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60 Ansichten203 SeitenSandro Marcel Wimberger- Chaos and Localisation: Quantum Transport in Periodically Driven Atomic Systems

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Binationale Dissertation der Fakultt fr Physik a u der Ludwig-Maximilians-Universitt Mnchen a u

DISSERTAZIONE IN COTUTELA

aus Passau

DOTTORATO DI RICERCA IN FISICA UNIVERSITA DEGLI STUDI DELL INSUBRIA DIPARTIMENTO DI SCIENCE CHIMICHE FISICHE E MATEMATICHE

Binationale Dissertation der Fakultt fr Physik a u der Ludwig-Maximilians-Universitt Mnchen a u

DISSERTAZIONE IN COTUTELA

aus Passau

1. Gutachter/Tutore: PD Dr. Andreas Buchleitner 2. Gutachter/Tutore: Prof. Italo Guarneri Tag der mndlichen Prfung: 14. Januar 2004 u u

O how much more doth beauty beauteous seem By that sweet ornament which truth doth give! The rose looks fair, but fairer we it deem For that sweet odour which doth in it live; The canker blooms have full as deep a dye As the perfumed tincture of the roses, Hang on such thorns, and play as wantonly, When summers breath their masked buds discloses; But for their virtue only is their show They live unwooed, and unrespected fade, Die to themselves. Sweet roses do not so; Of their sweet deaths the sweetest odours made; And so of you, beauteous and lovely youth; When that shall vade, by verse distils your truth. W. Shakespeare

Zusammenfassung

In dieser Arbeit untersuchen wir quantalen Transport im Energieraum anhand zweier Paradebeispiele der Quantenchaostheorie: hoch angeregte Wasserstoatome im Mikrowellenfeld, und gekickte Atome, die das Modellsystem des gekickten Rotors simulieren. Beide Systeme unterliegen aufgrund des aueren, zeitlich periodischen Antriebs einer komplexen Zeitentwicklung. Insbesondere werden zwei Quantenphenomne untersucht, die kein klassisches Analogon a besitzen: die Unterdrckung klassischer Diusion, bekannt unter dem Schlagu wort dynamischer Lokalisierung, und die Quantenresonanzen als dynamisches Regime, das sich durch beschleunigten Transport im -gekickten Rotor auszeichnet. Der erste Teil der Arbeit belegt auf neue Weise die quantitative Analogie zwischen dem Energietransport in stark getriebenen, hoch angeregten Atomen und dem Teilchentransport im Anderson-lokalisierten Festkrper. Eine o umfassende numerische Analyse der atomaren Ionisationsraten zeigt in Ubereinstimmung mit der Lokalisierungstheorie nach Anderson, dass die Ratenverteilungen einem universellen Potenzgesetz unterliegen. Dies wird sowohl fr ein eindimensionales Modell als auch fr das reale dreidimensionale Atom u u demonstriert. Auerdem werden die Konsequenzen aus der universellen Ver teilung der Ionisationsraten fr die asymptotische Zeitabhngigkeit der Uberu a lebenswahrscheinlichkeit der Atome diskutiert. Der zweite Teil der Arbeit klrt den Einuss von Dekohrenz induziert durch a a Spontanemission auf die krzlich im Experiment mit -gekickten Atomen beu obachteten Quantenresonanzen. Wir leiten Skalierungsgesetze ab, die auf einer quasiklassischen Nherung der Quantendynamik beruhen und die Form von a Resonanzpeaks beschreiben, welche in der mittleren Energie eines atomaren Ensembles im Experiment beobachtet wurden. Unsere analytischen Resultate stimmen mit numerischen Rechnungen ausgezeichnet uberein und erklren a die zunchst uberraschenden experimentellen Befunde. Darberhinaus weisen a u sie den Weg zur Untersuchung des wechselseitig konkurrierenden Einusses von Dekohrenz und Chaos auf die Stabilitt der quantenmechanischen Zeita a entwicklung gekickter Atome. Die Stabilitt lsst sich mittels des Uberlapps a a zweier anfnglich gleicher, aber unterschiedlich propagierter Zustnde charaka a terisieren. Dieser Uberlapp, bekannt als Fidelity, wird hier fr eine experiu mentell realisierbare Situation untersucht.

Riassunto

In questa tesi viene studiato il trasporto quantistico nello spazio dellenergia di due sistemi modello della teoria del caos quantistico: atomi di idrogeno altamente eccitati sottoposti ad un campo di micro-onde ed atomi calciati che simulano il modello -kicked rotor. Entrambi questi sistemi presentano una evoluzione dinamica complessa, derivante dallinterazione con una forza periodica esterna. In particolare vengono studiati due fenomeni quantistici che non hanno una controparte classica: la soppressione della diusione classica, conosciuta come localizzazione dinamica, e le risonanze quantistiche come regime dinamico del trasporto amplicato. La prima parte della tesi fornisce un nuovo supporto allanalogia quantitiva fra il trasporto di energia in idrogeno altamente eccitato sottoposto a un campo elettromagnetico intenso, ed il trasporto di particelle in solidi con localizzazione di Anderson. Unanalisi numerica completa dei rate di ionizzazione atomica mostra che questi obbediscono ad una distribuzione universale conforme ad una legge algebrica, in accordo con la teoria della localizzazione di Anderson. Questo risultato viene dimostrato sia per il modello unidimensionale che per latomo reale in tre dimensioni. Vengono inoltre discusse le ripercussioni della distribuzione universale dei rate di ionizzazione sul decadimento della probabilit` di sopravvivenza asintotica degli atomi. a La seconda parte della tesi chiarisce leetto della decoerenza causata dallemissione spontanea nelle risonanze quantistiche che sono state osservate in un esperimento recente con atomi calciati. Vengono derivate due leggi di scala basate sullapprossimazione quasi classica dell evoluzione quantistica. Queste leggi descrivono la forma dei picchi di risonanza nellenergia di un insieme sperimentale di atomi calciati. I risultati analitici ottenuti sono in perfetto accordo con simulazioni numeriche e motivano osservazioni sperimentali initialmente inspiegati. Aprono inoltre possibilit` di studio sulleetto coma petitivo della decoerenza e del caos sulla stabilit` dell evoluzione quantistica a degli atomi calciati. La stabilit` si pu` caratterizzare tramite la sovrappoa o sizione di due stati initialmente uquali, per` soggetti ad evoluzioni temporali o dierenti. Questa sovrapposizione, detta delity, viene studiata per una situazione sperimentale realizzabile.

Abstract

This thesis investigates quantum transport in the energy space of two paradigm systems of quantum chaos theory. These are highly excited hydrogen atoms subject to a microwave eld, and kicked atoms which mimic the kicked rotor model. Both of these systems show a complex dynamical evolution arising from the interaction with an external time-periodic driving force. In particular two quantum phenomena, which have no counterpart on the classical level, are studied: the suppression of classical diusion, known as dynamical localisation, and quantum resonances as a regime of enhanced transport for the kicked rotor. The rst part of the thesis provides new support for the quantitative analogy between energy transport in strongly driven highly excited atoms and particle transport in Anderson-localised solids. A comprehensive numerical analysis of the atomic ionisation rates shows that they obey a universal power-law distribution, in agreement with Anderson localisation theory. This is demonstrated for a one-dimensional model as well as for the real three-dimensional atom. We also discuss the implications of the universal decay-rate distributions for the asymptotic time-decay of the survival probability of the atoms. The second part of the thesis claries the eect of decoherence, induced by spontaneous emission, on the quantum resonances which have been observed in a recent experiment with kicked atoms. Scaling laws are derived, based on a quasi-classical approximation of the quantum evolution. These laws describe the shape of the resonance peaks in the mean energy of an experimental ensemble of kicked atoms. Our analytical results match perfectly numerical computations and explain the initially surprising experimental observations. Furthermore, they open the door to the study of the competing eects of decoherence and chaos on the stability of the time evolution of kicked atoms. This stability may be characterised by the overlap of two identical initial states which are subject to dierent time evolutions. This overlap, called delity, is investigated in an experimentally accessible situation.

Resumen

En este trabajo se investiga el fenmeno de transporte cuntico en el espacio de o a energ de dos sistemas paradigmticos de la teor del caos cuntico: por un a a a a lado, estados altamente excitados de tomos de hidrgeno en un campo de mia o croondas, y por otro lado atomos golpeados que simulan el modelo kicked rotor. Los dos sistemas presentan una dinmica compleja proveniente de la a interaccin con una fuerza externa peridica en el tiempo. En particular son o o estudiados dos fenmenos cunticos sin contraparte clsica: la supresin de o a a o difusin clsica, conocido como localizacin dinmica, y resonancias cunticas o a o a a como un rgimen dinmico de transporte amplicado. e a La primera parte de esta tesis proporciona una nueva demostracin de la anao log entre el transporte de energ en atomos altamente excitados sometidos a a a campos electromagnticos intensos y el transporte de part e culas en slidos o localizados de Anderson. Un anlisis numrico detallado de las ratas de ioa e nizacin atmica muestra que stas obedecen una distribucin universal cono o e o forme a una ley algebraica, lo cual est de acuerdo con la teor de localizacin a a o de Anderson. Esto es demostrado tanto para un modelo unidimensional como para atomos reales tridimensionales. Se discuten tambin las implicaciones de e las distribuciones universales de las ratas de ionizacin para el decaimiento o asinttico en el tiempo de las probabilidades de sobrevivencia de los atomos. o La segunda parte de la tesis clarica el efecto de decoherencia por emisin o espontnea en las resonacias cunticas observadas en un experimento reciente a a con atomos golpeados. Leyes de escalamiento son deducidas, basadas en una aproximacin cuasiclsica de la evolucin cuntica, las cuales describen la o a o a forma de los picos resonantes en la energ media de un ensamble experimena tal de atomos golpeados. Nuestros resultados anal ticos encajan perfectamente con sendos clculos numricos y explican observaciones experimentales que inia e cialmente fueron sorprendentes. Mas an, abren las puertas para el estudio de u los efectos competentes de decoherencia y caos en la estabilidad de la evolucin o temporal de atomos golpeados. Esta estabilidad puede ser caracterizada por el sobrelapamiento de dos estados iniciales idnticos, pero a la vez con distine tas evoluciones en el tiempo. Este sobrelapamiento es llamado delidad y es investigado para una situacin accesible experimentalmente. o

Contents

1 Introduction 1.1 1.2 Quantum chaos and experiments . . . . . . . . . . . . . . . . . . Quantum transport in periodically driven atomic systems . . . . 1.2.1 1.2.2 1.3 Anderson localisation and decay-rate statistics . . . . . . Quantum resonances with -kicked atoms . . . . . . . . .

1 1 2 4 8

13

Periodicity in time, position, and momentum . . . . . . . . . . . 13 2.1.1 2.1.2 Floquet theory . . . . . . . . . . . . . . . . . . . . . . . . 13 Bloch theory in position space . . . . . . . . . . . . . . . 14

2.2

The -kicked rotor . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2.1 2.2.2 2.2.3 The model . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Quantum resonances . . . . . . . . . . . . . . . . . . . . . 17 Particle vs. rotor: Bloch theory for kicked atoms . . . . . 19

2.3

Experimental realisation of the kicked rotor model . . . . . . . . 20 2.3.1 2.3.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . 20 Derivation of the eective Hamiltonian . . . . . . . . . . . 22 xi

xii

2.3.3 2.4

Contents

Experimental imperfections . . . . . . . . . . . . . . . . . 25

Quantum chaos and microwave-driven Rydberg states . . . . . . 27 2.4.1 2.4.2 Atomic hydrogen in a microwave eld . . . . . . . . . . . 27 Quantum-classical correspondence . . . . . . . . . . . . . 30

Part I

32

35

Universal statistics of decay rates . . . . . . . . . . . . . . . . . . 37 3.1.1 3.1.2 3.1.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . 37 Discussion of decay-rate distributions . . . . . . . . . . . 42 Algebraic decay of survival probability . . . . . . . . . . . 48

3.2

Experimental tests . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.2.1 3.2.2 3.2.3 Status quo . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Floquet spectroscopy . . . . . . . . . . . . . . . . . . . . . 55 Atomic conductance uctuations . . . . . . . . . . . . . . 57

Part II

Quantum resonances and the eect of decoherence in the dynamics of kicked atoms

63

65

Quantum resonances in experiments . . . . . . . . . . . . . . . . 65 Noise-free quantum resonant behaviour . . . . . . . . . . . . . . . 68 4.2.1 4.2.2 Momentum distributions . . . . . . . . . . . . . . . . . . . 70 Average kinetic energy . . . . . . . . . . . . . . . . . . . . 75

4.3

Contents

4.3.1 4.3.2 4.3.3 4.3.4 4.3.5 4.3.6

xiii

Stochastic gauge . . . . . . . . . . . . . . . . . . . . . . . 76 Theoretical model for randomised dynamics . . . . . . . . 78 Average kinetic energy . . . . . . . . . . . . . . . . . . . . 82 Asymptotic momentum distribution . . . . . . . . . . . . 86 Theoretical model vs. numerical results . . . . . . . . . . 89 Reconciliation with experimental observations . . . . . . . 93

101

quasi-classical approximation . . . . . . . . . . . . . . . . . . . 102 Classical scaling theory for quantum resonances . . . . . . . . . . 106 5.2.1 5.2.2 quasi-classical analysis of the resonance peaks . . . . . 106 Validity of the quasi-classical approximation . . . . . . 113

5.3

121

Stability of quantum dynamics and experimental proposal . . . . 121 Fidelity at quantum resonance . . . . . . . . . . . . . . . . . . . 124 6.2.1 6.2.2 Dynamical stability in absence of noise . . . . . . . . . . . 124 Fidelity in presence of decoherence . . . . . . . . . . . . . 129

6.3 6.4

Fidelity near to quantum resonance . . . . . . . . . . . . . . . . . 133 Fidelity with quantum accelerator modes . . . . . . . . . . . . . 134

xiv

7 Rsum e e 7.1 7.2

Contents

141

147 147

A.1 Proof of estimate (4.16) . . . . . . . . . . . . . . . . . . . . . . . 147 A.2 Proof of inequality (A.2) . . . . . . . . . . . . . . . . . . . . . . . 148 A.3 Proof of the asymptotic formula (4.17) . . . . . . . . . . . . . . . 148 B Statistics of the process Zm B.1 Independence of the variables zj 150 . . . . . . . . . . . . . . . . . . 150

B.2 Central Limit property . . . . . . . . . . . . . . . . . . . . . . . . 151 C Asymptotic distribution of the process |Wt| D Derivation of equation (5.26) E Extraction of delity from Ramsey fringes F Some formulas used in Part II G Publications 152 154 155 157 159

Bibliography

161

Chapter 1

Introduction

1.1 Quantum chaos and experiments

Continuous development of knowledge unavoidably leads to specialisation and dierentiation. This process in science tends to separate the specialties, and often pushes them so far apart that researchers working in one branch have a hard time in keeping their interest in the main questions of even related elds. A methodological integration is desirable to overcome language problems between dierent communities, and inter-disciplinarity between various branches in science (not only physics) in turn may foster new development. P. W. Anderson [1] noticed about 30 years ago that the study of complex systems where at each level of complexity new and interesting phenomena emerge oers a variety of connections between dierent branches of physics, chemistry, biology, and increasingly also economics and computer science [2, 3]. The eld of quantum chaos which sprouted a few years later [46] is an excellent example of a fruitful merger of ideas originating from many branches of physics. For the investigation of complex dynamical systems, classical and quantum physics were brought together. In particular, concepts from nuclear, atomic, molecular physics, nonlinear systems theory and statistics serve in the inter-disciplinary study of quantum systems which show signatures of classical chaos. Although a vast amount of eort has been undertaken in theoretical and numerical investigations on the quantum mechanical analogues of classically chaotic systems [710], clean experimental studies of quantal manifestations of classical chaos had been rather restricted up to the mid 1980ies [1118]. The only early experimental contribution, which in turn motivated the development of quantum chaos, came from nuclear physics, supplying a huge database of nuclear energy spectra [19]. Their statistical characterisation, without knowledge of exact solutions of the quantum many body scattering problem in heavy nuclei, is possible through the meanwhile well-established Random Matrix Theory [9, 17, 20, 21], which nowadays is used in many other branches of 1

Chapter 1. Introduction

physics [22, 23]. Several time-independent systems with (Hamiltonian) classically chaotic analogues were studied afterwards. The spectral properties of highly-excited Rydberg atoms subject to a strong magnetic eld [24, 25] have been extensively investigated providing insight into the inuence of classical dynamics on the corresponding quantum problem. This conservative system of perturbed Rydberg atoms with a large density of states, whose classical dynamics is highly chaotic, is a real complex system for which the experimentally measured spectra were found to match perfectly with quantum mechanical ab initio calculations [12, 2429]. More recently, also Rydberg states in crossed electric and magnetic (static) elds have been studied [14,3034]. Rydberg atoms in crossed elds may eventually provide an atomic realisation (with dominantly chaotic dynamics) of cross sections exhibiting Ericson uctuations [30]. The latter are well-known in nuclear (chaotic) scattering [3537]. A fashionable and direct way to illustrate wave functions as well as to study plenty of energy levels (up to very high energies) is oered by billiard systems [17, 18], where either particles (e.g. electrons) or light rays scatter o hard walls. Light-ray billiards, which are relevant, for instance, to the development of small micro-laser cavities [38, 39], are toy models for the study of wave chaos [17,4042]. There one resorts to the analogy between the two-dimensional Helmholtz equation for the electric eld modes and the stationary Schrdinger o equation [17]. All the above mentioned systems are governed by a time-independent Hamiltonian. In this thesis deceptively simple quantum systems are investigated which show a variety of complex transport phenomena induced by an external time-dependent driving force. The driving pumps energy into the unperturbed system and may turn even one-dimensional systems chaotic on the classical level, while for time-independent, autonomous problems at least two degrees of freedom are necessary for the occurrence of chaos [4345]. If one is able to control well the external forces and to isolate the composite systems from additional noise sources, such time-dependent, low-dimensional systems are good candidates for the experimental and theoretical study of quantum chaos.

1.2

in

periodically

driven

The interest in simple Hamiltonian systems with periodic time dependence was boosted by the ionisation experiments performed by Bayeld and Koch in 1974 [4]. An ecient multi-photon (the ionisation potential of the atomic initial state exceeded 70 times the photon energy) ionisation of atomic hydrogen Rydberg states subject to microwave elds was observed, and the highly non-perturbative nature of the process could be successfully explained by a classical diusion process in energy space [4649]. A great experimental breakthrough for the study of quantum transport in momentum or energy space was

then the demonstration of dynamical localisation in microwave-driven Rydberg atoms [13, 5053], and later also in cold atoms subject to pulsed standing waves [54, 55]. This phenomenon had been predicted theoretically [6, 5660], and its explanation ourished by summoning concepts of nonlinear dynamics and of solid state physics [61, 62] in an inter-disciplinary manner. Modern day experiments are able to control essentially isolated atoms with high precision. In this thesis we focus on two periodically driven atomic systems which are accessible to state-of-the-art experiments under large control of parameters, and thus of the underlying classical dynamical regimes. While the rst part of the thesis is devoted to the above mentioned ionisation of highly excited hydrogen Rydberg atoms, the second part concentrates on a model system which has found a reliable experimental realisation in the last decade, namely the kicked rotor a standard model in quantum chaology [6, 9, 17, 63, 64]. Both of these Hamiltonian systems are conceptually rather simple at rst glance. The kicked rotor is a free pendulum which is subject to time periodic kicks. Hydrogen is the simplest existing atom. However, the external driving force induces a complicated, yet deterministic dynamical evolution on the classical as well as on the quantum level. The experimental realisations of these two systems can to a great deal be viewed as isolated from possible noise sources which makes them experimentally clean, and permits a direct comparison between theoretical and experimental results. Thus, there is no need for further assumptions or simplications, which may be necessary, for instance, when models of mesoscopic chaotic systems are studied [18, 6570]. In particular, Rydberg states are experimentally controllable up to very high quantum numbers [11, 16, 7174]. This provides a large density of states and, hence, may allow for an approximate description by semiclassical methods (see, e.g. [7577] and references therein). Transport is by denition the change of location, i.e. the time evolution within a given system in an appropriate parameter space. In physical problems one may have in mind the phase space ow of classical densities [78] or temperature equilibration in conguration space. In our systems of interest, transport occurs classically also in phase space [78], but the essential coordinate is energy. While mesoscopic transport typically occurs as a ow of charge carriers in real (conguration) space, driven systems can exchange energy with the external eld and the natural view of transport must focus on momentum or energy space. The reader should keep in mind the ionisation of atoms where the initial bound state is coupled to the atomic continuum through the energy absorption from the external microwave eld. This is a real-life example of an open quantum system. The -kicked rotor has no continuum, yet its energy is unbounded from above (apart from unavoidable cutos in experiments or numerical computations). Therefore, we investigate transport in energy space, induced by the periodic driving force, on a microscopic scale of single atoms both in external (centre-of-mass motion of kicked cold atoms) and internal (electronic excitation in Rydberg atoms) degrees of freedom. Although this thesis focuses on quantum eects which do not have classical analogues, the knowledge of the classical evolution (obeying mixed regular-chaotic dynamics) and the use of semiclassical methods provide a deeper insight into

Chapter 1. Introduction

the physical mechanisms, which are otherwise dicult to extract from purely quantum data (e.g. the quantum spectrum). In the rst part of the thesis, welldeveloped concepts of nonlinear systems theory help us to understand eects lying beyond the analogy between the Anderson [9, 17, 79, 80] and the driven hydrogen problem. (Semi-)classical tools are extensively used in the second part of the thesis when the quantum resonances for -kicked atoms are studied. Our detailed mathematical and numerical analysis of the quantum resonances occurring with -kicked atoms is motivated by recent, equally puzzling and inspiring experimental results [8183].

1.2.1

The initially surprising observation of ecient multi-photon ionisation in periodically driven hydrogen atoms [4] had been explained by the classical analysis of the dynamics of the periodically driven Kepler problem [46]. Such analysis was supported later by quantum calculations [8487]. However, for driving frequencies larger than the ones used in the early experiments [4, 5], the quantum evolution starts to deviate substantially from the classical prediction [50, 51]. The classical diusive motion is then suppressed by quantum interference eects. They set in at frequencies at which the driving eld is able to resonantly couple by a one-photon transition unperturbed eigenstates of the atom in the vicinity of the initial state. This eect was qualitatively predicted by an appropriate one-dimensional description of the microwave-driven hydrogen problem using an approach very analogous to the -kicked rotor. To emphasise its dynamical origin as well as its anity to the problem of Anderson localisation [79, 88, 89] the corresponding formalism was baptised dynamical localisation theory [59, 60]. Anderson localisation occurs, for instance, in disordered solids and implies an exponential localisation of the charge carriers wave functions in conguration space [79, 80, 88, 89]. As illustrated in gure 1.1, the quantity of interest is the transmission of a (quasi-)particle across a random potential, at a given injection energy. At the potential humps, the particle can be either reected or transmitted with randomly distributed amplitudes, and a quantitative analysis formalising the transmission problem by a transfer matrix approach [9, 80] shows the existence of exponentially localised eigenfunctions along the solidstate lattice. The characteristic length scale, over which the eigenfunctions spread, is given by the localisation length . The measured conductance across the sample depends critically on the ratio of /L, L being the length of the sample. This ratio determines the population of the last lattice site at the edge of the sample, and hence the probability ux to the lead. The sketched scenario can be exported to the problem of energy absorption in periodically driven systems. While a formal mapping to the Anderson model is readily possible for the -kicked rotor, its application to the excitation and ionisation dynamics of atomic Rydberg states under microwave driving is not straightforward. However, both periodically driven problems can be formu-

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energy space

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configuration space

energy

L

Fig. 1.1: The Anderson scenario imported into the atomic realm: an initial population of the bound state |0 is transported to the atomic continuum in energy space, much in the same way as a particle is transmitted across a disordered potential in one-dimensional conguration space (dashed horizontal line). While the particle can be either reected or transmitted at each potential hump, with random probability amplitudes, in the atomic problem, absorptions/emissions of photons from/into the microwave eld of frequency lead to transmission into the atomic continuum (indicated by a chain of arrows). The atomic sample length L corresponds to the ionisation potential of |0 , measured in multiples of the photon energy . The one-photon transitions are slightly detuned from the unperturbed hydrogen levels. The resulting uctuations in the coupling matrix elements mimic the intrinsic disorder present in the Anderson model.

lated within the Floquet description (see chapter 2.4). In this way, the more complicated Rydberg system may be mapped onto the -kicked rotor locally in energy space [59, 60]. Doing so, the eect of dynamical localisation was predicted. The random features of disordered transport manifest on the atomic scale in the complex dynamical phase evolution of a large number of states, which constitute the time-dependent electronic wave packet. In the Rydberg regime, a high density of states is guaranteed since the energy splittings lie in the microwave frequency domain. Consequently, many states will be eciently coupled by the external driving through subsequent, near-resonant one photon absorption and emission processes. The anharmonic hydrogen spectrum necessarily leads to detunings of the one-photon transitions sketched in gure 1.1. Therefore, the coupling matrix elements uctuate in a (pseudo-)random manner [9, 87, 90] what mimics the intrinsic disorder of

Chapter 1. Introduction

F0(10%)

0.08 0.04 0.02

m quantu

thresh

old

dynamical loca

classical chaos bord er

lization

Fig. 1.2: Sketch of the scaled classical and quantum ionisation threshold F0 (10%) F (10%)n4 vs. scaled frequency 0 n3 for microwave driven hydrogen Rydberg 0 0 atoms, from theoretical predictions [50, 59, 60, 94] and experiments [11, 50, 51]. Above 0 1 the quantum threshold (full line) is considerably higher than the corresponding classical one (dashed): this is the regime of dynamical localisation. For 0 < 1, classical and quantum predictions approximately agree. The measured quantum thresholds are the central experimental evidence for Anderson localisation aecting the ionisation dynamics of driven Rydberg states (for alkali atoms the situation is very similar, with proper account for the locally reduced level spacing induced by non-vanishing quantum defects [9599]).

the Anderson model [87, 91]. The atomic ionisation scenario is then a perfect analogue of particle transport in the Anderson problem. It consists of the transport of electronic population, initially prepared in a well-dened bound state |0 , towards the atomic continuum. Anderson localisation manifests itself via the exponential decay of wave functions in the conguration space of disordered solids, and in the energy space of periodically driven systems and the experimentally accessible signature of localisation is the leakage (decay or ionisation) out of the sample. The characterisation of the latter is the main topic of the rst part of this thesis. The measurable quantity in the experiments with Rydberg atoms is the ionisation probability, for given eld amplitude F , frequency , and interaction time t. The microscopic transport problem can, therefore, be studied by measuring the macroscopic probability of ionisation, whereby signatures of complex nonlinear dynamics show up in the local structures of the ionisation signal [11, 92, 93]. While dynamical localisation has been directly measurable only in kicked-atom experiments (where it manifests itself through exponentially decreasing momentum distributions), the central experimental result for driven Rydberg states is the increase of the ionisation threshold with the scaled frequency 0 = n3 [13, 50, 51]. 0 corresponds to the microwave 0 frequency expressed in units of the Kepler frequency of the unperturbed electron 1/n3 . More precisely, the quantity extracted in the experiments is the 0 eld amplitude F0 (10%) F (10%)n4 , rescaled to the strength of the Coulomb 0

potential, at which 10% of the atoms ionise when launched from the initial state with principal quantum number n0 . The increase of F0 (10%) with n0 , at xed F , , and interaction time t, contradicts classical asymptotic (i.e. t ) estimates, which predict ionisation thresholds systematically lower than those measured in experiments [60, 72, 87, 100102]. The eld threshold F0 (10%) behaviour as a function of 0 is sketched in gure 1.2. The quantum suppression of classically chaotic diusion in the regime 0 > 1 is explained by dynamical localisation theory. Yet, the fundamental dependence of F0 (10%) on 0 provides only a rather indirect proof of Anderson/dynamical localisation in the atomic ionisation process. One may imagine other mechanisms which stabilise the atom against ionisation, such as semiclassical stabilisation eects [87, 103, 104]. These may be caused, for instance, by barriers in classical phase space, which hinder the quantum transport. Remnants of broken tori or even chains of nonlinear resonance islands are possible candidates for such processes [72, 77, 105107]. Moreover, also purely classical calculations predict a raise of F0(10%) with increasing 0 , for nite interaction times t [50, 72]. An accurate theoretical treatment of microwave driven one-electron Rydberg states has become available in the last decade, along with the necessary computer power for the numerical diagonalisation of the exact problem [95, 108]. Clear support for the hypothesis that Anderson localisation is indeed responsible for the F0 (10%) threshold behaviour comes from the fact that the amplitudes F0 (10%) show a universal scaling, for many atomic species investigated. This has been shown by heavy numerical calculations [9598] which are very helpful to correctly interpret experiment data [13, 99]. In this thesis, however, we pursue a dierent course, which goes beyond the threshold scaling, to investigate whether there exist additional, and more direct signatures of Anderson localisation.

Universal statistics of decay rates The scenario sketched in gure 1.1 indicates that the atomic ionisation process depends on the initially prepared electronic population, i.e. on the initial state |0 . One possible route to obtain a characterisation of the problem independently of |0 is to focus on the quantum spectrum of the atom in the microwave eld. The latter is independent of the initial state, and only determined by the eld parameters F and . Since we aim at the transport to the atomic continuum, the natural quantity for the comparison with Anderson models are the decay rates or the complex poles of the spectrum. They determine the decay of the eigenstates in the eld, but may locally depend strongly on the eld parameters [109113]. Precisely for that reason, we perform a comprehensive numerical analysis of the statistical properties of the decay rates, and confront our results with predictions from Anderson-localised solid-state models. Even beyond this direct comparison of the spectral properties of the microwave-

Chapter 1. Introduction

driven Rydberg atom and Anderson models, the decay-rate statistics is correlated with the long-time behaviour of the atomic survival probability. The latter also depends on the initial state of the atom, and one question to clarify is how the decay-rate distributions and the properties of |0 conspire to determine the time-dependence of the survival probability. The survival probability in a microwave eld has been extensively studied for real three-dimensional hydrogen as well as for alkali Rydberg states in numerical [95, 114] and laboratory experiments [73, 114]. The survival probability was found to decay asymptotically algebraically in time [95, 114]. Yet, dierent decay exponents have been extracted which depend on the initial atomic state, and on the eld parameters [95,114]. These ndings contradict recent predictions of a universal power-law decay for the survival probability [115]. The key point of our analysis is that the survival probability is determined by both, the set of decay rates of the individual eigenstates, and the projection of the specic atomic initial state on the eigenbasis of the full problem. Thereby the rates encode the global spectral information, while the projection contains the local distribution of the initial state in energy space. The clear separation of these two issues allows us to clarify the above mentioned contradiction, and to show that the universality only restricts to the decay-rates statistics. Moreover, the spectral properties of driven hydrogen Rydberg states may in turn be related to the rich underlying classical phase-space structure [112,113,116]. Our statistical analysis of the ionisation rates, therefore, provides not only a direct comparison to Anderson models but also an interpretation of the statistics by means of the phase-space localisation of the corresponding eigenstates. Doing so, dierent mechanisms which determine the statistical distribution of the decay rates can be discriminated. The thorough understanding of the decay-rate statistics and their mutual impact on the transport mechanisms in the regime of dynamical localisation 0 > 1 in gure 1.2, is the goal of the rst part of this thesis.

1.2.2

While the rst part focuses on the regime of dynamical/Anderson localisation, for which the energy absorption from the external driving is suppressed, the second part of this thesis is devoted to a dynamical regime for which quantum transport is enhanced with respect to the classical analogue. Such enhanced energy absorption, known as quantum resonance, occurs for the -kicked model at specic driving frequencies [64, 117, 118]. It leads to an unbounded energy growth of the rotor, which is quadratic in time, and it arises from a perfectly frequency-matched driving [64, 117, 118]. The -kicked rotor is a quantum pendulum with a potential which is pulsed on and o periodically in time [6, 9, 17, 63, 64, 80]. The potential depends on the excursion angle of the rotor. The model is sketched in gure 1.3, together with the train of periodic -like kicks. The experimental realisation of the -kicked rotor model builds on the tools of quantum and atom optics. Atoms provide a

|g(t)|

time t

F=g(t) sin()

Fig. 1.3: Periodically kicked quantum pendulum, where the kicking force F (t) is periodic in time (with period ) and in the angle variable .

multiple of allowed electronic transitions such that their internal structure can be used to impart momentum on them, and to trap, cool, guide, and also diract or reect them by means of optical light elds [119122]. Techniques which make it possible to manipulate atomic dynamics in a well-dened way, and which provide coherent matter wave sources are of high relevance for active research such as in atomic interferometry [123], for Bose-Einstein condensation [124126], atom lasers [127130], and in atomic lithography [131]. Integrated versions of these techniques may be used even to guide atoms along microstructures (atomic chips) [132134]. Although the experimental methods of atom optics are quite standard in modern laboratories, similar high precision manipulations of atoms appeared impossible when the quantum version of the -kicked rotor model was studied for the rst time [6]. The quantum resonances are very sensitive to variations in the driving frequency, which determines the resonance conditions [64, 117, 118]. Moreover, the quadratic energy growth of a resonantly driven rotor makes it necessary that a large energy window is accessible in experiments. Therefore, it is not surprising that experimental imperfections may lead to deviations from the predictions obtained for the idealised -kicked rotor model. Some systematic dierences between the model and the experimental realisation can be easily understood. First, the experiments always work with a large number of atoms whose centre-of-mass evolves quantum mechanically. Hence, the experiments necessarily average over many realisations rather than working with one single rotor. Second, the atoms move essentially along a straight line, which is dened by the kicking potential. This leads to an additional freedom which needs to be introduced when mapping the problem of kicked atoms onto the rotor model which in turn moves on a circle (cf. gure 1.3). These two eects, together with experimental imperfections are important when one wants to compare experimental results with theoretical predictions. In this thesis, a formalism for the exact treatment of an incoherent ensemble of kicked-particles which move on a line is developed, which takes care of the two above mentioned systematic dierences. In particular, we include in our theory the eect of decoherence on the quantum resonant motion of -kicked particles. The decoherence is introduced in a controlled way in the experiments by allow-

10

Chapter 1. Introduction

ing the atoms to emit spontaneously during the time evolution, and hence to randomly change their centre-of-mass momentum. Several experimental observations at quantum resonance conditions [8183, 135] which do not match with the standard theory of the -kicked rotor will be claried by taking care of the above stated dierences between ideal rotors and their experimental counterparts. Both parts of the thesis are devoted to the investigation of quantum transport for systems with a complex time evolution. Quantum chaos theory [710, 17] is concerned with quantitative measures for the complexity of quantum dynamics. For instance, the classical denition of chaos in dynamical systems the extreme sensitivity to the choice of initial conditions in phase space, characterised by an exponential divergence of phase space trajectories that were initially in close proximity is not operable for a quantum system. In a bound quantum problem, unitarity guarantees that the overlap of two wave packets remains constant for all times. Since the notion of trajectories is meaningless in the quantum world [136], the sensitivity cannot be characterised by an exponential divergence [4345], having in mind the nite resolution given by the uncertainty principle, in other words by the Planck constant. However, one may substitute sensitivity to initial conditions by sensitivity to changes in the Hamiltonian, an idea which goes back to the early days of quantum chaos [137, 138]. Instead of comparing two trajectories which start in close proximity in phase space, one may compare the time evolution of identical initial states which are propagated by slightly dierent Hamiltonians. A measure for the sensitivity of quantum dynamics is then the overlap of two such states [138140]. This overlap is dubbed quantum delity (see e.g. [141] and references therein), and it can in principle be accessed experimentally for the -kicked particle evolution [142, 143]. How the delity behaves when quantum resonance conditions are met for an experimental ensemble of kicked atoms will be addressed after the tools to handle the kicked atoms dynamics have been developed. The wish of a deeper understanding of the two time-dependent quantum transport problems introduced in this section is the central motivation for the following two parts of this thesis. While the rst part closely links the solid-state concept of Anderson localisation to the decay properties of microwave-driven hydrogen Rydberg states, the second part reconciles experimental observations with the here developed theory for -kicked atoms at quantum resonance.

