting behaviours and assuming that

β

(

g

)

is a ontinuousandmonotoniallyinreasingfuntiononeobtainstheresultdepited in Fig. (2.4). If

β

(

g

)

>

0

the value of the dimensionless ondutane inreases with the system size, one is therefore in the extended/onduting regime. Conversely for

β

(

g

)

<

0

the ondutane dereases with the system size and one ends up in the loalized/insulating regime where

g

→

0

. The presene of a xed point

g

c

where

β

(

g

c

) = 0

signals thepreseneofatransitionfromanextendedtoa loalized regime. One an see from Fig. (2.4) that suh a transition exits in the 3D ase. Thisisonsistent withthe resultsonthepreseneof amobilityedgeinthegeneral three dimensionalasethat we have previously disussed earlierinthis hapter. In the 1D and 2D ase the theory does not predit the presene of xed points and

β

(

g

)

is always smaller than zero. This means that no extended regime an exist for

d

= 1

,

2

and one hasalways Anderson loalization, no matter how small is the disorderstrength.

Models of disorder

Disorder an be introdued in a variety of dierent ways in a physial sys-tem. Here we justmention fewmodelsof disorderthat areloselyrelatedwiththe Hamiltonians that anbe experimentally realized usingultraold atoms.

Inthemost general ase, letus assumethata singlepartileis governed bythe Hamiltonian

H

=

−

~

2

2

m

∇

2

+

V

(

~r

)

(2.7) where

V

(

~r

)

is a quenhed disordered potential, i.e. a stati disordered potential that does not evolve in time. The random potential is dened by a probability distribution

P

(

V

)

andbyasetoforrelationfuntions

h

V

(

r

~

1

)

V

(

~r

2

)

. . . V

(

~r

n

)

i

. Here we indiated with

h

. . .

i

an average over many dierent disorder realizations. A disorder realizationis apartiular outome of theproessof hoosing thepotential value

V

(

~r

)

for all the values of

~r

. The disordered potential is usually assumed to be spatially homogeneous inthesense that itsstatistial properties do not depend on the spei position in the system. As a onsequene the average value of the potential

h

V

i

doesnot depend on

~r

andingeneral the

n

-point orrelation funtion dependsonlyon

n

−

1

relativeoordinates only

C

n

(

r

~

1

, ~

r

2

, . . . , ~r

n

)

. Inpartiularthe two point orrelation funtion, whih we simply indiate with

C

(

~r

)

, depends only on one variable:

C

(

~r

) =

h

V

(

~r

0

+

~r

)

V

(

~r

0

)

i

.

(2.8)

InatomigasesHamiltonian(2.7)anberealizedusinganoptialspekle poten-tial [43,44,45,24℄. Optial potentialsan be reated usinglaserlight thatindues an atomi dipole moment and a onsequent dipolar fore on the atoms whih is proportional to the intensityof the lasereld[21,22℄. Thespekle pattern,in par-tiular, isprodued byshining a oherent laser beam througha ground-glass plate whihisthenfousedonthe atomsusingaonverging lens. Theground-glassplate transmitsthelaserlightwithoutalteringtheintensity,butimprintsarandomphase prole ontheemerginglight. Then,theeletrield

E

(

~r

)

onthefoalplaneresults fromtheoherentsuperpositionofmanyindependentwaveswithequallydistributed random phases. This result is a random pattern of the transmitted light that di-retly translates in a disordered potential

V

(

~r

)

for the atoms. Both the modulus and sign of

V

(

~r

)

an be ontrolled experimentally by hanging the light intensity and the detuning of the laser frequeny with respet to the atomi transition. A detailed analysisofthestatistialpropertiesof atypialspekle potentialsused for ultraold gases experiments an be foundin[46,47℄

Disorder an be also introdued in a natural way by using a perfet lattie as a starting point. A typial example of a lattie Hamiltonian with ompositional disorder isprovided by

H

=

X

j

ε

j

|

j

ih

j

|

+

X

j,k

where

ε

j

arethe on-site energies while

J

j,k

desribe the hoppingbetween dierent sites of the lattie. The diagonal part of the Hamiltonian orresponds to the po-tential energy and the non-diagonal part to the kineti energy in the ontinuous spae desription(2.7). Let us notethatthe timedependent Shrödinger equation assoiated to Hamiltonian (2.9) orresponds to the model onsidered by Anderson inhisoriginalpaper(2.1)[1℄. Disorderanbeintroduedbytakingthesiteenergies or the hopping terms at random. Also in this ase one haraterizes the disorder by means ofa probabilitydistribution and aset of orrelationfuntions. A typial hoieinthe study of disorderedsystemis

P

(

ε

) =

(

1

/W

if

|

ε

|

< W/

2

0

otherwise

(2.10)

and onstant hopping

J

restrited just to nearest neighbouring sites. In this ase Hamiltonian (2.9)is indiatedasthe Anderson model.

