Quantum Chaos with Ultracold Atoms in Optical Lattices

Andrea Tomadin

Quantum Chaos with Ultra old Atoms

in Opti al Latti es

Relatore: Chiar.mo Prof. Ri ardo Mannella

1 Introdu tion 3

1.1 ModernExperimental Possibilitiesfor theStudy ofQuantum

Chaos . . . 3

1.2 The Opti al Latti e. . . 5

1.3 Outline of theThesis . . . 7

2 Fra tal Flu tuations inthe Opened Quantum Ki ked Rotor 9 2.1 Theoreti al Ba kground . . . 9

2.1.1 TheClassi alKi ked Rotor . . . 9

2.1.2 TheQuantum Ki ked Rotor . . . 12

2.1.3 Fra talFlu tuations ofRea tion Curves . . . 16

2.2 How to OpentheSystem . . . 18

2.2.1 Dynami alLo alizationandAbsorbingBoundary Con-ditions . . . 19

2.2.2 TheSurvivalProbabilityasthe Rea tion Curve . . . . 20

2.3 The Algorithms behindtheResults . . . 21

2.3.1 TheTimeEvolution . . . 21

2.3.2 TheFra talDimension . . . 22

2.4 The Resultsof theFra tal Analysis . . . 23

2.4.1 TheChoi eof the SamplingGrid . . . 24

2.4.2 Veri ation of theHypotheseson theSpe trum . . . . 27

2.4.3 Dependen eon Time . . . 28

2.4.4 Dependen eon the Ki king Strength . . . 30

2.4.5 Dependen eon theQuasimomentum . . . 33

2.4.6 Fra talProle andGridResolution . . . 34

2.5 ComputationsforanExperimentalDete tionofQuantum Fra -tals . . . 35

2.5.1 Parameters intheExperiments . . . 35

2.5.2 Experimental Ensemble Measurements . . . 36

2.5.3 Dependen eon theNumberofAtoms . . . 37

2.5.4 Dependen eon theQuasimomentum Spread . . . 39

3.1 TheMany-BodyModel. . . 44

3.1.1 TheGeneral Many-BodyHamiltonian . . . 44

3.1.2 TheComputationofthe Coe ientsintheHamiltonian 46 3.1.3 TheComputation oftheWannier Fun tions . . . 50

3.1.4 TheAnalysisof the Coe ients intheHamiltonian . . 55

3.1.5 Translational Symmetry andCy li Boundary Condi-tions . . . 57

3.1.6 Irredu ible Representation of The Floquet-Blo h Op-erator . . . 60

3.2 TheNumeri alComputation oftheSpe trum . . . 63

3.2.1 TheGeneration of theMany-BodyBases . . . 63

3.2.2 TheSparseness oftheHamiltonian . . . 65

3.2.3 TheTimeEvolution andthe Floquet Operator . . . . 66

3.3 TheResultsof theSpe tralAnalysisfor aOne-Band Model . 69 3.3.1 TheComputation oftheQuantum Spe trum . . . 71

3.3.2 The Statisti al Chara terization of the Spe trum of the Floquet-Blo h Operator . . . 74

3.4 ThePerturbative Ee tof the Se ondBand . . . 79

3.4.1 TheDerivation oftheDe ayRatesfor theFirst Band 81 3.4.2 TheEnergy Un ertaintyof theFo k States . . . 85

3.4.3 TheResultsof thePerturbative Openingof theSystem 87 3.5 Summary . . . 90

Introdu tion

Thepresen eofbothregularand haoti motionin lassi alsystemshasbeen

re ognizedand understoodfor a longtime, but onlyinthelast de ades the

manifestationsof haosinquantumsystemshavebeen investigatedand

sev-eraltheoreti aleortshavebegunto onvergetowardsthequantum haology

framework. The rst experimental tests for the methods of omplex

quan-tum dynami s ame in the Eighties, within the eld of ionization

experi-mentswith highlyex itedatoms. In thelast de ade,the impressive ontrol

grantedbyopti alte hniquesontheultra old atomi matterhaveprovided

the means to address more and more thoroughly the ee ts of haos in a

quantum system.

1.1 Modern Experimental Possibilitiesfor theStudy

of Quantum Chaos

Inthe lassi als enario,thedenitionof haosfo usesonthestabilityofthe

phase-spa etraje tories, i.e. onthe sensitive dependen eon theinitial

on-ditionsof thetimeevolution. The entral mathemati al toolto understand

howsto hasti ityenters thephasespa eof adeterministi onservative

sys-tem isthe Kolmogorov-Arnold-Moser theorem[1 ℄, whi h rystallized inthe

Sixtiesafterade ennialeort. Themainstreamofthistheoryistra edba k

tothepioneeringresear hofPoin aré[1℄,whore ognizedhownewquestions

hadtobeformulatedtoa hieveaglobalperspe tiveonthepropertiesofthe

motion.

Asthedynami sof omplexquantumsystemswasinvestigated,bothfor

open and losed geometries, for autonomous or driven setups, thene essity

aroseto hara terize theanalyzedsystemsintrinsi allybutalsowithrespe t

to the eventual lassi al ounterpart. Although the on ept of phase-spa e

traje tory islost inthe quantum realmbe ause oftheHeisenbergprin iple,

theavor of the lassi al denitionof haosis maintained. In fa tthe

Then, the possibilityto verifyinexperiments theee tsof the omplex

dynami sdependsonthe hoi eofimplementationswhereasubstantial

on-trol over the parameters of the systemis granted. The manipulationof the

ultra old atomi ensembles, developed inthe last de ade, is anideal

imple-mentation fa ility be ause it allows an unpre edented ontrol both on the

preparation andon theevolutionstage ofa many-body system. In

parti u-lar, the opti al latti e devi es have allowed sofar to test many predi tions

ofsolidstate systems,sin e analmost ideal realizationof model

Hamiltoni-ans an be a hieved, essentially freefrom impurities and other de oheren e

sour es.

A loud of old or ultra old atoms, produ ed in a Bose-Einstein

on-densate loadedinto aquasi-one dimensionalopti al latti e,is a many-body

intera ting system, butregimes are found where theatom-atom intera tion

an be negle ted and a single-parti le model is su ient for a des ription

ofthe system. Asingle parti le ina one-dimensional onservative potential

is a fully integrable system, and the addition of an external driving, whi h

ex hanges energy with the parti le, is ne essary to bring in omplexity. A

time-periodi for ing of the system an be treated formally with the

Flo-quet method [2, 3℄, whi h is analogous to theBlo h theorem for potentials

withspa eperiodi ity. Morepre isely,thetimeevolution operator overone

period is alled the Floquet operator

U

ˆ

FB

and the phases of its (unitary) eigenvaluesareusedto onstru tasetof onserved quantumnumber, alled

quasienergies. The quasienergies hara terize the one- y le time evolution

asthequasimomenta dene thetranslational properties oftheBlo hwaves.

Both dis rete translational symmetries  in time and in spa e  are often

onsideredinthis Thesis.

Asimpleexternal drivingofthe atomsinalatti eisamodulationofthe

opti al potential itself. In parti ular, ifthe latti e is ashed periodi ally,

thesingle-parti ledes riptionresultsintheQuantumKi kedRotormodel,a

paradigmati obje tofquantum haology,largelyaddressedintheliterature

[4,5℄. Themodelhasawell-denedsemi lassi allimit,asanee tivePlan k

onstant

~

eff

goes to zero. Inthe Quantum Ki ked Rotor, the denition of quantum haos an be onfronted with the dynami s of the lassi al limit

model,andthe studyofthephasespa eoftheStandardMap,theanalogof

theFloquetoperator,isusefulasapreliminaryanalysis. Thedeepquantum

regime is hara terized by ee ts originating in the oheren e of the wave

fun tion, and in parti ular a dynami al lo alization of the parti le in the

momentum spa e takes pla e, similar to the Anderson lo alization in solid

state sampleswithrandom impurities. The omparison withthesolidstate

theorygoesfurther,asatransportproblem anbeformulatedintheenergy

spa eandfra tal ondu tan eu tuationsgiveeviden eofthesensitivityto

parametri hanges.

a terization of a quantum omplex system was re ognized in the statisti s

of its energy (or quasienergy) spe trum. The spe tral analysis is essential

fora purelyquantumsystemwhere thesemi lassi allimit isnot dened,as

is the ase of intera ting identi al atoms in the opti al latti e: the

many-body s enario is formulated within a Hilbert spa e, and the entanglement

of the parti les due to the quantum statisti s is ru ial. Even in the

ab-sen e ofan external driving eld, a substantial energy ex hange among the

parti les is possible if several energy s ales are omparable in magnitude.