1.3

Chapter 2 preludes with the necessary prerequisites for the theoretical description of periodically driven atoms. The experimental realisation of the -kicked rotor and its relevant features for the observation of the quantum resonances are outlined.

11

Part I In Chapter 3 the statistical distributions of the decay rates of periodically driven Rydberg atoms are presented and analysed. Their implications on the time-decay of the survival probability of the atoms are emphasised. The chapter concludes with a discussion of possible experimental tests of the analogy between transport in driven Rydberg atoms and disordered solid states.

Part II In chapter 4 analytical formulas for the experimental observables at quantumresonance conditions are confronted with experimental and numerical data. While section 4.2 restricts to the absence of noise, section 4.3 presents an analytic theory for the stochastic dynamics, where external noise is modelled according to the eects of spontaneous emission. In chapter 5 we derive a scaling law in absence and in presence of decoherence (sections 5.2 and 5.3), respectively, that describes the shape of the resonance peaks observed in the mean energy of an atomic ensemble. The derivation is based on a quasi-classical approximation (of the quantum evolution) which was introduced in [144, 145]. Chapter 6 applies the machinery derived in the previous two chapters to the study of quantum delity, i.e. the overlap of two initially identical but distinctly evolved quantum states. Results at quantum-resonance conditions are presented, together with preliminary investigations of the time decay of delity at small detunings from quantum resonance.

Chapter 7 concludes the thesis with a brief summary of the obtained results and some comments on the direction of future investigations concerning quantum transport in periodically driven systems.

The appendices contain mathematical facts and formulas used in the second part; apart from the last one which collects the authors publications on the topics discussed in this thesis.

Chapter 2

In this chapter we lay the foundations for the presentation and discussions of the central results of this thesis. The atomic systems which we will study in more detail in the following chapters are introduced. We dene the basic notations and give the necessary theoretical background. In particular, the dierences between kicked atoms in experiments and the abstract -kicked rotor model, including relevant experimental imperfections, are discussed.

2.1

Verily I say unto thee. That this night, before the cock crow, thou shalt deny me thrice. Mt 26,34

2.1.1

Floquet theory

In the subsequent chapters, quantum transport in energy or momentum space is analysed for two open, non-autonomous systems. Both of these systems are subject to a time-periodic external driving force. This particular time dependence allows one to reduce the problems to stationary eigenvalue problems in an extended Hilbert space. More precisely, the Floquet theorem [146149] guar antees the following: for a time periodic Hamiltonian H(t + T ) = H(t), with 13

14

period T , we can write any solution of the Schrdinger equation in the form o |(t) = with c e i t | (t) , (2.1)

| (t) = | (t + T ) ,

where c C are time-independent expansion coecients. The generalised eigenstates | (t) and the corresponding eigenvalues (the quasi-energies) solve the stationary eigenvalue problem H | (t) = | (t) (2.2) for H H(t) i t . H acts in the extended Hilbert space of square-integrable, time-periodic functions L2 L2 (T), where T R/T Zis the unit circle [150]. H The spectrum of the Floquet Hamiltonian H is periodic in energy with period = 2/T . Therefore, one may restrict to a single Floquet zone of width ~ in energy when calculating the quasi-energies . This is of great use for the diagonalisation of the Floquet problem of microwave-driven Rydberg states (see section 2.4.1 below). For the latter atomic system, the time-periodic eigenstates | (t) are expanded in a Fourier series

| (t) =

m=

e i mt |

(2.3)

with time-independent components | m . This expansion allows us to recast the corresponding eigenvalue problem into a time-independent one, with the prize to pay that the matrix dimension is given by the product of the usual eigenbasis expansion and the dimension of the Fourier space. In theory, one has to deal with an innite dimensional matrix. In practice, it can be suitably reduced to nite size, which must be suciently large to guarantee numerical convergence [95, 108, 113]. Since the Floquet states | (t) form an orthogonal basis at any time t, the time evolution operator can be expanded in the periodic basis functions [151] U(t2 , t1) = e ~

i (t t ) 2 1

| (t2 )

(t1)|

(t1 , t2 R) .

(2.4)

In particular, the periodicity carries over to the evolution operator: U(t2 + m, for any integer m [152]. The T, t1 + T ) = U(t2 , t1), or U(t + mT, t) = U(t, t) is essential for the analytical and numerical treatment of latter relation for U the -kicked rotor problem. It means that the time evolution is reduced to a sequential application of the Floquet operator U U(t + T, t) to the initial state of the rotor.

2.1.2

In the experimental realisation of the -kicked rotor (see section 2.3), atoms move in a one-dimensional periodic potential in position space. Alike the Floquet theorem in the time domain, the Bloch theorem [153, 154] allows one to

15

reduce the problem to functions which are periodic in position. Explicitly, for H(x, p) = H(x + a, p), we write any solution of the Schrdinger equation as a o superposition of Bloch states exp( ix) (x): (x) =

with

is an arbitrary index which can be taken in [0, 2/a), because of (x + a) = exp( i a)(x) and the periodicity of the phase exp( i a). The periodic Hamil tonian H(x, p) = H(x+a, p) only couples eigenstates of the momentum operator with eigenvalues separated by integer multiplies of 2/a. For xed , such states form a momentum ladder with p = + 2m/a (m Z and constant spacings ) 2/a. Expressing p in units of 2/a, the fractional part of momentum therefore equals the quasi-momentum = p mod(2/a). For a free particle with unit mass, the solution of the Schrdinger equation is a plane wave with wave vector o 2 kw , and the eigenenergies follow the dispersion relation E(kw) = ~2kw /2. The Bloch theory guarantees that generic x-periodic potentials lead to similar extended solutions with a continuous dependence of the energies on momentum. The precise form of the dispersion relation is determined by the potential. The Bloch-state solutions (2.5) allow one to restrict the problem to a single Brillouin zone of widths 2/a in momentum, and the corresponding reduced zone scheme leads to the denition of continuous energy bands [153155]. In section 2.2.3, we will expand the solutions of the kicked problem in Bloch waves of the form (2.5). In the peculiar case of the quantum resonances, yet another periodicity occurs, now in momentum space besides the periodicity in time (Floquet theorem) and in position space (Bloch theorem). Then the momentum eigenstates of the Floquet operator are extended in momentum space. Details will be explained below, after the -kicked rotor has been introduced.

2.2

2.2.1

The model

In the sequel, we refer to the -kicked rotor as the quantum analogue of the famous Standard Map [43, 156] (also known as Chirikov-Taylor Map). The Standard Map is a two-dimensional (in phase space) Hamiltonian toy model which became important because of its simplicity and the possibility to use it as a local approximation of more complicated systems [43, 59, 156]. It provides a wide range of regular and chaotic types of behaviour which made it a natural

Exceptional cases are, e.g. an innite chain of potential wells of nite widths and innite height; then no tunnelling coupling between neighbouring sites is allowed. Or the so-called anti-resonance of the -kicked rotor with innitely degenerate eigenvalues, i.e. with an energy band of zero width [64].

16

subject of investigation when studying the quantum-classical correspondence [6, 63, 106]. The Standard Map reads in dimensionless action-angle coordinates [43] pj+1 = pj + k sin(j+1 ) j+1 = j + pj mod(2) , (2.6) (2.7)

with the kicking strength k , the kicking period , and pj , j the angular momentum and rotation angle just before the (j+1)th kick . The motion described by the map may be viewed as a free evolution in-between the integer times t = j and t = j +1 (equation (2.7)) followed by a momentum shift (kick) occurring at t = j + 1 (equation (2.6)). The Standard Map can be quantised [6] (with some freedom in the order of free evolution and kick [64]) and the state evolution from one kick to immediately after the next kick is determined by the unitary Floquet operator [6, 63, 64]

i U = e ~ k cos() i ~P 2 /2

(2.8)

2 U describes a free evolution given by e i ~P /2, followed by the kick operi ator e ~ k cos() . The time-dependent Hamiltonian of the quantum -kicked rotor may be written as

(2.9)

where , P are the angle and the angular momentum operator, respectively. After introducing the rescaled variables k k /~ and ~ , t ~t , the Hamiltonian divided by ~2 reads

+ P2 + k cos() H(t) = (t m ) . 2 m=

(2.10)

Both parameters k and are necessary in the quantum version of the rotor. The semiclassical limit corresponds to k , 0, for xed classical stochasticity parameter K = k = const. [63, 64, 80]. The latter constraint expresses that the classical dynamics is unchanged, while ~ 0. The function in (2.9-2.10) makes P 2 /2 negligible at the kick, and thus ensures that the free evolution and the kicking part factorise in the Floquet operator (2.8). The iterated applica tion of U yields the dynamics of the rotor in the discrete time given by the kick counter m Z We denote | the state vector of the rotor, and () = | , . (p) = p| the wave functions in the angle and in the momentum representation, respectively. The -kicked rotor describes a free quantum pendulum which is kicked periodically with an angle dependent strength k cos , c.f. gure 1.3. Since the rotor moves on a circle, the periodic boundary condition ( + 2) = () enforces that only integer angular momenta p = n Zare allowed.

The map may be rescaled such that it only depends on the stochasticity parameter K = k , however, this scaling does not carry over to the quantised map.

17

For the -kicked rotor model, the evolution operator is naturally decomposed in a free motion plus an instantaneous application of a kick, and the system evolves freely over the kicking period with phases that are identical to the unperturbed eigenenergies n2 /2 (n Z If the timing is such that after one period (for ). the fundamental resonances = 4 , a positive integer) the time evolution shows an exact revival without phase mismatch, then exp(i n2 /2) = 1 for all . n Z Hence, at these quantum resonances, the application of m kicks with a strength k is equivalent to the application of one single kick with strength mk. If we deal with kicked particles which move along a straight line, not like a rotor on a circle, then the momentum p may take any value in R. We can split , p = n + in an integer part [p] = n Z and a fractional part {p} = [0, 1). The free evolution part of the Floquet operator then reads e i 2 (n+) = e i 2 (n

2 2 +2n+ 2 )

(2.11)

The phase exp( i 2/2) is independent of n, and therefore always cancels when computing quantum expectations. We neglect it in the following. The kick operator can be expanded in the momentum basis [157]: e

i k cos()

=

n=

(i )n Jn (k)e in ,

(2.12)

where Jn is the Bessel function of rst kind and order n [157]. Thus, the kick does not depend on the fractional part , which, in fact, is a constant of motion. For = 2 ( N), 2 e i 2 (n +2n) = 1 , (2.13) if = 0 (usual rotor with periodic boundary conditions) and = 2, or in general, if = 1/2+j/ mod(1), with j = 0, 1, .., 1. The additional, adjustable parameter allows one to obtain the above mentioned conditions for the fundamental quantum resonances for all kicking periods = 2 . These are the values at which quantum resonances have been observed in experiments [8183, 135], and which we will study in detail in the second part of this thesis. Using a ) plane wave (0, ) = exp( in0 )/ 2 (n0 Z as initial state, it is easily derived that the rotor wave packet spreads ballistically if (2.13) is fullled, i.e. the mean value of its energy grows quadratically in time. The rotors energy is computed [6]: E(t) (t, )| 1 2 1 |(t, ) = 2 2 4

2 0

d (t, )

2 (t, ) 2 (2.14)

k2 t2 n2 + 0 , = 4 2

with (t, ) = exp (ikt cos()) (0, ) from (2.8) and (2.13). Other choices of the initial state may lead to additional terms growing linearly in time t, which is to be understood here as an integer that counts the number of kicks. Moreover,

18

if the free part of the Floquet operator equals the identity, we immediately see that the quasi-energy spectrum (see section 2.1.1) is continuous. It is given by () = k cos(). This follows from

U = e i k cos() e i ()

(2.15)

where all values () are allowed quasi-energies. For k 1, the eigenvalues of U cover the whole unit circle in the complex plane. The continuous spectrum originates from a translational invariance of U in momentum space. To make this clearer, we recall that the -kicked rotor problem can be mapped onto a one-dimensional tight-binding model of the form [6163, 80]: (W0 + Tm ) um +

r=0

Wmr ur = 0 .

(2.16)

) um (m Z are the Fourier coecients of 1/2[ +(, t) + (, t)] = 1/2[1 + exp ( i k cos())] (, t), where are the time-periodic Floquet states just before and after the kick. We denoted the on-site potential Tm = tan( /2 m2 /4), and the non-diagonal (hopping) terms Wr = 2 1/(2) 0 d exp(i r) tan (k cos()/2). A detailed derivation of (2.16) may be found in the literature on quantum chaos [9, 10, 17]. For irrational /4 , (2.16) corresponds to a one-dimensional disordered tight-binding model with pseudorandom numbers Tm [90]. Therefore, the -kicked rotor is mapped onto a standard Anderson problem with disorder, whereby the sites m are identied with integer angular momenta. It was conrmed numerically, that the quasienergy eigenstates are localised around some lattice site nj , and they decay exponentially away from that site with a characteristic localisation length , i.e. u(j) e n

|nnj |

(2.17)

A general wave packet also decays exponentially after some initial expansion period tbreak which is estimated to be tbreak k2 [64, 106, 109]. For the opposite case, /4 = s/q (s, q mutually prime integers), the on-site potentials Tm form a periodic sequence in m. Then the corresponding eigenstates of (2.16) are Bloch states of the form:

(0 ) (n)

= e i 0 n 0 (n), (2.18)

with

0 (n + q) = 0 (n) .

The quasi-positions 0 can be chosen within the interval [0, 2/q). Therefore, the -kicked rotor at quantum resonance is thrice periodic, in time, position space, and also in momentum space. With the phase in (2.11), an additional constraint on for the occurrence of quantum resonances is generally given by = m/2s, with 0 m < 2s an integer [64]. The higher-order quantum resonances, i.e. q > 1, support continuous energy

19

bands whose widths are believed to decay exponentially with increasing q [64]. Therefore, these resonances are much harder to resolve than the fundamental ones. This may be a good reason why only the resonances at = 2 , with = 1, 2, 3, have been detected in experimental realisations using atoms moving on a line [8183, 159], that is, with the additional freedom [0, 1) [145, 160].

2.2.3

The periodic boundary conditions for the wave function enforce a discrete ladder of integer (angular) momenta for the rotor, while for a kicked atom also fractional parts of momenta are allowed. The value of these fractional parts are crucial what concerns the quantum resonances, and rened resonance conditions depending on them have been stated in the previous section. The link between the kicked atom in the experiment, which will be described in section 2.3.2, and the idealised kicked rotor is generated by the spatial periodicity of the potential. The latter periodicity of the driving only allows for transitions (induced by (2.12)) between momenta that dier by integer multiples. Formally speaking, the Floquet operator (2.8) commutes with spatial translations by multiples of 2, and Bloch theory (section 2.1.2) enforces conservation of quasi-momentum , which corresponds to the fractional part of momentum p. For a sharply dened quasi-momentum, the wave function of the particle is a Bloch wave, of the form exp( i x) (x), with (x) a 2periodic function. The general particle wave packet is obtained by superposing Bloch waves parametrised by the continuous variable :

1

(x) =

0

d e i x (x),

(2.19)

where we can restrict to the Bloch zone x mod(2). The simple, yet important observation is now that the Fourier transform of (x), for xed , corresponds to the Fourier transform of (x): 1 (n + ) = 2 With (2.20) we have in turn 1 () = 2 1 (n + ) e i n = 2 (n) e in ,

n

(2.20)

(2.21)

20

which is the Fourier series of the 2-periodic function (). For modelling an experimental initial atomic ensemble, we will later use the special case when the initial state of the particle is a plane wave with momentum p0 = n0 + 0 , where 0 = {p0 } and n0 = [p0] are the fractional and integer part of p0 , respectively, i.e. (p) = ( 0)n,n0 . The wave function in this case reads 1 (2.22) () = ( 0)e i n0 . 2 For any given , () may be thought of as the wave function of a rotor with angular coordinate , henceforth baptised rotor. We denote the corresponding state of the rotor by | . From (2.8) and (2.21), it follows that | evolves into U | , with the Floquet operator

2 U = e i k cos() e i 2 (N +) ,

(2.23)

where N is the angular momentum operator, in the representation: N = id/d, acting on the Hilbert space of wave functions with periodic boundary conditions in . The Floquet operator (2.23) diers from the Floquet operator of the standard -kicked rotor [6, 63, 64, 80, 106] by the additional phase . This phase may be regarded as an external Aharonov-Bohm ux threading the rotor [161], which, for instance, induces strong uctuations when studying the parametric dependence of survival probabilities [162164]. The dierence between rotors and a particles dynamics owing to the presence of a continuum of quasi-momenta is indeed crucial for the observation of quantum resonances in experiments as well as in numerical simulations. Nearly all quasi-momenta involved in a particles wave packet (2.19) do not show resonant motion, as long as the initial momentum distribution is not prepared in very specic, narrow ranges of quasi-momenta [145, 160]. As stated in section 2.2.2, the occurrence of quantum resonances at = 2 ( N) requires also that = 1/2 + j/ mod(1), with j = 0, 1, .., 1. Only if both conditions on and are fullled, the Floquet operator commutes with translations in momentum space by multiples of q = 1, 2, respectively. Note that even if = 2 ( N), the number of resonantly driven -rotors is a set of measure zero in the continuum [0, 1). Therefore, only a tiny fraction of atoms from the initial Gaussian momentum distributions in the experiments reported in [8183, 159] does obey the second condition on .

2.3

Experimental setup

2.3.1

The quantum system studied in the laboratory to implement the kicked rotor dynamics is a dilute ensemble of laser-cooled alkali atoms (sodium, caesium). Following release from an atomic (magneto-optical) trap [121, 122, 165]

21

kT kL

|e |g

Fig. 2.1: Sketch of the experimental setup used to impart momentum by a standing wave (kL ) on laser cooled atoms (dots); the zoomed view shows the internal two level structure of the atoms with ground state |g and excited state |e . The cooling beams (kT ) may also be applied to induce spontaneous emission (see section 2.3.3).

these atoms (typically 106 [15, 83, 166]) are exposed to a spatially periodic potential applied by a standing wave of laser light. To create the standing wave, the output of a tunable laser is retro-reected by a mirror and the two resulting beams are aligned so as to counter-propagate. Figure 2.1 sketches the experimental setup. The beam intensity is controlled by a switch (in practice an acousto-optical modulator [167]) that allows the beam to be pulsed. This provides a pulsed, spatially periodic potential. Crucial for experiments with long interaction times with the standing wave is that the atoms are cold coming from a trapped and cooled dilute ensemble with a small initial spread in momentum. Following application of the standing wave, the atoms are allowed to expand freely. Then their spatial distribution is measured, and knowing the time of free expansion the atoms momentum distribution (immediately after the application of the standing wave) can be calculated [135]. The dynamics of a system subject to a time-varying potential, as provided by the pulsed standing wave, approximates that of the -kicked rotor. An initial Maxwell-Boltzmann, i.e. Gaussian, distribution of atomic momenta becomes exponential in form for /4 irrational, and k > 1. The exponential distribution sets in after an interaction time t larger than the break time tbreak (the time at which the quantum nature of the system begins to manifest). This led to the rst direct observation of dynamical localisation [54, 168, 169]. From the momentum distributions measured in the experiments one can immediately calculate the average kinetic energy of the atoms as function of the number of applied pulses. In the regime of dynamical localisation, the energy does not show a linear growth with the number of pulses as predicted by a classical diffusion argument [63,156], it rather saturates after the break time. In particular, in these cold atom experiments momentum distributions and mean energies can be resolved experimentally for very short times up to the break time tbreak, and

22

thus in the transition regime when dynamical localisation starts to manifest. Hence, the experimentally observed behaviour demonstrates the quantum suppression of classical diusion more directly than in experiments with driven Rydberg atoms. There, the only accessible signal is the ionisation yield of the atoms, and short-time observations are limited by the signal-to-noise ratio, and the nite switching time of the microwave eld [11, 53]. To allow for a better comparison with experimental literature, we use laboratory units in the subsequent section. There we will show that, under suitable conditions, it is possible to ignore the internal electronic structure of the atoms, and treat them as point particles [170]. The reduced atomic dynamics is induced by an eective centre-of-mass Hamiltonian, which in laboratory units reads P2 H(t) = + V0 cos(2kLX)f (t, Tp) . (2.24) 2M M denotes the atomic mass, kL the wave number of the kicking laser, V0 the potential amplitude (proportional to the laser intensity), and f (t, Tp) its modulation. In a restricted momentum region, indeed a periodic -like time-dependence + can be mimicked f (t, Tp) l= (t lTp), with the kicking period Tp . Hence, (2.24) governs the dynamics of a -kicked particle. To arrive at the form of the Hamiltonian (2.10) we have to rescale momentum in units of 2~kL, position in units of (2kL)1 , mass in units of M . Energy is then given in units of (2~kL)2/M , time in units of M/~(2kL)2, and the reduced Planck constant equals unity. In particular we obtain for the kicking period = ~Tp(2kL)2/M , and for the kicking strength k = V0 Tdur, where Tdur is the nite width in time of individual pulses. This procedure leads to the Hamiltonian (2.10) with the slight, but crucial dierence that X is an unbounded operator and its eigenvalues are not to be taken mod(2), as the angular coordinate of the conventional -kicked rotor. This problem has been addressed in section 2.2.3, and it is very important for the understanding of the experiments which work with atoms on a line rather than with rotors on a circle!

2.3.2

To derive (2.24) for the experimental realisation, we essentially follow standard arguments in atom optics as provided, for instance, in [122, 135, 171], where a more detailed and contextual description may be found. To capture the main features of the problem one usually considers a two-level atom moving in a standing wave of light: E(X, t) = zE0 cos(kLX) e i Lt + e i L t E (X, t) + E + (X, t) , (2.25)

with the amplitude E0 of the wave, z the unit vector in z direction, and L the laser frequency. Time t has to be understood as a continuous variable in this section. The free evolution of the atoms is determined by the Hamiltonian P2 HA = + ~0 |e e| , 2M (2.26)

23

where P denotes the (external) centre-of-mass momentum of the atom, and |e the internal excited state. 0 is the frequency of the atomic transition with the zero point of the internal energy chosen at the ground state level, which is denoted by |g , c.f. gure 2.1. The atom-eld interaction and its impact on the centre-of-mass motion of the particle is derived using several approximations, called dipole, rotating-wave, and adiabatic approximation, respectively, which we briey describe in the following. The dipole approximation states that the eld amplitude varies little over the atomic dimensions, an often used and valid assumption for atoms in the ground state and for optical wavelengths, as well as for Rydberg states in the microwave regime (see section 2.4.1). For near-resonant driving |L 0 | L + 0 , we may neglect fast rotating terms of the form exp(i (L + 0 )t). This results in the atom-eld interaction Hamiltonian: HAF = d+ E + d E ~ + i L t = + e i L t cos(kLX) . e 2

(2.27)

The atomic dipole operator splits up into two components d d+ + d ( + + ) e| d |g , where d is the vectorial dipole moment, and the operator nature is carried by = |g e| and + = |e g| . 2 e| dz |g E0/~ is the Rabi frequency with the dipole matrix element in z direction e| dz |g , and quanties the coupling between the atom and the external laser eld [172]. To simplify the equations of motion, one usually transforms to the rotating frame of the laser eld by dening the atomic excited state | exp( iL t) |e e e and the stationary eld amplitudes E exp( i L t)E . Notice that | is A , and in the rotating frame also an eigenstate of the internal component of H the internal energy equals ~L, L L 0 . Therefore, we arrive at the following complete Hamiltonian including the internal and external degrees of freedom: H = = P2 ~L | e| d+ E + d E e 2M ~ + P2 ~L | e| + + cos(kLX) . e 2M 2

(2.28)

We decompose the atomic state vector | explicitly into a product of internal and external states, the latter describing the centre-of-mass motion of the atoms: |(t) = |g |g (t) + | |e (t) . e (2.29) Separating the equations of motion induced by H into the coecients of |e and |g , we obtain the coupled pair of equations: i ~t |e i ~t |g = = P2 |e + 2M P2 |g + 2M

~

2

cos(kL X) cos(kLX)

|g ~L |e , |e .

(2.30) (2.31)

~

2

24

We attempt to solve (2.30-2.31) for the centre-of-mass motion of the atom, which occurs on a much slower timescale than those of the internal motion. Therefore, for |L| , we may assume that the internal motion damps instantaneously, i.e. t |e = 0, such that the excited state probability amplitude follows adiabatically that of the ground state:

~L

P2 2M

|e

(2.30)

~

2

cos(kLX) |g .

(2.32)

where we neglect the centre-of-mass energy term on the left-hand side of (2.32). We already assumed that |L | is large, and, therefore, we may also suppose that ~|L| |P 2|/2M . For L = 30 GHz [82] and caesium atoms this constraint corresponds to p/(2~kL) 800 in momentum p. Since such large momenta cannot be reached in experiments because of other reasons (see next section and [15, 83, 135,173]), the assumption ~|L | |P 2 |/2M is well justied. Moreover, L + 0 , and the rotating wave approximation is still applicable [15, |L | 83, 135, 173]. Then we obtain the Hamiltonian that describes the dynamics of a point particle in a sinusoidal potential: ~2 ~2 P2 P2 H= + + cos2 (kL X) = 1 + cos(2kLX) 2M 4L 2M 8L , (2.34)

where the constant component in the potential can be dropped, to arrive at the nal form P2 ~2 + V0 cos(2kLX) , with V0 H= . (2.35) 2M 8L The position dependent centre-of-mass potential in (2.35) arises from the position dependent shift of the atomic energy levels by virtue of the interaction with the standing wave (ac Stark shift) [172]. From the spatially periodic structure of H with period L/2 = /kL, we obtain now a clear physical interpretation of the fact that the kicking part of the Floquet operator (2.8) couples only momenta diering by integers [64] (see sections 2.2.2 and 2.2.3). The discrete ladder structure in momentum is imposed by coherent elastic scattering of photons from the standing wave: if the atom absorbs a photon that was travelling in one direction and re-emits it into the counter-propagating mode, the atom will recoil and change its momentum by twice the photon momentum ~kL . If the standing wave is now pulsed in the form of a regularly-spaced sequence in time with period Tp , (2.35) may provide a reasonable approximation to the ideal -kicked rotor dynamics induced by (2.10). To this end, the pulse width dur = Tdur~(2kL)2/M must be suciently short such that the distance travelled by an atom over Tdur is small compared with the spatial period of the standing wave L /2, i.e. Tdur L M/(2p), with the atomic momentum p.

The exchanged momentum must be in either direction of the two counter-propagating waves because of energy and momentum conservation.

2.3. Experimental realisation of the kicked rotor model 2.3.3 Experimental imperfections

25

Finite pulse width The most important experimental constraint, when one wants to mimic an idealised kicked system, is given by the fact that experimental pulses used to provide the kicking potential are always of some nite width dur in time. From the above arguments (see end of last section) it should be clear that the experimental dynamics fails to approximate ideal kicks for xed dur , in particular, at large atomic momenta. The experimental realisation is the better the larger the mass of the used atomic species. For this reason caesium, the heaviest stable alkali atom, is nowadays used in experiments [174]. The eect of non--like pulses has been studied extensively by Raizen and co-workers in experiments as well as in numerical simulations [15, 169, 173, 174]. Together with theoretical work [175] these results show that the eective potential (kicking strength) is substantially smaller in the region of large momenta (ke 0.75kmax at p/(2~kL) = 80 as compared to the centre n = 0, for Tdur 300 nsec [173]). Physically, if the atom is too fast it will start to average over the potential leading to smaller coupling, or, more precisely, the applied pulse (with a certain shape in time) enforces a window function in momentum space depending on the exact pulse shape. For large momenta this eect induces classical and quantum localisation [174176]. In this region beyond some momentum value nref , the classical phase space is lled by impenetrable barriers (tori), which survive small perturbations according to the Kolmogorov-Arnold-Moser (KAM) theorem [4345, 177]. For smooth pulses, the momentum nref is inversely proportional to the duration of the pulse dur, with a pre-factor which depends on the shape of the pulse [175]. Assuming a square pulse shape we obtain, for instance, ke 2kmax sin(ndur/2)/(ndur) [169, 173, 178], the window function being the Fourier transform of the pulse.

Other problems There are many more experimental diculties which are faced when an idealised one-dimensional -kicked particle dynamics [171, 173] should be simulated. For our purposes relevant problems are addressed briey in the following. A severe systematic restriction, connected to the discussion of the nite pulse width, originates from the experimental determination of the atoms momentum distribution. The latter is obtained by counting particles in some relatively small momentum interval centred around p = 0. Especially in the wings of the momentum distribution, the signal is weak, calling for an experimental threshold which decides whether to reject the counts or not. Practically two thresholds are applied: i) momenta are only counted in some xed window, and ii) a dark count threshold that discriminates the signal from background noise. For the kicked-rotor experiments reported in [82, 83], the eective mo-

26

mentum window was chosen 40 < p/(2~kL) < +40 (in most recent data 60 < p/(2~kL) < +60 [159]), and the value of the signal was set to zero for less than 20 counts [83, 135]. These two relevant thresholds are highlighted in gure 4.10 where experimental and theoretical momentum distributions are compared. The atomic density ( 1011 atoms/cm3 in [173]) must be small enough to avoid considerable atom-atom collisions which would spoil the model of independent, structure-less point particles used in the derivation of the eective Hamiltonian (2.35). Based on measurements of collision cross sections for caesium [179], the collision probability is estimated in [173] as 2%/msec. Considering kicking times t . 50 Tp , with typical Tp 20 . . .70 sec [8183], this corresponds to a maximal probability of 7% that one collision occurs during the experiment. Intensity uctuations in the laser beam producing the standing wave should be kept as small as possible to avoid what is known as amplitude noise [173, 180, 181]. Moreover, since the atoms are initially prepared in a threedimensional momentum distribution they are not always centred at the spot of the laser. The laser itself has a transverse Gaussian prole what leads to a potential which is the weaker the farther the atoms are away from the centre of the beam. Both of these two independent eects induce a variation of the kicking strength, which is experienced by the atoms [135, 182]. In particular, when additional momentum is imparted on the atoms by allowing them to emit spontaneously in all directions, the particles may move away from the spot of the laser beam. In the experimental situations of [8183], the transverse spreading (i.e. the deviation from the one-dimensionality of the motion) produced by SE events is smaller than the spread of the initial momentum distribution in the transverse plane to the kicking axis. The problem of spontaneous emission is for itself worthwhile to investigate: atoms are never two-level systems what makes a treatment necessary which includes the distribution of the atomic population over various sublevels, and also the process of dissipation by spontaneous emission. In the far-detuned case, assumed when deriving the eective Hamiltonian (2.35), the probability of absorbing a photon from the standing wave and emitting it in the vacuum mode is small. A good approximation for the steady-state scattering rate as a result of spontaneous emission (SE) is obtained by Rsc SE |e |2, where SE is the line width of the excited level with population |e|2 . Using (2.32), we may 2/(42 ) cos2 (kL x) (assuming that only the ground state is estimate |e|2 L 1, and ~|L| |P 2 |/2M as used above). signicantly populated, i.e. |g | SE 2 /(82 ) 1 for SE , |L | after averaging the This leads to Rsc L cosine. For the experiments performed by dArcy and co-workers, the mean number of SE events undergone by each atom due to one far-detuned standing wave pulse is estimated to be nSE RscTp . 2 103 [83, 135]. Apart from the unwanted eect described above, SE, indeed, provides a controllable way of adding noise to the evolution of the atoms [82,83,176,182184]. To this end, SE is introduced most exibly by an additional laser, which is independent of the standing wave (for instance, by the beams used to prepare and cool the atoms before the kicks are applied [82, 83], cf. gure 2.1). The

27

induced mean number of events can be scanned by varying the intensity (thus the Rabi frequency ), the interaction time or the applied detuning from the internal atomic transition [172]. The stochastic time evolution of kicked atoms in presence of SE at quantum resonance is theoretically modelled in section 4.3.2.