A disretized spae for ultraold atoms an be produed again using an optial eld [21, 22℄. In this ase, two ounterpropagating laser beams are used. Due to theinterferenebetweenthetwolaserbeams,anoptialstandingwaveisformed,in whihatomsanbetrapped. Inthiswaytheatomsfeelthepreseneofaperfetone dimensionaloptiallattie. Addingapairoflasersalsointheotherdiretionsgives the possibility to reate optial latties in 2D and 3D. If the laser eld is strong enough one reates a deep optial lattie and enters the so alled tight binding regime. In this regime thespae an be disretized and theatoms are governedby anHamiltonianwhihisverysimilarto(2.9)butwithontantonsiteenergies

ε

j

=

ε

and typially the hopping term is approximated to be onstant and dierent from zero only on nearest neighbouring sites

J

j,k

=

J

1

[21,48℄. At this point disorder an be introdued by randomly shifting the on-site energies. This might be done by superimposing a spekle potential to the optial lattie. Another possibility is to introdue another optial lattie with a dierent lattie spaing with respet to the rst one [49, 25℄. This realizes a bihromati optial lattie and introdues a shift of the on-site energy whih is not fullyrandom but still very interesting from thepoint ofviewoftheloalization properties. Wewill disussmoreindetailthese kind ofsystems,whih arealledquasiperiodi,inhapter 3.

Another interesting proposal for the reation of a disordered potential for ul-traold atoms is to use a mixture of two dierent atomi speies (or two dierent internal states of the same atom) [50, 51℄. The atoms of one of the two speies are trapped at random positions in the wells of a very deep optial lattie. As a onsequenetheir dynamiisfrozenandtheyannottunnelbetweendierentsites. These trapped atoms play the role of impurities. The other speiesinstead feels the presene of a weaker optial lattie or it does not feel thelattie at all and it is therefore free to move. This atomi speiesplay the role of test partile. Due to the interation between thetwo atomispeies, the test partilesfeel a random

1

potential formed by the impurities whih are trapped in the optial lattie. This model an be desribed with a free spae Hamiltonian similar to (2.7) if the test partiles do not feel the optiallattie. Conversely if also thetest partile feelthe preseneof thelattie (although muh shallowerthatthelattiefeltbythe impuri-ties) atight binding Hamiltonian(2.9) is usedfor thedesriptionof thesystem. A detailed analysisof animpuritymodelwill begiven inhapter6.

2.2 One dimensional disordered systems

Onedimensional systemsplayakeyrole inthe understandingof thephysisof dis-order [52, 53,2℄. First of all, itis the dimensionality where disorder have stronger eets, moreovermanypropertiesoftheeigenstatesandrelatedto transportanbe disussed rigorously. Finally numerial alulations are faster and easier to imple-ment.

In1D loalizationis always expetedno matter how strong therandom poten-tial is. Mottand Twose[54℄weretherstwhosuggestedthatallthesinglepartile eigenstatesmightbeexponentiallyloalizedin1Dbuttheyjustprovideda qualita-tive argument tosupporttheirstatement. Therstrigorousproofofthis resulthas been given byBorland [5℄ fewyears later. Nowadays theonlusion that all single partile statesare loalized ina 1D random potentials is well established as ithas been obtained withavarietyof dierent methods.

Astandard wayto prove loalization in1Disto userandom matrix tehniques developed by Oselede and Furstenberg in the sixties for the alulation of the Lyapunov exponent,whih istheinverse of theloalization length

Λ =

1

L

loc

(2.11) Consider theeigenvalueproblemassoiatedto theonedimensionalAndersonmodel (2.9)

−

J

(

ψ

j

+1

+

ψ

j

−

1

) +

ε

j

ψ

j

=

Eψ

j

(2.12) with

ψ

j

=

h

j

|

ψ

i

. Equation(2.12) isequivalent to

Ψ

j

=

M

j

Ψ

j

−

1

,

(2.13)

where

Ψ

j

represents atwo omponent vetor and

M

j

a2

×

2matrix

Ψ

j

=

ψ

j

ψ

j

+1

M

j

=

0

1

−

1 (

ε

j

−

E

)

/J

.

(2.14)

Figure 2.5: Loalized eigenstates of the onedimensional Anderson model obtained by diret numerial diagonalization. In the left panel we show an example of state with energy lose to the enter of the band

E

≈

0

for

W

= 4

. The state has a typial exponentially deaying prole. The loalization length obtained numerially isin agreementwiththe oneobtained withthe approximateexpression (2.16)(blak dashedline). Intherightpanelweshowthegroundstateofthesystemforinreasing disorder strength

W

.

random potential, an initial vetor

Ψ

j

grows or deays asymptotially as

e

±

Λ(

E

)

j

, where

Λ(

E

)

is a positive, non-random quantity that is known as the Lyapunov exponent. The solution at energy

E

is an exponentially loalized solution of the spetrumonly when thereare two vetors

Ψ

±

0

thatdeayrespetively for

j

→

+

∞

and

j

→ −∞

andthatoinideatsomesite. Thisassurestheexisteneofasolution ofenergy

E

thatdeaysexponentiallyonboth sidesofthesystemwithloalization length

L

loc

(

E

) = 1

/

Λ(

E

)

. If these two vetors do not exist for a given energy

E

, this energy doesnot belongs to the spetrumofthesystem.

theloalizationpropertiesoftheeigenstateswasrstderivedbyHerbertandJones [57℄intheaseoftheAnderson modelandsubsequently ithasbeen generalized by Thouless [58℄. Thisrelationis

Λ(

E

) =

Z

∞

−∞

ln(

E

−

E

′

)

ρ

(

E

′

)

dE

′

(2.15)

where

Λ(

E

)

istheLyapunovexponentand

ρ

(

E

)

thedensityofstates. Whenapplied to Eq. (2.12) with

ε

l

uniformly distributed in

[

−

W/

2

, W/

2]

and in seond order perturbation theory,Eq. (2.15) gives[59℄

Λ(

E

) =

(

W/J

)

2

24[4

−

(

E/J

)

2

]

.