Thismeansthat,intheBose-HubbardHamiltonianthatmodelsthesystem,

severaloperatorshave similar oe ients. Then, theevolution of thestates

over a typi al time s ale does involve a omplex mixing of the basis

ve -tors. The evolutor matrix is very large be ause of the exponential in rease

of the Hilbert spa e dimension, and so omplex that a statisti al analysis

is appropriate. The Random Matrix Theory, originally introdu ed to deal

withnu learenergylevels,predi tsthestatisti alpropertiesofthefull

spe -trum and in parti ular the strong mixing of the basis ve tors is eviden ed

in the repulsion of the levels as an external oupling parameter is tuned.

Inthe present ase, the opti allatti e isa elerated and the onsequent

ef-fe tive tilting potential is the ontrol parameter usedto probe theintrinsi

omplexityof themany-bodydynami s.

The problem of the transport through the tilted latti e arises naturally

and, in the single-parti le ase, it was onsidered several de ades ago by

Blo h, Wannier and Zener. The result, within the band theoryof periodi

latti es, involvesa verti al transport alongthe bandsto explain the

hor-izontal urrent along the uniform tilting eld. The approximation of one

band is then questionable, and the experimental possibility to a ess and

ontrol atomi matter inthe ex ited bands of thelatti e wel omes the

for-mulationof atheoreti alba kground for interband dynami s.

The motivation for the present work is the theoreti al analysis of the

regimeswhere,intheoptimally ontrolleds enarioof atom-opti s,itis

pos-sible to dete t experimental signatures of quantum omplex dynami s. In

parti ular, the fra tal ondu tan e u tuations oer the possibility to

ex-plorethe dynami allylo alizedsingle-parti le pi tureand thefullspe trum

in the many-body regime is ne essary for a omplete understanding of the

omplexinterband transport.

1.2 The Opti al Latti e

An opti allatti e is a onservative,periodi potential

V (x)

for atoms, gen-erated by laser beams. In the following Chapters we will not dis uss the

pra ti al realization of an opti al latti e but the nal result for

V (x) =

te hnologi al opportunitiesthatit oers[7,8℄.

Thebasi prin iple forthe onstru tion ofan opti allatti efor atoms is

thestrongly-detuned AC Stark shift of theele troni levels due to a

stand-ing wave of light produ ed bytwo superimposed ounter propagating laser

beams. In moredetail, the ele tri eld produ ed bytwo traveling waves of

frequen y

ω

L

, with the same amplitude and polarization

~e

, propagating in oppositedire tions alongthe oordinate

x

,is thestanding wave:

E

j

(x, t) = e

j

2E

0

cos (k

L

x) cos (ω

L

t).

TheHamiltonianforanatomintheele tri eldofthestandingwaveisgiven

by the kineti operator for thespatial degrees of freedom, by theenergy of

theunperturbedele troni levels andbythe oupling between theele trons

and the eld. If the eld varies slowly in time and spa e, ompared to the

frequen y of the ele troni transitions and to the spatial dimension of the

atom, respe tively, a dipole oupling between the eld and the ele trons is

appropriate. Ifwerestri tourselvestotwoele troni levels

|P i

and

|Si

,and dene the dipole operator as

d

ˆ

j

= d

j

(|P ihS| + |SihP |)

, the Hamiltonian reads:

ˆ

H =

p

2

2m

+ E

P

|P ihP | + E

S

|SihS| − E

j

d

ˆ

j

.

IntheRotatingWave Approximationtheele troni partoftheHamiltonian

redu es to the exa tly soluble Rabi problem [9℄ and the variation of the

energy for the groundstate ele troni levelis:

∆E

S

(x) =

[dE

0

cos (k

L

x)]

2

E

P

− E

S

− ω

L

.

TheACStarkshiftofthegroundlevelenterstheatomi Hamiltonianasthe

periodi , onservative potential

V (x)

alledopti al latti e. Ea h minimum of the periodi potential is alled a site and the period

a

of thelatti e is halfthe wavelength

2π/k

L

ofthe standing wave.

The solution is valid only if the denominator of the ground level shift,

i.e. thedetuning ofthelaserfromtheele troni transition,isnot toosmall.

Otherwise, the population transfer of ele trons from the ground to the

ex- itedstateis onsiderable. Moreover, bluedetunings

ω

L

> E

P

−E

S

redu e the spontaneous emissions of the atoms and limit to negligible values this

ee t of the de oheren e. The typi al momentum for the atomi motion

inthe latti e is

~

k

L

, that is alled re oil momentum be ause it is half of the momentum gained by a parti le at rest that ba ks atters a photon of

the eld. The typi al energy for an atom of mass

M

is the re oil energy

~

2

k

2

L

/2M

. These quantities are used to dene dimensionless variables in Se tion 3.1.3.

A fullthree-dimensional periodi trapping potentialfor theatoms is

posed. Inthefollowing,however, we analyzeonly one-dimensionalmotions.

Anatomi loud anbesqueezedtoaquasi-onedimensionalgeometryusing,

e.g. ylindri al traps that oupleto the magneti polarizationof theatoms.

The motionis regarded as quasi-one dimensional if thelevel spa ing of the

transverse onnement potentialismu hlargerthanthetypi allongitudinal

kineti energy, so that the transverse atomi motion remains frozen in the

groundstate [10, 11℄.

An opti al latti e isa very powerful devi e that an be tuned and

on-trolledat will[12 ℄. Theimplementation ofmanymodelHamiltonianis

pos-siblebe auseoftheunpre edented ontrol ontheatomi dynami s a hieved

in re ent years by means of opti al te hniques ( ompare with the

dis us-sionofSe tion 2.5.1ontheexperimentalrequirementsfortheveri ation of

ourtheoreti alpredi tions). The atomsloadedinto thelatti eusually ome

from a Bose-Einstein ondensate and the numberof atoms ina site an be

redu edtoorderunity. Coldandultra oldatomsloadedintoopti allatti es

areusedto implementsolidstate modelsinanee tiveenvironment almost

freeofimpurities;additionalpseudorandomosetsofthelatti esites anbe

produ ed to simulate a disordered environment; state-sele tive latti es are

used to ontrol atomi ollisions to realize quantum gates; a elerated

lat-ti esapplya onstantfor etotheatoms inanon-inertialframe(seeSe tion

3.1.1 where anexternal onstant eldisintrodu ed).

1.3 Outline of the Thesis

The Thesis is divided into two Chapters: in Chapter 2, a single-parti le

pi ture is used to study ondu tan e u tuations in a regime of negligible

atom-atom intera tions;inChapter 3,theintera ting many-bodyproblem

is addressed with a statisti al study of the full quantum spe trum of the

system.

InSe tion2.1, thesingle-parti le modeloftheQuantumKi kedRotoris

reviewed,alongwithatheoryforthefra talu tuationsof ondu tan e-like

observables. In Se tion 2.2, the work of [13℄ is followed for the denition

ofa weak opening of the system, probed bya survival probability fun tion.