2.4

Atomic hydrogen in a microwave eld

2.4.1

Like the -kicked rotor also the second atomic system which we want to investigate is time-periodic: hydrogen Rydberg atoms exposed to a monochromatic electromagnetic eld. A numerically exact method to treat this problem with a minimum of (irrelevant) approximations has been developed by Buchleitner and Delande [108, 112, 113, 185], and recently a rened version using large parallel supercomputers has been successfully applied to describe other non-hydrogenic one-electron Rydberg states [9598]. The highly excited Rydberg electron moves in three-dimensional conguration space in presence of the combined potential of the nucleus and the external time-periodic eld. The driving force further excites the electron and eventually may ionise it. Therefore, a complete theoretical treatment must account for the one-electron dynamics within the Coulomb potential, and the spectrum of the atom dressed by the (classical [172]) eld, including the coupling to the atomic continuum. The parameter space consists of the quantum numbers of the unperturbed initial electronic state |0 = |n0 0 m0 , the amplitude F , the angular frequency of the eld, and the interaction time t between atom and eld. n0 denotes the principal quantum number of the initial state, 0 and m0 its angular momentum and the projection of the latter onto the eld axis, respectively. In this thesis, we restrict to linearly polarised microwave elds, and assume that the eld is constant in space over the atomic dimensions (dipole approximation). We neglect the for our parameter values (n0 1, F . 1/n4, 1/n3) irrele0 0 vant relativistic, spin and QED eects, and further assume an innite mass of the nucleus [108]. In atomic units, the Hamiltonian may be written in several gauges [149, 186]: 1 H = p 2 + V (r) + r F cos(t) , 2 pF 1 sin(t) , H = p 2 + V (r) 2 length gauge velocity gauge (2.36) (2.37)

with the Coulomb potential V (r) = 1/|r|. Since the spectral properties are not aected by the choice of gauge, we have the freedom to choose the most appropriate one for our particular purposes. The numerical calculations use,

28

for faster convergence, the representation in the velocity gauge [108, 187, 188]. The Floquet theorem allows one to reduce the time-dependent problem given by either of the Hamiltonians (2.36-2.37) to a stationary eigenvalue problem. Using (2.36) and (2.3), the eigenvalue problem (2.2) is recast into a coupled set of time-independent equations: 1 2 p + V () m r 2 |

m

1 = rF (| 2

m+1

+|

m1

(m Z . ) (2.38)

The additional quantum number m counts the number of photons exchanged between the atom and the dressing eld [149]. The tridiagonal form of (2.38) in m is reminiscent of the structure of the tightbinding model which is used to describe particle transport in condensed-matter physics [153,154,189]. The tight-binding description encountered below in equation (2.16) is exact for the -kicked rotor, whereas (2.38) still contains the momentum and position operators, and hence leads to a more complicated matrix structure [95,113]. A tight-binding model will serve in section 3.1.3 to relate our results on the ionisation probability of the atom to predictions for Andersonlocalised solids. We emphasise again, (2.38) explicitly contains the coupling between nearest neighbour states | m in energy space. For vanishing eld amplitude F 0, the equations decouple, and any Fourier component | m is a solution of the Schrdinger equation with energy eigenvalue + m of the o dressed state. With increasing F , the external eld starts to couple the dierent photon channels labelled by m, as depicted in gure 2.2, and we have to solve the full Floquet problem (2.38) in order to obtain the dressed states of the atom. Since m runs from to +, the dipole term couples all bound states of the eld-free Hamiltonian to the atomic continuum. The spectrum of (2.38) no longer separates into two orthogonal subspaces (discrete bound states and continuum), but all bound states turn into resonance states embedded in the atomic continuum [108,150,190]. The projection of a resonances state onto a nite volume in coordinate space decays with a nite rate . The corresponding eigenvalue problem (2.38) can then be solved only for complex quasi-energies = Re( ) /2, because of the non-unitarity of the problem induced by the projection. Precisely the decay rates determine the experimentally measured ionisation yield of the driven Rydberg atom. To extract the complex quasi-energies , we use a complex scaling transformation [150, 190195], which allows us to separate the resonance states from the continuous part of the spectrum also for the driven atom. The method of complex scaling is perfectly suited for the present problem of (typically slowly) decaying states, which are non-square-integrable solutions of (2.38). The complex scaling transformation rotates the continuous spectrum away from the real axis into the lower half of the complex plane, and thus uncovers the resonance poles of the metastable states of the atom in the eld. The energies of these resonance states do not depend on the complex rotation angle, which must be suciently large to really uncover all the resonance poles [150,190,193]. A typical spectrum is plotted in gure 3.8; it illustrates the rotated continuum, and

29

r F 2

r F 2

m+1 m m1

Fig. 2.2: Illustration of the tight-binding structure of equation (2.38) in the photon index m, which labels the number of dressing photons: The external microwave eld couples the bound states of atomic hydrogen (horizontal lines) through the dipole operator r to the atomic continuum (dark areas).

the resonances between the real axis and the continuum. The complex dilated eigenvalue problem (2.38) is represented in a real Sturmian basis, such that a block-tridiagonal complex symmetric, sparse banded matrix [108, 113] can be diagonalised numerically. The diagonalisation supplies the quasi-energies , thus also the ionisation rates , and the associated eigenbasis of the microwave driven atom [113, 185]. Therefore we have everything at hand for the investigation of the quantum probability transport from an initially bound state |0 to the atomic continuum. In section 3.1.3, we are interested in the behaviour of the survival probability of the atom in the microwave eld. Psurv (t), i.e. the probability to nd the atom in a bound state after an atom-eld interaction time t = t2 t1 , is given by the projection of the propagated initial state |0 = |n0 0 m0 onto the subspace of all bound states | : Psurv (t) =

| U(t2 , t1) |0

(2.39)

where U is the time-evolution operator (cf. (2.4)), generated by the Hamiltonian (2.36-2.37). U(t2 , t1) propagates the wave function from time t1 to time t2 when the eld interaction is switched on and o, respectively. After averaging the initial and nal time t1 and t2 over one eld cycle T = 2/, respectively, while keeping the total interaction time t = t2 t1 xed , (2.39) can be shown to

The averaging over one eld cycle physically represents a phase average for the external eld, whose phase when the atoms start or stop to interact with the microwave is eectively averaged in state-of-the-art experiments [11, 108].

30

yield [108, 113]

Psurv (t) =

e t w .

(2.40)

The weight factors w m | m |0 |2 are the expansion coecients of the initial wave packet in the Floquet eigenbasis . The sum (2.40) runs over the entire spectrum within one Floquet zone of width in energy.

2.4.2

Quantum-classical correspondence

As mentioned in the introduction, the experimental ndings by Bayeld and Koch on hydrogen Rydberg atoms [4] fostered the understanding of the classicalquantum correspondence of classically chaotic systems. In [4] a very ecient, multi-photon ionisation was reported for eld intensities lower than the intensity necessary for a static electric eld to ionise the atoms. As evident from gure 1.2, the threshold value F0 (10%) F (10%)n4 at which 10% of the atoms 0 ionise, can be reproduced by classical calculations for microwave frequencies less then the classical Kepler frequency, i.e. < 1/n3 [11, 94, 100]. For the 0 range 1/n3 , the behaviour of F0 (10%) is best explained by dynamical lo0 calisation theory [59,60]. The scaling used in gure 1.2 is the natural scaling of the classical Hamiltonian equations induced by (2.36-2.37). Indeed, the classical equations of motions are invariant under the transformations highlighted in table 2.1 [25, 46]. r r/n2 0 F F0 F n4 0 p pn0 0 n3 0 t t/n3 0 H Hn2 0

Tab. 2.1: Scale transformations which leave the classical Hamiltonian dynamics of periodically driven hydrogen Rydberg states unchanged. n0 is identied with the principal quantum number of the initial atomic state.

These scaled variables are the basis for the comparison of the classical and the quantum evolution along the lines of the correspondence principle. The classical scale invariance induces an eective Planck constant ~e ~/n0 through the quantum commutator relation for position and momentum operators: i ~ = [r, p] r ~ 2 , pn0 = i n . n0 0 (2.41)

The weights are actually w = m m | 0 2 , i.e. complex numbers without the absolute square, since the eigenstates are solutions of a complex symmetric, non-Hermitian eigenvalue problem [108,113,185]. Here we use a simplied notation to avoid such technical complications.

31

Precisely this dependence on the initial principal quantum number n0 which corresponds to the classical action variable makes highly excited Rydberg states ideal objects for the study of the manifestations of mixed regular-chaotic classical dynamics in quantum mechanics. The nite ~e contains the information on the initial state, and the quantum system is able to resolve classical phase-space structures the better, the larger n0 . Indeed, Floquet states mainly concentrated on either regular regions, e.g. nonlinear resonance islands, or chaotic components of the classical phase space have been identied [77, 87, 93, 100, 112, 113]; see gure 3.11 on page 47 for typical phase-space plots of one-dimensional periodically driven hydrogen. In the following chapter, we focus on the parameter regime 0 > 1 to search for further unambiguous signatures of dynamical/Anderson localisation in periodically driven Rydberg atoms. The phase-space localisation properties of the quantum states will turn out to be essential for the statistical analysis of the ionisation rates [116].

Part I: Signatures of Anderson localisation in the multiphoton ionization of hydrogen Rydberg atoms

Ein historisches Kriterium fr die Eigenart der Prinzipien kann u auch darin bestehen, da immer wieder in der Geschichte des philosophischen und naturwissenschaftlichen Erkennens der Versuch hervortritt, ihnen die hchste Form der ,,Universalitt o a zuzusprechen, d.h. sie in irgendeiner Form mit dem allgemeinen Kausalsatz selbst zu identizieren oder aus ihm unmittelbar abzuleiten. Es zeigt sich hierbei stets von neuem, da und warum eine solche Ableitung nicht gelingen kann aber die Tendenz zu ihr bleibt nichtsdestoweniger fortbestehen. E. Cassierer, in [196]

Chapter 3

The survival probability (2.40), and hence the thresholds for the eld strength F0 (10%) F (10%)n4 , at which 10% of the atoms ionise owing to the interac0 tion with the external driving (cf. gure 1.2), convolutes the spectral (global) information provided by the decay rates as well as the local information about the initial conditions (i.e. the initial state n0 ). The latter is contained in the weight factors w in (2.40). To circumvent the mixing of the global spectral properties and of the local expansion coecients, the straightforward way to proceed is to analyse the ionisation rates of the Floquet problem. This allows us to identify unambiguous signatures of dynamical/Anderson localisation in the decay-rate distribution ( ) [116]. A clear indicator for Anderson localisation in transmission problems (i.e. in open quantum systems) is indeed provided by the decay-rate distribution of states exponentially localised within the sample. The distribution of decay rates obeys a power law () 1 [197]. Such a law is easily derived by assuming that the rates are proportional to the overlap of the corresponding states with the lattice site at the boundary. The boundary is to be identied with the lead in a solid-state transmission problem, or the atomic continuum of driven hydrogen. The tail of the wave functions determines the loss out of the sample in this simplied picture, which is sketched in gure 3.1. We assume a suciently large number of sites j = 1 . . . L, with L 1, and uniformly distributed states along the lattice, i.e. with constant density A (j) = 1 in the limit L . Then, if we suppose that the following relation is valid: n |n (j)|2 exp(2j/)|j=L , we obtain j=L (n ) = dn dj

1 j=L

2j

j=L

1 . n

(3.1)

This follows from the transformation formula for probability densities [198]. L 35

36

j=n j=L

Fig. 3.1: One-lead Anderson model with exponentially decaying state, which is localised at the lattice site j = n. On the left we assume a perfectly reecting wall. The open right end induces decay out of the sample, and the tail of the wave function 2 at the boundary j = L determines the loss rate n |n (j)|j=L . The localisation length characterises the width of the wave function. The same model will be used in section 3.1.3 to examine the time dependence of the survival probabilities.

denotes the index of the last site at the boundary and the localisation length along the lattice, over which the wave functions are exponentially localised. Using the above argument, the power law (3.1) was predicted also for dynamically localised, classically chaotic model systems [199]. For one-dimensional tight-binding models a similar behaviour () was found whereby the exponent 1 . . . 2 turned out to be slightly dependent on the model assumptions and on the degree of localisation [197,200]. Very recently, the distribution of resonance widths in multiple-light scattering systems was also shown to obey a power law with 1 in the localised regime [201]. In the subsequent sections, the statistics of the decay-rate distribution are elucidated for the atomic ionisation problem of strongly driven hydrogen Rydberg states. Generic (physical) Hamiltonian systems are neither completely chaotic nor integrable but show simultaneously both, chaotic and regular motion, which manifests in a mixed classical phase space. Atomic Rydberg states under microwave driving are paradigmatic real objects to investigate the quantum probability decay in presence of tunnelling and of quantum localisation phenomena (Anderson/dynamical localisation [59, 60] and semiclassical localisation in the vicinity of partial phase-space barriers [104, 105]). In addition to the good agreement with the predictions for Anderson-localised systems, our statistical analysis of the distribution of the ionisation rates makes it possible to systematically study the impact of classical phase-space structures on quantum transport in periodically driven Rydberg states.

37

3.1

3.1.1

Numerical results

The atomic ionisation process of Rydberg states subject to microwave radiation is mapped onto the Anderson model through the localisation parameter L = /L [59, 60], where the localisation length is measured in units of the photon energy . Both transport problems are sketched in gure 1.1. According to the theory of dynamical localisation [59, 60], L characterises the degree of localisation. For L 1, the electronic wave packet is strongly localised on the energy axis, while for L > 1 considerable coupling to the atomic continuum prevails. The sample length L is the ionisation potential of the initial state |0 = |n0 0 m0 , measured in units of : L= 1 2 1 1 n2 n2 c 0 . (3.2)

nc denes the eective ionisation threshold in the experiments [11,13,50] as well as in numerical calculations using a large but nite basis [108]. Provided that n0 identied with the principal action of the classical evolution is chosen within the chaotic component of phase space (assuming only tiny remnants of classically regular motion immerged in the chaotic sea), L measures the extension of the domain of complex transport along the energy axis. The mapping onto the original Anderson problem with sample length L implies a distribution of the decay rates () 1 , in the statistical average over many realisations of disorder at a xed value of L 1. In the atomic problem, statistically independent realisations of disorder with xed localisation parameter L are generated by simultaneously varying the eld amplitude F and its frequency . We use the following prediction of [59, 60] L 6.66F 2n2 3 0

7

n2 0 n2 c

(3.3)

from the original theory on dynamical localisation in periodically driven, onedimensional (1D) hydrogen atoms. While this theory, which is based on several approximations, has no quantitative predictive power, it provides at least a qualitatively correct picture, in particular when statistical averages are considered. Possible corrections to (3.3) for the real three-dimensional (3D) atom are discussed in [59], but (3.3) is assumed to hold qualitatively also for quasi-1D realisations of the initial Rydberg state (so-called extremal parabolic states [71]). In the sequel, (3.3) is used to guide our choice of the eld parameters F, for the statistical analysis of the atomic decay-rate distributions. Figures 3.2-3.5 show the probability densities of the ionisation rates of a 1D hydrogen model atom exposed to a microwave eld. The rates of the quasi-energies within one Floquet zone are presented. The zones are centred around the n0 = 40, 70, 100, 140 Rydberg manifolds, respectively, with

38

10.8

log(())

12

10

log()

Fig. 3.2: Distribution of the ionisation rates of 1D microwave-driven Rydberg states of atomic hydrogen, for dierent values of the localisation parameter L = 0.2 (plusses) L = 0.25 (diamonds), 0.5 (stars), 1.0 (circles), and 2.0 (pyramids). The distributions were generated, at xed L, by sampling the spectra within a Floquet zone of width centred around n0 = 40, over the frequency range 0 = n3 = 2.0 . . . 2.5. In laboratory 0 (SI) frequency units /2 = 205.63 . . . 257.03 GHz, with F chosen accordingly to x L (3.3) at the given values. The solid line represents the scaling ( ) 0.9 .

nc 2n0 . By (3.3), at xed L, each of these initial quantum numbers corresponds to a dierent range of eld parameters F, . In our model, the Rydberg electron is conned to 1D conguration space z > 0 dened by the polarisation axis of the eld, with the Coulomb singularity at the origin z = 0 [77,108]. The 1D approximation allows to produce vast sets of spectral data on up-to-date workstations in a reasonable amount of calculation time. Much more computer power is needed to simulate the full realistic 3D hydrogen atom. Data for the 3D case centred around n0 = 70 (nc = 105, see (3.3)), and for the selected localisation parameters L = 0.25, 0.5, 1 essentially reproduce the features of the 1D model. The dierent values of L = 0.2 . . .2 (1D), and L = 0.25 . . .1 (3D) are realised by sampling the spectra over frequency ranges 0 = n3 = 2 . . . 2.5 (1D), and 0 0 = n3 = 1.854 . . .1.883 (3D), and adjusting F accordingly. Because of the 0 dramatically enhanced spectral density of the 3D [95] as compared to the 1D problem a consequence of the additional angular momentum degree of freedom labelled by only ten equidistant frequency values are needed to generate an appropriate statistical sample in the 3D case. The total number of states contributing to the distributions is approximately 25000 (3D), in contrast to up to 100000 states for 500 equidistantly chosen frequencies of the 1D model. Surprisingly enough, only about 5% of the large number of states produce the same distributions. This shows the high stability of the observed statistics. Changes occur with increasing number of sample realisations only in the region

39

13 11

log(())

9 7 5 3 15

13

11

log()

Fig. 3.3: Same as in gure 3.2 for the initial atomic state n0 = 70, corresponding to the frequency range /2 = 38.37 . . . 47.96 GHz in laboratory units.

12 10 8 6 4 15

log(())

13

11

log()

Fig. 3.4: Same as in gure 3.2 for the initial atomic state n0 = 100, corresponding to the frequency range /2 = 13.16 . . . 16.45 GHz in laboratory units.

40

13

log(())

11

5 15

14

13

12

log()

11

10

Fig. 3.5: Same as in gure 3.2 for the initial atomic state n0 = 140, corresponding to the frequency range /2 = 4.8 . . . 6.0 GHz in laboratory units. L = 2.0 is not shown because of the high cost of numerical calculations at very large quantum numbers n0 > 100, even in the 1D case.

of very large decay rates > 107 a.u. for gure 3.2, down to > 109 a.u. for gure 3.5, because of the few lying in these regions (for L < 1). The robustness of the distributions (see gure 3.7) is observed either when using only few, but complete spectra (for instance, about 20 50 in the 1D case), or a random choice of the full list of decay rates, sampled over the entire frequency range. Our numerical technique using a complex scaling transformation [108, 113] rotates the continuum into the lower half of the complex energy plane. Doing so, the resonance poles of the resolvent operator are uncovered. The Floquet spectrum is periodic with period , and for the 1D model atom the continuum threshold has a well dened position on the real energy axis. It corresponds to the ionisation potential of the initial bound state modulo . By rejecting resonance poles in the vicinity of the threshold, we ensured that no states from the discretised continuum entered the statistics. Figure 3.8 shows the resonance poles in the complex energy plane, together with the rotated continua. The Floquet zone of width contains the poles from which the ( ) distribution is built up. Poles close to the thresholds are not considered. Figure 3.8 also shows poles which are rejected because of yet another criterion which is based on the values of the overlaps w with the initial bound state of the atom. This criterion helps to avoid continuum contributions as well, since these have tiny weights at the initial bound state with quantum number n0 . For the decay-rate distributions in gures 3.2-3.5, the rejection criterion chosen was that the overlaps had to be larger than 105 for < 108 a.u., and larger than 104 for 108 a.u. For smaller values of w down to 108 nearly no changes in the statistics were observed, apart from the rightmost part in the

41

12 10 8 6 4 15

log(())

13

11

log()

Fig. 3.6: Distribution of the ionisation rates of microwave-driven 3D Rydberg states of atomic hydrogen, with linear eld polarisation along the z axis, and with angular momentum projection m0 = 0 onto this axis. Localisation parameter are L = 0.25 (diamonds), 0.5 (stars), 1.0 (circles). The distributions were generated, at xed L, by sampling the Floquet spectra over the frequency (and corresponding amplitude, F ) range 0 = 1.854 . . .1.883, or in laboratory frequency units /2 = 35.5 . . . 36.1 GHz, within a Floquet zone of width centred around n0 = 70. The solid line represents the power-law scaling ( ) 0.9 . Data by courtesy of Andreas Krug [116].

distributions, where continuum states will appear if states with w < 104 are allowed (typically the rightmost two data points corresponding to the largest rates move to the right). On the other hand, eigenstates situated in the regular and elliptic region of classical phase space, deeply below the initial state with principal quantum number n0 , are also rejected if the overlap criterion is too restrictive. The contribution of such regular states is considerable for large localisation parameter L 1, while for small L 0.25, these states have non-resolvable ionisation rates . 1015 a.u. Figure 3.9 illustrates the eect of states attached to regular/elliptic regions, which is discussed in more detail in the next section 3.1.2. Since in the 3D calculations the nite numerical basis lifts the degeneracy of the states in angular momentum [113], it is more dicult to automatically exclude continuum states from the width distributions [95]. Contributions from the continuum eigenstates can, however, be excluded by the choice of the lowest allowed overlap value, similar to the procedure in the 1D case. For the distribution plotted in gure 3.6, the criterion was w > 105 . When lowering this threshold down to 1010 the overall relevant structure of the distribution at small and intermediate values of is unchanged, while at 107 a.u. new states appear very much as described above for the 1D case.

42

13

11

log(())

log()

9.8

8.8

7.8

Fig. 3.7: Illustration of the statistical robustness of the ionisation-rate distribution for the data set L = 0.25, from gure 3.4. Shown are the distributions for a random selection of decay rates of the full data set (diamonds), 5% (full squares), 10% (plusses), and for the complete spectra of the rst 50 equidistant values of the frequencies 0 = 2.0 . . . 2.05 (dash-dotted). No deviation from the full distribution is observable as long as the selected set of decay rates contains more than 3000 randomly chosen values. A similarly robust behaviour is found for all distributions plotted in gures 3.2-3.6.

3.1.2

The main observation in gures 3.2-3.6 is that in all cases the decay-rate distributions obey an algebraic law ( ) , (3.4)

with exponent 1. For small localisation parameters, both the 1D model atom (L = 0.2, 0.25) as well as the real atom (L = 0.25) exhibit distributions ( ) 0.9 over about six orders of magnitude, from 1015 a.u. to 109 a.u. This result is in good agreement with predictions of the decay in a disordered solid, where () , 1 . . . 2 is predicted in the parameter domain exp(L/) / 1, being the mean level spacing [197, 200]. However, as the localisation parameter is increased by systematically increasing F over the entire frequency ranges indicated in the previous section, the situation becomes more complex. We observe a depletion of the probability densities of small rates, balanced by a decrease of the decay exponent of ( ) in an intermediate range depending slightly on the chosen parameters, or on the chosen initial value of the principal quantum number n0 . For n0 = 40 (g1010 . . . 107 a.u., for n0 = 70 140 ure 3.2) this range is approximately (1D) (gures 3.3-3.5), and n0 = 70 (3D) 1010 . . . 108 a.u. (gure 3.6). Such a behaviour is incompatible with the simple assumption of exponentially

43

2.5e06

5e06

7.5e06

1e05 0.00033

0.00031

0.00029

Re()

Fig. 3.8: Resonance poles of the resolvent operator for the 1D hydrogen atom. The spectrum is periodic, and the continua are rotated into the lower half of the complex energy plane. Crosses represent the full spectrum obtained by the numerical diagonalisation of the Floquet operator (cf. (2.2) and (2.38)), the circles are the poles used to build up the distribution ( ). The regions close to the continuum thresholds (between the long and short dotted line) are not taken into account. In this way, we ensure that no continuum states enter the statistics of the decay rates. The spectrum is shown for F = 2.0901 108 a.u. and 0 = 2.0 (/2 = 205.63 GHz), in the vicinity of the initial state with quantum number n0 = 40. It corresponds to the distribution plotted in gure 3.2 for L = 0.2.

localised probability densities along the energy axis owing to dynamical localisation of the Floquet eigenstates over the chaotic component of classical phase space [115, 199]. For large localisation parameters L 1, a quantitative change is expected [59, 60], from a regime of localised quantum motion with very little ionisation to diusive ionisation, similar to the classically predicted diusion process. With increasing L, dynamical localisation is gradually destroyed, and alternative transport mechanisms gain importance. Therefore, deviations from the predictions based on purely Anderson-like models do not come as a surprise. For systems with mixed regular-chaotic phase space like the strongly driven atomic Rydberg states [11, 112114] and mesoscopic systems [202] it is known that the eigenstates of the corresponding quantum systems can be classied according to their localisation properties in classical phase space [102, 112, 113, 202205]. While dynamical localisation occurs for states situated in the chaotic domain of phase space (i), eigenstates can be localised also (ii) on regular or/and elliptic regions, and (iii) along remnants of regular motion immerged in the chaotic sea. In particular, elliptic regions as well as the nearly integrable component are rather robust under changes of some control parameter, such as the eld strength in our system [112, 113]. Accordingly, the localisation properties and the decay rates of the associated eigenstates tend to

44

12

10

log(())

4 15

13

11

log()

Fig. 3.9: Inuence of eigenstates with small decay rates ( 1012 a.u.), and small overlaps with the initial bound state of the atom. The data set L = 2 from gure 3.4 (pyramids; w > 105 ) is compared to dierent choices of the overlap criterion. The more stringent condition w > 104 (plusses) excludes many states with small ionisation rates ( 1012 a.u.) which typically are situated in the regular region of classical phase space, including states attached to the primary nonlinear resonance island. For w > 106 (full squares) continuum states with large > 107 a.u. appear, and no change in the distribution is observed any more for w > 107 (dash-dotted) or even smaller lower bounds for w . The vicinity of the continuum threshold (cf. previous gure) is not excluded from the distributions in this plot, and the same overlap criterion is chosen for the full range in 5 1015 . . . 105 a.u.

be less sensitive when F changes, while those eigenstates lying in the chaotic domain exhibit rapidly increasing rates as the eld strength grows. As stated in the previous section, the decay rates of Floquet states in the chaotic component are essentially determined by the localisation parameter L . Hence, deviations from the clean algebraic law ( ) 0.9 at large values of F are attributed to the rapid increase of the decay rates of class (i) eigenstates with L. This corresponds to the eective destruction of dynamical localisation, and the observed change in the decay exponent for & 1010 a.u. On the other hand, class (ii) and (iii) eigenstates move up to intermediate values . 1010 a.u., for the largest L shown in gures 3.2-3.6. The above classication of eigenstates is supported by a systematic study of the Floquet eigenstates of the 1D driven hydrogen atom [112,113]. Figure 3.10

This is the expectation for largely averaged quantities such as ( ). For a single realisation or sample, large uctuations, e.g. in the localisation length or in the ionisation probability, may occur around the approximate statistical prediction [109, 111, 206208]. Nonetheless one must keep in mind that the identication of the decay properties characterised by the ionisation rates and the localisation in classical phase space is a statistical statement, which is true when averaging over the contribution of many individual eigenstates.

45

presents generic Floquet eigenstates of the 1D model atom in the Husimi or Q-function representation [16, 112, 113, 211] in phase space. The above classication into (i-iii) states is veried by comparing their decay rates with their position in classical phase space. The latter is shown in gure 3.11 for the different localisation parameters L = 0.25 and L = 2. The phase space is spanned by action angle variables I, of a 1D harmonic oscillator [112, 212] . At small values of L < 0.5, the states of class (ii) and (iii) exhibit tiny ionisation rates below 1013 a.u., or even below the numerical precision 15 10 a.u. Therefore, in the regime of strong dynamical localisation, i.e. for small L 0.25, ( ) is dominated by the decay rates of class (i) eigenstates, lying in the chaotic component of classical phase space. The good agreement of the algebraic decay law with predictions from Anderson models suggests that our results are indeed a manifestation of Anderson localisation being at work in the chaotic ionisation process of microwave-driven Rydberg atoms. The argument at the beginning of this chapter leading to the power-law of the rate distribution is based on the proportionality of the decay rates with the tail of the corresponding eigenstates at the boundary. A similar argument is possible for states which leak out of a regular region in phase space because of tunnelling [199]. The assumption needed for () 1 is that the decay rates decrease exponentially along the lattice of quantised nearly integrable tori, i.e. regular exp(|j jmax|/e ). This is formulated on tori, where the integer j denotes the local, and jmax the maximal excitation of the quantised tori within the regular domain. e is an eective tunnelling length. Such a picture is applicable either to elliptic islands or to the regular regions, with only slightly perturbed tori with respect to the unperturbed hydrogen motion. The assumption of an exponential behaviour induced by tunnelling is certainly true on average. However, e may be subject to system specic uctuations which depend on the eective Planck constant [213217]. For the derivation of the law () 1 , a suciently large number of tori jmax must be present. This guarantees the necessary limit of many sites on the lattice as required in the Anderson problem. If the density of states is too low, nite size eects may spoil the above algebraic behaviour. Our results nicely conrm the power-law prediction for the decay-rate distribution with ( ) 0.9 in the region 1011 a.u. This part of the distribution is dominated by class (ii) and (iii) states localised along regular structures of phase space for L = 1 and 2 in gures 3.2-3.6. The derivation of ( ) 1 given above for the tunnelling regime, and at the beginning of this chapter for the Anderson scenario is rather crude. Yet the argument is valid for generic 1D Anderson models in a regime of small [197]. For large

There do exist states for which the better localised in phase space decay faster (larger ) than those which are more extended in phase space [112]. This reects the complexity of the problem [110, 112, 209, 210]. These specic variables are chosen because of the representation of the Floquet problem in a Sturmian basis which is characterised by a scale parameter n0 . The eigenstates are then naturally represented in a harmonic oscillator basis [112, 113, 212]. The action I may be mapped in a well dened way to the principal quantum numbers n [112, 113, 212]. For the 1D as well as the 3D problem, a quantisation procedure ` la Einstein-Brillouina Keller [8, 16, 44] is possible for regular regions locally in phase space [77].

46

150 150

action I

100

100

50

150 150

100

50

0 2 0 angle 2

0 2 0 angle 2

Fig. 3.10: Husimi distributions (contour plots) of four resonance eigenfunctions of the 1D hydrogen atom in a microwave eld with amplitude F = 1.073 109 a.u., and frequency /2 = 13.16 GHz. For an initial state n0 = 100, the localisation parameter (3.3) is L = 2. The widths of the resonances are 1.5 108 (top left), 9 11 310 (top right), 2.610 (bottom left), 51014 (bottom right). These quantum states represent typical eigenstates corresponding to the dierent regions in the distribution of gure 3.4 with L = 2. Comparing to the corresponding classical phase space in gure 3.11(b), we can classify the eigenstates according to their localisation in the chaotic domain (top left, top right), in the separatrix region (bottom left), or on the regular part of phase space (bottom right) [113].

150

47

(a)

(b)

100

50

Fig. 3.11: Classical Poincar surface of section for the 1D microwave-driven hydrogen e atom in harmonic oscillator action angle variables I, [112] for weak (a), and strong driving (b). The chosen parameters are /2 = 13.16 GHz, and F = 3.7741010 a.u. (a), F = 1.073 109 a.u. (b). For n0 = 100, this corresponds to a localisation parameter L = 0.2 (a) and 2 (b), respectively.

, the distributions do dependent on the specic models [197, 200]. A logarithmic correction to the power law () 1 , based on an additional simplied semiclassical transport mechanism, is derived in [68]. For the complex quantum transport in driven Rydberg states an analytical theory does not exist. But we notice that the universal decay exponent 0.9 found in our data matches surprisingly well with the predictions from Anderson localisation for L 0.25, or with the tunnelling model in the delocalised regime, respectively. To conclude the discussion of the decay-rate distributions we comment on a slight, but systematic discrepancy between the 3D and the 1D data. While a clear knee like structure develops at intermediate rates, with increasing L in the 1D model, for the real atom this knee is less pronounced. Let us compare in more detail the distributions for the localisation parameter L = 1: a change in the decay exponent occurs in the region 109 . . . 5 108 a.u. (gure 3.3), 10 8 11 10 . . . 10 a.u. (gure 3.4), 10 . . . 109 a.u. (gure 3.5), and 1010 . . . 108 a.u. (gure 3.6). This regions are domiin the 3D atom at nated by class (i-ii) eigenstates. The observed smoothing of the 3D distribution with respect to the 1D results arises from the additional angular momentum degree of freedom. In the same way as transport occurs along the radial degree of freedom (energy axis in 1D case), also the angular momentum ( ) states are coupled in the real atom. The third degree of freedom, the projections of onto the eld axis, is a constant of motion [71, 218], which is xed for the spectra plotted in gure 3.6. At small L 0.25, states largely composed of high- contributions have vanishing decay rates. On the other hand, for strong elds, i.e. L 1, such contributions do come into play because of the strong coupling. They manifest in the distributions ( ) in the range < 1010 a.u., they compensate for the reshuing of states which are mainly composed of lowcontribution, from small to intermediate values of . Therefore, the angular momentum degree of freedom is important for the decay-rate distributions, even

48

10 10

(a)

10 1 10

(b)

10

10 2 10 10 10

6

(c)

10

10

15

10

13

10

11

[a.u.]

10

10

Fig. 3.12: Expansion coecients w of the atomic initial state |0 = |n0 = 100 (1D), and |n0 = 70, 0 = 0, m0 = 0 (3D) in the Floquet eigenstates | vs. the associated decay rates ; for 1D L = 0.25 (a), 1 (b) and /2 = 13.16 . . . 16.45 GHz (500 spectra), and for 3D (c) with L = 1 and /2 = 35.6 GHz (one spectrum). (Qualitatively similar results are obtained for microwave driven alkali Rydberg states [95].) There is no denite correlation between the decay rates and w , apart from the general observation that with increasing coupling to the atomic continuum (from (a) to (b)), larger are assigned larger weights, implying faster decay. 3D data by courtesy of Andreas Krug [116].

3.1.3

The results on the ionisation-rate distributions presented in the preceding sections show how and to what extent chaotic transport mimics signatures of disorder, and induces amendments thereof arising from the peculiar structures of mixed regular-chaotic phase space. The underlying structure of the classical phase space is a feature of dynamical systems, and it has no counterpart in disordered solid-state models. The aim of the present section is condensed in the following question: how do the observed statistics of the ionisation rates carry over to the survival probability

49

of an atom in the driving eld? The survival probability or its complement, the ionisation probability, is the most directly accessible observable in laboratory experiments. Can one nd clear traces of the universal decay-rate statistics ( ) 0.9 in the survival probability of the atom in the eld? The answer is partly armative as we will see in the following. A power-law distribution of the decay rates means that there is a large number of them distributed over many orders of magnitude. If many of these channels are involved in the actual decay process an algebraic decay law Psurv (t) t (3.5)

is generally expected [114, 219, 220]. For microwave-driven rubidium Rydberg atoms, experiments have found clear evidence for a power-law with exponent 1/2 [114]. Corresponding numerical studies for alkali as well as hydrogen Rydberg states suggest that asymptotically the power law (3.5) indeed is generic. However, the strong parameter dependence of the exponent ranging from 0.5 . . . 2 [95] contradicts the prediction of a universal time decay for the survival probability [115]. The survival probability Psurv (t), i.e. the probability to nd the atom in a bound state after an atom-eld interaction time t, is given by (2.40). The ionisation process and the number of strongly contributing decay rates is determined also by the weights w . The latter represent the initial condition, and contain local information on the spectrum to the extent that the initial state |0 is localised in energy space (see gure 1.1). Therefore, the question whether a power-law like (3.5) holds and if it holds, over what range of interaction times depends crucially on the overlaps w . Starting from a general form ( ) , a universal time dependence Psurv (t) t2 has been suggested [115], leaning on the additional assumption w [115, 221, 222]. The proportionality of decay rates and weights allows an approximate derivation of the survival probability based on the distribution function ( ). Provided the density of states is suciently high, one may substitute the contribution of each resonance to Psurv (t) by an integral average over all possible with their weight function ( ):

Psurv(t)

0

d ( )w e t ,

(3.6)

what reduces to Psurv (t) t2 , (3.7) for w , and ( ) (within the physically relevant window of values). Whereas ( ) obeys a nice power law in the dynamically localised regime (see section 3.1.2), the assumption w does not apply for a generic situation of the ionisation problem. This assumption is justied only if the initial

Strictly speaking, ( ) is already the density averaged over many realisations/spectra, and for a single spectrum, Psurv (t) is determined by a density distribution of which may be amended by nite size eects and local uctuations. For this reason, the distributions shown in gures 3.2-3.6 collect the decay rates of many spectra.