(2.16) This relation is valid for small

W

and results ina loalization lengthat theenter of the band equal to

L

loc

(

E

= 0) = 96

J

2

/W

2

. States situated at the enter of the band, i.e. withenergy

E

= 0

,arethus loalizedon longerlength sales.

Theresultoftheperturbationtheoryanbeomparedwiththediretnumerial diagonalization. Thisisdone inFig.2.5where the twoblakdashedlinesrepresent an exponential deay with loalizationlength given by (2.16). The resultthat the loalizationlengthdivergesas

W

−

2

forsmall

W

isageneralresultinonedimensional systemand doesnot applyonly to theaseof theAnderson model.

2.3 Experimental observations of Anderson loalization Anderson loalizationwasinitiallyintrodued fornoninterating quantumpartiles [1℄, butits observation remainedelusive for manyyears. Itwas latelyrealized that Andersonloalizationisatuallyubiquitousinwavephysis,andthereforeitanbe applied alsotolassialwavessuhaslight orsound[9,36℄. Thispaved thewayfor the rst observations of Anderson loalization. Loalization of lassial waves has beenreportedsofarfor ultrasounds[12,60℄,foreletromagnetiwavespropagating in free spae in the mirowaves regime [10, 61℄ as well as in the optial regime [11,62,63℄andfor light inphotoni rystals[64,65,66℄.

Figure 2.6: (a) Shemati representation of the expansion of an ultraold atomi gas in a bihromati optial lattie, as realized in Ref. [25℄. The ondensate is initially onned in a nite region of spae (left) and then its released along the quasiperiodipotential. Asthestrengthof theseondarylattie (whih playsthe role of disorder strength) is inreased the size of the ondensate after a xed expansion timeisredued (right). (b)Axialsizeofthe ondensateafter

700

msofexpansionas afuntionofthestrengthoftheseondarylattiefordierentvalues ofthetunneling energy

J

. Inset: typial exponentially deaying prole of the atomi loud in the regime of loalization. Figure takenfrom Ref. [27℄.

Figure2.7: (a),(b)Cartoonofthetypial experimentalproedure of Ref.[24℄. The atomi loud is initially onned by an harmoni onnement and then it is sud-denly released into the spekle potential. () Density proles of the loalized atomi loudoneseondafteritsrelease, theexponential nature oftheloalization islearly observed. (d)Loalization length

L

loc

tted fromthemeasured proles asa funtion of the disorder strength. The shaded area represents the theoretial predition with therelativeunertaintyderivingfromtheestimationof theexperimentalparameters. Figure takenfromRef. [24℄

we show a typial experimental measure ofthe widthof theexpanding ondensate after a xed expansion time as a funtion of the disorder strength. One an see thatforlargevaluesofthedisorderstrengththeexpansionisompletely frozenand theabsene of diusionpredited byAnderson is observed. Another key featureof Anderson loalization is the exponential shape of single partile eigenstates. This reetsinanexponentialshapeoftheatomiloudthatanalsobeobservedwithin situabsorptionimaging. Intheinset ofFig2.6(b)andinFig.2.7()two examples of measured exponentially loalized proles are shown. Fitting the exponentially loalized prolesone an alsoobtain a measureof theloalization lengthas shown inFig.2.7(d).

Inhapters 3,4and5ofthisthesiswewillextensivelyfousontheexperimental setup realized in Ref. [25℄, namely a one dimensional bihromati optial lattie. Thishoie is motivated by thefatthat inthis experiment, not only the disorder strengthanbeontrolledatwill,butalsotheinteratomiinterationanbetuned, makingthisongurationpartiularlysuitableforthestudyoftheinterplaybetween interation anddisorder indued loalization [67,68,33,69℄.

More reently Anderson loalization of matter waveshas been reported also in 3Dwith both fermions [70℄ and bosons [71℄using a similar proedure withrespet to theone thathasbeenused in1D.

Noninterating partiles in

quasiperiodi potentials

Quasiperiodisystemsareaspeiallassofnon-periodisystems. Theypossesstwo ormore periodiitieswhoseperiodsareinommensuratewitheahother. Although thesesystemsarenotrandomintheusualsense,theylakoftranslationalsymmetry sine there exist no translations whih will leave the periods of all the periodi strutures invariant. Nevertheless, there exist translations that leave the system almostinvariant. Thisleadstoquiteunusualbehavioursinquasiperiodisystems. It is well known that in a perfetly periodi system all the eigenfuntions are extended Bloh waves [40℄ while for a one dimensional random potential all the eigenfuntionsareexponentiallyloalized[53,2℄. Thesepropertiesarestritly on-neted with the spreading behaviour of an initially loalized wavepaket, in the former aseit expands ballistiallywhilein thelatteritremains loalized.

In between these two extreme ases we nd quasiperiodi systems that show an intermediate behaviourbetween the two [77, 78℄. Inpartiular itis known that quasiperiodi systems an have both extended and loalized states already in one dimension. Furthermoreritial stateswhih mayberegardedasbeing intermedi-atebetween loalized andextendedan appear. Asaonsequene thedynamis of awavepaketan rangefromloalization to ballistiexpansion andalsoanomalous diusion an be observed [79, 80℄. These quantum properties are often related to thequiteanomalous transport propertiesof quasirystals[81,82,83℄.

onsidering modelsinontinuous spae[89℄.