The Se tion 2.3 is devoted to the des ription of the numeri al algorithms

thatwe have implemented to derivetheresults presentedinthesubsequent

Se tions of Chapter 2. In Se tion 2.4, we ompute the fra tal dimension

of the survival probability as a fun tion of the parameter that is best

on-trollableinexperiments; the dependen e ofthe results onthe tuningof the

remaining parameters is analyzed thoroughly. In Se tion 2.5, the

single-parti le approa h is extendedto a ount for statisti al ensembles of atoms,

the average survival probability is introdu ed and the attention is fo used

In Se tion 3.1, the onstru tion of a many-body dynami almodel is

re-viewed;westartfromageneralbosoni Hamiltonian, weexpandthebosoni

eldinWannier fun tionsand we explainthealgorithm to ompute the

o-e ientsfortheexpansion,alongwiththeresultsforasele tedsetofterms;

theFloquetoperatorforthesystemisintrodu ed. InSe tion3.2,the

ompu-tation of the Floquetoperator is addressed; we des ribe thepro edure that

wehaveimplementedto omputetheFloquetoperator forlargesystems;we

dis uss the pre ision, the onvenien e and the limitations of the algorithm.

In Se tion 3.3, we use the given pro edure to analyze a system of atoms

in the rst band of the opti al potential; the full Floquet spe trum of the

systemis omputed andthestatisti alanalysisofthequasienergiesgivenin

[14℄isextended. InSe tion3.4, theproblemofboth verti al and

horizon-tal transport of atoms through the latti e is ta kled; two de ay hannels

fromtherstto these ond band,derived inSe tion 3.1,arestudied

pertur-batively; the results are used within the framework of Se tion 3.3 and the

de aystatisti sis ompared withtheresultsofsingle-parti le Landau-Zener

Fra tal Flu tuations in the

Opened Quantum Ki ked

Rotor

The goal of the present Chapter is to dene and study observables that

allowoneto dete texperimentallythe omplexdynami sarisingina

single-parti le quantum system driven by an external for e. The single-parti le

modelisthe quantum ki ked rotor,aparadigmati topi inquantum

haol-ogy,of vasttheoreti al andpra ti al interest. Anon-destru tive opening of

thesystemispossible be auseof itsdistin tive dynami alproperties, whi h

are illustrated after the orresponding lassi al and quantum system are

brieyreviewed.

Weextendpreviousworkpredi tingafra tals alingofa ondu tan e-like

quantityfor the open system. We present a omprehensive investigation of

thefra taldimensionofthisobservableasafun tionofthebest ontrollable

experimentalparameter,in ontrastto[13℄wheretheparametri dependen e

ontheinitial onditionswasstudied. Anexperiment,however, willtypi ally

average over various initial onditions. This naturally favors our approa h,

whi hexpli itlypredi tsthesurvivalof hara teristi fra talsignatureseven

with respe t to averaging over a typi al experimental ensemble of initial

onditions.

2.1 Theoreti al Ba kground

2.1.1 The Classi al Ki ked Rotor

A lassi alrotor isan elongatedobje t,fastened at oneend to afri tionless

pivot, with a moment of inertia

I

and whose position is parameterized by anangle

θ

. Thetrivialmotionofthisobje tbe omesstrikingly ompli ated ifanimpulsive for eof moment

k

and period

τ

is exertedperiodi ally from

momentum after one appli ation of the for e is

∆J = k cos θ

and in the vi inity of

t = lτ

the equation of motion for the angular momentum is

˙

J = k cos θ δ(t − lτ)

. Sin e we dene the angular momentum as

J = I ˙θ

, the Hamilton Equations are given, and the orresponding 

δ

-ki ked rotor Hamiltonianis:

H(t) =

J

2

2I

+ k cos (θ)

+∞

X

l=1

δ(t − lτ) .

(2.1)

Thenumberoffreeparametersinthemodelisoverabundant,soweintrodu e

dimensionlessvariableswheretheperiodisredu ed tounityandthesystem

istunedbythe produ t

kτ

only:

k ← k/I,

E ← τ

2

E/I,

_{t ← t/τ,}

_{J ← τ J/I.}

H(t) =

J

2

2

+ kτ cos (θ)

+∞

X

l=1

δ(t − l) .

(2.2)

Thetime evolutionofa given traje tory,startedat

(θ

0

, J

0

)

,isdes ribed pi torially by plotting in the same two-dimensional phase spa e

(θ, J)

the anoni al variables just after ea h ki k at time

t = lτ

, and the resulting sequen eobeysa re ursive relation alledStandardMap [1 ℄:

θ

l

≡ θ(t = lτ

+

), J

l

≡ J(t = lτ

+

)
.

J

l+1

= J

l

+ kτ sin(θ

l

),

θ

l+1

= θ

l

+ J

l+1

mod 2π.

(2.3) Thesequen eofthe anoni alvariablesofasingletraje tory

(¯

θ, ¯

J )

des ribes the dynami s of the whole region of the phase spa e where it is spread,

be ause thesamesetofpointsinthephasespa eissharedbyall the

traje -toriessu hthat

(θ

0

, J

0

) = (¯

θ

l

, ¯

J

l

)

forsome

l

. So,ifmanytraje tories,spread inseveralregionsofthephasespa e,areplotted,asinthephasespa e

por-traits or Poin arése tions ofFigure2.1,thesingletwo-dimensional phase

spa e is su ient to understand the asymptoti dynami s indu ed by the

dierent initial onditions

(θ

0

, J

0

)

.

Letussket hthephasespa eportraitoftheStandardMapasthe

param-eter

kτ

isin reased. If

k = 0

theHamiltonian(2.2)isintegrable,theangular momentum

J

isa onstant ofthemotion, theangle

θ

in reaseslinearlywith time and there ursive relation (2.3) redu es to:

J

l

= J

0

,

θ

l+1

= θ

l

+ J

0

mod 2π.

The sequen e of the anoni al variables for a given traje tory lies within

linearsets of onstant angular momentum and ollapses into a nite set of

sket h, where thethi k line isthe rotor,the ir ular dashedline is thetraje tory

and the angle

θ

parameterizes the position of the rotoron the ir le. The large arrow represents the impulsive for e of strength

k

, whi h a ts, periodi ally, in a xeddire tion. Panels(a,b, )showthePoin arésurfa eoftheStandardMap(2.3)

for the in reasing values of the parameter

kτ ≃ 0.17, 1.17, 7

, respe tively. The plots show

30

traje tories evolved for

t = 500

ki ks and sin e the phase spa e portrait is invariant for

J → J + 2π

, only the stripe

J ∈ [0, 2π]

is shown. The originsof the traje tories are hosen, a ordingly to the symmetry properties of

the Map, to a hieve asigni ant sampling of the phase spa e. The dynami s is

almost regular in panel (a), while in panel (b) a substantial sto hasti layer is

seenandthe riti alfor e

(kτ )

⋆

≃ 0.97

fortheunbounded diusionofmomentum through antoriisex eeded. Inpanel( )eventhestableislandsofthefundamental

The phase spa e portraits for a small external for ing

kτ ≪ 1

, as in the panel (a) of Figure 2.1, are quasi-regular and an be explained by the

Kolmogorov-Arnold-Mosertheorem[1℄. Themajorityofthelinearsetsofthe

unperturbed systembendsslightly, but an innite hierar hy of smallerand

smallerislands ofellipti shapeappearinthevi inityof theresonan es.

When the external for ing is in reased, as in the panel (b), sto hasti

layersarebornbetween dierent hainsofstableislands. Thephasespa eis

alledmixed,sin equasi-regularstru turesand haoti regionsarepresent.

Ifa traje tory isstarted inasto hasti layer, itsdynami s is unstablewith

lo alexponential divergen e within the layer, whiletheinnite hierar hy of

stable islands a ts as a barrier and indu es an algebrai de ay rate. If the

for ing ex eeds the riti al value

(kτ )

⋆

_{≃ 0.97}

all thelinear sets of the

un-perturbed systemdisappear andturn into fra talremnants alled antori;

dierent sto hasti layers join together and traje tories an diuse freely

allowing for an unbounded growth ofthe angular momentum.

The further in rease of thefor ing

kτ

redu es the spatial extent of the stableislandsandenlargesthesto hasti layersthatendupto overthe

en-tirephasespa e, asshowninpanel( ). The approximatedtimedependen e

of the kineti energy average, overall the possible traje tories of therotor,

iseasily omputed inthe fully haoti pi tureand yieldsa lineargrowth[1 ℄

hJ

l

2

/2i ≃

(kτ )

2

4

l.