50

10 10 10

wj

10 10

10 10 10

11

13

15

10

15

10

13

10

11

10

j [a.u.]

10

10

10

10

Fig. 3.13: Expansion coecients wj vs. decay rates j for initial states prepared at the sites jinit = 900 (plusses), 999 (circles), 1000 (crosses clustered along the diagonal) of a 1D Anderson model of sample length L = 1000 (see (3.8) and main text); results for 10 realisations of the random on-site potential are gathered together. Only in the special case of the initial state placed right at the edge of the sample, jinit = 1000, there is a one-to-one relation between decay rates and expansion coecients!

state |0 is very close to the boundary of the atomic sample, i.e. to the continuum threshold (see gure 1.1). For a generic choice of the initial state within the sample, not very close to the last nearly resonantly coupled bound state below threshold using the picture of Rydberg levels coupled via multi-photon chains to the continuum there is no support for a proportionality w . This is illustrated in gure 3.12, where the distribution of the weights in the ( , w ) plane is shown, on a doubly logarithmic scale, for |0 = |n0 = 100 in 1D (a,b), and |0 = |n0 = 70 0 = 0 in 3D driven hydrogen (c). w and are essentially uncorrelated. This is consistent with the observation that phase space localisation properties and decay rates of strongly driven Rydberg atoms are not unambiguously related [112, 113]. To understand better the role of the proportionality of decay rates and weights, we confront our results for the microwave-driven hydrogen problem with a simple, 1D Anderson model [116, 197, 221] which is sketched in gure 3.1. This model easily allows for an arbitrary shift of the initial population inside the

sample. It is dened by the Schrdinger equation o i dj (t) = Vj j (t) + j+1 (t) + j1 (t) , dt

51

(3.8)

where the j (t) are the probability amplitudes for a particle to reside at site j of a 1D sample of length L. The on-site potentials Vj are chosen as a random sequence of uniformly distributed values [1.5, 1.5]. This choice models disorder. The coupling to the lead is mediated by requiring absorbing boundary conditions at the right end, i.e. by adding a small imaginary part i = i 0.31 to VL on the last site j = L. All these choices of parameters are not crucial for the results collected in gure 3.13. They were made such as to mimic a localisation scenario a la Anderson, with signicant leakage into the lead at j = L. ` The initial excitation is a -like wave packet launched at some site within the sample: j (t = 0) = 1, j = jinit 0, otherwise. (3.9)

We put the initial site systematically closer to the end j = L, and the decay rates of the eigenstates of the open tight-binding model are calculated by a diagonalisation routine for symmetric, non-Hermitian matrices (NAG F02GBF routine). Figure 3.13 shows that the generic situation met in the ionisation of driven hydrogen atomic states (gures 3.2-3.6) continuously evolves into one for which expansion coecients and decay rates are indeed strongly correlated . If the initial state is placed well inside the sample, expansion coecients and decay rates do not show any correlation. However, the distribution in the (, w) plane collapses onto an almost straight line wj j when the initial state approaches the open end at site j = L. It is the latter situation which was modelled in recent numerical calculations for periodically driven hydrogen atoms additionally subject to an intense static eld. Such a eld eectively shifts the energy of the initial state closer to the atomic continuum by lowering its ionisation potential. The absorbing boundary conditions used in [115] make contact to the above introduced Anderson model. Choosing the initial state close to the continuum basically reduces the problem to a perturbative relation between the decay rates j and the expansion coecients j of the site amplitudes with the lead giving wj j [221]. The strength of the static eld required by [115] to put the threshold in the vicinity of the initial state (n0 = 60 with nc = 64 [115]) would be of the order of the microwave amplitude F . In such a situation, the number of quasi-resonantly coupled states, in the spirit of gure 1.1, is extremely small, what leads a large localisation parameter L 3.5 [115]. The sample length (3.2) then is L < 1 [115], and the limit of many lattice sites necessary for the Anderson scenario is not fullled. Therefore, a dynamically localised distribution of the wave

The correlation indeed was conrmed numerically by standard tests from section 14.5 and 14.6 of [223]: while for the case j = jinit the tests were positive, the correlation found otherwise was not signicant (as in the case of hydrogen decay rates and weights).

52

10

0

Psurv(t)

10 10 10

(a)

10

0

10

10

10

10

10

10

10

10

10 10 10 10 10 10

1 2 3 4 5

(b)

10

10

10

10

10

10

10

10

t [field cycles]

Fig. 3.14: Survival probability Psurv (t) obtained from equation (2.40) for 1D hydrogen. Initial states |n0 = 40 (a), and |n0 = 100 (b); localisation parameters and driving frequencies as follows: (a) L = 0.2, 0 = 2 (solid line), 0 = 2.25 (dashed), 0 = 2.5 (dash-dotted); (b) L = 0.5, 0 = 2.5 (dotted), L = 1, 0 = 2 (solid line), 0 = 2.5 (long dashed), and L = 2, 0 = 2 (dash-dotted), 0 = 2.5 (short dashed).

functions in energy space cannot develop, and the only hope for a decay law Psurv (t) t1 arises from states in the regular region of phase space. After an initial fast drop of the survival probability, which is induced by the one-photon transition to the continuum, the regular states might lead to a t1 -decay. Given the results in gures 3.12 and 3.13, for a general choice of the initial population, we can no longer expect a universal time decay for the survival probability Psurv (t) of microwave-driven Rydberg atoms, in spite of the universal features of the decay-rate distributions ( ). Psurv (t) is shown in gure 3.14 for the 1D model atom, and in gure 3.15 for real 3D atomic hydrogen. The algebraic decay of Psurv(t) t is generic, arising from the broad distribution of w and over many orders of magnitude. Even for one single realisation of eld parameters, the numerically computed spectrum consists of about 100 . . . 200 eigenstates for the 1D, and of approximately 2500 for the 3D case. All these states are summed in (2.40) to obtain Psurv(t). However, the decay exponent varies considerably from one realisation to the other. We also averaged the survival probability, for xed L, over 500 realisations of the parameters , F in the 1D case. The averaging smoothes the uctuations which arise from the sensitive dependence on , and F [110112, 208210]. In

53

10 10 10

Psurv(t)

10 10 10 10 10

10

10

10

10

10

10

10

10

t [field cycles]

Fig. 3.15: Survival probability Psurv (t) (2.40) of 3D atomic hydrogen, n0 = 70, m0 = 0, 0 = 0 (solid line), 15 (long dashed), 45 (short dashed), 0 = 1.856 (/2 = 35.6 GHz), L = 1.0 [116]. Dierent w -distributions are realised by changing the angular momentum 0 of the initial atomic state, always leading to near-algebraic decay (like in 1D), though with distinct (and non-universal) decay exponents. Only ctitious expansion coecients w / < > induce asymptotically a universal power-law decay Psurv (t) t (dash-dotted line), 2 1.1 (indicated by the full line). , < >, and 0.9 are extracted from the . 1010 a.u. part of the L = 1.0 distribution in gure 3.6. The power-law sets in at ton 104 eld cycles. This time corresponds to the inverse of the maximum up to which ( ) 0.9 prevails, i.e. ton 1010 a.u.

the derivation of (3.7), these uctuations are eectively averaged as well. The results are found in gure 3.16, and they show a much smoother decay in time than the survival probabilities in gure 3.14. Moreover, the averaged curves have a systematically increasing decay exponent with increasing L, and is clearly not determined solely by as suggested by (3.7). The universal exponent = 2 (3.7) is observed only in accidental cases [95], or if we articially reshue the expansion coecients according to w w / , with being the mean ionisation rate averaged over the entire Floquet spectrum [116]. In the latter, unrealistic case, illustrated in gure 3.15, Psurv(t) asymptotically decays with exponent 1.1. This perfectly matches the prediction = 2 with = 0.9 for the small . 1010 a.u. part of the corresponding distribution ( ) in gure 3.6. The time ton when the power-law decay Psurv (t) t1.1 sets in corresponds to rates . 1010 a.u. Up to these rates ( ) 0.9 is observed in gure 3.6. In atomic units

54

Average P(t) over frequency 0=2.02.5 500 values

10 10 10

Psurv(t)

10 10 10

10

10

10

10

10

10

10

10

t [field cycles]

Fig. 3.16: Frequency averaged survival probabilities Psurv (t) of 1D hydrogen. Initial state |n0 = 100 , frequency range 0 = 2.0 . . . 2.5, in 500 equidistant steps. Localisation parameters: L = 0.5 (dotted), L = 1 (solid), L = 2 (dashed). A systematic increase of the decay exponent with increasing L prevails, and the curves are smoother than the data for solely one xed frequency. For better comparison, the dash-dotted line repeats the survival probability for L = 2 and 0 = 2.0 from gure 3.14. The thin line shows a power-law scaling t0.9 , which, in general, does not match well with the data.

ton 1010 a.u., which corresponds to 104 eld cycles in gure 3.15. Since by construction w , it is not surprising that the asymptotic decay of Psurv (t) is solely determined by the contribution of small in ( ). A similar expectation for the real survival probabilities is bound to be valid at most approximately, because the weights w are then not correlated with the decay rates (see gure 3.12). This leads to a more complex mixture of time-scales induced by the nature of the prepared initial state of the atom. To summarise the discussion of the decay properties of strongly driven Rydberg atoms, we can state that the universal features of the decay-rate statistics (of classically chaotic quantum transport) generally do not carry over to the time decay of the survival probability of an initial state prepared at an arbitrary location in phase space. This observation expresses the essential decorrelation between the phase space localisation properties of quantum states, and their asymptotic coupling to the continuum. The ionisation is mediated by both sets, the ionisation rates and the overlaps w in (2.40). These two uncorrelated sets determine only together the decay in an unambiguous manner.

55

3.2

Experimental tests

Citius, altius, fortius. The Olympic motto

Our numerical analysis of the previous section clearly supports the analogy between transport in Anderson-localised models and the ionisation process in driven hydrogen Rydberg atoms. An experimental verication of our results seems feasible. Two experimental options which may quantify the mentioned analogy beyond the threshold scaling in gure 1.2 are discussed in the following.

3.2.1

Status quo

As stated in the introduction (section 1.2.1), the behaviour of the eld threshold, at which a certain percentage of atoms ionises, as a function of the scaled microwave frequency 0 = n3 , may not necessarily originate from dynamical 0 localisation. The experimental results [13, 50, 51] are consistent with the theory of dynamical localisation, but also other mechanism, like the mentioned semiclassical eects, may stabilise the atom against ionisation for 0 & 1 [72, 77, 87, 103, 104, 106, 107]. Similarly, the experimental data [114] showing an algebraic decay of the survival probability of the atoms in the eld, is fully consistent with our ndings. In the preceding section, we saw that the slow power-law decay Psurv(t) t may be caused by the algebraic decay-rate statistics ( ) 0.9 . For L . 0.25, ( ) was shown to be determined by Floquet states localised in the chaotic component of phase space. Therefore, we can interprete the power-law decay of Psurv(t) as a consequence of dynamical localisation (e.g. the data shown in gure 3.14). However, this nding is again an indirect experimental proof of Anderson localisation. We cannot predict an exact exponent from arguments based on the ionisation-rate distribution alone. Moreover, also classical model calculations, for systems with mixed regular-chaotic phase space, show an algebraic decay of the asymptotic survival probability [219]. The power-law then results from the multiple time scales involved when trajectories, which are launched in the chaotic component, get trapped in the vicinity of regular regions in phase space [114, 224234].

3.2.2

Floquet spectroscopy

To conrm the presented numerical results on the decay-rate distributions in experiments, one needs to measure a broad range of ionisation widths. Resonance poles in the complex energy plane were studied experimentally for the wave propagation in microwave cavities [236]. A large number of resonances was obtained from measuring reection coecients vs. the initial excitation

56

Fig. 3.17: Photo-absorption cross section from the ground state of three-dimensional atomic hydrogen to an energy interval between the n = 21 and n = 22 manifolds dressed by a microwave eld of frequency 710 GHz, and amplitude F 1000 V/cm [235]. The arrow marks a very narrow resonance line corresponding to a Floquet state that is situated mainly within the primary resonance island of classical phase space [77, 235].

energy of such cavities [42, 236, 237]. For beams of Rydberg atoms, which interact with static elds (electric and/or magnetic), photo-excitation spectra were measured by means of laser spectroscopy [26,29,34,71]. From the photo-excitation cross sections one can deduce the decay rates if the resonance peaks do not overlap. This criterion is typically fullled for the decay rates of hydrogen Rydberg atoms which are subject to a time-periodic microwave. The distributions in gures 3.2-3.5 show that . n3 , with the average spacing of the energy levels. 0 For atoms interacting with a microwave eld, the spectrum may be probed by an additional laser, which excites atoms starting from a low lying state |0 , e.g. the ground state |n0 = 1 , to some nal highly excited Floquet state. This method of Floquet spectroscopy has been proposed in [235, 238241]. The photo-absorption cross section is given by [185, 241]:

+ m| T

(probe) Im

m=

|0

+ m E0 probe

(3.10)

where | m are the Floquet states (see section 2.1.1), and T = er the electric dipole operator, depending on the polarisation e of the probe laser. E0 is the energy of the initial state |0 , which basically does not couple to the microwave

57

since the latter is chosen to be negligibly weak, for states with principle quantum numbers n0 1. Scanning (probe) over the energy range (of one photon of the dressing microwave eld), a reaction curve as presented in gure 3.17 is obtained. Individual peaks hardly overlap what allows one to extract the resonance widths. The coupling matrix elements in the numerator of (3.10) lead to a biased selection of possibly observable resonances. This bias may be controlled by varying the polarisation of the probe eld, and the initial state |0 to focus on dierent parts of the spectrum according to their overlaps with |0 [71]. From the data of the 1D and 3D decay-rate statistics, we saw that the distributions are quite robust with respect to the dimensionality of the problem, the eld parameters at xed L, and also when selecting only a fraction of the complete spectra (gure 3.7). Hence, we conclude that the decay-rate distributions may be constructed from experimental spectra, given that measurements are performed for a suciently large number of realisations, as done in the numerical experiments.

3.2.3

Further quantitative support for the analogy between transport in energy space and the conductance across Anderson-localised one-dimensional wires has been provided recently. In [110] an atomic conductance was dened by g 1 w , (3.11)

with the mean level spacing . Comprehensive numerical data [111,208] showed that g obeys a log-normal distribution for localisation parameters L < 1. The statistics was performed in the same manner as for the decay-rate distributions, i.e. by varying the eld parameters and F , while L is kept constant. The log-normal distribution was predicted for one-dimensional Anderson models in the localised regime [242244]. The conductance (3.11) may be interpreted as the complete ionisation rate, dened by the weighted sum over the individual decay rates of the Floquet states. Therefore, it contains alike the ionisation probability the local information on the initial state. The statistical distributions of the atomic conductance indeed show a systematic dependence on the initial principal quantum number n0 . For n0 . 60, nite size eects lead to worse ts of the data with lognormal distributions. The sample length scales like L n0 for xed 0 = n3 0 (cf. (3.2)), and hence L(n0 60, 0 2) . 10 is too small to quantitatively compare the ionisation problem to the Anderson scenario as suggested by gure 1.1. The decay-rate distribution proved to be rather insensitive to such nite size eects. For n0 = 40 in gure 3.2, a systematic dierence as compared to the cases n0 70 cannot be observed. Consequently, ( ) is a more direct

I.e. the logarithm of g is distributed with a Gaussian density. Hence the corresponding uctuations are huge, and typically range over several orders of magnitude [111, 208].

58

10

3 2 1 0

w(w)

10 10 10

(a)

10

1 5 4 3 2 1

10

10

10

10

10

10 10 10 10 10

3 2 1 0

(b)

10

10

10

10

10

w

Fig. 3.18: Distribution of the weights w (see denition after (2.40)) of the spectral expansion corresponding to the data shown in gures 3.2 and 3.4, respectively; for the initial states n0 = 40 (a), and n0 = 100 (b), and for the localisation parameters L = 0.2 (plusses), L = 0.5 (stars), and L = 1 (circles). The dash-dotted lines show the scaling w (w ) w1.1.

indicator for the analogy to the Anderson problem, because the Floquet spectrum does not depend on the initial bound state. In both statistics, ( ) and of the atomic conductance g, the signatures of localisation a la Anderson are lost ` for too large localisation parameters L & 1. This manifests in the knee-like structure which appears in ( ), for 1010 . . 108 , and in the increasing deviation of distribution of g from the log-normal one. In section 3.1.3 the weights w were found to be uncorrelated with the decay rates (gure 3.12). Now the w enter in the denition of the atomic conductance (3.11), and their statistical behaviour may be extracted from the known distributions of the conductance g (log g) [110, 111, 208] and of the decay rates ( ). Figure 3.18 presents numerically obtained distributions w (w ) corresponding to the data shown in gures 3.2 and 3.4. We observe that w (w ) does not signicantly depend on the initial state. Moreover, there prevails a slight but systematic dependence on the localisation parameter L: while, for L = 0.2, w (w ) w1.1 over about 3 orders of magnitude, the distribution changes with increasing L such that intermediate weights with 103 . w . 102 gain more and more signicance. The power-law distribution for the well localised case L = 0.2 may be motivated in the following way. Assuming that log g and log

59

are independent random variables, as implied by gure 3.12, and furthermore that g w it follows [198]: (log w ) = d log ,g (log , log w log ) d log (log )g (log w log ) d log G(log w log ) 1 , (3.12)

where (log ) = ( ) |d /d log | 1, for ( ) 1 . G denotes a Gaussian probability density, arising from the log-normal distribution of g. This nally yields d log w 1 w (w ) = (log w ) , (3.13) dw w neglecting that g (3.11) has to be summed over the entire spectrum. The simplied assumption g w leads, however, to a power-law decay as observed in 3.18. The above analysis shows that the distribution of the weights are the same as obtained if w , together with ( ) 1 , is fullled, in spite of the fact that the decay rates and the weights are uncorrelated for our system! This suggests that there is more information in the system than it is contained in the statistical distribution of the rates, of the weights, and of the atomic conductance alone. What is missing is a clear experimental verication of the numerical results on the conductance uctuations! Experiments with rubidium Rydberg atoms were performed, however, considerable uctuations have not been detected [245]. The ionisation probability was measured as a function of the scaled frequency, for xed interaction time t, and approximately constant eld amplitudes. The latter depend on the frequency in a waveguide or in a microwave cavity, and must be exactly calibrated. On the other hand, the numerical conductance uctuations proved to be quite robust with respect to relative uncertainties in the eld strength up to about 5%. Therefore, the experimental data most probably did not show uctuations because the scanned frequency range was too small to observe uctuations over several orders of magnitude [111]. In the following, we derive a simple estimate for the minimal range in over which strong uctuations are expected. We start from the atomic transport scenario sketched in gure 1.1: the ionisation of the initial state is mediated by a subsequent chain of one-photon transitions connecting resonantly lying levels. A change in the driving frequency may be viewed as a shift of the photonic ladder of quasi-resonantly coupled states [91], whereby the most sensitive region is the one closest to the continuum threshold. For small enough changes of the frequency , all states on the ladder remain the same, apart from the last unperturbed level nex just below threshold. The mean level spacing scales as 1/n3 (for the 1D atom; for the 3D case, and if the eld is strong enough to considerably lift the degeneracy in angular momentum, the scaling is closer to 1/n4). Hence, must be larger than n3 such as the last ladder level ex does change when varying by . Simultaneously, & 1/2n2 , so that we ex

60

continuum threshold

n3 ex

1 2n2 ex nex

Fig. 3.19: Quasi-resonantly coupled states (long horizontal lines) close to the continuum threshold. The short lines represent the non-coupled unperturbed hydrogen Rydberg states. nex is the last ladder state from which a single one-photon transition leads to ionisation.

arrive at the estimate for the minimal frequency change, which may produce a signicantly dierent ionisation scenario, and hence a large variation in the ionisation probability or the atomic conductance: & (2)3/2, or in scale units 3/2 0 & n0 (20 )3/2. For n0 = 100, and the lower bound 0 = 2, this gives 0 & 8 103 . Actual numerical computations of the atomic conductance show that dramatic uctuations over several orders of magnitude occur down to a scale 0 103 for n0 = 100 [111, 208]. For decreasing initial quantum numbers n0 < 100, 0 becomes systematically larger. Therefore, an experimental investigation must be performed over a frequency range 0 0 which is suciently broad in order to detect atomic conductance uctuations (at a resolution of 0 ). The precise value of 0 depends on the atomic species used in the experiment, as well as on the chosen initial state. The reason is that the local level spacing depends on the atom (hydrogen, rubidium, etc.), and the ionisation dynamics depends on the prepared initial state via the overlaps w in (2.40). Scanning the frequency in a waveguide is simple, but to keep good control over the eld amplitude calibration to fulll the constraint of constant L is experimentally challenging [53, 73, 246]. The extreme uctuations of the atomic conductance as a function are predicted in a range of microwave amplitudes and frequencies for which the classical transport is suppressed by dynamical localisation. In dierent contexts it was noticed that classical chaos plays an important role in the theory of quantum scattering, including mesoscopic conductance uctuations [18,37,247]. Self-similar fractal uctuations have been detected for transmission processes through mesoscopic nanostructures [248250]. They were explained based on a semiclassical theory which is built on the hierarchical structure of mixed regular-chaotic phase space [67, 68, 251, 252]. Another approach [253] predicts well dened conditions on the statistical properties of resonance poles which aord fractal uctuations, independently of semiclassical arguments: statistically independent sequences of strongly overlapping resonances, which obey a power-law () ( > 0) at small , are proven to be sucient for the

61

appearance of fractal uctuations. Microwave-driven hydrogen Rydberg atoms are characterised by (a) mixed regular-chaotic phase space, as well as by (b) a power-law distribution of the decay rates. The latter is observed in the regime of dynamical localisation, and for states in the regular region where tunnelling determines the distribution (see section 3.1.2). Nonetheless, self-similar structures of the atomic conductance uctuations over many orders of magnitude in 0 are not observed. One reason is that the resonances are typically non-overlapping, i.e. . , in the range where a clear power-law ( ) dominates. For instance, in gure 3.4, 1015 . . . 109 a.u., while 106 a.u. In ( ) 0.9 is obeyed for addition, the above argument based on the photonic ladder picture provides an estimate for a lower bound of the frequency variation below which a signicant change in the ionisation signal is not expected. This suggests that a fractal structure over many scales in the microwave frequency cannot appear.

Part II: Quantum resonances and the eect of decoherence in the dynamics of kicked atoms

Creator Deus mathematica ut archetypos secum ab aeterno habuit in abstractione simplicissima et divina, ab ipsis etiam quantitatibus materialiter consideratis. J. Kepler (see [196])

Chapter 4

4.1 Quantum resonances in experiments

In the two periodically driven systems studied in this thesis, the dynamical evolution may substitute for the intrinsic disorder present in Anderson models. For the -kicked rotor, this can be seen directly from the mapping on the tight-binding equations (2.16). Experiments with cold and dilute atomic ensembles, as described in section 2.3, demonstrated the eect of dynamical localisation that manifests (a) in a stationary momentum distribution, and (b) in the cessation of the linear growth of energy as a function of time. Both of these signatures occur after the quantum break time tbreak k2 [63, 80, 106]. On the other hand, the quantum resonances of the -kicked rotor are much more dicult to access experimentally. A avour of the experimental limitations has been provided in section 2.3.3. The expected ballistic motion of the atoms (i.e. their energy should increase quadratically in time) is fast. This makes resonant atoms escape the experimentally observable, restricted momentum window on a relatively short time scale. Moreover, while dynamical localisation is a rather robust phenomenon, the quantum resonances are very sensitive to slight detunings in the kicking period , and, in addition, they depend on the quasi-momentum (see end of section 2.2.3). As a consequence, the correspondence between theoretical expectation and experimental data is much less convincing in the resonant case than in the localised one. In the following, three experimental observation are discussed which do not match directly with the theory of the -kicked rotor.

65

66

47 27

(a)

mean energy

7 48

28 8 50 30 10

(b)

(c)

10

15

20

Fig. 4.1: Mean energy (in units of (2~kL )2 /M , c.f. (2.24)) vs. the kicking period for an experimental ensemble of kicked atoms, after 30 kicks, and with kicking strength k 0.8. (a) no decoherence, mean number of spontaneous emission events per : nSE 0, (b) nSE 0.1, (c) nSE 0.2. The duration of the kicking pulses was dur = 0.047. The momentum window used to compute the energies from the momentum distributions was bounded by ncut = 40 (solid) [82], or ncut = 60 (dashed) [159], respectively. The shown range of corresponds in laboratory units to 6.5 . . . 210.5 sec; any signal above a threshold of 20 counts/atoms in the momentum histogram was included to obtain the energy, and for each value of the energy results from an average over 10 (solid) and 3 (dashed) repetitions of the experiment, respectively. The vertical uctuations as well as the oset with respect to to the numerical data shown in gure 4.6 originates in various experimental problems (see partly section 2.3.3 or [135, 173]). Comparing the two data sets for each value of nSE shows quite plainly how sensitive the mean energy depends on the chosen thresholds and experimental realisations [83,159]. Experimental data by courtesy of Michael dArcy and Gil Summy.

Linear instead of quadratic growth of mean energy?

67

The rst mystery, what concerns the experimental observation of quantum resonances, is related to the momentum distribution of the atomic ensemble. Ballistic peaks were reported by Oskay and co-workers [81], arising only from a tiny fraction of the atoms, which indeed shows a quadratically increasing energy. This is in contrast to the bulk of the atomic ensemble that is instead frozen in a rather narrow momentum distribution. In other words, no quadratic growth in the average energy of the atomic ensemble has been observed at = 2, 4 in [81], because only a few atoms with the correct quasi-momenta do follow the resonant motion. The resonant values are = 1/2, and = 0, 1/2, for = 2 and = 4, respectively (cf. sections 2.2.2 and 2.2.3). Quantum resonances occur whenever the kicking period is a rational multiple of 4. In experiments [82, 83, 135,159], only at the values = 2, 4, 6, a special behaviour has been found. Figure 4.1(a) shows experimental data of the mean energy of a large atomic ensemble ( 106 atoms [83, 135]) as a function of the kicking period . The quantum resonances correspond to the tiny peaks at the above mentioned values of . Only the dashed curve clearly resolves the peak at = 2. In the present and the next chapter, we are interested in a detailed understanding of these resonance peaks.

Higher-order quantum resonances? The overall global structure, apart from signal-to-noise uctuations, in the mean energy vs. is well understood. Time-dependent correlations between subsequent kicks of the -kicked rotor evolution induce an eective diusion constant De k2 sin2 ( /2) [57, 227, 254256], and the mean energy saturates after the quantum break time, i.e. for t > tbreak De [6, 56, 63, 64, 80, 256], for all irrational values /4. The ne peaks in the experimental data on top of this global structure do change from one experimental run to the other, and hence no higher-order resonances are visible. We suspect that the higher resonances are too sensitive in order to be resolved by the experiments, because their quasi-energy spectra () have a much smaller bandwidth than (2.15) for the fundamental resonances [64]. This means that they manifest clearly only after a relatively long evolution time, which is larger than the typical times of the published experimental data t . 30 kicks [8183, 135, 159]. In addition, the widths in of the higher-order resonance peaks are expected to be even smaller than the ones visible at fundamental resonances = 2, 4, 6.

Enhancement of resonance peaks by decoherence? The third puzzle occurs when adding noise that destroys the phase coherence in-between the imparted kicks. The noise is introduced in a controlled way by forcing the atoms to emit spontaneously in the presence of an additional laser which is near-resonant to an internal electronic transition of the atoms.

68

This laser may induce a transition, followed by spontaneous emission (SE). The spontaneously emitted photon can depart in an arbitrary direction in position space [211], and the whole process results in a random shift of the centre-ofmass momentum of the atom. The additional laser beams are switched on immediately after each kick, for a duration SE 0.067 [82, 83, 135]. The mean number nSE of SE events per atom and per kicking period was varied between nSE 0 . . . 0.2. The surprising experimental result was that decoherence led to a stabilisation and an enhancement of the resonance peaks in the mean energy vs. . Instead of destroying the peaks, decoherence helps them to form as can be seen in gure 4.1, for nSE 0.1 in (b), and nSE 0.2 in (c). Qualitatively, we may understand this enhancement eect in the following way. SE events shift the atomic centre-ofmass momenta, as a result of the photon recoil imparted along the relevant axis dened by the kicking potential. These random shifts destroy the conservation of quasi-momenta, see section 2.2.3. Therefore, SE amends the dynamics in a much more intriguing manner than other types of noise (e.g. amplitude noise as random, time-dependent, but spatially uniform uctuations in the kicking strength k [180,181]) that do preserve the discrete momentum ladder structure. The latter roots in the spatial periodicity of the kicking potential. Since the quantum resonance conditions depend on quasi-momentum (see end of section 2.2.3), SE shues atoms in and out of the resonant motion. That this reshufing actually leads to an enhancement of the mean energy peaks originates from the form of the full atomic momentum distributions and the experimental imperfections discussed in section 2.3.3. Our goal is to clarify the above mentioned experimental observations, and to identify quantitatively the physical origin of these puzzles. To this end, we analyse in mathematical detail two characteristic quantities of the dynamical evolution of an atomic ensemble: the momentum distribution of the atoms and their average energy. The particle dynamics is studied at the fundamental quantum resonances = 2 ( N), both in the absence of decoherence, and when external noise is added in the form of SE. The last section of this chapter then tries to reconcile our analytical and numerical results with the experimental ndings that inspired the present work.

4.2

With the help of the Bloch decomposition (2.19), we can rst restrict to study the dynamics of a single rotor with xed quasi-momentum , and average over all quasi-momenta at the end to obtain the experimental observables. At quantum resonance with kicking period = 2 ( N), the wave function of an individual rotor can be explicitly derived for all times t. With some mathematical gymnastics, we then obtain the momentum distributions and the average energy of an ensemble of rotors from the exact wave functions.

69

Given a xed [0, 1), the Floquet operator (2.23), at = 2 ( > 0 integer), reads 2 U = e i k cos() e i N e i , (4.1) where we used the identity exp( i n2 ) = exp( i n ), and we dened (2 1) mod(2) to be taken in [, ). The n-independent phase exp( i 2 ) is neglected in the following, since it always cancels when computing quantum expectations. The second operator on the right-hand side of (4.1) will be denoted R(). In the representation it acts as a translation [155] according to: (R() )() = ( ) . (4.2) The state of the rotor after the tth kick is then given by the iterated application of (4.1), where R() only shifts the angle coordinate at each stage: t (U )() = e i kF(,,t) ( t) , with F (, , t) =

s=0 t1

(4.3)

t1

(4.4)

where Wt = Wt ()

e i s .

s=0

(4.5)

We denote by n the eigenvalues of the angular momentum N . Then, in the representation, the state (4.3) reads (after changing variable from to N +arg(Wt )): t n |U = e i n arg(Wt )

0 2

If the initial state of the particle is a plane wave (2.22) of momentum p0 = n0 + 0 , takes the constant value 0 = (20 1) mod(2). Substituting (2.22) in (4.6), and computing the integral by means of formula (F.1), the momentum distribution for the 0 rotor at time t is:

2 P (n, t|n0, 0) = Jnn0 (k|Wt |) ,

(4.7)

where Jn (.) is the Bessel function of rst kind and order n. Using the identity 2 (F.2), and Jn (.) = ()n Jn (.) together with n Jn (.) = 1 [157] one computes 2 the expectation value of p (or of the energy by dividing by 2): p2 (n0, 0, t) =

n

2 (m + n0 + 0 )2Jm (k|Wt|) m 2 m2 Jm (k|Wt |) + m m 2 (n0 + 0)2Jm (k|Wt |)

= =

(4.8)

70

Already at this stage, we arrived at a very important result. The momentum distribution or its second moment, the mean energy, is determined by the function Wt . In the noise free case, Wt is completely deterministic, and one must average solely over all initial conditions n0 , 0 to obtain the experimental observables. As will be shown section 4.3, decoherence turns the function Wt into a stochastic process, whose behaviour again completely determines the momentum distribution and the mean energy of the atomic ensemble. Without noise, and for the resonant quasi-momenta, = 1/2 + j/ mod(1), with j = 0, 1, .., 1, and = 0, the function Wt = t, i.e. all terms in (4.5) add up in phase. For non-resonant and irrational /2, the terms in (4.5) add up to a quasi-periodic walk in the complex plane. The periodicity is perfect for rational /2. Now for t , the eect of noise, which randomly changes quasi-momentum, and hence , leads to a random walk in the complex plane given by the summation of the phases in (4.5). The resulting asymptotic momentum distribution will be derived in section 4.3.4. Explicit computation of the geometric series (4.5) yields if 0 = 0: |Wt| = e i 0 t/2 e i 0 t/2 e i 0 t 1 = e i 0 (t1)/2 i /2 e i 0 1 e 0 e i 0 /2 = sin(t0 /2) . sin(0/2) (4.9) As stated above, for 0 = 0 one obtains |Wt | = t. Then the distribution (4.7) spreads linearly in time, and the average kinetic energy increases like k2 t2 /4. 0 = 0 corresponds to the resonant values of quasi-momentum 0 = 1/2 + j/ mod(1), j = 0, 1, .., 1. For any other value of , the distribution changes in time in a quasi-periodic manner. It oscillates in time with the approximate 1 period 0 , inverse to the detuning of 0 from the nearest resonant value. At any time t, the distribution is negligibly small at |nn0 | > k| csc(0/2)| > k|Wt |, since the Bessel functions decay faster than exponentially in this case [157,257].

4.2.1

Momentum distributions

With the conditional distribution (4.7), we can derive the average momentum distribution of an atomic ensemble for a given initial momentum distribution of the atoms. The latter is given by the experimental realisation, and in the following we map it to a distribution for the rotors with [0, 1).

Incoherent ensemble of atoms If the initial state of the particle is a wave packet, then it is a coherent superposition of continuously many plane waves with dierent quasi-momenta, which are non-resonant except for a nite set of values 0 = 1/2 + j/ mod(1), j = 0, 1, .., 1. It can then be proven that the asymptotic growth of energy in time is proportional to k2t/4 [145]. Here we consider the case when

71

the initial atomic ensemble is an incoherent mixture of plane waves. Numerical simulations based on such choices of an initial state have shown satisfactory agreement with experimental data [83,176]. The initial momentum distribution shall be described by a density f (p). We can equivalently consider an ensemble of rotors, with distributed in [0, 1) with the density

+

f0 () =

n=

f (n + ).

(4.10)

In the case when f (p) is Gaussian with standard deviation (a reasonable assumption for the experiments reported in [8183]), the Poisson summation formula [258, 259] yields

f0 () =

m=

f (m)e i2m

2 2

cos(2) + O(e8

2 2

).