Duetotheirpeuliarity,theloalizationpropertiesofquasiperiodisystemshave always reeived a lot of attentions, espeially after the disovery of quasirystals [81,82℄and the observationof their anomaloustransportproperties[83℄. However, few yearsago, a newboost hasbeen given to the study ofthis topi afterthat two experiments with ultraold atoms have reported the rst observation of Anderson loalizationofmatterwaves. Infatoneofthetwoexperimenthavebeenperformed using a 1Dquasiperiodipotential and realized anexperimental implementation of the Aubry-André model[25℄. Oneyear later another experimental implementation of the Aubry-André modelhas been realized using photoni rystals [65℄. For this reasoninthishapterwewillfousontheloalizationpropertiesoftheAubry-André modelandon its onnetionwithatomigases experiments.

Thishapter is organized asfollows. In setion 3.1we explain how the Aubry-André model an be realized experimentally using ultraold atoms in bihromati optial latties. Insetions 3.2and 3.3we disusstheloalization properties ofthe model, rst onsidering thenature of the eigenstates, as originally done by Aubry and André, and then studying the dynamis of an initially loalized wavepaket. Thisseondmethodreproduesthetypialexpansionexperimentthatisperformed with ultraold atoms. For this reason we fous on two questions whih an be relevant from the experimental point of view,namely therole played bytheinitial shape of the wavepaket and the dierene between the inommensurate and the ommensurate ase. Finally in setion 3.4 we disuss the loalization properties of theAubry-André model in momentum spae and we propose a possible way to detetthetransitionfromextendedtoloalizedregimeinafeasibleexperimentwith ultraold atoms by measuringthe momentum distribution oftheatoms.

3.1 From bihromati optial latties to the Aubry-André model

One-dimensional bihromati latties are realized in experiments with Bose-Einstein ondensatesbysuperimposing two optiallattiesof dierent wavelengths

[

89

,

90

,

49

,

25

]

,produingan externalpotential ating onthe atoms inthisform:

V

b

(

x

) =

V

1

(

x

) +

V

2

(

x

)

=

s

1

E

R

1

sin

2

(

k

1

x

) +

s

2

E

R

2

sin

2

(

k

2

x

+

ϕ

)

,

(3.1) where

k

j

= 2

π/λ

j

(

j

= 1

,

2)

is the wavenumber of the laser light that reates the optial lattie,

E

R

j

=

~

2

k

2

V

b

(x)=s

1

E

R

1

sin

2

(k

1

x

1

)+s

2

E

R

2

sin

2

(k

2

x

2

)

V

1

(x)=s

1

E

R

1

sin

2

(k

1

x

1

)

Primary lattice

V

2

(x)=s

2

E

R

2

sin

2

(k

2

x

2

)

Secondary lattice

d

d/

α

~

Figure 3.1: Representation of a bihromati optial lattie. The superposition of a deep primary lattie (blue line) and of a shallower seondary lattie (green line) produes the quasiperiodi potential represented by the red line. The blak dots in-diate the on-site energies within a disretized desription of the system. The blak lineshowsthatthe modulation introduedby theseondarylattie hasaosinusoidal form. The two blak arrows represent the two key length sales of the system: the lattie spaing introdued by the primarylattie,

d

,and theperiodiity of the modu-lation introdued by the seondary lattie,

d/

α

˜

.

antunnelfromonesitetotheotherwithagiventunnelingrate

J

[91℄. Theseond lattieissigniantlyshallower(

s

2

≪

s

1

)andperturbsweaklythestrutureformed by the primary lattie; in pratise, the presene of the seondary lattie does not modifysigniantlythepositionoftheminima ofthepotentialbut produesonlya shiftofthe onsite energies,introduingadeterministi"disorder,orquasi-disorder [25,90,49℄.

Noninterating atoms inthe presene of a one-dimensional bihromati optial lattie aredesribed bythe Hamiltonian

H

=

−

~

2

2

m

∂

2

∂x

2

+

V

b

(

x

)

(3.2)

leftwitha simpleperiodisystem,where the spetrumisharaterized bybandsof allowedenergiesandenergygapsandtheeigenstatesareBlohfuntionsdeloalized over the whole lattie [40℄. In the tight binding regime the energy gap between the lowest band and the rst exited band,

E

G

, is so large that the physis of the systeman be well desribed byonsidering only thelowest energyband. Thisis a good approximation as longas allthe energy sales involved inthe problemunder onsideration are muh smallerthan

E

G

. Let us introdue a set of Wannier states

|

w

j

i

labelled bythe site index

j

(see appendix A for an introdution on Wannier funtions). Eah of them, onsidered in real spae,

h

x

|

w

j

i

=

w

j

(

x

) =

w

(

x

−

x

j

)

represents a funtion entered around the lattie site

j

, at position

x

j

=

jd

. In partiular, as previously mentioned one an onsider as a basisof thesystem just the Wannier funtions assoiated to the lowest energy band. One an therefore express wavefuntionsand operatorsprojeting on thebasisof Wannier states

|

ψ

i

=

X

j

ψ

j

|

w

j

i

,

H

=

X

i,j

|

w

i

i

H

i,j

h

w

j

|

,

(3.3)

where

H

i,j

=

h

w

i

|

H

|

w

j

i

,

ψ

j

=

h

w

j

|

ψ

i

and

n

j

=

|

ψ

j

|

2

representstheprobability of nding a partile in the lattie site

j

. Let us evaluate expliitly the matrix elements

H

i,j

:

H

i,j

=

Z

w

i

∗

(

x

)

Hw

j

(

x

)

dx

=

Z

w

i

∗

(

x

)

H

(0)

w

j

(

x

)

dx

+

Z

w

∗

i

(

x

)

H

(1)

w

j

(

x

)

dx,

(3.4)

where

H

(0)

=

−

~

2

2

m

∂

2

∂x

2

+

s

1

E

R

1

sin

2

(

k

1

x

)

isthe part of theHamiltonian formed by thekinetitermandbytheprimarylattie,while

H

(1)

=

s

2

E

R

2

sin

2

(

k

2

x

+

ϕ

)

isjust formed bythe seondarylattie. Negletingthe overlapbetween Wannierfuntions beyond nearest neighbours for

H

(0)

and retaining only theon-site ontribution for

H

(1)

one nds thattheonly non-zeromatrix elements are

H

i,j

=

E

0

δ

i,j

−

Jδ

i,j

±

1

+

δ

i,j

Z

|

w

i

(

x

)

|

2

H

(1)

dx

(3.5)

where

E

0

=

Z

w

∗

i

(

x

)

H

(0)

w

i

(

x

)

dx

;

J

=

−

Z

w

∗

i

(

x

)

H

(0)

w

i

+1

(

x

)

dx.

(3.6)

an be written ina muh simpler form using thetrigonometri relation

sin

2

(

k

2

x

+

ϕ

) = [1

−

cos(2

k

2

x

+ 2

ϕ

)]

/

2

. Usingthesymmetryof theWannier funtionsonean showthat

Z

−

s

2

E

R

2

2

cos(2

k

2

x

+ 2

ϕ

)

|

w

i

(

x

)

|

2

dx

= ∆ cos(2

παi

+

ϕ

′

)

(3.7)

where we have used the fat that

x

i

=

id

=

iπ/k

1

, we have redened the phase

ϕ

and introdued

α

=

k

2

/k

1

=

λ

1

/λ

2

and

∆ =

s

2

E

R

2

2

Z

cos(2

k

2

y

)

|

w

(

y

)

|

2

dy.

(3.8)

Finallynegleting all onstant termsone endsup withthefollowing simple expres-sionfor the matrix elements

H

i,j

=

−

Jδ

i,j

±

1

+

δ

i,j

∆ cos(2

παi

+

ϕ

)

.

(3.9)

Substitutingthisexpressionin(3.3)andexpressingalltheenergiesinunitsof

J

one nds the Aubry-André Hamiltonian

H

=

−

X

j

(

|

w

j

ih

w

j

+1

|

+

|

w

j

+1

ih

w

j

|

) +

λ

X

j

cos(2

παj

+

ϕ

)

|

w

j

ih

w

j

|

(3.10)

where

λ

= ∆

/J

. In this last expression we expliitly see that the modulation introdued bytheseondarylattie hasaosinusoidalform anditan beseenasa potential inthe disretespae:

V

j

=

λ

cos(2

παj

+

ϕ

)

.

(3.11)

Let us notie that the disrete potential is quasiperiodi as long asthe parameter

α

, whih is the ratio between the wavelengths of the two latties, is an irrational number. Infat,onlywhen

α

isirrationalthepotential

V

j

addsaseondperiodiity whih is inommensurate with respet to the underlying periodiity given by the disreteness of the system. Let us notie that

V

j

is invariant under a shift of

α

by an integer numberand therefore, without any lossof generality,one an hoose

α <

1

.

Ingure3.1weshowanexampleofabihromatioptiallattieandwe shemat-ially illustrate the disretizationproedure. We onsidered

α

= (

√

˜

α

= (

√

5

−

3)

/

2

and therefore we enounter a minimum in the lattie modulation approximately every

2

.

62

lattie sites.

Writing down the time independent Shrödinger

H

|

ψ

i

=

E

|

ψ

i

equation for thetight binding Hamiltonian(3.10) one obtains

−

ψ

j

+1

−

ψ

j

−

1

+

λ

cos(2

παj

+

ϕ

)

ψ

j

=

Eψ

j

.

(3.12) Thisequationistheonewhihisusually alledAubry-André orHarpermodel[85℄. This model is of partiular importane beause, despite its simpliity, is very rih from the point of view of the loalization properties and those are known exatly. Thekeyparameterthatdeterminestheloalization properties,when

α

isirrational, is

λ

whih quanties how strong is the quasi-disorder ompared to the tunneling energy. Inthefollowing withaslight abuseofnotation wewillsometimes all

λ

the disorder strength.

3.2 Loalization properties of the Aubry-André model Theloalizationpropertiesofmodel(3.12)havebeendisussedfor thersttimeby AubryandAndré[85℄. Lateranumberofnumerialandanalytialstudiesonrmed theirresults[77,86,92,79,88,93℄. Here,followingtheoriginalalulationof Aubry-André, we show how one an derive the loalization properties of the model using the self-duality of Eq. (3.12) and theThouless formulafor the Lyapunovexponent (2.15).