(2.4)

2.1.2 The Quantum Ki ked Rotor

Asannoun edintheintrodu tion,the s ope ofmodel(2.2) goesfar beyond

the des ription of a ki ked pendulum. We fo us on the realization of the

model by old atoms in an opti al latti e of period

a

and depth

V

0

, when the potential is not onstant in time, but is periodi ally turned on for a

time extent short with respe t to all the other hara teristi time s ales.

The ashing has a period

τ

and we approximate the shape of the time dependen e ofthepotential with a

δ

fun tion. We thus re overtheexternal impulsivefor eoftheki kedrotormodel. Anensembleof oldneutralatoms

is ne essary to set up any experimental realization, but pro edures an be

devised to redu e the atomi intera tions to negligible values, asexplained

in Se tion 2.4.5. Then we an resort to a single-parti le model to des ribe

the dynami s of the atomi ensemble. The Hamiltonian for an atom in a

ashedlatti e potential isreadily written [7℄:

ˆ

H(t) =

p

ˆ

2

2M

+ V

0

cos



2π

x

ˆ

a



+∞

X

l=1

δ(t − lτ) .

(2.5)

The hara teristi momentum of the latti e is the re oil momentum

k

L

≡

π/a

,and the hara teristi energy is there oilenergy

E

R

≡ k

2

k ≡ V

0

/8E

R

,

E ← E/8E

R

,

t ← t · 8E

R

,

p ← p/2k

L

,

x ← x · 2k

L

ˆ

H(t) =

p

ˆ

2

2

+ k cos(ˆ

x)

+∞

X

l=1

δ(t − lτ).

(2.6)

The range of the variables in the quantum ki ked rotor model (2.6) is

dierent from the

δ

-ki ked rotor(2.2) , where the spatialdomain isa ir le: inthe new model thepotential isperiodi , but the wave fun tion isnot, so

we annot represent it within thelimited spa e domain of a potential well

[−π, π]

. Nonetheless, we an useBlo h's theoremto redu e the problemto a periodi one with one more parameter, the quasimomentum

β

. In fa t, Blo h's theorem asserts that the wave fun tion of a system, invariant for

dis rete translations, is the produ tof a traveling wave

e

iβx

and a periodi

fun tion

u(x)

:

ψ(x) = e

iβx

u(x),

_{u(π) = u(−π),}

_{x ∈ R.}

(2.7) The wave number

β

is the quasimomentum of the Blo h wave

ψ(x)

and, sin eitisa onservedquantity,it anbetreatedasaparameterofthesystem.

The quasimomentum represents the fra tional part of the momentum

p =

n + β, n ∈ N,

and its onservation suggests to use an integer momentum operator

n

ˆ

so that

p = ˆ

ˆ

n + β

. The quasimomentum an be hosen in the Brillouin Zone

[0, 1[

, in dimensionless units, and the wave fun tion

ψ(x)

is periodi only when

β = 0

.

Though notperiodi ,the Blo h solution(2.7) hasa periodi fa tor

u(x)

andthetraveling wave prefa tordoesnot hangeintime. Itis onvenientto

writeanee tiveHamiltonianfortheperiodi partofthewavefun tion,and

to omputeallthephysi alquantitiesofinterestusing

u(x)

. Theelimination of a phase fa tor from the fun tional form of a wave fun tion is done bya

gaugetransformation, whi h yields:

ˆ

H

β

(t) =

(ˆ

p + β)

2

2

+ k cos(ˆ

x)

+∞

X

l=1

δ(t − lτ) ,

(2.8)

i∂

t

u(x, t) = ˆ

H

β

(t)u(x, t),

π ≤ x < π.

The periodi ity of the solution

u(x)

of this ee tive S hrödinger equation impliesthat the spa e variable

x

an be restri ted to one ell of thelatti e potential and that theperiodi boundary onditions of (2.7) are impli itly

imposed. Finally,thegeometri alanalogyofthequantumki ked rotorwith

the

δ

-ki ked rotor is omplete, but the dynami s of thequantum system is ri herbe ause ofthe additional parameter

β

[15℄.

The time periodi ity of the system is exploited in the lassi al ase by

a time

τ

. In the quantum ase we pro eed analogously, by omputing the one- y leoperator or Floquet-Blo hoperator

U

ˆ

FB

, i.e. thetime evolution operator

U(∆t, 0)

ˆ

for atime

∆t = τ

. The evolution between adja ent ki ks issimply a freemotion but some are isneeded to a ount for theee t of

thesingular instantaneous potential turned on at

t = lτ, l ∈ N

. The result for the Floquet operator

U

ˆ

FB

withthegiven hoi e of gauge(2.8) is:

ˆ

U(τ

−

, 0

+

) = e

−i(ˆ

n+β)

2

τ /2

,

_U(τ

ˆ

+

, τ

−

) = e

−ik cos ˆ

x

ˆ

U

FB

≡ ˆ

U(τ

+

, 0

+

) = e

−ik cos ˆ

x

◦ e

−i(ˆ

n+β)

2

τ /2

(2.9)

Itisworthnotingthatthe Floquetoperator (2.9)isgiven inamixed

repre-sentation, where neither the position nor the integer momentum operators

are

c

-numbers.

Theliteraturefo usesonthelong-timebehaviorofthewavefun tionand,

sin ephysi alquantitiesare onstantbetweenadja entki ks,itis onvenient

to onvert the time variable

t

into an integer ounter for the ki ks number byletting:

t ← l

if

t = lτ

+

_.

Thelong-timebehaviorofthequantumki kedrotoris(a)stronglydependent

onthe hoi e of

τ

and

β

and(b) omprisestwophenomena whi h strikingly ontrast with the lassi al analogs: dynami al lo alization and quadrati

growthof theenergy.

Dynami alLo alization Thegeneri  timedependen eofthemean

en-ergy is an initial linear growth followed by stationary, errati u tuations,

as shown in Figure 2.2 (a). Even for arbitrary high values of the ki king

strength

k

the system eventually eases to absorb energy from the pulsed potential and the mean energy u tuates about a onstant value. In the

momentum spa e, the shape of the wave fun tion

|ψ(n)|

2

, shown in

Fig-ure 2.2 ( ), hasa maximum around a momentum

n

˜

and tails that de rease exponentiallyas

|ψ(n)|

2

∝ e

−|n−˜

n|/ ξ

.

(2.10) We saythat thewave fun tion is lo alized and, sin e thespa e where the

lo alization takes pla e is not the real spa e but the momentum spa e, a

morespe i designationforthephenomenon isdynami allo alization [5℄.

Thetypi al length

ξ

oftheexponential tails is alledlo alization length. The lo alization of a wave fun tion in real spa e is a well known and

general ee t in solid state physi s that arises, in parti ular, if a latti e

potentialisae tedbyon-siteu tuations. Thismeansthatadistributionof

randomenergyshifts

ε

i

issuperimposedonthelatti edepth

V

0

,anddierent latti e sites have dierent energies

V

i

= V

0

+ ε

i

. It is possible to re ast the quantum ki ked rotor into an ee tive tight-binded solid state system,

0

25

50

t

0

100

200

300

400

500

E(t)

1 k

2 k

0

50

100

150

200

t

0

100

200

300

400

500

600

700

-200

0

200

n

-15

-15

-10

-10

-5

-5

0

0

Log

10

|

ψ

(n)|

2

a)

b)

c)

Figure2.2: Inthe panels(a)and (b)theaverageenergyisshownasafun tion of

time. Theparametersof theevolutionare

k = 5, β = 0

,

τ = 1.4

forthe lo alized ase(a),

τ ≃ 4π · 17/29

fortheresonant ase(b). Thepanel(a)isdividedintotwo regionsto better appre iateboththeinitial lineargrowth,

E(t) ∝ t

, analogousto (2.4)for

t ≤ t

break

≃ (kτ)

2

/4 ≃ 12

,andtheasymptoti onstantu tuations. The wavefun tion in themomentum spa efor

t = 200

is shown in panel ( ) and the exponentialshape(2.10)is learlyvisible.

where pseudo-randomness arises ifthe ki king period

τ

is in ommensurate to

4π

. The omprehensionof thedynami al lo alization relies on the proof oflo alization inthe realspa e for theee tivesystem.