(4.11)

Here f denotes the Fourier transform of the density f . For > 1, that is relevant for the experiments referred to in section 4.1, it is practically indistinguishable from the uniform distribution f0 () = 1. Each rotor is described by a statistical state, which attaches the probability f0 ()1 f (n + ) to the momentum eigenstate |n . The momentum distribution P (p, t) of the particle at time t is obtained as follows. For any given 0 [0, 1), averaging (4.7) over the dierent n0 of the ini tial distribution yields the momentum distribution P (n, t|0) for the 0rotor. Weighted by f0 (0), this is the same as the momentum distribution P (p, t) of the particle over the ladder p = n +0 (0 xed, n variable). The on-ladder distributions corresponding to dierent 0 combine like a jigsaw puzzle in building the global momentum distribution for the particle. The result is a complicated function of p which is plotted after t = 50 kicks in gure 4.2. The distribution oscillates on the scale 1/t, which follows from the 0 dependence of (4.7) and (4.9). Nevertheless, on average, this distribution evolves into a steady-state distribution. This may be shown either by time-averaging [160], or by coarsegraining. In the following section, we present the latter approach in detail, and derive the steady-state distribution that is obtained in the asymptotic limit t .

Coarse-grained distribution Because of the above mentioned complicated local (i.e. within intervals n < p < n + 1) structure of the momentum distribution as a function of p, it is hard

Recently, kicked rotor dynamics showing the inuence of timing noise (random uctuations in the kicking period) on dynamical localisation have been reported with < 1 [166].

72

10 10 10 10 10 10 10

0

P(p,t=50)

150

100

50

50

100

150

p

Fig. 4.2: The momentum distribution of an ensemble of 50 particles, with an initial Gaussian distribution of p0 = n0 + 0 (for the experimentally reported standard deviation 2.7 [82, 83, 135]). Plotted as dots are the individual 0 rotor distributions |0 (t, n)|2 (over the corresponding, distinct integer momentum ladders n = [p]) vs. p = n + 0 ; after t = 50 kicks, k = 0.8, = 2. The 0 rotors with the smallest detuning from the resonant quasi-momenta follow the resonant motion for longer times, an example is presented by the solid line. The ones with larger detunings stop following the resonant acceleration, after some time t 50, see dashed line for such a case. For comparison, the dotted line shows the normalised coarse-grained distribution of a larger ensemble taken from gure 4.3.

to characterise the distribution, and apart from the global structure, it is not useful for the comparison with experimental data. One way of removing the fast oscillations is replacing P (p, t) in each interval n < p < n+1 by its integral Pn (t) over that interval. This corresponds to using a bin size 2~kL (c.f. (2.24)) for the observed distributions, which is also the typical resolution in the determination of the momentum distributions in the experiments reported in [82, 83, 159, 260]. Assuming f (p) to be coarse-grained itself, the new distribution is approximately computed in the form:

Pn (t) =

m

(4.12)

where

1

Mn (t) =

0

2 d Jn (k|Wt|) .

(4.13)

73

2 Substituting variables and noting that Jn is an even function of its argument we may rewrite

Mn (t) = =

d 2 J k sin 2 n

t 2

csc

y=xt

dy (2)2

t1 r=0

2 2 y 2r Jn k sin(y) csc + t t t

where in the last step we used the periodicity of the sine. In the limit when t and 2r/t , the sum over r approximates the integral over , and (4.14) converges to

Mn =

1 (2)2

dy

0

2 d Jn (k sin(y) csc()) .

(4.15)

The steady-state coarse-grained distribution Pn is then obtained by replacing (4.15) in (4.12). We do not know if the double integral may be computed in closed form. In appendix A the following (non-optimal) estimate is derived, for the wings of the momentum distribution, valid for any integer N > k: Mn |n|N

ke 16

2N 2N +1

N 1+2N

12N

2+

1 N

(4.16)

Using this estimate it is easy to compute that for k > 1 the total probability carried by states |n| > 4k is not larger than 0.31. Therefore, at large kicking strength k, the distribution is rather narrow as compared to the exponentially localised distribution which is observed far from resonance, because the width of the latter scales like k2 [63, 64, 106]. In appendix A it is further proven that the distribution (4.15) has the following large-|n| asymptotics:

Mn

4k as |n| . 3n2

(4.17)

Such an algebraic decay 1/n2 carries over to the coarse-grained momentum distribution Pn whenever the initial momentum distribution f (p) is fast decaying (e.g. like a Gaussian):

Pn (t) Pn m

4k 3 (n m)2

f (m) as |n| .

(4.18)

The convergence of the coarse-grained distributions to the steady-state distribution is illustrated in gure 4.3, where the evolution of statistical ensembles of particles with an initial Gaussian momentum distribution was numerically simulated. The central part of the distribution quite early stabilises in the nal form of a narrow peak of width k. Away from this peak, the algebraic tail n2 develops over larger and larger momentum ranges as time increases in the wake of two symmetric, tiny ballistic peaks that move away linearly in time. The fall of the distribution is quite steep past such peaks. This is easily

74

10 10

1 2 3 4 5 6

(a)

Pn(t)

10 10 10 10

0

130

260

390

520

10 10 10 10 10

(b)

10

100

1000

n

Fig. 4.3: Evolution of coarse-grained momentum distributions (4.12) at quantum resonance with = 2, and kicking strength k = 0.8, for an ensemble of 104 atoms, without decoherence. The initial momentum distribution is centred Gaussian, with root-mean-square deviation 2.7. (a) Distributions for t = 30 (solid lines), t = 50 (dotted), t = 100 (dashed), t = 200 (dash-dotted). (b) Doubly logarithmic plot of the distribution at t = 1000 (solid line), compared to the asymptotic formula 4k/( 3 n2 ) (dash-dotted line) (4.17). No cutos are used, so the distributions characterise the ideal behaviour of an ensemble of -kicked particles.

understood from the rst equation in (4.14): since | sin(tx) csc(x)| t/2, at n > kt/2 the integrand decays faster than exponentially [157, 257]. The distribution in gure 4.3 (b) has stabilised to the limit distribution over a broad momentum range. Apart from the far tail, where the moving peak structure is still apparent, the distribution follows the asymptotic decay (4.17) already for |n| & 15. Hence, using (4.17), the total probability on states |n| > 40 is 8 103 . From the above analysis, we obtain that all moments of p of order 1 diverge as t , in spite of the onset of the stationary distribution, owing to the slow algebraic decay (4.18) of the latter. For the case of the second moment, which is (apart from a factor 2) just the mean energy of the ensemble, the growth is actually linear in time, as we shall presently show.

75

The mean kinetic energy of an ensemble of rotors at time t is obtained by averaging (4.8) over the initial momentum distribution. In the following, the expectations obtained by averaging over a classical (incoherent) ensemble will be denoted E{.}, while quantum expectations of the energy of one individual -rotor will be denoted E. The mean energy then reads k2 E{E(t)} = E{E(0)} + 4

1

df0 ()

0

(4.19)

As t , the fraction in the integrand, multiplied by /t, tends to a periodic function of ( 1/2) with period 1/ . Thus (4.19) has the following t asymptotics: 1 k2 t E{E(t)} E{E(0)} + f0 ( (j) ) , (4.20) 4

j=0

where (j) = 1/2 + j/ mod(1) are the resonant quasi-momenta. In the case when f0 () 1, i.e. the initial quasi-momentum distribution is uniform in [0, 1), this formula is exact for all times t (by (F.13)): E{E(t)} = E{E(0)} + k2 t . 4 (4.21)

With (4.11), (4.21) is practically exact at all times for an initial Gaussian distribution with width > 1 around the origin. Such a distribution corresponds to the initial momentum distribution in the experiments reported in [8183]. Higher-order energy moments may be likewise computed. For large but nite t, we observe from gure 4.3(a) that the n2 decay of the distribution is truncated at n kt, since the speed in momentum of the resonant rotors is determined by square root of (2.14). Consequently, the variance of energy increases like t3 . The increase of other moments may also be estimated in this way. Higher moments are important for the comparison with the noisy evolution, which we will discuss in the following section.

4.3

One major incentive for the present analysis was the rather counter-intuitive experimental observation that, for a given interaction time, the mean energy was found to increase when spontaneous emission (SE) was added, with respect to the absence of external dissipative noise (see gure 4.1 and [82, 83]). The function (4.5) provides some intuition about what will happen when the quasi-momenta are reshued in a random manner by SE events. By virtue of the randomisation of the phases (4.5) will turn into a random process, whose

76

statistics determine the long time asymptotics of the momentum distribution. The statistical assumptions which we use to model the stochastic -kicked particle evolution are a compromise between physical adherence and mathematical convenience. By comparing our results with more realistic numerical simulations we demonstrate that their validity does not crucially depend on our technical assumptions. In the following analysis we neglect the delay between absorptions and subsequent SEs. We hence assume that the atom undergoes random momentum changes by virtue of SE at a discrete sequence of random times. Both the SE times and the corresponding momentum changes are modelled by classical random variables, independent of the centre-of-mass motion of the atom. This assumption is reasonable as long as the mean number of SEs nSE is relatively small; otherwise, the atoms may, in the average, be slowed down or cooled a velocity-dependent eect, which cannot be accounted for under this assumption. On the other hand, at the high eld intensity of a quasi-resonant laser beam, which is required for large nSE , stimulated emission would prevail. Stimulated emission would induce biased momentum changes along the axis of the SE inducing laser, and hence act in a similar way as the kicking pulses. The statistical atom dynamics may be studied by investigating the stochastic Hilbert-space evolution of the atomic state vector [176, 184, 261264]. More specically, we assume an initial incoherent mixture of plane waves with a distribution f (p0). We further assume that all the variables describing SE events are given and xed. Their specication denes a realisation of the random SE events. Then we compute the deterministic evolution of the atomic state vectors up to time t. The quantum probability distribution of an observable in the nal state depends on the chosen realisation and on the initial state as well; averaging over both yields the nal statistical distribution for that observable. We will see that the dissipative inuence of SE on the centre-of-mass motion is typically small and can be described independently from the eect on the phase evolution in (4.5). These two eects of the coupling to the environment are separated by a stochastic gauge transformation, which is introduced in the next section.

4.3.1

Stochastic gauge

This section shows that, although the random SE events destroy the true quasimomentum conservation, the problem can be formulated in an appropriate gauge for which momentum is not aected by SE. This formally implies the conservation of quasi-momentum, while the random momentum shifts are in This approach was rst implemented for the study of randomised -kicked rotor dynamics in [265]. Please note that in references [176, 184] SE occurs usually during the kicking pulse while in our model corresponding to the experiments of [82, 83] SE is induced by a second, independent light eld applied after each kick.

77

corporated as additional time-dependent phases in the Floquet operator. Let |(t) be the state vector of the atom immediately after the tth kick. Let the integer t denote the number of SEs during the subsequent SE-inducing window. Such events are assumed instantaneous. Denote sj1 the delay (in physical time) of the jth event with respect to the (j 1)th one, and dj the momentum change of the atom induced by the jth SE. For notational convenience a 0th ctitious SE is assumed to occur immediately after the tth kick, and a (t + 1)-th one immediately before the (t + 1)-th kick, with d0 = dt +1 = 0. The state vector immediately after the (t+1)th kick is, apart from an inessential phase factor,

2 |(t + 1) = S(t)e i k cos(X) e i (P +t ) /2|(t) ,

(4.22)

where:

t

t =

j=1

sj

dk , t =

k=0 j=0

dj , S(.) = e i (.)X .

(4.23)

Note that the rst and the second operator on the right-hand side of (4.22) commute because both only depend on X. In addition, the translation operator S has the property S(+) = S()S(). The conservation of quasi-momentum t), which shifts the momentum ladders of the is broken by the operator S( Bloch decomposition (2.19). However, conservation of quasi-momentum may be restored by means of the substitution: |(t) = S(0 + 1 + ... + t1 )|(t) (4.24)

which is a time-dependent, momentum shifting gauge transform; the resulting gauge will be termed the stochastic gauge in the following. Replacing (4.24) in (4.22), and using e i (P +)

2 = S(t )e i k cos(X) e i (P +t ) /2 S(0 + 1 + ... + t1 )|(t)

P s i P +t + i k cos(X) s=0

t1

/2

|(t) (4.25)

Therefore, in the stochastic gauge the state is propagated in the following manner: 2

|(t + 1) = e i k cos(X) e

P i P +t + s

t1 s=0

/2

|(t) .

(4.26)

This evolution does preserve quasi-momentum (in the same way as does (2.8) by (2.12)), and the reduction to rotor dynamics may be performed as described in section 2.2.3, leading to | (t + 1) = U (t)| (t) ;

2 U (t) = e i k cos() e i (N +t ) /2 ,

(4.27)

78

where

t1

t = + t +

s=0

s .

(4.28)

In this way the stochastic evolution (4.23) has been separated in two parts. One of these is described in (4.24) by the operator S; the other is the evolution (4.27) in the stochastic gauge. The former is just a translation (in momentum) by the total momentum imparted by SE during the considered time. It is the latter part that encodes the stochastic phase evolution.

4.3.2

In experiments, SE occurred at random times within SE-inducing time windows, and there is one such window immediately after each kick. We shall say that a SE event occurs at the integer time t, whenever at least one SE occurs in the window following the tth kick. In this way, the number of SEs may be larger than the number of SE events. The probability of a SE event is denoted pSE . The assumptions (S1-3) below build on the separation of dierent time scales. The experimental time window for SE, SE = 0.067 = 0.424, for = 2 (or 4.5 sec in [82, 83]), is on the one hand very large compared to the time scales given by the Rabi frequency of the atomic transition and by the inverse SE damping rate [82,83]. On the other hand, SE is very small compared to the kicking period. It is therefore reasonable to neglect memory eects inherent to SE processes [266]. A complete randomisation of quasi-momentum after each SE-inducing cycle occurs when the mean number of SEs per period is large, nSE 1: then eective averaging leads to quasi-independent t in (4.28). If nSE is small, then complete randomisation requires that the distribution of the momentum shifts t mod(1) is uniform in [0, 1). On the other hand, t is the sum of a random number of momentum changes owing to single SEs. If these are assumed independent and identically distributed, then each of them has to be uniformly distributed in some interval of integer width. The assumption for the following analysis are now specied as follows: (S1) SE events occurring at dierent times are statistically independent. Hence, the random variables t (t = 0, 1, 2, ...) specifying the total (projected onto the axis of the kicking potential) momentum change produced by SE in the tth kicking period (equation (4.23)) are independent, identically distributed random variables. (S2) The nite duration of the SE-operating windows is negligibly small compared to the kicking period. Hence, st while sj 0 for j < t , so that t t in (4.23), and t 0 + t s in (4.28). Dierent SEs may occur s=0 in the same kicking period, each separately contributing to the total momentum change t recorded in that period; nevertheless, their separation in time is neglected.

79

(S3) The occurrence of a SE event results in total randomisation of the quasimomentum. Given assumption (S2), this is equivalent to assuming that the conditional distribution of the variable t mod(1), given that a SE event occurs at time t, is uniform in [0, 1) (in units of two photon recoils, see section 2.3.1). We further assume a zero mean for the distribution of t . While no further specication is needed for the formal elaborations below, in numerical simulations we shall in fact use a uniform conditional distribution in [1/2, 1/2] (in units of two photon recoils). (S4) As in the SE-free case (section 4.2.1), we assume the initial statistical ensemble to be an incoherent mixture of plane waves. With (S1-3) we derive now the equivalent of (4.7) in the presence of SE. (S4) is used when computing the average kinetic energy and the asymptotic momentum distributions of an atomic ensemble (section 4.3.3 and 4.3.4). Let SE events occur after t0 0 at integer waiting times t1 = 0 , t2 = 0 + 1 , . . ., tj = tj1 + j1 , ... . The variables j (j 0) are integer, independent random variables. Under assumption (S1) they are distributed on the positive integers n with probabilities: (n) = pSE (1 pSE )n1 for n > 0 , (0) = 0 , (4.29)

where pSE is the probability that at least one SE takes place in one kicking period. For all integers t > 0 we dene Nt max{j : tj t}, the number of SE events occurring not later than time t, and N0 = 0. The integer random variables Nt, t 0 dene a Bernoulli process [267]. After such preliminaries, we set out to study the evolution in the stochastic gauge, as dened by equations (4.27). The quasi-momentum of a rotor is constant in time, and for each rotor 0 = (SE events are allowed immediately after kicks, and no kick occurs at t = 0). In the stochastic gauge, the random propagator from time 0 to time t for the rotor is given by the ordered product

t1

US, (t) =

s=0

U (s) .

(4.30)

The subscript S on the left-hand side refers to the stochastic gauge. The one step propagators U (t) are dened in equation (4.27). Note that the U (t) depend on the time elapsed between s = 0 and s = t 1 through the accumulated shifts in t (4.28). Similar to what was done in section 4.2.1 at exact resonance = 2 one may write (cf. equation (4.1)):

U (t) = e i k cos() e i t N e i k cos() R(t ) ,

(4.31)

where t = (2t 1). R(s ) again acts as a translation in the -representation. Although t is not restricted in the interval [0, 1) in equation (4.27), the resonance condition allows for t to be taken in [, ) in (4.31). Under assumption

This process must not be confused with the continuous Poisson process [267] that may be used for the SEs occurring within one and the same kicking period; see section 4.3.5.

80

j

m=0

m mod(2) , (4.32)

where m is the total momentum imparted by the mth SE event. Hence, t = Nt in-between the tth kick and the (t + 1)th one. A realisation of the SE events is assigned by specifying the values of all the SE random variables just dened, which we collectively denote by the shorthand notations for the random momentum shifts and for the random times of SE events. Once the nal observation time t and the realisation are xed, for notational convenience we re-dene Nt1 = t tNt1 . Replacing (4.32) in (4.31), and then in (4.30), we get

t1

US, (t) =

s=0

e i k cos()R(Ns )

(4.33)

By repeated use of

e i k cos R() = R()e ik cos(+) ,

all the translation operator are shued to the left in (4.33). Then the evolution operator up to time t (4.30) may be rewritten in the form:

US, (t) = R 0 + N1 + . . . + Nt1 e i kF(,,,t) ,

(4.34)

where we dened

t1 s

F (, , , t)

s=0

cos +

r=0

Nr

(4.35)

F (, , , t) contains all the shifts in the angle variable in the sum within the cosine. Alike in (4.6) it solely represents a phase in the representation, with the important dierence that F (, , , t) depends on the random variables Nr (0 r < t). To arrive at the analogue expression of (4.7) in the case without SE, we next dene j = j1 m m . Replacing s in (4.35) by s = tj + l with m=0 j = Ns , and summing over j, l separately, we obtain

Nt1 j 1

F (, , , t) =

j=0 l=0

cos( + j + l j ) .

(4.36)

j 1 m

zj =

r=0

i j + i r j

; Zm =

j=0

zj ; Wt = ZNt1 ,

(4.37)

81

Note that Wt here diers from Wt in equation (4.4). Wt is now a random process, which itself depends on the realisation of the random variables (, ). The latter encode the random momentum shifts and the random times of SE events, respectively. By (S4), the initial state of the atom is assumed to have a denite initial momentum, i.e. to be a plane wave of momentum p0 = n0 + 0. Given a realisation (, ), we operate on the corresponding rotor state (2.22) with the propagator (4.34). The (random) state of the rotor at time t is given, in the momentum representation, by: n |US,0 (t)0 t = e i t d i (nn0 ) i k|Wt | cos() e , 2 0 = (n n0 )arg(Wt ) nN (t1) .

2

(4.38)

2 P (n, t|n0 , 0, , ) = Jnn0 (k|Wt|).

(4.39)

This is formally identical to (4.7), but now Wt depends on the initial quasimomentum 0 and on the realisation of the SE events as well. It is a stochastic process and the random state (4.38) performs a random walk in the rotors Hilbert space. Computation of statistical averages requires averaging over the SE random variables (, ), and for an initial atomic ensemble also over the initial momenta p0 = n0 +0 . Under our assumptions all such variables are classical random variables. Expectations (respectively, conditional expectations) obtained by averaging over such classical variables are denoted E{.} (respectively, E{.|.}). For instance, E{.|p0} stands for the average over the SE variables alone, given the value of p0 , or equivalently of n0 and 0 (alternatively 0 = (20 1)). The larget behaviour of the stochastic process Wt that drives the stochastic rotor evolution is ruled by the large m behaviour of the process Zm (4.37). The properties of the latter process are completely determined by the assumptions (S2) and (S3). Together with equation (4.32) assumption (S3) entails that the j are mutually independent random variables, uniformly distributed in [, ) (with the possible exception of 0 , whose distribution is dened by the initial ensemble). This fact has the following consequences, which are derived in the next section and in appendix B: the complex variables zj , zk are independent whenever |j k| 2. Moreover zj , zk are pairwise uncorrelated whenever j + k > 1: (4.40) E{zj zk |} = jk j , E{zj zk |} = 0 . These properties of the zj allow us to compute the expectations and higher moments of the random process |Wt|. Furthermore, this shows that the process Zm is a random walk in the complex plane, which originates from the summation of random phase terms in (4.37). Equivalently, the distribution of Zm approaches an isotropic Gaussian distribution in the complex plane as m (see appendix B). On account of the properties of the Bernoulli process, the process Wt at large t has quite similar features: in appendix C we show that as t , the distribution of |Wt| approaches an isotropic Gaussian distribution

82

in the complex plane centred at 0. The random walk property manifests in a linear growth of the mean energy in time t, as well as in a Gaussian momentum distribution for large t [160]. The asymptotic momentum distribution will be derived in section 4.3.4 below. In the next section, we will proof that the mean energy indeed increases almost alike without SE (4.21), for a uniform initial distribution of quasi-momentum, i.e. for f0 (0) = 1: 1 1 E{E(t)} = E{E(0)} + k2t + D(t 1). 4 2 (4.41)

The additional term D(t 1)/2 arises from the heating eect of SE events, which do not only change quasi-momentum, but also the integer part of momentum. In experiments, this heating is typically very small, and its diusion constant D depends on the mean number of SEs per kicking period (see section 4.3.5). We conclude that the growth of the mean energy is but weakly aected by SE, in contrast to what is observed in the experiments, cf. gure 4.1. This discrepancy between the experimental data and the theoretical prediction will be discussed in detail in section 4.3.6. In the sequel, a more general expression than (4.41) is proven rigorously for an arbitrary initial momentum distribution.

4.3.3

The statistical moments of the random process Wt may be explicitly computed at all t. For instance,

Nt1

E |Wt|2 | =

j,k=0

E {zj zk | }

(4.42)

is calculated by using the relations (4.40), which are proven in the following. We denote:

j 1 r=0

j = e so that

i j

, j =

r , j

zj zk = e i(j k )

j1

j 1 k 1 r=0 s=0

r s j j

m m

k1 l=0

!

j k (4.43)

l l

=e

m=0

j1

= j k

l=0

k1

l l

m=0

m . m

Let j > k, j + k > 1. Then j > 1, and the rst product has the factor j1 j1 . Hence (4.43) depends on j1 via this factor alone (if j = k + 1), or via

83

this factor multiplied by (if j = k + 1), leading to a factor e i j1 (j1 l) j1 with l j1 1. In both cases averaging over the uniformly distributed random variable j1 yields zero (by (4.32) and assumption (S3) j1 is indeed uniformly distributed whenever j > 1). The case j < k then follows by complex conjugation in (4.43). For combinations of the form zj zk , we observe from (4.43) that the factors dependent on j,m can never cancel each other, and therefore they average to zero. This proves the second claim made in (4.40). If, on the other hand, j = k, then 2 j 1 sin2(j j /2) |zj |2 = , (4.44) r = j sin2(j /2) r=0 similarly to (4.9). This gives for the expectation

E{|zj | |} =

2

dP (j )

(4.45)

where dP (j ) is the distribution of j . For j > 0, this distribution is uni form: dP (j ) = dj /(2), and the integral is computed according to (F.13): 2 E{|zj | |} = j . For j = 0, we dene

E{|z0| |} = M(0)

2

dP (0)

(4.46)

l1

f0 (j ) , j

j=0

1 j 0 + + mod(1) 2 2 l

(4.47)

is the distribution of 0 , and f0 is the probability density of the initial quasimomentum (4.10). With these results (4.42) becomes

Nt1 E |zj |2| + 2(Nt1) Re (E{z1z0 |}) , j=0

(4.48)

where (.) is the characteristic function of the strictly positive integers. For the last term one obtains from (4.37) by averaging over the uniformly distributed 1

z1 z0 = 1 1 s=0

e i s1 e i 0 0

r=0

0 1

e i r0

average 1

e i s1

0 s=0

e ir0 ,

r=1

(4.49)

and hence:

E{z1 z0 |} = N (0) 0

dP (0)e i r0 .

r=1

(4.50)

84

Nt1 j=0

The variables j were dened such that (4.46) and (4.50) we nd:

Nt1

E |Wt |2 |

= M(0) +

j=1

To obtain the mean energy of an atomic ensemble subject to SE, we must average over the random SE times . The special case when no event happens up to time t, i.e. 0 = t if t1 > t (with t1 the time of the rst SE event), implies Nt1 = 0, and this gives E |Wt |2 | = M(t) . (4.52)

In general, Nt1 = 0 is equivalent to t1 t, and t1 is distributed according t to (4.29). Therefore, using that the probability for t1 t is given by qSE = t (1 pSE )t , and its compliment by 1 qSE the mean energy equals:

t t E |Wt|2 = qSE M(t) + t(1 qSE ) + C(t, pSE )

(4.53)

n=1

(4.54)

In the case of a uniform initial quasi-momentum distribution, M(t) = t by (F.13) and (4.50) vanishes, hence also C(t, pSE ) = 0, and E{|Wt|2 } = t , (4.55)

like in the case without SE. With a smooth initial quasi-momentum distribution (4.10) it follows from the denitions (4.46) and (4.50) of M and N that

t t

|C(t, pSE )|

n=1

n=1 t

n(n)

3(f0)

n=1

n(n) ,

(4.56)

where (f0 ) is the maximum of |f0() 1| in [0, 1). In the last step, we used for (4.50) the upper bound

0 0 0 1

r=1 r=1

e i r0 (f0)

r=0

e i r0 (f0 )0 .

With a non-uniform initial quasi-momentum distribution, letting pSE 0 at xed t causes the second and the third term on the right-hand side of (4.53)

85

to vanish. If instead t at xed pSE > 0, then the second and the rst term on the right-hand side approach t and zero respectively, exponentially fast; the third term remains bounded according to (4.56). So the result which is obtained with a uniform quasi-momentum distribution is always approached asymptotically. Assuming that the initial state and the SE realisation are given, and denoting t = t1 s , the quantum expectation of the energy of the atom at time t may s=0 be written as: E(t) = = 1 2 1 2 1 dp (p + t )2| p|(t) |2 2 1 2 dp p2 | p|(t) |2 + t + t dp p| p|(t) |2 , 2 dp p2 | p|(t) |2 =

(4.57)

where (4.24) was used. This expression has to be averaged over the initial statistical ensemble and over all SE realisations. Then the standard random walk result (applicable because of the assumptions (S1-3)) is 2 E{t } = D(t 1) , E{t } = 0 , (4.58)

2 where D = E{t } is the mean square momentum change per period owing to spontaneous emission. For an initial plane wave (2.22) of momentum p0 = n0 + 0, with the help of (4.39) one nds

dp p| p|(t) |2 =

n 0

d (n + )| n| (t) |2

=

n

(4.59)

2 n Jn (.)

dp p2| p|(t) |2 =

n

Replacing (4.59) in (4.57), the expectation of the last term in (4.57) vanishes by virtue of (4.58). The expectation of (4.61) is found with the help of (4.53). Thus the nal result for the mean energy is 1 1 t t E{E(t) E(0)} = k2 [t(1 qSE ) + M(t)qSE + C(t, pSE )] + D(t 1). (4.62) 4 2 (4.62) reduces to the SE-free one equation (4.19) for qSE = 1. The term on the right-hand side which includes k2 as a factor is the mean energy in the

86

stochastic gauge. With a uniform quasi-momentum distribution, (4.62) reduces to (4.41), which is identical to the result obtained in the SE-free case, except for the last term in (4.62). However, a similar, albeit cumbersome computation of higher-order moments would reveal sharp dierences, which reect totally dierent ways of the spreading of the momentum distribution in the two cases. While, in the noise free case, the second moment of energy, for instance, increases with t3 in time (see discussion after (4.21)), it would grow like t2 in the asymptotic limit where a Gaussian momentum distribution develops (see next section). Under assumption (S3), the quasi-momentum distribution of the atoms is immediately turned uniform by the rst SE event. The time scale for uniformisation of the quasi-momentum distribution is then tc = 1/ ln(1 pSE ). Equation (4.62) shows that for t tc the growth of energy is linear with the coecient k2/4 + D/2, like in the case of a uniform quasi-momentum distribution. On the other hand, since C(t, pSE ) is bounded in time, for (f0)tc t tc the growth of energy is dominated by the term M(t), which is the same as in the SE-free, non-uniform case.

4.3.4

In this section we assume an initial momentum distribution with n0 = 0. For initial distributions with values n0 = 0, we may average the nal result over the initial integer grid points similarly as done in (4.12). We denote P (p, t) the momentum distribution at time t, and show that, as t , P (p, t) approaches a Gaussian distribution with mean value 0; in the sense that, for an arbitrary smooth function (p), lim t lim = dp P (p, t) p t = lim t dp P p t, t (p) (4.63)

dp (p)GD+k2 /2(p)

where G2 (p) denotes the normal distribution with zero mean and variance 2. To this end we compute by applying the stochastic gauge (4.24), with t = t1 s as dened in the previous section: s=0

t

= =

(4.39)

E E E

n

dp | p|(t) |2 dp | p|(t) |2

2 Jn (k|Wt |)

p t p + t t n + 0 + t t . (4.64)

& 0.

87

For t 1 one may neglect corrections of order 1/ t, hence 0 / t in the argument of the smooth function , so tE

n 2 Jn (k|Wt|)

n + [t] t

(4.65)

where [.] denotes the integer part. Asymptotically as t , the statistics of Wt are determined by the fractional parts of sums of many s (cf. (4.32) and (4.37)). Such sums of a large number of independent terms have a broad distribution, so their integer and fractional parts tend to be independent of each other as the number of terms in the sums diverges. The squared Bessel functions and the function in the last equation may then be separately averaged. Denoting t(p) E we may write t

n 2 E Jn (k|Wt |) t

p + [t] t

(4.66)

n t

(4.67)

As shown in appendix C, the distribution of = |Wt| is asymptotically at large t given by dFt () = 2t1 d exp(2/t). Consequently,

t

dFt()

n

2 Jn (k)t

2

= 2

0

dx xex

n t 2 Jn (kx t)t

n t

(4.68)

The integral over is a classical expectation, but the sum over n is a quantum expectation instead. With (2.12) it may be formally written as It (kx)

n

2 Jn kx t t

n t

t cos()

(4.69)

where K = e i kx

(4.70)

If we regard t1/2 as the Planck constant, then t is equivalent to a classical limit. In that limit t1/2 N corresponds to (angular) momentum p, and K corresponds to p p + kx sin(), by comparison with (2.6) and (2.8). Therefore, the classical limit (t ) for the momentum distribution in the state K |0 is given by the distribution of kx sin(), with uniformly distributed in [0, 2]. Replacing the quantum expectation (4.69) by the average over the related classical distribution yields

t

lim It (kx) =

kx kx

dp 2

(p) k2 x2 p2

(4.71)

88

Substituting this in (4.68), interchanging the integrals, and computing the integral over x gives

t

lim t = 2

0

dx xex I (kx) =

2

2

p2

dp (p)

dx

|p|/k 1 2

xex

k2x2 p2

p2

y2 =(kx)2 p2

1 k

dp (p)e k2

p2

ey dy = y k

dp (p)e k2 (4.72)

1 = k

dp (p)e k2 ,

(p) =

dp (p p )GD (p ) ,

(4.73)

where GD is the limit (t ) normal distribution of t / t. Recalling (4.72) and the denition of t given in (4.63), we immediately obtain the result claimed there. Hence, P (p, t) is asymptotically equivalent to a Gaussian with zero mean and variance k2 t/2 + Dt. Being just the leading term in the asymptotic approximation as t , this misses those terms in the exact result (4.62) which are bounded in time. The way (4.72) was derived from (4.67) shows that decoherence turns the dynamics classical by causing the eective Planck constant to decrease with time. An exact derivation of (4.72) from (4.67), which does not refer to the semiclassical argument from above, is obtained by starting from (4.68), and replacing t n/ t by its Fourier transform t (u): t n t 1 = 2 du t(u)e i nu

t

(4.74)

x=/ t

dFt ()

0

du t (u)J0 2k sin

2

2 2

dx xex

du t (u)J0

u 2 t 2kx t sin

u 2 t

(4.75)

2 = 2

dx xex

du (u)J0(kxu) .

(4.76)

1 (u) = 2

0

dp (p)e ipu

(4.77)

1 = 2

dx xex

dy

1

1 1 y2

2(p kxy) ,

(4.78)

and nally

89

dx xex

kx

dp

kx

(p) k2 x2 p2

(4.79)

Interchanging integrals, we obtain the same result as above (equation (4.72)) by dierent means.

4.3.5

In the preceding sections, we derived the mean energy growth and the asymptotic momentum distribution for an ensemble of -kicked atoms in the presence of SE. Our theoretical model was based on the assumptions (S1-S3) stated in section 4.3.2. We now test the result thus obtained by a direct comparison to numerical simulations that mimic a more realistic stochastic evolution of the initial states. In general, for a single transition in a three-dimensional atom, the probability distribution of momentum shifts produced by SEs is not isotropic [269]. This in particular implies that the distribution of single SE, projected momentum shifts p is not uniform. In the case when the SE-inducing beam is orthogonal to the kicking direction, it has the parabolic form: P0(p) = C 0

9 8

3(p)2 2

, |p|

kT 2kL

(4.80)

, otherwise,

where C is a normalisation constant, and kT kL are the (assumed to be orthogonal) wave vectors of the SE-inducing light and of the kicking light, respectively. This distribution is derived for a situation where SE from a m = 1 atomic transition is induced by circularly polarised light [269]. The allowed change in momentum p is restricted within the interval [kT /2kL, kT /2kL], with kT /2kL 1/2 (resulting in C 1) in [82, 83]. With a non-uniform distribution such as (4.80) a correlation in time may be established between quasi-momenta in dierent periods of the evolution. The mean momentum change as a result of absorption followed by SE is ~kT for a single SE-inducing beam with wave vector kT . Our assumption of zero mean (along the kicking direction) is justied either when kT kL , or when the experimental arrangement uses two or more appropriately directed beams, whereby the atoms may be excited with equal probability. In such cases, the distribution of the projected p is more complicated (and closer to uniformity) than (4.80). Since the experiments use a large ensemble of caesium atoms, and SEs involve several hyperne sublevels [83, 135], the assumption of a nearly uniform distribution of momentum changes seems, however, most appropriate. We shall nonetheless use (4.80) as a term of comparison in type (II) simulations (see below) in order to test the eects of deviations from uniformity. Such numerical data demonstrate that our assumption of a uniform distribution in an exactly integer interval of allowed momentum changes does not aect the results, for experimentally relevant times at least.