The self-duality property of equation (3.12) an be found by introduing the following transformations

ψ

j

=

e

iθj

∞

X

l

=

−∞

d

l

e

il

(2

παj

+

ϕ

)

d

l

=

e

−

iϕl

∞

X

j

=

−∞

ψ

j

e

−

ij

(2

παl

+

θ

)

.

(3.13)

Using these transformationsinequation (3.12) one an showthat thenewvariable

d

l

satisesthe dualequation

−

d

l

+1

−

d

l

−

1

+

4

λ

cos(2

παl

+

θ

)

d

l

=

2

E

λ

d

l

,

(3.14) whih hasexatlythe same formasequation (3.12) ifwe set

4

λ

→

λ ,

d

l

→

ψ

j

,

2

E

λ

→

E ,

θ

→

ϕ .

(3.15) The symmetry of Eqs (3.12) and (3.14) has an important onsequene. One an note that,if

ψ

n

isa loalizedsolution of(3.12), thatis

∞

X

j

=

−∞

then

d

l

willbe anextendedsolution solutionof (3.14),that is

∞

X

l

=

−∞

|

d

l

|

2

=

∞

,

and vie versa. This tell us that the dual transformations (3.13) exhanges the loalization properties of the eigenfuntions. However, a priori we ignore whih eigenfuntions are loalized and whih are extended. To go further we need to usethe Thouless formula (2.15) whih relates the Lyapunov exponent

Λ(

E

)

to the density of states

ρ

(

E

)

[58℄. This formula was originally introdued for random systemsbutitan be usedwithoutanyhange alsofornon-randommodels suhas (3.12). Whenever

α

is an irrational number, making useof thedual property, one an relate the integrated density of states of the Aubry-André model

N

λ,α

(

E

)

to theoneofitsdualounterpart

N

4

λ

,α

(

E

)

[85℄. Thesamean bedone forthedensity of states

ρ

(

E

) =

∂

∂E

N

(

E

)

andone nds

N

λ,α

(

E

) =

N

4

λ

,α

2

E

λ

;

ρ

λ,α

(

E

) =

ρ

4

λ

,α

2

E

λ

2

λ

.

(3.16) Usingtheseexpressionsandthe Thoulessformulaoneobtainsthedualtransformof theLyapunovexponent

Λ

λ,α

(

E

) = Λ

4

λ

,α

2

E

λ

+ ln

λ

2

.

(3.17)

Starting from this expression itis now possible to infer theloalization properties of the Aubry-André model with few simple onsiderations. First of all, we note that the Lyapunov exponent

Λ(

E

)

assoiated to a given eigenstate is neessarilya positivenumber and that

Λ(

E

)

vanishesonly whenever this state is extended. Let us also reall the result that we have derived earlier in this setion that the dual transformation inverts the loalization properties; therefore whenever

Λ

λ,α

(

E

)

is non-zero it follows that

Λ

4

λ

,α

2

E

λ

is zeroand vie-versa. Therefore assuming that

Λ

4

λ

,α

2

E

λ

= 0

itfollows that

Λ

λ,α

(

E

) = ln

λ

2

and the positivityof theLyapunov exponent impliesthat

λ >

2

. Conversely,when

Λ

λ,α

(

E

) = 0

Λ

4

λ

,α

2

E

λ

= ln

2

λ

and

λ <

2

.

WeanthereforeonludethattheAubry-Andrémodel(3.12)undergoesa tran-sitionfrom extendedtoloalizedeigenstatesat

λ

= 2

. Alleigenstates areextended for

λ <

2

and exponentiallyloalizedfor

λ >

2

. Moreover all the eigenstateshave thesame loalizationlength

L

loc

= 1

/

Λ =

1

Figure 3.2: Numerial study of of the loalization properties of the Aubry-André model. In the left panel we show that the ground state of the system is extended for

λ

= 1

(green line) and

λ

= 1

.

9

(blue line) while it is exponentially loalized for

λ

= 2

.

1

(magentaline) and

λ

= 2

.

5

(red line). In the loalized regime we also show theloalization lengthpreditedby theanalytial formula(3.18)(blak dashedlines). In the right panel a olor density plot shows the ground state as a funtion of the disorderstrength. The transition fromthe extended to the loalized regime at

λ

= 2

is learly visible.

Theoppositepropertiesholdsforthedualmodel(3.14). Thesimplederivationthat wepresentedheredoesnot giveanyinformation about thenatureoftheeigenstates for

λ

= 2

. Itisknownthattheyareneitherplanewavesnorexponentiallyloalized. It isonjetured thatthey aredereasingfuntions withapower law[85℄.

The exponential loalization that takes plae for

λ >

2

has been identied by Aubry andAndré[85℄asAndersonloalization inaquasiperiodipotential, analog to Anderson loalization in a purely random potential. A dierent interpretation, based ona semilassialanalysis, hasbeen reently proposedinRef.[94℄.

In gure 3.2 we present a numerial alulation that onrms the results that wehavealreadyobtained on theloalization properties oftheAubry-Andrémodel. Weshowthe behaviourofthegroundstate ofthesystemfor dierent values ofthe potential strength,

λ

. In the right panel it is learly observed that for

λ

= 1

and

1

10

100

0.1

1

10

10

2

10

3

10

4

10

5

Width

Time

λ

=1.5

1.7

1.9

2

2.1

3

1

10

10

2

10

3

10

4

10

5

λ

=1.5

1.7

1.9

2

2.1

3

Figure 3.3: Expansion of a noninterating loud of atoms in the Aubry-André modelwith

α

= (

√

5

−

1)

/

2

. The timeevolution ofthe widthof the wavepaket

w

(

t

)

is shown for dierent values of the disorder strength,

λ

= 1

.