The dynami al lo alization of thequantum ki ked rotor is the essential

ingredient in the rest of thepresent Chapter, sin e itallows us to open the

system in a perturbative way, and produ es the orre t spe tral statisti s

thatimpliesthe fra talu tuations studied subsequently.

Quadrati Energy Growth Thegeneri hoi eof

τ

and

β

setsthe quan-tumki kedrotorinaregimeofdynami allo alization,buttherearepe uliar

values of the two parameters that ausea quadrati , unbounded growth of

themeanenergy,shortly aquantumresonan e, asshowninFigure2.2(b).

The onditions for quantum resonan esare[16℄:

τ ∈ {4πs/q; s, q ∈ N},

β ∈ {m/ 2s, 0 ≤ m < s; s ∈ N}.

(2.11)

The denominator

q

inthe expression

τ = 4πs/q

is alled theorder of the resonan e. Weseethataquantumresonan etakespla eiftheki kingperiod

τ

is ommensurateto

4π

,i.e. theoriginofthepseudo-randomnessgeneration inthe dynami allo alizations enario breaksdown.

The quantum resonan e ee t an be explained by resorting to a

spe -tral analysis [16℄, but a simple example is su ient to understand how the

hoose

β = 0

, the resonan es of the lowest order

τ = 4πm

and the initial ondition

ψ(n, t = 0) = δ

n,0

, i.e.

ψ(x, t = 0) ≡ 1

. The Hamiltonian (2.8) is invariant under

τ → τ + 4π

, sowe an just set the ki king period equal to the fundamental resonan e

τ = 4π

. The kineti fa tor of the Floquet operator simplies to

U(τ

ˆ

−

_{, 0}

+

_{) = I}

, the evolutor operator is given by the

powers of

U(τ

ˆ

+

_{, τ}

−

₎

andthemean energy an be omputed analyti ally:

ψ(x, t) = e

−ikt cos(x)

E(t) = −hψ(t)|∂

x

2

|ψ(t)i/2 =

k

2

4

t

2

_.

Moregenerallywestatearesultfrom[17 ℄whi hpredi tsthat,apartfrom

errati u tuations, the timedependen e oftheenergy is:

E(t) ≃ ηt

2

+ O(t),

with

η ≃

 k

q



2q

.

(2.12)

2.1.3 Fra tal Flu tuations of Rea tion Curves

In this subse tion, we review briey the ne essary onditions for observing

fra tal u tuations in a general de aying problem. We follow losely [6 ℄

where the spe tral onditions implying fra tal ondu tan e u tuations in

thequantumrealm wererst dened.

S atteringisawidelyusedte hniquetoprobethepropertiesofasystem.

Theresultofas attering analysisoftenmaybeexpressedasa rossse tion,

or adierential rossse tion urve

T (E)

ifthe s anoveranenergy interval is onsidered. More generally,if anobservable thatdes ribeshowa system

rea tsto a perturbation is plotted asa fun tion of aparameter, area tion

urve isobtained.

The s attering te hnique applies to open systems, where an external

probe is laun hed and dete ted very far from the bulk of the system, or

the de ay of a metastable state is analyzed asymptoti ally in spa e. A

metastable state that de ays in a typi al time

1/Γ

ontributes a omplex energy

E − iΓ/2

to thespe trum ofthe system. The rossse tion nearthe energy

E

hasa typi al Breit-Wigner shape of width

Γ

, ifthevarious states do not intera t. A ross se tion urve for an open systemis mainly

deter-minedbythemetastablestatesandits prole an bethus approximatedby

thesuperpositionof severalBreit-Wigner shapes:

T (E) =

X

j

c

j

Γ

2

j

(E − E

j

)

2

+ Γ

2

j

.

The rea tion urve of a system with many metastable states, or de ay

hannels, experien es a orrespondinglarge number of u tuations be ause

erties of the u tuation pattern using a few indi es. If the rea tion urve

T (E)

isvery roughand sensitive to small hanges intheenergy parameter, a fra tal dimension analysis investigates if the statisti al properties of the

urve areself-similaron dierent energy s ales.

We note immediately that the fun tion

T (E)

for a nite system, ap-proximately given by anite sum of smooth fun tions, is smooth itselfand

annot be a fra tal in the mathemati al sense. Nonetheless, it was proved

that an algorithmi and numeri al approa h to the fra tal dimension

om-putation within ertain s ale ranges is meaningful and the approximated

fra taldimension

D

f

isagoodstatisti altooltodes ribethegraph of

T (E)

. The index

D

f

is amenable of theoreti al and numeri al omputation, and to analysis of possible experimental data, so thepresent theory is a ru ial

playground to verifyour understandingof thedynami s of agiven system.

Therequirementson thesystemto yielda fra talrea tion urve arethe

following three onditions on thestatisti al propertiesof thespe trum:

(i) the distribution

ρ(Γ)

of de aywidthshasa power-law form,

ρ(Γ) ∝ Γ

−α

;

(2.13)

(ii) the realparts

E

i

oftheenergy spe trumareun orrelated,

hf(E

i

)f (E

j

)i

ij

= hf(E

i

)i

i

hf(E

j

)i

j

;

(iii) the average de aywidthis mu h larger than themean level spa ing,

hΓ

i

i

i

≫ hE

i+1

− E

i

i

i

.

Letusremarkthattheexponent

α

isthefundamentalquantityofthetheory andthe fra taldimension of thegraph of

T (E)

is

D

f

= 1 + α/2.

Thethree onditionsdeterminethepropertiesofthede ay hannelsofthe

systemandhowdierent hannels ontributetogether tothetimeevolution.

The key point is that the de ay in time is not simply exponential, driven

mainlybyadominant hannelbutmany hannels inuen ethedynami s of

thesystem.

The theory is based on the proof that the three onditions enfor e a

fra tal prole by an appropriate superposition of many independent

Breit-Wigner shapes. The rst ondition guarantees that there are many tiny

peaks that produ e ne s ale u tuations; these ond ondition states that

thede aying hannels ontributeun orrelated os illations, and thereareno

Breit-Wigner shapes strongly overlap despite their non-intera ting nature.

The total de ay pro ess is slow, i.e. non exponential, and an be roughly

approximated as

|ψ(t)|

2

∝ t

α−2

.

(2.14)

The same analysis arries over to rea tion urves at xed energy

T (E)

thatu tuatewhensomeotherparameter

τ

ofthesystemistuned. Provided that the given onditions on the spe trumare fullled, a fra tal dimension

for the prole

T (τ )

is expe ted. The same onditions are stated for the quasienergy spe trum, if the system is subje ted to an external periodi

for eand theFloquetmethodis used.

The theory is entirely quantum, grounded on the spe tralproperties of

the system, and gives the approximated algebrai de ay (2.14) as a

onse-quen e. Ontheotherhand,ananalogous theorywasgiven forsystemswith

mixed phase spa e in the semi lassi al regime. The hierar hy of stable

is-lands of the mixed phase spa e a ts as a trap that prevents the system to

enter the haoti region,where the de ayisgenerally exponential [18℄. The

result is again an algebrai de ay in time, but this feature is now thevery

startingpointof thetheory,and not a onsequen e. So,ifthesystemunder

analysishasa lassi alanalogwhosephasespa eismixed,thefra talnature

ofthe rea tion urve mayhave twoexplanations, andapreliminary he kis

ne essary to de ide whether the leading ee t is a semi lassi al wandering

around the hierar hy of stable islands, or an appropriate statisti s of the

purelyquantumspe trum. Se tions2.4.2and 2.4.4addressthisproblemsin

moredetail forthe quantumki ked rotor studied here.