90

192

196

192 196

(a)

mean energy

192 196

(b)

0 300

200 100 0 0 50

(c)

100 150 200

t [number of kicks]

Fig. 4.4: Average energy vs. time t at exact resonance = 2 for the same ensemble of atoms as in gure 4.3, for k = 0.8, in the presence of SE events simulated in dierent ways as described in the text: (a) type (I) simulation: a random number of SEs occur immediately after kicks, each causing a momentum change p uniformly distributed in [1/2, 1/2]. (b) type (II): SE times are Poisson-distributed in a window SE = 0.067 , with free evolution in-between them; p is distributed as in (a). (c) type (III): SE times as in (a), but p has the parabolic distribution (4.80) with kL /2kT 0.476. Rates of spontaneous emission pSE = 0.05 (solid), pSE = 0.1 (circles), pSE = 0.2 (plusses). The theoretical prediction (see text) for the coecient of linear growth Ddec is approximately 1.59, whereas the data lead to Ddec 1.58 1.60 except in (b) for pSE = 0.05 where it takes the value 1.55 (strong uctuations arising from a too small statistical sample at this smallest SE rate). The insets zoom into the region close to t = 200. No momentum cutos are used.

91

For the uniform distribution of p in the interval [1/2, 1/2], p2 = 1/12, so the coecient D in (4.58) is D = nSE /12. With the distribution (4.80) p2 = 3/40, and D = nSE 3/40. In the theoretical model based on assumption (S2) the distribution of the random times at which single SEs occur within one kicking period is totally irrelevant, so pSE and nSE enter as independent parameters. They have to be related to each other in order to make contact with experiments. A seemingly natural way assumes a Poisson distribution for the SEs occurring within one operating window, at least for not too large pSE . In that case, pSE = 1 exp(nSE ). We have performed numerical simulations of three types: for type (I) we used all assumptions (S1-3), whereas for type (II-III) the assumptions (S2) and (S3) were replaced by more realistic ones. Type (II) allowed for free evolution inbetween successive SEs occurring in the same kicking period. For type (III) the distribution (4.80) of p was used. Type (I) simulations serve as a demonstration of the theoretical exact results, and much more as a term of comparison with type (II-III) simulations. The essential agreement between the three types demonstrates that our theoretical conclusions remain valid, under less stringent premises. Both types of numerical results were obtained by independently evolving rotors in a given initial Gaussian ensemble, and by incoherently averaging the nal results. Random SE events were simulated as follows. After choosing values for SE and pSE = 1 exp(nSE ), random SE times were generated in each kicking period from a Poisson distribution with the characteristic time SE /nSE within the time window (t, t + SE ). To each random time a random momentum jump was associated, from the chosen distribution (uniform or parabolic). In type (I) simulations, such jumps were added to the quasi-momentum the rotor had at (integer) time t. The integer part of the result determined a corresponding shift in the computational basis of angular momentum eigenstates (i.e. if the sum of quasi-momentum and the shift exceeded one or was negative then (n) (n 1), respectively, with integers n). The fractional part was used as quasi-momentum for a full one-period free rotor evolution. In all cases the computational basis of momentum eigenstates was chosen as large as possible in order to model as faithfully as possible the ideal models analysed in previous sections. Figures 4.4 shows a long-time plot for dierent rates pSE = 0.05 . . .0.2, and for the two cases: type (I) with SEs happening immediately after the kicks (a), and type (II) with SEs within a nite time window (SE = 0.067 = 0.424 [83]) (b). For pSE = 0.2 data is given in gure 4.4 (c) for the parabolic distribution (4.80), i.e. for the type (III) model. The energy growth is in all cases linear with the predicted slope Ddec k2/4 + D/2, as discussed above (see equation (4.62)). Figure 4.5 presents the coarse-grained momentum distributions Pn (t) (see section 4.2.1) dened as the probability that the momentum p of an atom at time t lies in [n, n + 1) (in units of two photon recoils). They are computed for = 2 and dierent SE rates; SE events are modelled after type (I-III). With added decoherence, the distribution keeps spreading as a whole all the time, looking more and more Gaussian-like while it attens out. Apart from statistically induced uctuations in the wings of the distributions no signicant dierence between the dierent simulations (gures 4.4-4.5) is detectable, and

92

10 10 10 10

2 4 6 8

(a)

250

2

125

125

250

10 4 10 6 10 8 10 200 10 10 10 10

2 4 6 8

Pn(t)

(b)

100

100

200

(c)

140105 70 35

35

70 105 140

n

Fig. 4.5: Evolution in time of coarse-grained momentum distributions for the same initial ensemble as in gure 4.3, and for k = 0.8, in the presence of SE. (a) pSE = 0.1, (b) pSE = 0.2, (c) pSE = 0.8, for t = 30, t = 50, and t = 200. The SE events are simulated in dierent ways. Solid lines were computed like in (a) in the previous gure; diamonds, circles and squares, like in (b); plusses and crosses, like in (c) there. Dierences in the wings, in particular at t = 200, where Pn (t) . 107 , are attributed to the nite statistical sample in the simulation of the SE events. For strong noise in (c), the momentum distributions are already similar to a Gaussian distribution as predicted for t in section 4.3.4.

93

our conclusions from the preceding sections remain valid in all cases. We conclude that the results obtained in the present work are not very sensitive to assumptions (S1-3) that made the analytical treatment possible. In the sequel, the analytical and numerical ndings presented in this chapter are confronted with the experimental results of [82, 83, 135]. In particular, the puzzle about the observed enhancement of the quantum resonance peaks when adding SE will be resolved [159].

4.3.6

Omnis actio et omnis mutatio est de contrario in contrarium. G. Bruno, La cena de le ceneri

In the presence of decoherence induced by spontaneous emission, experimentally measured energies at xed observation time tobs were found to exhibit resonance peaks near the resonant values = 2, 4, 6 that were higher than in the SE-free case (gure 4.1). On the other hand, our analytical analysis (equations (4.21) and (4.62)), together with numerical data (gures 4.4 and 4.6), predicts in both cases, with and without added noise by SE, a linear increase of the average energy as a function of time, and in addition with an almost negligible dierence in the pre-factor for the experimentally used parameters (D/2 . 0.2/24, for the data in gure 4.1). Such observations may have been suggestive of an enhancement of quantum resonances owing to decoherence: however, such a phenomenon has no match in the theory developed in the previous sections. This paradox will be resolved in this section. It will be shown that certain restrictions, that are unavoidably present in real experiments, depress the ideal resonant behaviour in a way that is most severe in the absence of SE. So the explanation rather lies with the experimentally measured, SE-free peaks being lower with respect to the ideal case, than with the SE ones being higher. The most important experimental features not taken into account in the foregoing theoretical analysis have been mentioned already in section 2.3.3: (EXI) experimental kicks are not like. The ideal model is then only valid as long as the distance travelled by the atoms over the nite duration dur of the kick is much smaller than the spatial period of the potential (Tdur = 0.5 sec in [82, 83]). In dimensionless units of (2.10) dur = Tdur~(2kL)2 /M 0.047. At large momenta this requirement is violated, and the atomic motion starts averaging over the potential, while the small momentum regime is practically not aected by the replacement of the function by a pulse of nite width. The resulting dependence of the kicking strength on momentum destroys the translational invariance in momentum space required for quantum resonant motion. At large momenta, the eect of the nite pulse width induces a momentum

94

45 30 15

(a)

mean energy

0 40 25 10 45 30 15 0 5 10 15 20

(b)

(c)

Fig. 4.6: Mean energy (in units of (2~kL )2 /M , c.f. (2.24)) vs. kicking period after t = 30 kicks, and for a type (I) simulation starting with an ensemble of 105 kicked atoms, with Gaussian initial momentum distribution ( 2.7) and k = 0.8. (a) no decoherence pSE = 0, (b) pSE = 0.1, (c) pSE = 0.2. The shown range of corresponds in laboratory units to 21.2 . . . 254.7 sec (a,c), and to 10.1 . . . 254.7 sec (b). The peaks appear lower in (a) because the used computational grid in had a too low resolution to hit the value = 2, 4, . . . exactly, while the peaks in (b,c) are broader and therefore less sensitive. (The sensitivity decreases with increasing pSE , see section 5.3.)

boundary which we denoted nref in section 2.3.3. The atom dynamics mimics the ideal lowest-order resonances for a (possibly long) while [175], but not the higher-order ones whose period (in momentum) is not very small compared to nref [64,117,118]. This is an additional reason preventing experimental detection of high-order resonances, no matter how long the observation time (cf. discussion in section 4.1). Figure 4.7 shows a simulation for an ensemble of rotors, with a rectangular pulse shape of width p. This does not include the smooth switching on/o of pulses, as described e.g. in [180]; no substantial dierence is however expected in the dynamics on relatively small time scales. In each kicking interval the rotors freely evolved over a physical time dur. During the remaining time p they evolved according to the pendulum Hamiltonian (N + )2/2 + k cos(). The latter evolution was computed by a Trotter-Kato discretisation [270] of the Floquet operator (this split-operator method [16] is equivalent to replacing the pulse by a thick sequence of subkicks).

40 10 10

1

95

100 10 10 10 10 10

1

20

20

100 50

50

(a)

(b)

Pn(t)

10 10

10 10 10 10 10

1 2 3 4 5 6

(c)

(d)

10 10 10 10

140 70

70

180 90

90

10 180

n

Fig. 4.7: Coarse-grained momentum distributions for the same initial ensemble as in gure 4.3 and without SE, for k = 0.8 and = 2. The ideal case of kicks (solid line) is compared to the case of rectangular pulses with dur = 0.047 (open circles). Times are t = 10 (a), 30 (b), 50 (c), 200 (d).

(EXII) The experimental signal-to-noise ratio allows only a nite reliable interval of momenta to be observed; in [83] this border was ncut = 40 (data with counteracted gravity; in most recent data ncut = 60 [159]). Momenta with n > ncut are not included in the mean energy data of [83], cf. gure 4.10. Therefore, the theoretical momentum distributions have to be appropriately weighted prior to the computation of the mean energy and to the comparison with experimental data. The crudest way is cutting the theoretical distributions beyond ncut and renormalising the probability to 1. The eect of (EXI) and (EXII) on the ideal behaviour discussed in the previous section is easily understood in qualitative terms. We start with the SE-free case. The resonant growth of energy is stopped as soon as the ballistic peak in the tail approaches the closest of the two borders that are the eective cutos: (EXI) nref and (EXII) ncut . If this happens earlier than the observation time, then the resonance peak is signicantly depressed in comparison to the ideal case. We shall presently argue that such depression mainly arises from the cuto (EXII) for the experimental data in [83, 135]. In the case of a rectangular pulse, nref is not a precisely dened quantity because of the slow decay of the Fourier harmonics of the pulse. It has to be meant in an eective sense. We hence resort to numerical simulations. In gures 4.7 numerically computed momentum distributions are compared with those obtained in the ideal -kicked rotor case; according to such data, the eective nref should be located in the momentum range 70 120. The second cuto ncut

96

80

mean energy

(a)

60 40 20 0

(b)

60 40 20 0 0 10 20 30 40 50

t

Fig. 4.8: Eect of nite pulse width and of momentum cuto on the growth of the mean energy for the same initial ensemble as in gures 4.3 and 4.7, for kicking strength k = 0.8 and period = 2, without SE (a) and with pSE = 0.2 (b). Solid lines are for the ideal kicks and no momentum cuto; dashed lines for rectangular pulses, no cuto; dash-dotted lines for rectangular pulses and momentum cuto at ncut = 40. In (a) the energy is signicantly depressed by the cuto after t > 20, not so in (b).

is simulated by not counting momenta higher than ncut when calculating energies (the computational basis of momentum eigenstates is however much larger than ncut). Following experimental parameters [83,135] we choose ncut = 40, the distributions are renormalised to unity after disregarding states with momenta larger/smaller than n = 40. In gures 4.8 the eect of this cuto on the growth of the mean energy is shown. In the presence of ncut the deviation from the ideal case appears somewhat earlier, as expected from ncut < nref ; moreover, the deviation at t & 20 is strongly enhanced in the presence of ncut (and in the absence of SE). As shown in gures 4.9 (a), the momentum distributions including both cutos (EXI) and (EXII) are stable in time, not moving at all in the centre around n = 0. The slight enhancement at |n| 15 40 as compared to the case without cutos (shown in gures 4.7) is only due to nref which to some extent acts like a reecting boundary. The ballistic peak, however, which moves in momentum like n kt/2 (see discussion after (4.18)) is lost already after about t 40/k 16 kicks, cf. gure 4.8. The peak is then beyond the cuto (EXII). The estimated loss after about 16 kicks is consistent with the saturation of the mean energy vs. time at quantum resonance which has been

1

97

10

10

(a)

(b)

10 10 10

2

Pn(t) [renormalized]

10 10 10

(c)

10 10 10

2

(d)

10 10 10

2

40

20

20

40

20

20

40

n

Fig. 4.9: Evolution of coarse-grained momentum distribution for the same ensemble as in gures 4.3 and 4.7, for = 2 and k = 0.8, with rectangular pulses (dur = 0.047) and momentum cuto at ncut = 40. The distributions are renormalised within the shown momentum window. (a) pSE = 0, (b) pSE = 0.1, (c) pSE = 0.2, (d) pSE = 0.8, after t = 10 (solid lines), t = 50 (dashed), t = 200 (dash-dotted).

observed in [82, 135], for t > 15, in the experimental results as well as in the theoretical modelling (gure 4 in [82]). The dependence of the mean energy on the kicking period is strongly inuenced by the cutos at exact resonance. This dependence in the absence of SE, with rectangular pulses and cuto at ncut is shown in gure 4.11(a). By comparing to gure 4.6(a) (ideal case without cutos), we directly see that the only substantial dierence is at resonant values = 2, 4, 6: cutos lead to lower resonance peaks. When the cuto (EXII) ncut = 40 is applied in the ideal case of kicks, no dierences can be detected from the results plotted in gure 4.11(a), again conrming that cuto (EXII) is the crucial one. The resonance peaks are smaller, because the resonant growth of energy stops, as soon as the ballistically moving rotors hit the boundary (cf. gure 4.8). Then the mean energy very quickly falls below its ideal value (after about 16 kicks in the plotted case), as can be seen in gure 4.8. Added SE totally changes this picture. The energy growth originates now in the overall broadening of the distribution, and not just in the ballistic peaks in the tail, as can be seen comparing the various parts of gure 4.5, where kicks are like, and no cuto is present. The distributions with weak SE are broader in the tails as compared to those with strong SE; the latter are however atter in the centre, which is why they have roughly the same root-mean-square deviation. As already commented, in the SE-free case the quasi-momentum is constant in time, and atoms with quasi-momenta close to 1/2 travel faster, thus

98

producing the long tails and the ballistic peaks at their edges. In the presence of SE, no atom may persist a long time in the fast-travelling quasi-momentum range, whence it is removed the sooner, the larger pSE . Therefore with SE, the cutos are felt much later by the evolving distribution (gures 4.5 and 4.9). Whereas the cutos still prevent observation of the fastest atoms, they do not signicantly aect the growth of energy until large times. Even then, the momentum distribution normalised within |n| < ncut approaches the at distribution in |n| < ncut (gure 4.9(b,c,d)), which has a limit value for the second moment signicantly higher than the SE-free steady-state distribution in the presence of the cuto. Figure 4.10 compares the ideal momentum distributions which are taken from gures 4.3(a) and 4.5(a) to recent experimental results at t = 30 kicks. Note that the numerical simulation shows the ballistic wings of the distributions at pSE = 0, which are swamped by the noise background in the experimental data. Nonetheless, both experimental and theoretical data agree very well in the central (stationary) part of the distribution. The same is true for the case with SE where we chose pSE = 0.1 (nSE = ln(1 pSE ) 0.105) which compares best to the experimental data for which nSE 0.14 0.04 [159] is estimated. The latter estimate is rather rough and depends on several absorption processes in the experimental setup, and on uctuations in the intensity of the SE inducing laser [159,260]. SE occurs not only in the direction of the kicks (cf. discussion in section 4.3.5). Hence, we cannot exclude a possible enhancement of the eect that dierent atoms may also experience dierent intensities of the kicking pulses (see section 2.3.3). The atoms are then subject to a range of kicking strengths which has not been taken into account numerically. Those atoms which do experience a low intensity yield an enhanced population near to zero momentum which might look like a numerical distribution in which the SE rate is lower. In contrast to the SE-free case, the dependence of the mean energy on the kicking period after 30 kicks is but slightly aected by the cutos when SE is present. This is shown in gure 4.11 (b,c), to be compared to gure 4.6(b,c). In the experiments (see gure 4.1 here, and gure 2 in [82], gure 6 in [83]), the peaks for all cases (a-c) are still smaller than in our gure 4.11, which can be explained by the extreme sensitivity of the energy at exact resonance to all sort of perturbations besides those included in our present analysis, and also by diculties in experimentally tuning to the exactly resonant values of . Additional experimental restrictions, e.g. the experienced uctuations of the potential depth and the resulting averaging over slightly dierent experimental realisations [83, 135, 182], may lead to a further reduction of the peak, especially in the case without decoherence which is most sensitive to any kind of disturbance. Our analytical results of this chapter are exclusively obtained for the exact resonance condition in the kicking period = 2 ( N). The experimental and, in particular, the numerical data of the mean energy vs. also show that the widths of the resonance peaks changes systematically when SE is introduced (cf. gures 4.1 and 4.6). The shape of these peaks is responsible also for their experimentally observed stabilisation with increasing noise. An analytical

99

10 10 10 10

(a)

Pn

10 10 10 10

1

(b)

80 60 40 20

20

40

60

80

n

Fig. 4.10: Coarse-grained momentum distributions at quantum resonance = 2 (type (I) simulations) from gures 4.3(a) and 4.5(a) (dotted), as compared to experimental data [159] (full lines), for pSE = 0 (a) and pSE = 0.1 (nSE = ln(1 pSE ) 0.105), experimental estimate nSE 0.14 0.04 (b); kicking strength k = 0.8, at t = 30 kicks. The arrows mark the ballistic wings in the case without SE. The dashed line in (a) shows the steady-state distribution (for t = 1000) taken from gure 4.3(b). The dash-dotted lines show the signal threshold and momentum cuts (ncut = 60) imposed on the experimental data when calculating mean energies as plotted in gure 4.1. The asymmetry in the experimental distribution around n = 0 is caused by the nonideal compensation of gravity which acts in the direction of the standing wave, in the experimental realisation [82, 83, 135, 159].

100

40 20 0

(a)

mean energy

40 25 10 45 30 15 0

(b)

(c)

10

15

20

Fig. 4.11: Numerical simulation of the mean energy as a function of the kicking period for the same ensemble of SE and the same initial distribution of atoms as in gure 4.6, after 30 kicks, and with kicking strength k = 0.8. (a) no decoherence pSE = 0, (b) pSE = 0.1, (c) pSE = 0.2. The width of the rectangular pulse is dur = 0.047 and ncut = 40, as for the experimental data [82, 83] shown in gure 4.1.

theory for arbitrary irrational /2 does not exist, but in the next chapter we develop an approximate description for the vicinity of the resonance peaks for = 2 + , with | | 1.

Chapter 5

In the previous chapter, we derived analytical results for the momentum distribution and the average kinetic energy of an initial atomic ensemble, and compared them to numerical simulations and the experimental data. In the following, we go beyond the case of the exact resonance condition for the kicking period , and study the vicinity and, in particular, the shape of the resonance peaks, which are observed in the mean energy after a xed observation time tobs . For irrational values of /(4), dynamical localisation sets in (cf. section 2.2.2), that is, on increasing tobs beyond a break-time tbreak , the observed energy values should not increase any more. If tobs is signicantly larger than tbreak , a scan of the measured energy vs. the kicking period yields plots alike in gure 4.1(a), and gure 4.6(a). In 4.6(a) peaks are clearly observed at the resonant values = 2, 4, 6. For continuity reasons, the resonance peaks have a width, determined by the nite value of tobs . In the ideal case, they would shrink on increasing tobs , and further, narrower peaks associated with higher-order resonances would appear for large tobs . In this chapter, we derive a description of the structure of the peaks around = 2 ( N), based on a nite time, small- asymptotics, where = 2 + . This technique was introduced in [144, 145], and partially anticipated in [271, 272], where the special role of the combination k| | of the two parameters k, the kicking strength, and , the detuning from the resonance, was rst realised. 2 In particular, we nd that the width of the resonance peak scales like (ktobs )1, so that, at large tobs , the peak is much narrower than the naive expectation 1/tobs . Such sharp sub-Fourier resonances reect the high sensitivity of the quantum-chaotic kicked rotor with respect to slight variations in the detun-

Unbounded growth was also proven for a dense set of close-to-commensurate values of [158]. Extremely long times are however required to resolve such arithmetic subtleties.

101

102

ing from resonance, and they may be relevant for high-resolution experiments.

5.1

quasi-classical approximation

The most elementary classical resonances are met when a system, such as an oscillator, is driven by a force that is periodic in time with frequency equal to the natural frequency of the system. In linear systems, a classical resonance leads to unbounded energy absorption, e.g. for a resonantly driven harmonic oscillator (see, for instance, section 1.6a in [44]). For nonlinear systems, however, this is typically not the case because the unperturbed frequencies depend on the systems energy, so the system is rapidly driven o resonance by the initially resonant excitation itself. Similarly, for a quantum system the resonant excitation is eventually stopped owing to the anharmonicity of the unperturbed spectrum. The quantum resonances of the -kicked rotor are rare examples of unbounded excitation in the deep quantum regime. For instance, the experimental and numerical data presented in the previous chapter were obtained for ~ = O(1), and k ~1 = O(1), cf. section 2.2.2 for the denition of the parameters k and in the quantum model. As discussed there, the quantum resonances arise for particular kicking periods which are commensurable with 4. They have no direct counterpart in the corresponding classical system, and are totally unrelated to resonances of the classical rotor. In the following, we present a quasi-classical analysis of the quantum resonances of the -kicked particle and their vicinity. This seemingly self-contradictory task is accomplished by establishing a direct correspondence between the quantum resonances and the classical nonlinear resonances of a related model. This classical system is not obtained in the conventional classical limit of vanishing Planck constant ~ 0, but rather in the limit when the detuning from the resonant periods = 2 ( N) approaches zero. In our quasi-classical approximation, the role of Plancks constant is played by the detuning from exact resonance in the kicking period . This approximation allows us to describe the near-to-resonant quantum motion in terms of a Standard Map [156], which is dierent in parameter values from the one that is obtained in the conventional classical limit of the -kicked rotor, i.e. when the real eective Planck constant 0, and k simultaneously [63]. The quantum resonances of the -kicked rotor then correspond to the primary nonlinear resonance of this

In a similar context, the sub-Fourier sensitivity of the kicked-rotor dynamics on periodicity conditions was used to sharply discriminate dynamical localisation from diusive quantum transport [273]. There do exist classical accelerator modes of the -kicked rotor that lead to ballistic motion [43]. E.g., for k = 2, and (p0 = 0, 0 = /2), it is easily checked from (2.6) and (2.7) that momentum grows linearly in time pt = 2t. However, these modes are of a dierent nature, and occur for very specic initial values (p0 , 0 ), while the quantum resonances do not depend on the initial conditions [43], if there is no additional degree of freedom, like the quasi-momentum for the kicked particles.

5.1.

quasi-classical approximation

103

quasi-classical Standard Map. The stable elliptic island associated with the nonlinear resonance [4345] accounts for the structure of the resonance peak in the quantum mean energy vs. curves (see gures 4.1 and 4.6). To derive the quasi-classical approximation [145, 160], we rescale the kicking strength k = k/| |, and dene d I = | |N = i | | . d (5.1)

The free evolution part of the Floquet operator for the rotor (2.23) may be rewritten as follows: e i 2 (n+) = e i

2

n2 i 2 n2 i n i 2 2

=e

I | |

i sign(

I2 ) 2| |

i |I| i 2 2

(5.2)

where the last factor does not depend on I and may be dropped. Similarly, we obtain for the kick operator in (2.23) e i k cos() = e and then the Floquet operator reads

k cos() | | H e , U (t) = e | | i i | i | k cos()

(5.3)

(5.4)

1 H (I, t) = sign( )I 2 + I( + ) . (5.5) 2 If | | is regarded as the Planck constant, then (5.1) together with (5.4) is the formal quantisation of either of the following classical maps: It+1 = It + k sin(t+1 ) , t+1 = t It + + mod(2) (5.6)

with

where has to be chosen according to the sign of . We stress that classical here is not related to the ~ 0 limit but to the limit 0 instead. The small| | asymptotics of the quantum rotor is thus equivalent to a quasiclassical approximation based on the classical dynamics (5.6), that will be termed -classical in the following. Changing variables to J = I + + , = + (1 sign( ))/2 turns the maps (5.6) into a single Standard Map (c.f. (2.6-2.7)), independent of the value of : Jt+1 = Jt + k sin(t+1) t+1 = t + Jt . (5.7) (5.8)

This will be called the -classical Standard Map ( SM) in what follows. In gure 5.1(a) quantum energy curves vs. in a neighbourhood of = 2 are compared with energy curves computed using the classical map (5.6). For any given particle in the initial ensemble, the map (5.6) with equal to the quasi-momentum of the particle was used to compute a set of trajectories started at I = n0 | | with homogeneously distributed 0 [0, 2). The nal

104

45 35

(a)

mean energy

45 35 25 15 5 2

(b)

12

17

22

Fig. 5.1: (a) shows the mean energy vs. kicking period , after t = 30 kicks, and for kicking strength k = 0.8; magnication near the quantum resonance = 2. Quantum data taken from gure 4.6(a) (for better comparison repeated in the lower panel (b)) (solid lines) are compared with the mean energies of an ensemble of 106 classical atoms (circles) with the same initial momentum distribution, evolving under the classical dynamics (5.6). The value of corresponding to the small peak on the right of the resonant spike is marked by an arrow for reference to gures 5.4 and 5.6.

energies 2 It2 /2 at t = tobs of the individual trajectories were averaged over 0 , , n0 with the appropriate weights. This is equivalent to using the SM in all cases, with dierent initial ensembles J0 = const. = n0 | | + + 0. As 0 is varied, such ensembles sweep the full unit cell of the SM, so sampling dierent 0s amounts to probing dierent regions of the classical phase space as illustrated in gure 5.2. The average energy Et = 2 It2 /2 is plotted vs. = 2 + in gure 5.1(a), along with results of the corresponding quantal computations. The main qualitative features emerging of gure 5.1 are: (i) on a gross scale the curves are shaped in the form of a basin with a high, narrow spike in the centre, closely anked by a much smaller peak on either side. (ii) quantum and classical curves nicely agree at small | |, in particular the structure of the spike is the same. Their behaviour at large | |, i.e. for & 6.6 and . 6 is qualitatively similar but quantitatively dierent (see gure 5.8 in section 5.2.2). This overall qualitative behaviour may be explained in classical terms, and an approximate scaling law for the t, k, dependence of the mean energy close to resonance can be obtained, as shown in the next section.

5.1.

quasi-classical approximation

105

3 2 1 0 1

(a)

(b)

6 3 0 3 6

p=I/||

2 3 24 16 8 0 8 0 0 2

(c)

(d)

8 0 8 16

Fig. 5.2: Poincar surface of sections for the map (5.6), and k = 0.8, = 0.05, = 0 e (a), = 0.35 (b), = 0.4 (c), = 0.5 (d). Plotted are the true momenta p = I/| | to emphasise the much larger contribution to the mean energy from the (nonlinear) resonant zone for 1/2, whose width is approximately given by p 4 k/| |. For = 0.5, the period-1 xed point of the map (5.6) is close to 0 (in fact at p = 1/2), so all trajectories launched at p0 = , move along the island, leading to large excursions in p. As decreases to 0, the period-1 xed point [43] moves away from p = 0, and is not hit by trajectories started at p0 = (n0 = 0). The trajectories in (b) and at p 8, = 0 in (c) are rotational orbits, corresponding to the regular regions outside the primary resonance island in gure 5.3 (right panels). (a) contains a higher-order period-2 resonance embedded in the otherwise rotational motion. In the coordinates of the map (5.6), the eective Planck cell 2 0.314 is larger than the area of the period-2 islands, which is estimated by k 0.126 (see [43]). Higher-order resonances are neglected in our analysis since they aect much smaller regions in phase space than the dominant primary island (in (d)). The full phase space in the coordinates of the SM is shown in the next gure (right panels).

106

5.2

5.2.1

quasi-classical analysis of the resonance peaks

The classical standard map is dierent from the map obtained in the classical limit proper ~ 0 of the kicked rotor. In particular, if k > 1, then the classical and the -classical dynamics are at sharp variance whenever k < 1. In the former unbounded diusion occurs, while in the latter the dynamics is quasi-integrable instead. In this quasi-integrable system, the classical trajectories remain trapped forever in-between impenetrable phase space barriers, which survive small perturbations according to the Kolmogorov-Arnold-Moser (KAM) theorem [4345, 177]. It is exactly the deep changes which occur in the classical phase space (cf. gure 5.3) as is varied at constant k that account for the energy vs. dependence at xed time t = tobs . In the following discussion, we assume for simplicity an initially at distribution of p0 [0, 1); then I0 = 0, and J0 = + 0 with 0 uniformly distributed in [0, 1). Without loss of generality we also consider = 1. Hence if | | 1 then J0 is uniformly distributed over one period (in action) (, 3) of the SM. Since Jt = It + + , and I0 = 0, the mean energy of the rotor at time t is: Et, =

2

It2 /2 =

(Jt)2 , Jt = Jt J0 . 22

The exact quantum resonance = 0 corresponds to the integrable limit of the SM, where Jt = 0. However, Et, is scaled by 2 , so in order to compute it at = 0 one has to compute Jt at rst order in . This is done by substituting the 0-th -order of (5.8), i.e. t 0 , into (5.7). This leads to

t1

Jt = | |k

s=0

sin(0 + J0 s) + r( , t)

(5.9)

where r( , t) = O( ) as 0 at any xed t. The energy at time t is found from (5.9) by taking squares, averaging over 0 , J0 , dividing by 2| |2, and nally letting 0: Et, = 1 8 2

2 3

d0

0 2

dJ0

3

(Jt )2

2 t1 2

k2 2 8

0

d0

0 3

dJ0

s=0

sin(0 + J0 s) (5.10)

(4.9) k2 = 8

dJ0

107

The small contribution of the initial quasi-momentum in the atoms energy was neglected. Apart from that, (5.11) is the same result as was found by the exact quantum mechanical calculation performed at = 0 in section 4.2.2 for the case of a uniform quasi-momentum distribution. Thus the -quasi-classical approximation reproduces the quantum behaviour at exact quantum resonance. The integral over J0 in (5.10) collects contributions from all the invariant curves J0 = const. of the SM at = 0. Of these, the one at J0 = 2 leads to quadratic energy growth because it consists of (period 1) xed points [43]. This is called a classical nonlinear resonance. It is responsible for the linear growth of energy (5.11), because the main contribution to the integral in (5.10) comes from a small interval 2/t of actions around J0 = 2. Note that J0 = 2 corresponds to 0 = 1/2, the quantum resonant value of quasi-momentum. It is hence seen that the quasi-classical approximation explains the quantum resonances of the kicked rotor in terms of the classical resonances of the Standard Map. In the sequel, we estimate Et, for | | > 0, where the dynamics is maximally distorted (with respect to the = 0 one) for J0 in the vicinity of the elliptic xed points [43] of the SM, i.e. of the 2n, in the very region which is mostly responsible for the linear growth of energy at = 0. Being formed of period1 xed points, the J0 = 2n, = 0 invariant curves break at | | > 0 as described by the Poincar-Birkho theorem [43]. The motion is then strongly e distorted inside regions of size (in action) Jres astride J = 2n. Such regions are termed the primary resonances of the SM, and a well-known estimate is Jres 4(k| |)1/2 [43]. Inside these resonances, the approximation (5.9) fails quite quickly, so their contribution Et, res to the mean energy has to be estimated dierently. The dynamical situation is illustrated in gure 5.3 by Poincar surface of sections for the dynamics induced by (5.7-5.8). e In the remaining part of the classical phase space the motion mostly follows KAM invariant curves [4345], slightly deformed with respect to the = 0 ones, still with the same rotation angles (see gure 5.3). The contribution of such invariant curves to the mean energy is therefore roughly similar to that considered in the integral (5.10), provided J0 is therein meant as the rotation angle. On such grounds, in order to roughly estimate Et, we remove from the integral (5.10) the contribution of the resonant action interval near J0 = 2, and replace it by Et, res: Et, where (t) = k2 8 k2 t (t) + Et, 4

Jres /2 res

(5.12)

dJ

Jres /2

(5.13)

and J is the deviation from the resonant value 2. The contribution Et, res may be estimated by means of the pendulum approximation [43, 44]. The penHigher resonances appear near all values of J0 commensurate to 2. At small | | such higher-order resonances aect regions of phase space that are negligibly small with respect to the primary resonance, see gures 5.2 and 5.3. The higher resonances are altogether ignored in the present discussion. Also note that structures that are small compared to the Planck constant | | are irrelevant for the purposes of the classical approximation.