5

,

1

.

7

,

1

.

9

,

2

,

2

.

1

,

3

. In the left panel, the starting wavepaket is a

δ

-funtion loalized in a single site. In theright panel weusean initialGaussian wavepaketof width

σ

= 5

. In both ases, one learly observes the transition from extended to loalized states that ours at

λ

= 2

.

3.3 Spreading of wavepakets in theAubry-André model Inthis setionwe disussthe problemofquantum diusionof aninitially loalized wavepaket in the Aubry-André model. This is of partiular relevane for exper-iments with ultraold atoms where the expansion of an atomi loud is the main toolusedfor the detetion ofAnderson loalization [24,25℄. Boththewidthof the expandingloud andits shape areofgreat interests.

Theexpansionofanoninteratingwavepaketisdesribedbythetimedependent Shrödingerequation

i

~

∂

∂t

|

ψ

i

=

H

|

ψ

i

thatintheaseofHamiltonian(3.10)takes thefollowing form

i

∂ψ

j

∂t

=

−

ψ

j

+1

−

ψ

j

−

1

+

λ

cos(2

παj

+

ϕ

)

ψ

j

,

(3.19) wherewehaveabsorbedthePlankonstant

~

inthetimevariablesothat

t

beomes adimensionlessquantity. Theatualtimeinseondsanbeobtainedbymultiplying thedimensionless parameter

t

by

~

/J

.

We investigate theevolutionstarting from two dierent lasses of initial ondi-tions,namely a

δ

-funtion loalizedina singlelattie site,

and a Gaussianwavepaket ofwidth

σ

,

ψ

j

(0) =

Ce

−

j

2

2

σ

2

,

(3.21)

where

C

isanormalizationfatorthathastobedeterminedinordertohaveanorm of the wavepaket equal to one

P

j

|

ψ

j

|

2

= 1

. Thehoie of Gaussian wavepakets is onvenient if one wants to simulate realisti experimental ongurations; it also allows one to explore the behavior of sharp to broad wavepakets in a ontinuous manner. Owing to arbitrarinessof the phase

ϕ

, here we have hosen, without any lossof generality,to plae theinitial wavepaketaround thelattie site

j

= 0

.

As a measure of the loalization we onsider two quantities: the width of the wavepaket measuredasthe squareroot oftheseondmomentof thespatial distri-bution

|

ψ

j

(

t

)

|

2

,

w

(

t

) =

p

m

2

(

t

) =

s

X

j

(

j

−

X

)

2

|

ψ

j

(

t

)

|

2

,

(3.22)

and thepartiipation number

P

(

t

) =

P

1

j

|

ψ

j

(

t

)

|

4

,

(3.23)

whihmeasures thenumberofsigniantly oupiedlattie sites[95℄. Thequantity

X

representsthe average position ofthewavepaket, dened as

X

=

P

j

|

ψ

j

|

2

. The loalization transition of the Aubry-André model at

λ

= 2

, whih has beenintroduedintheprevious setion,anbedeteted inthedynamis(quantum diusion), by looking for example at the width of thewavepaket as a funtion of time[79℄. Inpartiular the asymptotispreadingof thewavepaket width

w

(

t

)

an be parametrized as

w

(

t

)

∼

t

γ

,and one nds three dierent regimes asthe value of

λ

is varied:

(i)

λ <

2

: ballistiregime,

γ

= 1

(ii)

λ

= 2

: subdiusiveregime,

γ

∼

0

.

5

(iii)

λ >

2

: loalizedregime,

γ

= 0

.

We solve Eq. (3.19) by using a standard fourth order Runge-Kutta (RK4) algo-rithm for the numerial integration. The auray ofthe integration is heked by monitoring the onservation of the norm of the wavepaket and of the energy of the system. A standard hoie for the value of

α

onsists of hoosing the inverse golden mean

α

= (

√

1

10

2 3 4 5 6 7 8 9 10

L

loc

λ

Figure3.4: Loalizationlengthof thewavepaket

L

loc

asa funtionofthe disorder strength

λ

in the loalized regime. We measure the loalization length by ttingthe tails of the loalized wavepaket after the expansion. We ompare the values of the loalization lengthextrated fromthe tting(red points)withthe analytipredition

L

loc

=

ln(

λ/

1

2)

(blak line).

As regards theshape of thewavepaket we fouson theloalized regime

λ >

2

where spreadingstopsafteratransienttime. Sine,for agivendisorderstrength

λ

, all eigenstates areexponentiallyloalized withthe same loalization length we ex-petthatalsothewavepaket, whihisformedbyalinearsuperpositionofdierent eigenstates, has exponentially deaying tails with the same harateristi loaliza-tion length. By tting the density proles of the wavepaket after the expansion we extrat a value of theloalization length. In Fig.3.4 we show theresult of our tsasa funtionofthe disorderstrength

λ

(redpoints)andweompare themwith the theoretially expeted value

L

loc

= 1

/

log (

λ/

2)

(blak line) showing a perfet agreement.