2.2 How to Open the System

The theoreti al des riptionof a physi al system is omplete only if it

om-prehends the intera tion ofthe systemwithsome external perturbation. In

fa t,the measurement itselfis aperturbation ofthesystem, whi h must be

a ountedforifthe theoryisof someuse. Butameasurement pro ess must

be devised su h that we probe the system itself, not the system and the

probing devi e: the systemmust then be su iently robust to preserve its

mainpropertiesevenwhen oupled tothe external environment. Theresult

ofthe measurementshouldneatlyseparateintothe ontribution ofthe

un-perturbed systemand the  ontribution of the perturbation,as thetheory

of the linear response postulates. In the present ase, it is the dynami al

lo alization of thesystem that allows us to introdu e a onvenient probing

pro edure. Thepro edureistointrodu elossesinthesystemandstudythe

tions

We simulate losses by negle ting the momentum omponents thatex eed a

given momentuminterval

]n

1

, n

2

[

. Thismeansthatafterea hki kthewave fun tionis modiedbyletting

ψ(n) = 0

if

n ≤ n

1

< 0

or

0 < n

2

≤ n.

(2.15) The wave fun tion is not renormalized after the trun ation is applied, and

hen e the evolution is not unitary. Be ause of momentum lo alization, see

Equation(2.10) ,thewavefun tion omponentsofmomentumstatesfaraway

fromthe momentum

n

˜

where thewave pa ketis entered areexponentially small, and the perturbation introdu ed by erasing these values is indeed

small. If we denote by

P

ˆ

the proje tor on the momentum interval

]n

1

, n

2

[

, theevolutionof thewave fun tionis:

ψ(t) =

 ˆ

P ˆ

_U

FB



t

ψ(t = 0).

(2.16)

The omposition of the Floquet operator and the momentum proje tor is

a new ee tive, non-unitary Floquet operator. Its spe trum

E

i

− iΓ

i

/2

is omplex and the imaginary parts give a distribution

Γ

i

of de ay rates. A power-lawstatisti aldistribution(2.13) ofde ayrates isthekeyhypotheses

inthe theoryoffra tal u tuationsof rea tion urves.

Theleakage pro essisthusintrodu edbyasharptrun ationofthewave

fun tion at its borders. This kind of boundary onditions are alled

ab-sorbing boundary onditions. In general, it is a rude way to ouple a

system to an external environment, sin e any feedba k from the

environ-ment is negle ted. A better model should a ount for the probability that

awave pa ketre-enters thesystemandinuen esthesubsequent evolution.

Possible experimental realizations oftheabsorbingboundary onditions are

dis ussedbelow inSe tion 2.5.1.

The hoi e of the values

n

1

and

n

2

must be done a ordingly to the other parameters of the system to guarantee that dynami al lo alization is

at work. If thewave fun tion is not substantiallylo alized, well within the

absorbing borders, a fast de rease in time takes pla e that invalidates the

probing approa h outlinedpreviously and, moreover, thestatisti sof de ay

rates does not follow a power-law distribution. A onservative hoi e of

largevalues for

n

1

and

n

2

hasthedisadvantagethata transient initial time isneededbeforethe systemrea tsappre iablyto theperturbation.

In the following, the asymmetri hoi e

n

1

= −1

and

n

2

= 200

is made and the initial wave fun tion is

ψ(n) = δ

n,0

. Thesystem is driven immedi-ately to a regime of onsiderable leakage through the negative momentum

0

50

100

150

200

n

-10

-8

-6

-4

-2

Log

10

|

ψ

(n)|

2

0

50

100

150

200

n

-10

-8

-6

-4

-2

a)

b)

Figure2.3: Thewavefun tion in momentum spa efor

k = 5, τ = 1.4, β = 0

after

t = 100

ki ksisshowninpanel(a): thevaluesarepresentedas ir les,togetherwith alo alsmoothingapproximation. Inthepanel(b)onlythesmoothapproximations

areshownafter

t = 200, 300, 400, 500, 1000, 2000, 10

4

ki ks. Thelo alizationlength

(2.17)isttedintheinterval

n ∈ [¯n = 75, 190]

forthetimesshowninpanel(b);a onstantslope

ξ ≃ 0.036

isobtainedwithapre isionof

10%

andthe orresponding tisshownin panel(b).

permits the existen e of an exponential tail as shown if Figure 2.3. The

exponential tail

Log

₁₀

_{|ψ(n; t)|}

2

_{≃ −n/ξ(t),}

for

n ≥ ¯n,

(2.17) is a signature of the lo alization of the asymmetri wave fun tion and its

extension

n

2

−¯n

isdependentonthe hoi eof

n

1

. Largervaluesof

n

1

in rease theextension ofthe exponential tail but lengthen thetransient initial time.

Thelo alizationlength

ξ(t)

anbereadimmediatelyinasemilogarithmi plotandoughttobe onstantintimeafteratransient

t < ˜

t

. Intheabsorbing boundary s enarioweuse

ξ(t) → ξ, t ≫ ˜t

asa lo alization riterion.

2.2.2 The Survival Probability as the Rea tion Curve

Thenaturalrea tion urvetomeasurethe ee tsoftheleakagethroughthe

boundaries isthequantumsurvivalprobability fun tion:

P

surv

(t; τ, β, k, n

1

, n

2

) ≡

X

n

1

<n<n

2

|ψ(n; t)|

2

.

(2.18)

This ondu tan e-like quantity,introdu edin[13℄ anddened in lose

anal-ogytothetransportproblema rossasolid-statesample,representsthe

fra -tion of an atomi ensemble remaining within the momentum range

]n

1

, n

2

[

upto time

t

. Thesurvivalprobability isdependent on alltheparameters of thesystemand, inthefollowing, we will analyzethe u tuations drivenby

tuning the ki king period

τ

, and we will establish thefra tal nature of the graphof

P

surv

(τ )

.

2.3.1 The Time Evolution

Nowweillustratethealgorithmto implement numeri allytheone- y le

evo-lution dened in the Equation (2.16) . First, theperiodi fa tor

u(x)

of the wavefun tion isrepresentedasthe array

u

n

ofexpansion oe ientsonthe integer momentum basis:

u(x) =

n

max

X

n=n

min

u

n

e

i2πnx

.

(2.19)

Of ourse,the basisisniteandtheexpansionisrestri tedwithintherange

n ∈ [n

min

, n

max

]

. Theresults areinsensitive tothisne essaryapproximation as far as the range is mu h larger than the interval where the absorbing

boundaries are dened:

n

min

≪ n

1

and

n

max

≫ n

2

. In all the following simulations, we use

n

min

= −1024

and

n

max

= 1023

,fora totalof

Z = 2048

points. Doublingthedimension

Z

oftheexpansionbasisdidnot hange the resultsas he ked for paradigmati ases.

The rightmost fa tor

U(τ

ˆ

−

_{, 0}

+

₎

of the evolution operator is diagonal

inthe momentum representation, so its a tion redu es to the element-wise

multipli ationof twove tors:

 ˆ

_U(τ

−

_{, 0}

+

_{) u}



n

= exp [−iτ(n + β)

2

_/2]u

n

.

Theperiodi fa tor

u(x)

ofthewave fun tionisthenrepresentedinthereal spa e, on a mesh of

Z

points in the ell

[−π/2; π/2[

of the latti e. A Fast Fourier Transform isusedto implement (2.19) and obtainthearray

u

j

= u(jπ/Z),

with

j = −1024, . . . , 1023.

The leftmost fa tor

U(τ

ˆ

+

_{, τ}

−

₎

of the evolution operator is diagonal in the

realspa e representation and again:

 ˆ

_U(τ

+

_{, τ}

−

_{) u}



j

= exp [−ik cos (jπ/Z)]u

j

.

TheinversionoftheFastFourier Transform aststhewavefun tion ba kto

the momentum representation and the absorbing boundaries trun ation is

applied byletting

u

n

≡ 0

for

n ≤ n

1

and

n ≥ n

2

. The array

u

n

is usedfor theimmediate evaluationof theaverage momentum, of theaverage energy,

or theveri ationof the exponential lo alization onditionasinFigure2.3.

In the following simulations, the initial ondition of the time evolution

is set to

u

n

(t = 0) = δ

n,0

, i.e. the wave fun tion of a parti le at rest with denite momentum.

TheFloquetoperatormatrix inthe integer momentum basisisobtained

momentum

n u

n

m

= δ

m,n

. The olumnsof the matrix are theve torsof the evolved states:

 ˆ

_{P ˆ}

_U

_FB



mn

=

X

n

′

 ˆ

_{P ˆ}

_U

_FB



mn

′

δ

n

′

,n

= u

n

m

(t = 1).