108

9.14 7.14 5.14 3.14

6.55

(a)

J0

7.2

1/2

(b)

4(k)

3.14

Fig. 5.3: Poincar surface of sections for the Standard Map (5.7-5.8), and k = 0.8, e = 0.01 (a), = 0.1 (b). As increases, the invariant curves (right panels) become more and more distorted, and the contribution of trajectories around the primary resonance island (left panels), with a width of Jres 4(k| |)1/2, to the energy must be calculated separately (see equation (5.12)). The black boxes in the right panels represent the Planck cell 2| |. In the coordinates of (5.7-5.8) sweeping from 0 to 1 is equivalent to scanning J0 from to 3, with the primary resonance island centred at J0 = 2. The four plots (a-d) in the previous gure eectively correspond to a scan from J0 (gure 5.2(a)) upwards to J0 2 (gure 5.2(d)).

dulum Hamiltonian emerges from the Fourier analysis of the kick perturbation: | |k cos() m (tm) = | |k m cos(2mt) (note that for the SM (5.7-5.8) the kicking period is 1). In the time average, higher Fourier components may be neglected near the SM resonance [43, 44], and the motion is thus described (in continuous time) by the following pendulum Hamiltonian in the canonical coordinates J , : 1 Hres = (J )2 + | |k cos() . 2 (5.14)

The resonance width Jres is estimated by the separation (in action) between the separatrices of the pendulum motion. The period of the small pendulum oscillations is 2tres where tres = (k| |)1/2 [43, 44], so we use tres as a characteristic time scale for the elliptic motion in the resonant zone. One may altogether remove | | from the Hamilton equations, by scaling momentum and time by

109

G(t/tres)

1.5

0.5

10

20

30

40

50

t/tres

Fig. 5.4: The function G (5.16) vs. the scaled time variable t/tres. It increases quadratically as the momentum of a harmonic oscillator for short times, then the continuum of frequencies of dierent periods of the nonlinear pendulum motion leads to dephasing, and G saturates with damped oscillations for large argument t/tres 1. The arrow marks the position of the horn in gures 5.1 and 5.6.

factors (k| |)1/2 = 4/Jres, (k| |)1/2 = 1/tres respectively. Therefore, (Jt )2 = (Jt J0 )2 k| |G(t k| |) , (5.15)

for an ensemble of orbits started inside the resonant zone, where G(.) is a parameter-free function, whose explicit expression involves elliptic integrals [44]. G(.) represents the average energy contribution from trajectories in the primary island of the SM, i.e. G(t k| |) 1 8

2 2

d

0 2

dJ0 J

t k| |, 0 , J0

(5.16)

Hence, this function results from averaging over nonlinear pendulum motions with a continuum of dierent periods, so it saturates to a constant value when its argument 1. At small values ( 1) of the argument, it behaves quadratically. This behaviour is illustrated in gure 5.4, where G is plotted vs. the scaled variable t/tres . The contribution to the total energy is then obtained on multiplying (5.15) by | |2 Jres/(4), because only a fraction Jres/(2) of the initial ensemble is trapped in the resonant zone. As a result Et,

res

Jres (Jt )2 k2 G t k| | 4 22 k | |

(5.17)

When Jres is small, sin2(J /2) may be replaced by J 2 /4 in the integrand in (5.13), leading to k2 (t) t 0 (t k| |) 4

110

175 150

175

(a)

(b)

mean energy

0 12.535

12.555

12.575

12.595

Fig. 5.5: Mean energy (in units of (2~kL )2 /M , c.f. (2.24)) vs. the kicking period 2 (a), and 4 (b), after t = 20 (solid), t = 30 (dotted), t = 50 (dashed), t = 100 (circles connected by dash-dotted line) kicks; for an ensemble of 105 kicked atoms with Gaussian initial momentum distribution ( 2.7) and kicking strength k = 0.8.

with 0 (x) 2

x

ds

0

sin2 (s) . s2

Replacing in (5.12), we obtain the central result of this chapter: R(t, k, ) Et, 4 H(x) 1 0(x) + G(x) , Et,0 x (5.18)

x = t k| | = t/tres .

Hence R(t, k, ) depends on t, k, only through the scaling variable x = t/tres . This is natural, since the main ingredient in the derivation of (5.18) is the pendulum motion around the nonlinear island of the SM. The pendulum dynamics for (5.14) is governed by a single parameter which is the coupling strength k| |, or the frequency of small oscillations around the elliptic xed point res(k| |) k| |, respectively [43, 44]. Because of (5.18) the width in of the resonance peak scales like (kt2 )1 . This implies that for xed k, the width of the resonance peak shrinks very quickly in time, as illustrated in gure 5.5 for an experimental relevant ensemble of -rotors. The scaling law (5.18) is demonstrated by numerical data shown in gure 5.6. The function H(x) was calculated numerically; in particular, G(x) was computed by a standard Runge-Kutta integration of the pendulum dynamics (5.14). The scaling function H(x) decays proportional to x1 at large x, because so do 1 0 (x) and 4G(x)/(x); the latter owing to the saturation of G. From

111

gure 5.6 it is seen that 0 is quite slowly varying at x > 4. The structures observed in that region are then due to G(x), which describes the resonant island. Figure 5.4 shows that G(x) saturates via a chain of oscillations of decreasing amplitude around the asymptotic value. These give rise to three local maxima in the graph of x1 G(x), followed by a chain of gentle oscillations in the tail. The rst and most pronounced maximum lies in the small-x region, and is not resolved by the scaling function H(x), apparently because it is eaced by the rapid decay of 1 0 (x). The subsequent maximum and its symmetric partner at < 0 are instead resolved and precisely correspond to the horns, the right-hand one of which is marked by the arrow in gures 5.1(a) and 5.4. The oscillations in the tail of x1 G(x) are also well reproduced in the tail of H(x). The scaling law shows that at given k, t the energy curve Et, vs. decays proportional to | |1/2 past the horns. As t increases the horns rise higher while moving closer and closer to = 2 , because they are located at a constant value of x = t k| |. Thus an overall | |1/2 dependence eventually develops (cf. gures 5.1 and 5.6). In the case when the smooth initial momentum distribution includes values n0 = 0 and/or is appreciably non-uniform in quasi-momentum, the statistical weights of the various phase-space regions are dierent. Scaling in the single variable t/tres still holds, but the scaling functions 0 and G may be dierent. For a Gaussian initial distribution with mean square deviation 2.7, as used in the experiments reported in [82, 83, 135, 159], a scaling like in gure 5.6 was found numerically. The initial energy, which is negligible in the case of p0 [0, 1), must be subtracted such that the scaling holds, i.e. Et, p2/2 / Et,0 obeys the scaling law (5.18), as shown in gure 5.7 for 0 a Gaussian initial momentum distribution. Our analysis shows that the structure of the resonant peak is essentially determined by the primary resonant island of the SM. It neglects higher-harmonics resonances of the SM, higher-order islands, and especially the growth of the stochastic layer surrounding the primary resonance. Such structures grow with | | and are expected to introduce deviations from the scaling law (5.18). Hence this analysis is valid only if | | is suciently small, such that k| |cr 1. On trespassing the threshold for global chaotic motion k| |cr 1, the critical regime of the SM is entered. No isolating KAM curve survives [43], so the energy curve rises in time for | | > | |cr (see gure 5.8). Estimating the mean energy at relatively short times and below the threshold for global chaos in the SM, i.e. | |k < 4.5 [43], is dicult, because unbounded, non-homogeneous diusion and elliptic motion inside residual stable islands coexist. The increase of the curve with | | at constant t is a result of the decreasing size of the latter islands, and of the rapid increase of the diusion coecient (proportional to (| | | |cr ) , 3 at large enough t [225, 274]).

112

R(t,k,)

H(x)

0.1

10

x=t(k||)

0.5

100

Fig. 5.6: Demonstrating the scaling (5.18) of the resonant peak, in a right neighbourhood of = 2. Open circles correspond to dierent values of the parameters , k, t, randomly generated in the ranges 1 < t < 200, 0.001 < < 0.1, 0.1 < k < 50 with the constraint 0.01 < k < 0.2. In each case an ensemble of 2 106 classical rotors was used to numerically compute the scaled energy R(t, k, ) (5.18), with a uniform distribution of initial momenta in [0, 1) and a uniform distribution of initial in [0, 2). Full squares present quantum data for k = 0.8, t = 50 and t = 200. The solid line through the data is the scaling function H(x) of (5.18) obtained by direct numerical computation of the functions 0 (x) and G(x). The dashed line represents the function 0 (x); the dash-dotted line has slope 1 and emphasises the x1 decay described in the text. The arrow marks the value of the scaled detuning x which corresponds to the arrow in gure 5.1.

1

113

R(t,k,)

0.1

10

x=t(k)

0.5

100

Fig. 5.7: Same as gure 5.6 for Gaussian initial momentum distribution with mean square deviation 2.7 (open circles, crosses), similar to experimental initial distribution in [82, 83, 135,159]. The circles present -classical data where the initial energy of an individual atom was subtracted before computing the average energy value, while the crosses show -classical data where the mean initial energy was subtracted from the average energy value of the atomic ensemble after some xed time t. The latter is the experimentally accessible method, and no dierence between the two methods can be observed what concerns the scaling of the data. The solid line is the scaling function H(x) from (5.18).

5.2.2

The quasi-classical approximation is exact at all times for = 0, as shown above. At nonzero , it is valid for not too large times t, and it is in the long run spoiled by quantum, non -classical eects. At | | < | |cr the -classical motion is bounded by KAM curves, so the main quantum mechanism leading to non- -classical behaviour is tunnelling across the regular regions. Estimating the related time scales is a non-trivial problem, because the 2-periodicity in action of the -classical phase space may enhance tunnelling, and even result in delocalisation, depending on the degree of commensuration between 2 and the Planck constant | |. For instance, if | |/2 is rational, then the quantum motion will be ballistic asymptotically in time. This is just the ordinary quantum resonance of the quantum kicked rotor (see section 2.2.2). In order that one such resonance with | | = 4s/q exists at | | less than some | 0 |, it is necessary that q > 4/| 0|. It will show up after a time roughly estimated by | | times the inverse bandwidth. The bandwidth is estimated to decrease faster than exponentially at large q [64,118], so one may infer that the time of validity of the -quasi-classical approximation is at least exponentially increasing with

114

50 45 40

mean energy

35 30 25 20 15 10 5 4.5 5 5.5 6

6.5

7.5

Fig. 5.8: Same as gure 5.1, however plotted in a broader window in around the quantum resonance peak at = 2. For larger detuning | | > 0.25, the -classical approximation (circles and thin line) leads to larger energies than the quantum simulation (thick solid line). At | | > 0.25 classical diusion occurs, and the classical curves rise as a function of time t, while the quantum energies basically remain unchanged (for not too large times for which higher-order quantum resonances may be important).

1/| | as the exact resonance at = 0 is approached. At | | > | |cr the classical motion is unbounded, and the dierence between -classical and quantal energy curves vs. is basically set by various quantum localisation eects, including localisation by cantori close to the | |cr [275, 276]. As a consequence, if t is large enough, then the -classical curve lies much higher than the quantum one (cf. gure 5.8). Nevertheless the latter still rises with | | at constant t, because of the growth of the localisation length De k2 sin2 ( /2) [57, 227, 254256], with = 2 + ( N).

5.3

The scaling law (5.18) derived in the previous section shows that the only relevant time scale for the evolution of the quantum motion suciently near to = 2 ( N) is given by tres = 1/ k| |. If we allow for a noisy time evolution as discussed in section 4.3, an additional time scale comes into play that characterises the strength of the noise [277]. In the following, we indeed nd an equivalent of the scaling law (5.18) in the presence of noise, which is based on the two time scales tres and tc , where tc is the mean waiting time between two spontaneous emission (SE) events. The quasi-classical approximation introduced in section 5.1 for the study of

115

the coherent nearly resonant quantum motion may be adapted to include SE, because the eects of SE were modelled by a totally classical noise. In the stochastic gauge (section 4.3.1), the classical approximation may be implemented in the rotor propagators (4.27) much in the same way as in section 5.1. The resulting classical map corresponding to (5.6) is: It+1 = It + k sin(t+1 ) , t+1 = t It + + t . (5.19)

We now exploit assumption (S2) of section 4.3.2 and write t = + t , where t t = s=1 s is the total momentum imparted by SE up to time t. In order to turn o the stochastic gauge, we need to recover the accumulated SE momentum change, hence we change variables to It = It + | |t . The momentum of the atom at time t is then | |1 It + , where is the initial quasi-momentum. Denoting t = t + /(2 ), and changing variables from (It, t) to (It, t ) in 5.19, we obtain the following map:

It+1 = It + | |t+1 + k sin(t+1 ) , t+1 = t It + + 2 t , t+1 = t + t+1 , . 0 = 2

(5.20)

The t are independent random variables, whose distribution is determined by the statistics of SE. Numerical simulations of such noisy classical maps are shown in gure 5.9, and very well match with the quantal computations at small | |. Under the substitution Jt = It + +2 t , the map (5.20) reduces to a noisy SM, which diers from the SM by a random shift t of the action J at each step. We assume an initially uniform quasi-momentum distribution. At any SE time tj , the distribution of the ensemble in the phase space of the SM is reshued by the random action change. Under the assumption of homogeneous distribution of single SEs in an interval of integer length (assumption (S3) in section 4.3.2), the resulting distribution of Jmod(2) is approximately homogeneous over the unit cell of the SM. Such randomisation may be assumed to wash out correlations between the past and the subsequent random dynamics. Hence the scaling (5.18) may be used to write the energy at time t as Et, k2 4

Nt1

j H(j /tres )

j=0

1 + D nSE t , 2

(5.21)

where . stands for the average over all the Bernoulli realisations of the times of SE events, according to the discussion after (4.29). nSE is the average number of 2 SE per period, and D = nSE 1 t is the mean square momentum imparted by a single SE. For an individual realisation, (5.21) states that the energy is given by the sum of the SE-free scaling function H of the evolutions over time intervals j . The SE resets the evolution after each event at times j , apart from the momentum shift contained in the second term in (5.21). If tc is suciently large

116

52 42

(a)

(b)

76 56

32

mean energy

36 16

(c)

(d)

65 45 25

Fig. 5.9: Analogue of gure 5.1(a), with the same initial ensemble, for k = 0.8, in the presence of SE. Results of full quantum calculations (circles) and of classical ones (solid lines) in the presence of SE are compared near the resonance = 2, for dierent times and dierent rates of SE: (a) pSE = 0.1, t = 30, (b) pSE = 0.1, t = 50, (c) pSE = 0.2, t = 30, and (d) pSE = 0.2, t = 50. The quantum simulation was type (I) (see section 4.3.5), while the classical simulations used the map (5.20).

compared to 1, one may replace the Bernoulli process by the continuous time Poisson process with the characteristic time tc = 1/(ln(1 pSE )) = 1/nSE . This process has the delays distributed with the density t1 exp(/tc ). c Its statistics reduces to that of the unit Poisson process (with tc = 1) by just rescaling all times by the factor 1/tc . This entails

Nt1

j H(j /tres )

j=0

(5.22)

where 1 Q(u, v) 4

1 Nu

1 H(1v) j j

j=0

(5.23)

The superscript 1 species that the average is now over the realisations of the unit Poisson process: each realisation has the continuous time interval [0, u] 1 divided in subintervals 1 by a random number Nu of Poisson events. We are j

hence led to the following scaling law: Et, D or, equivalently, 2 Et, D t/tc Q(u, v) , u = t/tc , v = tc /tres . 2k2tc t t tc + k2 tc Q( , ) 2tc tc tres

117

(5.24)

(5.25)

The scaling function Q(u, v) may be explicitely written in terms of the function H(x) as reported in appendix D: 4Q(u, v) = uH(uv)eu +

0 u

dx ex xH(xv)(2 + u x) .

(5.26)

Limiting behaviours of the scaling function Q(u, v) immediately follow from this 1 the right-hand equation, or from (5.23) itself. On one hand, for u = t/tc side in (5.23) is a sum of a large number t/tc of terms. In that limit, such terms are quite weakly correlated and may be independently averaged, leading to: 1 u 1 : Q(u, v) u dx H(vx)xex . (5.27) 4 0 On the other hand, for t/tc 1 = t/tc ; hence 0 u 1, the sum reduces to the single term j = 0, with 1 :

1 (5.28) Q(u, v) uH(uv) . 4 In particular, (5.28) shows that (5.24) coincides with (5.18) in the SE-free limit tc . In the opposite limit, (5.27) shows that, if k is xed, then the width in of the resonant spike will not shrink any more with time when t tc , and its width thereafter scales like (t2 k)1 . The spike is therefore erased (that c is, it is absorbed in the background) in the strong noise limit tc 1. In the latter limit, the method developed above breaks down, because on average after each kick a SE event happens, which does not let the time evolution recover for some time interval. The result is then a completely random motion which does not depend on the system specic dynamics, and hence not on the value of the kicking period . The spreading of the resonance peaks with increasing noise, as can be seen nicely in gure 5.9, explains why they are more stable and easier to observe experimentally than in the case without noise. Together with the results of the last chapter, this resolves now the puzzles put forward by the experimental data in gure 4.1 or in [82, 83, 135]. An intuitive argument for the spreading of the resonance peaks is that for a xed value of , due to SE there is an enhanced chance to nd a quasi-momentum , such that the free evolution part of the Floquet operator is approximately the identity. This is the condition for quantum resonant motion at = 2 ( N), as discussed in section 2.2.2. Numerical simulations in gure 5.10 satisfactorily support the scaling law (5.24). Data were obtained in a similar manner as for the case without SE; however, one of the parameters u, v is varied, while keeping xed either the other parameter or the ratio u/v. The theoretical scaling function Q(u, v) was calculated

118

3 2 1 0

(a)

0 5

10

15

Q(u,v)

1 0.5 0 0

(b)

5

10

15

(c)

0 1 2

Fig. 5.10: Demonstrating the scaling law (5.24) in a right neighbourhood of = 2. In (a), (b) the quantity on the left-hand side of (5.25) is plotted vs. one of the parameters u = t/tc or v = tc/tres while keeping the other xed: (a) v = 2, (b) u = 4. In (c) the ratio u/v = 4 is xed. Open symbols correspond to dierent values of the parameters t, tc, k, , randomly generated in the ranges 1 < t < 200, 5 < tc < 60, 0.001 < < 0.1, 0.1 < k < 20, with the constraints 0.001 < k < 0.2 and tc k = 2 in (a), t/tc = 4 in (b), t = 4t2 k in (c). In each case an ensemble of 2 106 classical rotors was used, c with a uniform distribution of initial momenta in [0, 1) and a uniform distribution of initial in [0, 2). The random momentum shifts at each step of the classical evolution (5.20) were generated from the uniform distribution in [1/2, 1/2]. Full squares represent quantum data for k = 0.8, and = 0.01 in (a), = 0.05 in (c), and t = 50 and t = 100 in (b). The solid lines correspond to the theoretical formula (5.26).

119

numerically using in (5.26) the function H(x) computed as described in section 5.2. If f0 (0) is smooth but not uniform, the scaling in absence of SE of the form (5.18) holds but with a dierent scaling function H as explained in section 5.2. Therefore the arguments of the present section leading to (5.24) should hold also in this case.

Chapter 6

What a fuss people make about delity! O. Wilde, The picture of Dorian Gray

In the previous two chapters, a powerful machinery was developed for a comprehensive understanding of the dynamics of -kicked atoms, at the fundamental quantum resonances and in their vicinity. In the present chapter, the theory of chapter 4 as well as the quasi-classical approximation of chapter 5 are applied to calculate and/or estimate the overlap of two initially identical states which are subject to two distinct evolutions. The overlap function, the quantum delity, is studied at exact resonance conditions (with kicking period = 2 , N), and for small detunings at = 2 + . In contrast to the derivation of the momentum distributions, and their second moment, the average energies, the delity depends crucially on the phases of the evolved wave functions. In section 6.2.2 the destruction of coherence by added decoherence in form of spontaneous emission (SE) builds on the stochastic wave function (4.38), and the method developed in section 4.3 is a useful tool to calculate the evolution of wave packets, including their phases.

6.1

When discussing the atomic conductance uctuations (section 3.2.3), a characteristic feature emanating from classical chaos displayed: the extreme sensitivity 121

122

with respect to changes of the systems parameters [109111, 140, 278]. In the case of the atomic conductance the control parameter was the driving frequency. How can one quantify the sensitivity with respect to parameter variation, i.e. to changes in the Hamiltonian? A measure for the stability of quantum dynamics was introduced in [138]: the overlap of two initially identical quantum states which are exposed to two dierent time evolutions [138140]. The overlap, or the quantum delity, is dened as 2 F (t) = |e i H t/~e i H0 t/~| (6.1) where H0 , and H = H0 + V represent the Hamiltonians of a reference system and of its slightly perturbed variant, respectively, with the perturbation parameter . The decay of delity in time was studied for classically chaotic quantum systems, usually for relatively small perturbations 1. Depending on the strength of the perturbation, several regimes of Gaussian and exponential decay were identied [279283]. For a classically quasi-regular system, a power-law decay has been found [284]. The quantum-classical correspondence, however, is still subject of some controversy [141]. For an initial statistical mixture of pure states |n , appearing with probabilities n , the delity may be dened in terms of the statistical density operator n |n n | [155, 285]: =

n

F (t) = Tr Ut Ut

(6.2)

Ut and Ut represent the evolution operators corresponding to the Hamiltonians 0 and H , respectively. This denition will be used below when dealing with H incoherent initial ensembles of momentum states (see equation (6.7) and (E.8)). An experiment which accesses the delity for the kicked oscillator was proposed in [286, 287], and investigations with kicked atoms were reported in [143, 288]. The experiment is based on the actual sublevel structure of the atoms ground and excited state which are used to induce the kicking potential (see section 2.3.2). Before the relevant dynamics starts, two ground state sublevels are equally populated, and couple dierently to the external driving because of the dierent detuning with respect to the transition to the corresponding excited level (V0 1/L in (2.35)). The splitting of the populations and the reversal after the dynamical evolution is performed by a Ramsey-type interferometer technique [172,289]. In the absence of coherence-destroying spontaneous emission, the contrast of the measured interference fringes is related to the overlap of two initially identical states evolved under dierent Hamiltonians. The dierence occurs in the coupling parameter, the kicking strength. The experimental setup is illustrated in gure 6.1: caesium atoms are initially prepared in the hyperne level F = 3, mF = 0 of the 62 S1/2 ground state. The rst microwave (Ramsey) pulse creates an equal superposition of the atoms internal states, F = 3, mF = 0 and F = 4, mF = 0. Now the atoms are exposed to the kicked evolution, before a second Ramsey pulse reshues the population of the two relevant ground state levels. The phase delay between

first Ramsey pulse phase shift

111 000 111 000 111 000 111 000

123

k2 k1 SE

k2 k1 measurement time

TRamsey

t

kicked particle dynamics

Fig. 6.1: Scheme of the experimental setup of [135, 143, 288]. Each atom is prepared in the upper state (left), then a /2 Ramsey pulse is applied before the time evolution starts (kicked-particle dynamics with/without gravity) for a coherent superposition of the upper (with corresponding kicking strength k = k2 ) and lower (k = k1 ) level. The dynamics lasts for a time interval t, afterwards a second /2 Ramsey pulse (with time delay to the rst one TRamsey ) is applied which reshues the population in the upper and lower level according to the value of the Ramsey phase. We allow additional spontaneous emission (SE) events to happen during the time interval t in our scheme.

the two Ramsey pulses, which determines the nal population of the levels, is experimentally tunable [288]. Finally, the (centre-of-mass) momentum distribution of one of the ground state levels is measured corresponding to a state projection onto this level. From the experimental signal, the overlap of the two distinctly evolved states may be extracted as an average over all atoms used in the experiment. To increase the eciency of the experiment, when focusing on certain dynamical regimes, and, in particular, to minimise the eect of averaging, the actual proposal [135, 143, 288] was to use a variant of the kicked atom dynamics. This variant is subject to an additional linear potential which is provided by gravity. The constant eld modies the dynamics considerably because it alters the phases of the states accumulated between consecutive kicks. Very robust quantum accelerator modes have been observed in specic parameter ranges around the quantum resonances [83,290,291] discussed in the previous chapters. Using a slightly amended quasi-classical approximation with respect to to the one presented in chapter 5, a large number of these modes could be identied [144] and measured [288]. Apart from the last section 6.4, where preliminary results on the quantum delity in the kicked accelerator are reported, we restrict to the conventional kicked particle dynamics at the fundamental quantum resonances ( = 2 , N) and in their vicinity.

124

6.2

6.2.1

Dynamical stability in absence of noise

Before the discussion of the delity, which will be dened in equation (6.7) in a form useful for our purposes, we show how to extract it from the data obtained by experiments [143, 288]. The following arguments are based on the intuitive interferometric picture suggested by gure 6.1. A more elegant presentation of (6.3-6.7) may be found in appendix E. Let us assume we start the experiment at time T = 0. One /2 Ramsey pulse [172, 289, 292] acts on the upper atomic level |2 (|1 + |2 )/ 2, and on the lower level |1 (|1 |2 )/ 2, respectively. E1, E2 are the energies of the atomic levels (~ 1 is assumed). After a time delay TRamsey , the second pulse produces transformations |2 (exp( i)|1 + |2 )/ 2, |1 (|1 the exp(i )|2 )/ 2, with being a controllable phase of the Ramsey pulse source [172,288,292]. The rotor states are expanded in the momentum basis, where the expansion coecients between the two pulses are dierent for the two arms of the interferometer, because of the two dierent time evolutions, characterised by k1 and k2 , respectively. In the sequel, (t, n, k1,2) denotes the time-dependent wave function of a -rotor in the momentum representation (the bar in (2.20) is dropped; the integer t denotes the number of applied kicks). Initially, a momentum distribution is prepared in the level |2 , and the free evolution after the rst Ramsey pulse leads to T =0:

n0

1 (n0 ) |n0 |2 2

(n0 ) |n0 ( |1 + |2 )

n0

(n0) |n0 e i E1 T |1 + e i E2 T |2

n0

At time T = TKR (corresponding to t = 0) the kicked-particle evolution starts with the kick counter t. The kicks eectively act only on the external centre-ofmass motion, not on the electronic degrees of freedom (see section 2.3.2), and hence the state vector up to the second Ramsey pulse reads TKR < T < TRamsey : 1 (t, n, k1) |n e i E1 T |1 + (t, n, k2) |n e i E2 T |2 . 2 n At time T TRamsey the distribution is reshued according to the above mentioned rule 1 |,end(t,n) = e i E1 T |1 (t, n, k1) + (t, n, k2)e i Ramsey + 2 e i E2 T |2 e i Ramsey (t, n, k1) + (t, n, k2) . (6.3)

The Ramsey phase Ramsey (E2 E1)TRamsey describes the phase dierence accumulated between the pulse source and the internal atomic phase evolution

125

in the time interval TRamsey [172,292]. We now assume that the projection onto a specic momentum state and onto the lower internal level |1 is measured. We average incoherently over a given initial momentum distribution of independent rotors. The probability of nding the atom in a momentum eigenstate with eigenvalue p n (i.e. a coarse-grained integer value, see section 4.2.1) depends on the number of applied kicks t, and on the two kicking strengths k1 and k2: Pn (t, k1, k2) = 1 4 + 1 2 df0() | (t, n, k1)|2 + | (t, n, k2)|2

df0 () (t, n, k1) (t, n, k2) (n)

cos k1 ,k2 + Ramsey , where k1 ,k2 is the phase of the interference term, i.e.

df0 () (t, n, k1) (t, n, k2) (n)

(6.4)

i k

(n) 1 ,k2

(6.5)

The initial momentum distribution (typically Gaussian with 2.7 [83]) implies an ensemble of rotors with density f0 () (see section 4.2.1). The third (interference) term in equation (6.4) is responsible for the appearance of oscillatory fringes in the visibility. The latter is dened by the dierence between the maximum and minimum of the experimental signal as a function of the Ramsey phase, for all other parameters xed [288]. We denote the amplitude of the modulation in Pn (t, k1, k2) by A(k1, k2, n) = | df0 () (t, n, k1) (t, n, k2)|. In [288]

+

O(k1, k2) =

n=

(6.6)

was identied as the quantum stability measure of Peres [138]. For reasons of proper normalisation, the quantity O(k1, k2) is problematic, because of the absolute square taken before the average over all momenta. In the following, we dene the quantum delity as

+ 2 df0() (t, n, k1) (t, n, k2)

F (t, k1 , k2)

n=

(6.7)

In our denition, the absolute square is taken after all averages, and it coincides with the standard denition (6.2) of delity involving ensemble averages (see appendix E). In the experiment, only (6.4) is directly accessible, and therefore only A(k1, k2, n). However, from the form of (6.4) is is clear that scanning the Ramsey phase over a suciently wide range (maximally over an interval of 2) (n) it should be possible to reconstruct the unknown phase k1 ,k2 , and hence to

For identical evolutions in the two arms of the Ramsey interferometer, the delity as dened in (6.7) keeps the value 1 for all times.

126

access the delity (6.7). To compute the delity (6.7) we must calculate the product of two states of the form (4.6), for two dierent values of the kicking strength, and then sum over all integer momenta n:

+ n= + (t, n, k2) (t, n, k1) 2 2

= =

1 (2)2 n= 1 2

2 0

d

0 0

de i (k1 k2 ) cos())|Wt | ,

where we used | ( t arg(Wt))|2 = 1/(2) for the special case when the initial state of the particle is a plane wave with momentum p0 = n0 + (cf. equation (2.22)). With (F.1) the overlap (6.8) reads

+ n= (t, n, k2) (t, n, k1) = J0 (|Wt |(k2 k1)) ,

(6.9)

+ 2 (t, n, k2) (t, n, k1)

F (t) = =

n= 2 J0 (|Wt|(k2

k1 )) .

(6.10)

For k k1 k2 = 0, this gives F = F (t) = 1. For the resonant values of quasi-momentum res = 1/2 + j/ mod(1), with j = 0, 1, .., 1, |Wt| = t, and we can use the asymptotic expansion formula (F.8). The delity for large times at quantum resonance is then: Fres (t) 2 cos2 tk tk 4 . (6.11)

This shows that the delity falls o like 1/t for the resonantly driven rotors. A numerical computation of the quantum evolution (gure 6.2) agrees with this asymptotic result already at times t 2. The situation is less clear for general values of the quasi-momentum, because (6.10) sensitively depends on through Wt in (4.9). Another particular case is given for = 0 or 0 = :

2 F=0 (t) = J0

sin t

(k2 k1)

(6.12)

which at t = 1 depends only on the dierence k of the two kicking strengths. Then (6.12) can be made arbitrarily small by choosing k z0, where z0 2.40 is the rst zero of J0 . To compute the full delity

1 2

0

dJ0 (|Wt|(k2 k1 ))

(6.13)

10

0

127

10

F=1/2(t)

10

10

10

10

10

10

number of kicks t

Fig. 6.2: Numerically computed evolution of one resonant -rotor for = 2, and initial momentum p0 = = 1/2. The corresponding delity F=1/2 (t, k1 = 0.8, k2 = 0.6) vs. time t is shown (circles) together with the asymptotic formula (6.11) (solid line). The dot-dashed line indicates the 1/t decay.

for a uniform distribution of quasi-momenta with f0 () 1, we must perform the average over the dierent quasi-momenta, i.e.

1 0

t1 r=0

As shown in section 4.2.1, in the limit when t and 2r/t , the sum over r approximates the integral over , and (6.14) converges to

1 0

2

1 (2)2

dx

0

d J0 (k sin(x) csc()) .

(6.15)

With (F.4) we obtain the nal result for the asymptotic value of the delity: F (k) 1 (2)2

2 2 d J0 0

k csc() 2

(6.16)

The dependence on k of the formula (6.16) is shown in gure 6.3 together with explicit numerical calculations. There are local maxima, which means that the asymptotic delity F does not fall o monotonically as a function of

128

10

0

10

F (k)

10

10

10

10

15

20

Fig. 6.3: Asymptotic delity at quantum resonance (6.16) vs. k (solid line), compared with numerical data obtained by evolving ensembles of 104 rotors with uniform initial momentum distribution in [0, 1), and = 2 (open squares). The data is shown for all k at t = 50, when it typically has reached its asymptotic value (see gure 6.4).

the perturbation, which is given by k. Note that a nite saturation value F is always reached, i.e. the delity at exact quantum resonances = 2 always saturates, where the saturation time may depend on the choice of k1, k2. In gure 6.4 numerical data for the delity F (t, k) is presented for various values of the dierence k between the two kicking amplitudes. Experimentally feasible should be values of k/k 0.05 . . . 0.5 [260], and the results reported in [288] were obtained for k1 0.8 and k2 0.6 (the laser amplitude depends on the position of the atoms with respect to the beam centre; this introduces some spread in the kicking strength, cf. section 2.3.3). The latter values are used as a guidance in the following discussions. The minima in gure 6.4 are given by the minima of (6.13). Dierentiating with respect to k gives dF = d(k)

1 0

0

(6.17)

For t 1, |Wt |2/t approaches ( 1/2) (see (4.19)), and hence, we approximately obtain: dF d(k)

1 0

(6.18)

Without loss of generality we restricted to = 2, with only one resonant value res = 1/2 (for = 2 , we obtain the same result for all > 0, see section 4.2.2). Therefore, the rst zero of the Bessel function J1 (z) (z = z1 3.83) is responsible for the minimum in gure 6.4. This is demonstrated by plotting

129

F(t,k)

0.1

0.01 0

10

20

t k

30

40

50

Fig. 6.4: Numerically computed delity (6.13) at quantum resonance ( = 2) vs. tk, with reference kicking strength k1 = 0.8 and xed k = 0.6283 (squares) 1.257 (circles) and 1.885 (diamonds), and the same initial ensemble as in gure 6.3. The position of the minima corresponds to the time tmin 3.83/k (marked by the arrow). The delity saturates for times t & 25 at a constant value, which is indicated by the horizontal lines.

F (t, k) vs. tk. On the other hand, zero, was found numerically only at t excluded for integer kick counters t.

1 0

6.2.2

We start with the scenario in which SE can happen only in one of the two arms of the Ramsey interferometer (gure 6.1). SE is induced by an additional near resonant (to one interferometer path, i.e. the k1 arm) laser, which is switched on for a time SE immediately after each kick. As described in section 4.3, the additional laser induces transitions to the excited state of the atom from which it may decay spontaneously, leading to random momentum shifts. If a SE event happens, we in principal know along which path of the interferometer the atom went, and a measurement of the emitted photon would correspond to a which-path detection [211]. This information destroys the coherence of the Ramsey evolution [172], and the delity is expected to decay exponentially. For calculating the delity of a state subject to SE with the deterministic state (4.6) the phases t n arg(Wt) in (4.6) are substituted by random processes (in the case without SE the phases cancelled because they do not depend on the kicking strength). If now a SE event happens, we must perform an ensemble average over the randomised phase t in the state (4.38) subject to SE, what gives zero. The probability that up to time t no event occurs is given by

130

10 10 10

0

F(t)

10 10 10 10

10

15

20

25

30

35

t

Fig. 6.5: Fidelity vs. number of applied kicks t at = 2, for the same initial ensemble as in gure 6.3, and with added spontaneous emission (type (I) simulation, cf. section 4.3.5) in only the k1 -arm of the Ramsey interferometer (gure 6.1); kicking strenghts: k1 = 0.8, k2 = 0.6 (circles), k2 = 0.75 (crosses), and for an event probability per kick pSE = 0.2. The decay exponent (solid line) 2/tc = 0.45 2 ln(1 pSE ) = 0.446 is independent of k. The horizontal oset of the two curves arises from the initial, k-dependent drop of delity (see previous gure), and is not a feature of noise.