3.3.1 Inommensurate vs. ommensurate ase

It isworth stressingthat a trulyquasiperiodipotential an not berealized inany realisti experiment, sine the wavelengths of the lasers are always known with a nite number of digits and therefore their ratio will always be a rational number. Moreover realexperiments have always a nite size. It is thus important to larify to whih extent the preditions of the Aubry-André model are relevant for the desriptionof experiments withultraoldatoms in bihromatioptial latties.

1

10

10

2

1

10

10

2

10

3

10

4

Width

Time

n=3

5

7

9

11

∞

Figure 3.5: Time evolution of the width of the wavepaket

w

(

t

)

of noninterating partiles, starting from a single-site

δ

-funtion, for

λ

= 2

and for dierent orders,

n

, of the approximants in the Fibonai sequene. The blak arrows represent the values of

t

at whih we observe the transition from the behaviour predited for a quasiperiodi potential (inommensurate lattie) to the diusion expeted in a peri-odipotential.

rational approximation

α

n

of order

n

of the irrational number. In partiular we onsider a sequene of rational numbers

α

n

,that onverges to the irrational value

α

as

n

→ ∞

[96,78℄. The sequeneofapproximants

α

n

an befoundbysuessive trunations of the ontinued-fration expansion of

α

. For the ase of the golden mean

α

= (

√

5

−

1)

/

2

[95℄ the approximants are given by

α

n

=

p

n

/q

n

, where

p

n

and

q

n

=

p

n

+1

are two onseutive terms of the Fibonai sequene (

p

1

=

p

2

= 1

,

p

n

=

p

n

−

1

+

p

n

−

2

for

n >

2

).

It turns out that the inommensurate ase an thus be onsidered as the limit of a sequene of ommensurate Hamiltonians, whose eigenvalues

E

ξ,m

and eigen-funtions

φ

ξ,m

j

an be labelled by the quasi-momentum

ξ

and the band index

m

, sine thespatialperiodiityofthesystem,withperiod

q

n

,permits tousetheBloh wave deomposition. One nds that, for suiently large

n

and for

λ >

2

, the eigenfuntions are indeed haraterized by periodi replia of exponentially loal-ized funtionswithineah period ofthe potential,that inthe limit

n

→ ∞

tendto a singleloalized funtion[91℄.

Letus nowonsider the sameproblem fromthe point of viewof thedynamial properties. The time evolution for

α

= (

√

10

-70

10

-60

10

-50

10

-40

10

-30

10

-20

10

-10

1

-80

-60

-40

-20

0

20

40

60

80

|

ψ

j

|

2

Lattice site, j

n=5

6

7

8

9

10

∞

Figure 3.6: Modulus square of the wavefuntion

|

ψ

j

|

2

for dierent values of

n

, plotted at a xed evolution time

t

= 1000

, for

λ

= 7

and

β

= 0

. The initial wavepaket at

t

= 0

isa

δ

-funtionloalized at

j

= 0

. The vertial arrowsare drawn atthe positions

q

n

/

2

.

limit

n

→ ∞

onemustreovertheresultsoftheAubry-Andrémodel, witharitial behaviour for

λ

= 2

and loalized states for

λ >

2

. The approah to this limit in nontrivialand involvestheharateristi timeand lengthsales ofthesystem.

In Fig. 3.5 we rst show our results for the diusion of a

δ

-like wavepaket in a lattie with the ritialvalue

λ

= 2

. For any nite

n

thewavepaket exhibitsa subdiusive spreading (

w

(

t

)

∼

t

γ

with

γ

≈

0

.

5

), as in the inommensurate ase, within an initial time interval. Then, at time

τ

, the width starts growing as in a ballisti expansion in a periodi lattie. The transition between the two regimes turnsout to ourwhenthe widthof thewavepaketbeomesofthesame orderof thespatialperiodiityofthelattie. Thetransitiontime,

τ

,indiatedbythearrows in Fig. 3.5, inreases with the order

n

of the approximants and theorresponding width,

w

(

τ

)

exhibits a linear dependene on the periodiity of the system,

q

n

1 . The role of the spatial periodiity is even more evident if one plots the density distributionintheregimeofloalization,asshowninFig.3.6for

λ

= 7

and

t

= 1000

. In this gure the arrows are drawn at the positions

q

n

/

2

. As one an see, the

1

0.1

1

10

10

10

1

10

2

10

3

10

4

Width

Time

n=3

4

5

6

7

∞

Figure 3.7: Time evolution of the width of the wavepaket

w

(

t

)

of noninterating partiles, startingfroma single-site

δ

-funtion,for

λ

= 3

andfor dierent values of

n

.

deviations from the density distribution of the inommensurate ase (

n

→ ∞

) are aused bythe spreadingofthelateral omponentsof thedistribution, i.e., thoseat a distaneof theorder of,or largerthan

q

n

/

2

. Theasymptoti behaviour(

t

→ ∞

) is always ballisti. However, for a nite

t

and for

λ >

2

the entral part of the densitydistribution(withinawidthoforder

q

n

)exhibitsanexponentialloalization, independent of

n

, and is almost indistinguishable from the one predited for the inommensurate lattie. Thespreadingofthelowdensitytailsaetsthebehaviour ofthewidthdenedinEq.(3.22). AnexampleisshowninFig.3.7. Forshorttimes theontribution ofthe expandingtails isnegligible, whilefor later timesthewidth inreasesasinaballistiexpansion. Itisworthstressing,however,thattheseeets of thelowdensitytails areexpetedto behardly detetableinatual experiments, due to t