(2.20)

The dimension of the momentum basis is

Z = 2

11

, be ause the Fast

Fourier Transform a hieves the optimal asymptoti omplexity

O(Z log Z)

if

Z

is a power of

2

. The ode for the time evolution was written in the C language and the FFTW3 library was used for the Fast Fourier Transform

evaluation. Thealgorithm doesnot require massivememory allo ation,and

the evaluation of

10

4

ki ks takes several se onds on a desktop omputer.

The algorithm is intrinsi ally non-parallel, somultiple pro essors ma hines

donot give anyspeed-up.

2.3.2 The Fra tal Dimension

Thetheoryof fra talrea tion urves presentedabovewasdevelopedon the

groundsofthebox- ountingalgorithm, whi his ommonlyusedtoevaluate

numeri ally theHausdordimensionofageometri set. Inthe asethatthe

geometri set isthe graphof a real fun tion

P

surv

(τ )

dened onan interval

[τ

min

, τ

max

]

, the box- ounting algorithm be omes parti ularly simple. We divide the interval in subintervals

I

i

= [τ

i

, τ

i+1

]

of width

τ

i+1

− τ

i

= δ

and ompute

N (δ) =

X

i

eil

((max − min){P

surv

(τ ); τ ∈ I

i

}/δ) .

Ifthegraphofthefun tionisafra talwithdimension

D

f

,thenthepower-law s aling

N (δ) ∼ δ

−D

f

holdsandtheplot

Log

10

[N (δ)]

v.s.

Log

10

[1/δ]

produ es a line whose slope is just the fra tal dimension. The grid

G

introdu ed in (2.22) where thefun tion isevaluated mustbe equidistant, andthewidth

δ

mustbe amultiple of the grid spa ing

δτ

. Moreover, the power-law s aling holdsonly inan interval:

δτ ≪ δ ≪ τ

max

− τ

min

.

(2.21)

The box- ounting algorithm tends to underestimate proles with

D

f

>

1.5

[19℄and thet ofthepower-lawon thedoublelogarithmi plots isoften ae tedbyerrorsdue to an un ertain sele tion of thes aling region (2.21) .

We omplement theresults ofthebox- ountingusing amorerenedversion

of the same method, alledvariational algorithm introdu ed in [19 ℄. The

range

[τ

min

, τ

max

]

isdividedinto

R

subintervals;thenaninteger

1 ≤ l ≤ R/2

isxed and the total variation of

P

surv

(τ )

is omputed on ea h group of

2l

adja entsubintervals,evenifthegroupsoverlap;theaverageoverthegroups

ofthe totalvariationis alled

V

R

(l)

andapower-law s aling

V

R

(l) ∼ l

−D

f

expe tediftheanalyzedproleisafra tal. The

R

parameteris hosensu h that we obtain the learest s aling law, and the variational minimization

on

R

ofthelineart errorsinthe doublelogarithmi plotgivesthenameto the algorithm. If the number of subintervals

R

is too large, we obtain the box- ountingmethod again; ifit istoo smalltheprole isseen asablurred

loudofpointsand the dimension saturatesto

D

f

→ 2

. In theintermediate regime the benets over theplain box- ounting an be appre iated by eye,

asillustratedbythe appli ations inthefollowing.

Both the box- ounting and the variational algorithm are quite fastand

numeri ally inexpensive;the orresponding odeswerewritten intheC and

FortranlanguagesandweretestedagainstarandomwalkandaW

eierstrass-Mandelbrot urves of dimensions

D

f

= 1.5

and

D

f

= 1.6

respe tively, ob-tainingthe orre tdimensionsup to therstde imal digit.

Inthefollowing,weattributetheerrorofonede imaldigit tothefra tal

dimensions. Seeking a higher pre ision is not only very di ult, be ause

of some un ertainties in the tting pro edure, but also meaningless for the

physi alinterpretationofthefra talityintheproleof

P

surv

(τ )

. Thefra tal dimensionsobtained by thebox- ounting and thevariational algorithm are

denoted

D

bc

and

D

va

,respe tively.

The fra tal dimension an be also estimated by an independent route,

relyingontheanalysisofthelo al propertiesoftheproles. Forthispurpose

we introdu e the orrelations

C

and thevarian es

V

:

C(∆τ ) = hP

surv

(τ ) · P

surv

(τ + ∆τ )i

τ

,

V (∆τ ) = h|P

surv

(τ + ∆τ ) − P

surv

(τ )|

2

i

τ

.

Inthelimit

∆τ → 0

,bothfun tionsareexpe tedtoyieldapower-laws aling withanexponent

a

determined bythefra taldimension

D

f

1 − C(∆τ)/C(0) ∼ c ∆τ

a

,

_{V (∆τ ) ∼ c ∆τ}

a

,

D

f

= 2 − a/2.

These relations follow from the Brownian-motion-like nature of the

u tu-ating proles,and an bedemonstrated inthesemi lassi al ase[20℄, while

their appli ation in a deeply quantum problem is justied by analogy and

numeri al experien e. Thepower-laws alingturnsinto alinearrelationship

when the fun tionsare plotted in a doublelogarithmi s ale,and theslope

of the line is the exponent

a

whi h gives the fra tal dimension. The s al-ing exponents of orrelations and varian es will be denoted

a

corr

and

a

var

, respe tively.

2.4 The Results of the Fra tal Analysis

InthepresentSe tion,wepresentourmainresultsobtainedfromthe

1.75

2.00

2.25

2.50

P

surv

(

τ

)

1.40929

1.40930

1.40931

1.40932

1.40933

1.40934

τ

100

1 k

10 k

E(

τ

)

0

500

1000

1500

2000

t

0

2500

5000

7500

10000

12500

E(t)

0 10 20 30 40

0

50

100

150

200

250

a)

c)

x

10

-3

b)

t

₀

t

₁

Figure2.4: Theimpa tofthelowerorderresonan e

τ

⋆

/4π = 12/107

isstudiedafter

t = 2000

ki ks,for

k = 5

,

β = 0

. Inthepanel(a)thesurvivalprobabilityisplotted on the grid given in (2.24). The panel (b) shows the average energy, without

absorbing boundary onditions, on a dierent grid, entered on the resonan e.

Panel ( ) showsthe average energy,without absorbingboundary onditions,asa

fun tion oftheki knumber

t

bothforthe resonan e

τ

⋆

(dashedline)and forthe

point in the grid(2.24) losest to

τ

⋆

(solidline). Thelo alized urvefollowsthe

quadrati energygrowthofthe resonan ein theshort timeinterval

[t0

≃ 20, t

1

≃

450]

(seealsoinset).

as a fun tion of the ki king period

τ

. First we hoose an appropriate grid in

τ

for the evaluation of the fra tal dimension. Then we investigate the dependen e oftheresults on the parameters of thesystem.

2.4.1 The Choi e of the Sampling Grid

The omputation of the fra taldimension of thegraph of

P

surv

(τ )

requires the hoi e of asampling grid

G

,equidistant along the

τ

axis:

G(τ

0

, δτ ) ≡ {τ

i

= τ

0

+ i · δτ, i ∈ {0, . . . , m − 1}}.

(2.22)

Theuseof

m = 10

4

pointsinthegrid onjugatesgoodqualityinthe

ompu-tationof the fra taldimension withareasonable expenseof omputer time.

The hoi eof

τ

0

and

δτ

isnot immediatelyobvious,be auseofthepresen e of the resonant values of

τ

. If the onditions (2.11) for a resonan eare ful-lledat

τ

⋆

,theasymptoti lineargrowth ofthemeanmomentumdrivesthe

wavefun tionthroughtheabsorbingboundariesandthesurvivalprobability

P

surv

(τ

⋆

)

experien esafastdrop. Sin ethefun tion

P

surv

(τ ; t)

is ontinuous on

τ

foraxed,nitevalueof

t

,thedropduetoaresonan eat

τ

⋆

ae tsthe

survival probabilitygraph with avalley whose shape isun orrelated to the

andthat ouldprodu e deviationsfrom the expe ted fra taldependen e.

The setof theresonan eshas zeroLebesgue measure,but it isdenseon

the

τ

axis, so resonan es annot be entirely avoided and their ee t must beestimatedsomehow. Therstobservationisthathigherorderresonan es

need along time to showup, be ause theprefa tor

η

ofthequadrati term in(2.12) issuppressedmorethanexponentiallyfor

q ≫ 1

. The hoi eofthe interval

[τ

min

, τ

max

] ≃ [1.40, 1.41]

(2.23) guaranteesthatthefasterresonan ehastheorder

q = 107

,whi hisexpe ted notto inuen eappre iablytheevolutionofthewavefun tion forthetimes

t . 10

4

of on ern here. Moreover, the distribution of the resonan es with

q ≤ 2000

isalmostuniformandthereisno rowdingofresonan esthat ould modifythe dynami s inan extendedrangeof

τ

values.

Theresonantvaluesof

τ

are ommensurateto

4π

,sowedeviseatri kto hoose the points of the grid in ommensurate to

4π

up to their signi ant digits. Indoing so we re all that thegolden mean value

√

5 − 1

/2

is the most irrational number sin e its ontinued fra tions expansion is the one

withslowest onvergen e. Thedenitionof

τ

0

isgiven inaformthatmakes expli ittheadopted onstru tion:

τ

0

4π

=

7

5

6142

95403



√

5 − 1



≃ 1.40,

δτ = 9.98 × 10

−7

.

(2.24)

Throughout thesimulations thatwe ondu ted,the ondition(2.17) for

thelo alizationofthewavefun tion wasoftensubje tedtoveri ation,and

nosignaturesofaresonantbehavior(2.12)onthe hosengridwaswitnessed,

asexempliedinFigure2.4forthe lowest orderresonan e. Inthepanel(b)

ofthe Figure,theaverageenergy is omputedon agrid enteredon the

res-onan e, andthehighpeakintheaverage energy learly showstheresonant

value

τ

⋆

, but no valley is appre iated in the survival probability prole of

panel(a) omputedonthegrid(2.24) . Inthepanel( ),we onfrontthetime

dependen e of the average energy for the resonant

τ

⋆

and for the point of

thegrid that lies losest to the resonan e. The two urves oin ide for the

short times

t < t

1

, where the growth is rst diusive and then quadrati . For

t & t

1

the resonan e undergoes an indenite quadrati growth, while the non-resonant value of

τ

produ es a lo alized wave fun tion. The non-resonant urve at

τ

followsthe resonant prole at

τ

⋆

ina timerange

[t

0

, t

1

]

dependent onthe detuning

τ − τ

⋆

,andminimizedwiththegiven hoi e of

the sampling grid. Within this short time range, the interpretation of the

u tuationsofthesurvivalprobability urveisquestionable,sin e the

inter-play of the intermediate quadrati growth and the presen e of the

absorb-ingboundary onditions is still a problemopen for atheoreti al treatment.

0

1

2

3

4

5

s

0

0.2

0.4

0.6

0.8

1

CDF

(s)

-12 -10

-8

-6

-4

-2

0

Log

₁₀

Γ

4

6

8

10

12

14

16

Log

10

ρ

(

Γ

)

0

1

s

2

3

4

0.0

0.2

0.4

0.6

0.8

P

(s)

a)

b)

Figure2.5: Inthepanel(a), the umulativedistribution fun tionofthe

renormal-izedenergyspa ings isshownas ir les,togetherwiththe umulativedistribution

fun tion of the Poisson distributionfor un orrelated energies (solid line). In the

insettheprobabilitydensitiesofthetwodistributionsare shown. Theprobability

distributionfun tionofthede aywidthsisshowninpanel(b),togetherwithaline

of slope

−1

that is anappropriate t overeightorders of magnitude. There are about

2.5 × 10

5

levelsin thespe tralstatisti s, orrespondingto about

10

3

values

of

τ

in theinterval

[1.40, 1.41]

, shown in the panels(d-f)of Figure 2.6, orto

250

levelsforea h

τ

. Theaverage levelspa ingis

2π/250 ≃ 0.025

, orrespondingto

1

intherenormalizedunitsofpanel(a). Inthepanel(f)ofFigure2.6weseethatthe

dataisquitesparseonthequasienergys ale

5 × 10

−

3

andinfa tasmalldeviation

from thePoissondistributionis appre iatedfor

s . 0.25

in theinsetof thepanel (a).

within the interval (2.23) , for most intera tion times of interest in the

fol-lowing, an be safely onsidered to be due to the spe tral onsequen es of

thelo alizedregime, asexplained inSe tion 2.1.3.

We also undertook a more quantitative approa h to the hoi e of the

grid, where theoptimal grid

G

is to be sele ted by themaximization ofan appropriate fun tion

F (G)

on the grids spa e. The fun tion

F

denes and measures the total distan e of the grid from the set

S

of the resonan es ontained in the hosen interval

[τ

min

, τ

max

]

and of maximum order

q

max

. Higher values of the fun tion

F

orrespond to grids whose points areaway fromthe resonan es. Formally, we dene

S

and

F

as:

S = {4πs/q} ∩ {q ≤ q

max

} ∩ [τ

min

, τ

max

]

F (G(τ

0

, δτ )) =

X

τ

r

∈S

f

r

(min{|τ

r

− τ

g

|; τ

g

∈ G})

f

_r

(1)

(∆τ ) = ∆τ ;

f

_r

(2)

(∆τ ) = ∆τ

2

;

f

_r

(3)

(∆τ ) = ∆τ /q

r

.

The e ien y of the method depends on the hoi e of the weight fun tion

f

r

that an be used, as in the example

f

(3)

r

, to give more weight to the resonan eswithsmallerorder

q

r

,thatinuen e awiderneighborhood along

Figure 2.6: (a,b, ) show the survival probability as a fun tion of

τ

for

k = 5

,

β = 0

after

t = 10

4

ki ks at dierent magni ations. For the sameparameters,

(d,e,f)showtherealparts ofthequasienergiesasafun tionof

τ

(obtainedasthe eigenphasesoftheevolutionoperator(2.20)),whi hwasrepresentedinthebasisof

momentumstatesasanitematrixintherange

n ∈]0, 250[

andthendiagonalized). Weseethattheu tuationsonnerandners alesarea ompaniedbyubiquitous

avoided- rossingsintheeigenvaluespe trum(notethatforbettervisibilityin(d-f)

onlyasmallpartofthefullspe tralrange

[−π, π]

isshown).

the

τ

axis. Our omputations of the fun tion

F

, for randomly sele ted ensemblesofgrids

G(τ

0

, δτ )

,returnaatprole,whereanoptimalgriddoes notemergenorasharpminimumrevealsthespoilingpresen eofaresonan e.

2.4.2 Veri ation of the Hypotheses on the Spe trum

Thefra talityoftheu tuationsofthesurvivalprobabilityreliesonthethree

hypotheses(i-iii)on thespe trumstated inSe tion 2.1.3. We omputed the

non-unitary Floquet operator matrix as in (2.20) and we diagonalized it

for about

10

3

representative values of

τ

in the grid (2.22) , obtaining the eigenphases

E

τ,j

− iΓ

τ,j

/2

,withthequasienergies

E

τ,j

andthede aywidths

Γ

τ,j

. The quasienergies are dened in

[0, 2π[

and are sorted

E

τ,j

≤ E

τ,j+1

su hthatthequasienergydieren esare

∆E

τ,j

= E

τ,j+1

−E

τ,j

. Theaverage quasienergy spa ingis

h∆E

τ,j

i

τ,j

≃ 2π/(n

2

− n

1

− 1)

.

TheFigure2.5, panel(b),showsthatthe key ondition(i)onthe

distri-bution

ρ(Γ)

of the de ay widths is veried over eight orders of magnitude, with

α ≃ 1

. A ording to (2.1.3) ,

D

f

≃ 1.5

for the fra taldimension of the graphof

P

surv

(τ )

an beestimated. The averagede ay is dire tlyobtained