(1 pSE )t , with the event probability per kick pSE . Hence, we estimate F (t, k1, k2) e2t/tc with tc 1 , ln(1 pSE ) (6.19)

for t & tc , what is in excellent agreement with numerical data for pSE = 0.2 in gure 6.5, and for pSE = 0.05 . . .0.8 in gure 6.6. It shall be emphasised that the delity decays at a rate 2/tc which does not depend on the dierence k, which on the other hand characterises the separation of the two distinct evolutions. For a nite number of rotors and a nite number of SE events, the phase average over exp( i t) does give a typically small, but nite value (j) 1/2 exp( i t) = 1/Ntot Ntot exp( i t ) Ntot , and Ntot can be estimated by j=0 Nrot t/tc , for independent realisation of t . In gure 6.6(a), the estimate for the statistical error is 1/(Nrott/tc ) 1.7 105 which agrees with the data for t 30. If SE occurs in both arms of the Ramsey interferometer, our model [160] which builds on the stochastic wave function (4.38) cannot be used directly because possible SE entangles the time evolutions between the Ramsey pulses. Only for the special case, when in both arms the SE rates pSE are identical, i.e. both arms experience the same random process (which leads to the cancelling of the phases responsible for the fast exponential decay (6.19)), we obtain an estimate

(j)

131

10 10 10

-1

-2

F(t)

10 10

-3

-4

10

-5

0 10 20 30 40 50 60 70 80 90 100

t

Fig. 6.6: Same as previous gure 6.5 for k1 = 0.8, k2 = 0.6, and pSE = 0 (full line), pSE = 0.05 (squares), pSE = 0.1 (circles), pSE = 0.2 (diamonds), pSE = 0.8 (plusses). The slopes (dash-dotted) follow the prediction (6.19): 1/tc = 0.05 (pSE = 0.05), 1/tc = 0.1 (pSE = 0.1), 1/tc = 0.23 (pSE = 0.2), 1/tc = 3.2 (pSE = 0.8).

for t . In appendix C, the random process |Wt| is shown to obey the distribution 2 exp(2/t)/t for t , which it does not depend on quasimomentum. Then we may average (6.13) over the random process to obtain the asymptotic (t ) result:

SE

tk2 2

(6.20)

where we used the integral formula (F.12). Hence, for this special case, we expect the delity to decay algebraically, i.e. much slower than the exponential decay (6.19). Up to now all results focused on the regime of the fundamental quantum resonances = 2 ( N). In the next section we use the -quasi-classical approximation introduced in chapter 5 to investigate the delity in the vicinity of the quantum resonances.

132

0.005

(a)

(b)

F(t)

0.1

0.01

0.001 0 1

(c)

Tsmall

0.1

(d)

Fig. 6.7: Numerical simulations of the delity (6.7) as a function of the applied number of kicks t, for an ensemble of 104 atoms with initial uniform momentum distribution in [0, 1); parameters are k1 = 0.8, k2 = 0.6 and = 2 + with = 0.1 (a), and = 0.025 (c). The corresponding Fourier transforms are shown in (b) and (d), respectively. The peaks in (b,d) show essentially a constant spacing, which is related to the lowest frequency small in the data (a,c), with small = 2/Tsmall 0.066 for = 0.1, and small 0.033 for = 0.025. The highest frequencies . 0.63 are related to a period Tfast 10 which occurs for short times t < 100 in the delity oscillations (a,c), c.f. gure 6.8(a). The arrows in (b,d) mark the position of = small, but the peaks are not resolved, only in (b) there is a tiny trace of the Fourier peak at the lowest frequency which is predicted by the estimate (6.21).

Fourier transform

133

6.3

For computing the delity as dened in (6.7), one must calculate the scalar product of two wave functions. In general, these are not known analytically for the kicked particle dynamics. An exception are the quantum resonances studied in the previous sections. For kicking periods = 2 + with nite = 0, generally a direct calculation of (6.7) is not possible. However, we can restrict to the -quasi-classical model introduced in chapter 5. Figure 6.7(a,c) shows data for the delity of an atomic ensemble (uniform initial momentum distribution in [0, 1)) for = 2 + . We observe characteristic uctuations which we study by calculating the Fourier spectrum of the signal in the frequency domain, see gure 6.7(b,d). For large times, the lowest frequency in the Fourier spectrum dominates. The latter is related to the dierence of the two characteristic frequencies of the nonlinear resonance of the the -quasi-classical phase space. In section 5.2.1 the important role of the characteristic time scale tres = (| |k)1/2 1/res was pointed out. It determines the motion for the quantum system for suciently small detunings . Since two dierent kicking strengths have to be compared this corresponds to compare the motion of two -classical systems with slightly dierent tres . In the harmonic approximation, the basic frequency is dened by the beating frequency, i.e. the dierence of the two xed-point frequencies: small = res,1 res,2 = | |( k1 k2 ) . (6.21)

The dominant delity oscillations for large times t > 100 are indeed determined by small as one can see in gure 6.7. Table 6.1 collects values of small obtained from gure 6.7 and similar cases, showing the correspondence between the estimate (6.21) and the numerical simulations. res,1 0.2507 0.3545 0.4342 0.5013 res,2 0.2171 0.3070 0.3760 0.4342 small 0.0336 0.0475 0.0582 0.0671 Tsmall = 2/small 187 132 108 94

data Tsmall 190 130 105 . 100

Tab. 6.1: Table of periods of delity oscillations (cf. equation (6.21)) for k1 = 0.8, k2 = 0.6 and = 2 + , and for large times t > 100. Presented are the estidata mated periods Tsmall , and the mean periods Tsmall obtained from the data in gure 6.7, and from similar simulations for = 0.05, 0.075.

At small times 10 < t < 100, we observe in gure 6.8(a) (similar for data in gure 6.7(c)) oscillations with a varying amplitude for increasing time. However, we do not know yet, how dierent harmonics or frequencies conspire in the delity oscillations in the region 10 < t < 100. Taking only rotors close to the resonant value of quasi-momentum = 1/2, i.e. in the range [0.4, 0.6], we observe that the highest frequencies dominate

134

10

0

F(t)

10

10

2 0

(a)

10 10

1

10

10

10

Frestricted(t)

10 10 10 10

1 2 3 4

(b)

0 20 40 60 80 100 120 140 160 180 200

t

Fig. 6.8: (a) reproduces data from gure 6.7(a) to show the short time behaviour on a log-log scale. (b) shows delity as in (a) but with the restriction [0.4, 0.6] (solid line), and [0.4, 0.6] (dashed). While the resonant values of quasi-momentum lead to fast oscillations with periods corresponding to the maximum frequencies observed in the Fourier spectrum (gure 6.7), the non-resonant ones show a delity decay without oscillations. The curve in (a) results from the average over all [0, 1), c.f. denition of delity in (6.7).

in the delity oscillations, while for non-resonant [0.4, 0.6] the delity decays without considerable oscillations (see gure 6.8). The maximum frequency max 0.62 in gure 6.7 seems to be universal for all studied = 0.05 . . .0.1, and its origin is still to be claried.

6.4

Since the quantum accelerator modes mentioned in section 6.1 provide a useful tool to prepare atoms in a certain region in momentum space or even in phase space [135, 143, 145, 288], we present some preliminary results on the theoretically more complicated problem with added linear eld. The quantum accelerator modes are found in the vicinity of the fundamental quantum resonances studied in this thesis, i.e. for kicking periods = 2 + ( N). Therefore, a similar qualitative analysis as in the previous section may be used to describe the behaviour of the delity as a function of time, and the parameters k1, k2, , where characterises the linear constant potential (the gravity in appropriate units [145]). The accelerator modes are explained by the islands

2.86 1.86 2.86 1.86 0.86 0.14 1.14

135

0.86 0.14 1.14 2.14 3.14 3.14 2.86 1.86 0.86 0.14 1.14 2.14 3.14 3.14 1.14 0.86 2.86 1.14 0.86

2||

2.86

2.14 3.14 3.14 2.86 1.86 0.86 0.14 1.14 2.14 3.14 3.14 1.14 1.14 0.86 2.86

0.86

2.86

Fig. 6.9: Poincar surface of sections determined by the map (6.22) for = 5.86, e k = 0.5 (upper left), k = 0.6 (upper right), k = 0.75 (lower left), k = 0.8 (lower right). The gravity parameter is chosen = 0.01579 (corresponding to the gravity acceleration in laboratory units [145]). The size and the position of the main island, corresponding to the accelerator mode (1,0) [145], depends on k [145]. Higher-order modes, represented by the island chains around the main (large) islands, are neglected in our discussion. This is partly justied by the large value of the eective Planck constant 2| |.

of stability in the phase space of an appropriate classical map, which can be reduced to a two-dimensional map dened on the 2-torus yielding for negative [145]: Jt+1 = Jt + k sin(t+1 ) t+1 = t Jt . (6.22)

This map is the analogue of (5.7-5.8) with the additional term (remember that k = | |k). Figure 6.9 shows typical phase-space plots generated by iterating (6.22). The islands support the accelerator modes much in the same manner as the nonlinear resonances support the quantum resonances in chapter 5. The delity is obtained by numerical calculations of the quantum evolution of an ensemble of rotors subject to the kicked accelerator. It is presented in gure 6.10 as a function of time t. As in the previous section, we observe characteristic uctuations which are related to the structure of the -classical phase space.

136

10 10 10 10 10 10

0

F(t)

100

200

300

400

500

t

Fig. 6.10: Numerical simulation of the delity vs. the number of applied kicks t for the quantum evolution of the kicked accelerator. The initial momentum distribution was chosen as in gure 6.7, parameters are = 5.86, = 0.01579 , k1 = 0.8, k2 = 0.75 (solid line), k2 = 0.7 (dotted), k2 = 0.65 (dashed), k2 = 0.6 (thin line), k2 = 0.5 (dot-dashed). The thin dashed lines show the overall exponential decay in the regime of not too large times.

The xed-point frequencies FP1 and FP2 (corresponding to res in the previous section) are not directly related to the oscillations observed in gure 6.10. However, a stability analysis around the elliptic xed point (whose position depends on k, see gure 6.9) like in the case without gravity (6.21) must be amended by the relative shift of the island centres of the main accelerator modes visible in gure 6.9. In [145] it was shown that the velocity of the modes is independent of the kicking strength, which is the only dierence in the two dynamical situations that we compare in the present discussion. A crude estimate for the frequencies of oscillations is given by acc |FP1 FP2 | + FP , (6.23)

where FP is the distance in the angle coordinate of the corresponding island centres. This shift of the frequency can be a understood in the following way: for the elliptic motion in one island the trajectory is ahead, whereas for the elliptic motion in the other island the trajectory is behind, so they get closer to each other and rephase after a shorter period than in the case when both islands lie concentric. The shift, which we estimate by FP , depends on the precise geometry of the two islands, and the overlapping area. A better estimate can certainly be derived for a specic island conguration. The periods

In the delity, one of the distinctly evolved states enters in complex conjugated form, hence the phase has a dierent sign with respect to the second state; therefore one trajectory is ahead while the second, corresponding one lacks behind in the classical picture.

10 10

0

137

2.86 0.86 1.14 3.14 300 3.14 2.86 0.86 1.14 3.14 300 3.14

1 2 3 4 5

F(t)

10 10 10 10

0

0 1 2 3 4 5

100

200

1.14

0.86

2.86

10 10 10 10 10 10

2||

1.14

100

200

0.86

2.86

Fig. 6.11: Upper left: delity vs. number of kicks t, for = 5.86, k1 = 0.8, k2 = , and a prepared initial coherent state centred at J0 = 0, 0 = 0.5 with a spread in the angle coordinate of d = 0.1 (dash-dotted), J0 = 0, 0 = 0.5, d = 1 (dotted), J0 = 0, 0 = 3, d = 1 (solid) (the evolution is averaged over 100 quasi-momenta uniformly distributed in [0, 1)). Lower left: Fidelity for same parameters but with uniform initial ensemble of 104 plane wave rotors with p0 = 0 [0, 1) (like in gure 6.10). The slope (dashed) is 0.03, and it is determined by the states having strong overlap with the chaotic component of classical phase space plotted for k1 = 0.8 in the upper right, and for k2 = in the lower right panel. The eective Planck constant 2| | 2.66 is indicated by the square.

won by the numerical data from gure 6.10 compare well (yet not perfectly) with the prediction from (6.23). To further investigate the overall decay of the delity vs. time we focus on a case where the islands are as large as possible for the given values = 5.86, k 0.8, namely k1 = 0.8 and k2 = in gure 6.11. The delity shows beating arising from several frequencies. On the other hand, the decay can be shown to originate solely from the chaotic motion in classical phase space. Launching a coherent state wave packet [16, 211] centred at either the classical island or in the chaotic sea, we observe that in the rst case, the oscillations are dominated by one frequency only, and after some short decay the mean value saturates (probably decaying further owing to tunnelling, see short discussion below), while the delity decays exponentially for much longer times in the latter case. We thus conclude that the initial decay of the delity is governed

138

by states in the chaotic component of classical phase space and/or by states sitting only partly at the accelerator mode island. These states get lost for the accelerator mode, possibly on dierent time scales, but much faster than states sitting mainly within the island. The latter can only escape via tunnelling out of the island which is a much slower process, and we assume that the delity decay after the rst stage after the plateau sets in is indeed determined by such tunnelling processes. The tunnelling regime is very interesting from the theoretical point of view, however, in the experimental setup we referred to in section 6.1, this long-time regime (t 100) is hardly attainable. The slope of the initial delity decay is by a factor of 6 smaller than the Lyapunov exponent [43] calculated for the map (6.22). Therefore, we assume that the quantum coarse-graining of the phase space dynamics smears out the fast decay to arrive at a much smaller decay rate than expected by purely classical means. Note that to observe the structure (clear exponential decay amended by more or less well-dened oscillations) in the delity in gures 6.10 and 6.11, it is necessary to average over some nite (possibly small) interval of quasi-momenta around the values which support the accelerator mode [145]. Calculating the delity in a window in momentum space around the origin n = 0 (not counting the contributions from outside the window), the overall decay in the initial stage is not aected, as can be seen from gure 6.12. However, the oscillations with frequencies estimated by (6.23) are destroyed as soon as the accelerator part of the momentum distribution is outside the window. This stresses again the role of the nonlinear island in classical phase space, corresponding to the accelerator mode, which is responsible for the uctuations. We also used a window in momentum around the accelerator mode peak in the momentum distribution. The window is moving with the peak at a speed /| | [145], and its eect is that there no longer prevails a clear initial exponential decay. Only the second stage occurs earlier, with exponential decay but much smaller slope (which is the same as for the data without any window, see gure 6.12). The latter decay is amended by fast oscillations, which are seen for all cases plotted in gure 6.10 using a window around the accelerator mode peak. The frequency of these fast oscillations is close to the two individual xed point frequencies FP1 or FP2 , but so far we have no explanation for its occurrence yet. The presented results on the the delity in presence of the additional linear potential await a proper theoretical analysis. Such a treatment should include the mentioned problems of determining the decay slopes in both exponential regimes, and the development of a semiclassical theory for the tunnelling out of the accelerator mode island in classical phase space.

139

10

10

F(t)

10

10

10

25

50

75

100

125

150

175

200

t

Fig. 6.12: Fidelity vs. number of applied kicks t for = 5.86, k1 = 0.8, k2 = 0.6: data from gure 6.10 (thick solid) compared with data when the sum in momentum n extends only over a nite window in equation (6.7), 30 < n < 30 (dashed) and nacc 30 < n < nacc + 30 (dotted). The centre of the accelerator mode peak in the momentum distribution nacc moves with the velocity /| |, which is independent of k [145]. The thin solid line shows data like the dotted line for restricted initial momenta p0 = 0 [0.49, 0.51]. While the cuto around zero momentum destroys the oscillations as soon as the accelerated peak has reached the boundary (at t 22), the overall exponential decay (up to t 100) is the same as in the data without cuto. The window around the accelerator mode peak has a completely dierent eect: it leads to a huge drop in the delity because of probability loss for t & tloss 16 (dotted line), and what remains decays with the same slope (dot-dashed) as the thick solid line for t & 100, probably related to tunnelling. Puzzling are the fast oscillations of the dotted curve for t & 25; their amplitude is increased using only a small range of initial quasi-momenta around the one supporting the accelerator mode (thin solid line).

Chapter 7

Rsum e e

7.1 Summary of results

In this thesis transport in energy or momentum space, arising from the energy absorption from an external time periodic driving force, is studied focusing on quantum eects relevant on the atomic scale.

The rst part of the thesis presents novel quantitative support for the hypothesis that the energy transport in periodically driven hydrogen Rydberg atoms is determined by dynamical localisation. The latter implies the quantum suppression of classical diusion, and is based on the mechanism of Anderson localisation [61, 88, 89]. The distributions of ionisation rates of the driven Rydberg states allow for a clear identication of signatures of Anderson localisation. In the statistical average, the ionisation rates follow a universal algebraic distribution ( ) 0.9 (cf. section 3.1). This result is in good agreement with quasi-one-dimensional Anderson models [197]. The corresponding eigenstates are dominantly located in the chaotic component of phase space, which is exactly the domain where dynamical localisation suppresses the transport to the atomic continuum [60]. In addition to the identication of Anderson localisation through the ionisation-rate statistics, the latter provide evidence for other mechanisms hindering quantum transport such as tunnelling out of the regular regions in phase space. We were able to separate dynamical localisation and localisation induced by regular phase-space structures by investigating the localisation properties of the Floquet eigenstates in phase space. Driven Rydberg atoms are real systems perfectly suited for the investigation of the mixed regular-chaotic phase-space dynamics, and the classical-quantum correspondence is essential for a clear understanding of the underlying physical phenomena. Our statistical analysis of the decay rates may be experimentally veried by 141

142

Chapter 7. Rsum e e

spectroscopic techniques similar to the ones extensively used in measuring the (dc) Stark spectrum of Rydberg atoms [71]. In contrast to recent predictions [115], the ionisation probability as a function of time does not behave universally in spite of the universal decay-rate statistics. The more or less algebraic, but highly non-universal decay arises from the local property of the expansion coecients in (2.40) contrary to the global spectral information carried by the ionisation rates. Since the latter and the corresponding coecients are essentially uncorrelated (see gure 3.12), the universality in ( ) does not imply universality in the ionisation or survival probability as a function of time. Experiments with rubidium Rydberg states [114] as well as extensive numerical simulations on lithium and rubidium Rydberg atoms [95] support our results on hydrogen. In the second part of this thesis, we have claried several experimental ndings on the quantum resonant dynamics of atoms subject to time periodic like pulses. Thereby the particle nature of the atoms, which move on a line, and not on a circle as assumed for the kicked-rotor model, plays a crucial role. The atoms in the experiment are non-interacting, and thus they are modelled in our theory as a classical ensemble of quantum particles. The puzzling enhancement of resonance peaks in the mean energy (of the atomic ensemble) vs. the kicking period [82, 83] is identied as an experimental artefact [159]. The latter arises from cutos which inevitably impair the quantum resonant motion in absence of decoherence by spontaneous emission. Our analytical results show that the mean energy increases in both cases with and without spontaneous emission linearly with the number of kicking pulses (equations (4.20) and (4.62)). However, the momentum distributions of the atomic ensemble behave completely dierently in the two cases: without decoherence the main contribution to the energy growth originates from a tiny fraction of resonantly driven atoms (the residual ones are strongly localised around the initial momenta close to zero). On the other hand, the energy increase in presence of spontaneous emission emerges because the momentum distribution spreads as a whole. Decoherence acts as expected and destroys the coherent dynamics, what makes the distribution become Gaussian in the asymptotic limit of large times, characteristic of a diusive random walk in momentum space. Based on a quasi-classical approximation, we have derived a scaling law for the resonance peaks in the mean energy in the absence and presence of decoherence. From these scaling laws (equations (5.18) and (5.25)) the widths of the peaks as function of the detuning from the resonance follow, including their time dependence. The experimentally observed stabilisation of the peaks by adding a small amount of noise corresponds to their broadening arising from the reshuing of atoms in and out of the resonance conditions. This enhances the probability of following the resonant motion, also for small detunings from the exact resonant kicking period. The presented scaling allows the experimentalist to collect data from dierent parameter ranges, and average them in order to enhance the experimental resolution. Our treatment of the spontaneous emission by a quantum Monte Carlo method,

143

or equivalently by a random walk in the rotors Hilbert space, not only permits an exact analysis of the kicked atoms dynamics in presence of noise but also shows that periodically kicked atoms provide an interesting system for the study of decoherence. Most recent experiments [142] make it possible to compare two slightly distinct time evolutions, and to study the inuence of the underlying dynamical regime on the overlap function of two initially identical but dierently evolved wave packets. Using the results derived at the exact quantum resonance conditions as well as the quasi-classical approximation which describes the system close to the resonances, the overlap function, i.e. the delity, has been investigated. In the absence of decoherence, the delity characterises the stability of the quantum motion subject to slight perturbations. Varying the dynamical regime (e.g. quantum resonant, dynamically localised, or dynamics with added linear potential) alters the time dependence of delity. Since the -kicked rotor aords a variety of dynamical scenarios building on its mixed classical phase space, its atom optical realisation provides an exciting system to study the inuence of decoherence and noise competing with classical chaos or its quantum manifestations [293, 294]. Our preliminary results support the intuition that the delity oers a route for the quantitative study of the destruction of quantum coherences via its behaviour on time and on the underlying dynamics.

7.2

Future perspectives

This thesis describes the dynamics of atomic degrees of freedom which are directly accessible to experiments: electronic states in driven Rydberg atoms and external centre-of-mass motion in kicked atoms. The presented discussions gathered tools of classical and quantum transport theory to describe experimental systems prepared and controlled by quantum and atom optical techniques. New experimental methods are oered by cooling atoms even below the temperatures as used so far for the kicked-particle experiments [55, 81, 83, 135, 143, 166, 174, 184, 290]. The coherent atomic ensemble of a Bose-Einstein condensate [124126, 295] may be used to prepare very well dened initial conditions, for instance, initial momentum distributions of atoms. It is possible to reach spreads in momentum of p0 = 0 0.05 (in units of two photon recoils), which are suciently small to strongly populate atoms in the conditions required for the quantum-resonances (see section 2.2.3). The ballistic peaks in the momentum distribution may be clearly visible in such an experiment if in addition the problem of nite-pulse widths in the kicks is minimised (cf. section 4.3.6). Moreover, one may hope to observe higher-order quantum resonances which have not been unambiguously resolved in experiments yet [159]. To observe them it is necessary to guarantee a high resolution in the kicking period as well as a large population of the quasi-momenta near to the corresponding resonant values [64]. The inuence of the atom-atom interaction in the coherent Bose-Einstein con-

144

Chapter 7. Rsum e e

densate [124] on the dynamical evolution is certainly an exciting problem to investigate. The particle nature (as compared to the rotor moving on a circle) is important for the quantum resonances but also for initial coherent superpositions of momentum eigenstates what concerns the kicked-atoms dynamics in presence of an additional linear potential. The eect of interferences on the quantum accelerator modes occurring when such an additional potential is present is partly discussed in [145], and a more complete theoretical understanding together with an experimental exploration of coherent ensemble dynamics would be of great interest. The presented quasi-classical approximation not only allows one to grasp the main features of the kicked-atom dynamics near to the fundamental quantum resonances. It oers a new tool for the study of quantum eects in the quasiclassical limit of small detunings from the resonant kicking period. In section 6.4, the decay of delity was partly related to the chaotic component, and to tunnelling out of the main nonlinear island in the quasi-classical phase space. As the latter is typically nicely separated in a regular island, corresponding to the quantum accelerator mode [145], and a chaotic motion, the quasi-classical approximation provides an ideal playground for the investigation of tunnelling phenomena, such as dynamical [296299] or chaos-assisted tunnelling [210, 216, 217, 235, 300, 301]. On the experimental side, these are challenging perspectives since very well-dened initial conditions are an essential prerequisite to observe relatively weak signals such as tunnelling oscillations [297, 298]. The noisy transport problem in microwave-driven atomic Rydberg states has been explored in several experiments with focus on the destruction of dynamical localisation [13, 52, 53, 72]. The inuence of spontaneous emission on onedimensional Rydberg wave packets, which otherwise show no or little dispersion owing to very weak decay to the continuum, was investigated [77,302,303]. Such studies may be extended to a comprehensive analysis of the survival probabilities, as studied in section 3.1.3. These are certainly aected by noise arising from spontaneous emission, in particular for the large interaction times plotted, for instance, in gure 3.15. A merger of the two systems studied in this thesis, which can also be realised experimentally [304, 305], is provided by periodically kicked Rydberg atoms [306, 307]. Here, the kicking pulses excite the internal electronic degrees of freedom in the same way as for monochromatically driven Rydberg states (as opposed to our kicked rotor realisation, where the atoms centre-of-mass motion is aected). In addition, the kicks contain all harmonics of the basic frequency = 2/ , with the kicking period . Therefore, many decay channels to the atomic continuum, including direct one-photon ionisation, are present. Recent results [308, 309] show analogies to the Rydberg problem discussed in this thesis, and in particular, uctuations in the ionisation probabilities which are reminiscent of the atomic conductance uctuations mentioned in section 3.2.3. Moreover, since direct one or few-photon ionisation is possible, very large decay rates appear besides the rates connected to multi-photon transport. This implies that parametric fractal uctuations in the survival probability or in an

145

analogue of the atomic conductance (3.11) may be observable in an experimentally accessible situation. The presented new aspects of the analogy between periodically-driven Rydberg states and Anderson-localised solids may also be investigated in more complex systems than one-electron problems. More precisely, one may study systems in which more (strongly coupled) degrees of freedom are present. Can one identify transport mechanisms similar to the ones discussed in this thesis also for systems such as the driven three-body helium problem [76, 310, 311]? The additional particle-particle interaction oers new perspectives in the analogy between atomic problems with many degrees of freedom and many-body solid state systems.

Appendix A

A.1 Proof of estimate (4.16)

Mn (t) = |n|N

1 2 2

/2

dx

/2

d

|n|N

2 Jn (z)

with z k sin(x) csc() , for any positive integer N . In A.2 we show that

2 Jn (z) 2 |n|N

(A.1)

ez 2N

2N

(A.2)

With this inequality, we derive an upper bound for the sum in the inner integral in (A.1) when || > /2, for 0 < < . If || does not fulll this condition, 2 we use the upper bound 1 for Jn 1 (cf. F.3), and that by (A.1) leads n= to the rst term on the right in (A.3). Noting that |z| < k| csc()| < k/(2 ) whenever /2 > || > , one obtains altogether

Mn (t) |n|N

+2

ke 4N

2N

(A.3)

We now minimise the right-hand side by requiring 1 4N Solving for , we obtain ek = 4N 16N 2 ek 147

1 2N +1

ke 4N

2N 1

=0.

(A.4)

(A.5)

148

(which is indeed not larger than whenever N > k 1.03 . . .). Replacing (A.5) in (A.3) yields the estimate (4.16):

Mn |n|N

ke 16

2N 2N +1

N 2N +1

12N

2+

1 N

(A.6)

A.2

Using twice the bound (F.7) and the power series expansion of Bessel functions [157],

2 Jn (z)enr n=0

n=0

|z|er/2 2

2n

(A.7)

n=N 2 Jn (z) eN r e|z|e

r/2

(A.8)

Inequality (A.2) follows upon optimising, i.e. minimising, with respect to r what gives r = 2 ln(2N/|z|). Then eN r e|z|e

r/2

r=2 ln(2N/|z|)

|z| 2N

e2N ,

and since z R, and taking into account a factor of two for the equivalent of (A.8) with negative n, the inequality (A.2) follows.

A.3

/2 /2 2 d J0 (kz csc()) for z = 0 ; f (0) =

1 . 2

(A.9)

The integrand in (A.9) is meant = 0 for = 0, z = 0. Using the integral identity (F.3) for Bessel functions, (4.15) may be rewritten as:

Mn =

dx cos(2nx)f (sin(x)) ,

0

(A.10)

so, for |n| > 0, Mn is the 2nth coecient in the cosine expansion of f (sin(x)). The function f (z) is dierentiable in [1, 1] \ {0}. It will be presently shown that 4k f (0+) = lim f (z) = 2 = 0 . z0+

149

Since f (z) is an even function, it will follow that its rst derivative is discontinuous at z = 0. We choose > 0 and write f (z) = f (z) + g (z) , f (z) 1 2

2 d J0 (kz csc()) .

(A.11)

Then g is dierentiable around 0, with g (0) = 0. Hence, f (0+) = f (0+). Next we note that if z > 0, f (z) = 1z 1 d F (kz csc()) ,

where F (x) xJ0 (x)J0(x). Noting that | csc() 1 | < c1 for 0 < || < /2 and some numerical constant c1 , one easily nds f (z) = 1 z 1 = 2k

1

d F (kz1 ) + O( )

du u1 J0 (u)J1(u) + O( ) ,

(A.12) 0+ we 4k . 2

kz/

where J0 (z) = J1 (z) was used. Letting z 0+ and thereafter obtain: f (0+) = lim f (0+) = 2k 1

0+ 0

du u1 J0 (u)J1(u) =

The integral was computed by using (F.6) and then formula 11.4.36 in [157]. Next we recall from (A.10) f (sin x) = 1 M + M cos(2nx) . 2 0 n=1 n

According to the above analysis, the derivative of this function jumps by 8k/ 2 at x = j (j any integer). Hence the second derivative has the singular part 8k 2 j (x j), leading to the asymptotic value 16k 3 for the coecients in its cosine expansion. This yields

Mn

4k as n . 3n2

(A.13)

Appendix B

B.1 Independence of the variables zj

We show that, for any integers n, m, (m > n), the variables (zn , ..., zm) are independent of the variables (z0 , ..., zk) whenever k n 2. It suces to show that (zn , ..., zm) are independent of (0, ..., k, 0, ...k ). To see this, let f be an arbitrary (Borel) function of m n + 1 complex variables, and consider Mk E{f (zn , ..., zm)|0, ..., k, 0, ..., k}. (B.1)

Looking at (4.37), and recalling that the j are mutually independent, one notes that (zn , ..., zm) depend on j , j (0 j k) through the factor k k exp( i 0 j j ), hence only through j=0 j j mod(2). Therefore, (B.1) k is a function of the variable k 0 j j mod(2) alone, i.e. Mk = Mk (k ). Furthermore, since k + 1 < n, Mk (k ) = = dP (k+1 , k+1 ) Mk+1 (k+1 ) dP (k+1 , k+1 ) Mk+1 (k + k+1 k+1 ) (B.2)

because k+1 , k+1 are independent of past variables; here dP (., .) is their joint distribution. Now k+1 is independent of the integer k+1 , and it is uniformly distributed in [, ). Then the integral does not depend on k , so E{f (zn , ..., zm)|0, ..., k, 0, ..., k} = E{f (zn , ..., zm)}, what proves the announced independence property. 150

151

B.2

The properties of the process Zm allow to conclude that its distribution is asymptotically Gaussian, thanks to known results about the Central Limit Theorem for weakly dependent sequences [312]. Isotropy of the limit Gaussian distribution follows from computing the mean square displacement along an arbitrary direction in N steps, i.e. N E Re2 e i zj

j=0

since

Using E{zj zk |} = jk j for any j, k 0 such that j + k > 1 (see (4.40) and section 4.3.3), the result is independent of , hence the limit distribution is isotropic in the complex plane.

Appendix C

We show that as t , the distribution of |Wt | approaches a Gaussian distribution in the complex plane centred at 0. To this end we assume for simplicity that Wt and Zm (4.37) are real; the argument carries through by replicating it for the real and the imaginary parts separately. We dene Wt = Wt/ t, and prove that the characteristic function [259, 267] of Wt , i.e. (Wt) E e i yWt

(C.1)

is a Gaussian for t . From the denition of the process Nt after equation (4.29), it follows that Nt for t with probability 1 if pSE > 0. The previous appendix shows further that the conditional distribution of ZNt , at given Nt , is asymptotically Gaussian as Nt with variance Nt. Therefore, one obtains E e i yWt Nt = E

yZN t t

Nt

c1e

c2 y 2 Nt t

(C.2)

with constants c1, c2. To obtain (Wt) we must average (C.2) over the distribution of Nt which is Bernoullian (see its denition after (4.29)). Consequently, (Wt ) c1E

t

c2 y 2 Nt t

= c1

m=0

(1 pSE )tm .

(C.3)

With the binomial formula (F.10), this gives (Wt) c1 1 pSE 1 e c1 1 pSE c2 y 2 t 152

t

c2 y 2 t

c1 epSE c2 y ,

2

(C.4)

153

where (F.11) was used. Since the Fourier transform of a Gaussian is again a Gaussian with inverse variance, it follows that asymptotically Wt obeys a Gaussian distribution with variance 1, hence also Wt with variance t. From the discussion in section 4.3.3, i.e. with (4.55) which is in all cases approached for t , the variance is exactly t. This nally tells us that = |Wt| is asymptotically distributed with a density dFt () = 2t1 d exp(2/t), i.e. a two-dimensional isotropic Gaussian in the complex plane.

Appendix D

Let 1 , (j = 0, 1, ...) be real nonnegative, independent random variables exj ponentially distributed with E{1} = 1. For j a nonnegative integer denote j 1 sj = j k , and s1 0. For given u > 0 let Nu max{j : sj < u}. We k=0 shall compute the expectation of the random variable

1 Nu

fu

j=0

f (1) + f (u sNu ) , 1 j

(D.1)

where f (x) is a given nonrandom function; the sum in equation (5.23) is of this form, with f (x) = xH(xv). We write fu = fu,j , where j=0 fu,j (u sj1 )[(u sj1 1 )f (1) + j j + (1 + sj1 u)f (u sj1 )] , j and (.) is the unit step function. Then, denoting G(x) =

x s 0 dsf (s)e ,

(D.2)

(D.3)

E{fu } =

j=0

u 0

+

j=1

(D.4)

where dPj (x) = dx ex xj1 /(j 1)! is the distribution of sj1 for j > 0. Summing over j and replacing the denition of G(x) we nally obtain E{fu } = 2

0 u

dx ex f (x) + f (u)eu +

0

dx ex f (x)(u x) .

(D.5)

Appendix E

Let us consider a two-level atom. The state space is H C 2 where H is the state space of a point particle on a line. The states are then represented by spinors (1, 2) where 1,2 H represent the internal levels of the atom. E1 , E2 are the energies of the levels (~ 1 is assumed). Applying to the atomic transition between these levels two successive /2 Ramsey pulses at a frequency E2 E1, one has a Ramsey separated eld interferometer [172, 289, RA 292], see gure 6.1. We denote with the phase of the Ramsey pulse source. For notational convenience, we assume that there is no time delay between the Ramsey pulses and the start and the end of the kicked-particle evolution, respectively. A Ramsey pulse then produces the following instantaneous change of the state vector ( = 0 for the rst pulse at time 0): 1 2 1 2 1 e i e i 1 1 2 R 1 2 . (E.1)

The evolution operator from time 0 to time t which includes the kicked-particle dynamics then reads Ut, R Wt R0 , where Wt = e i E1 t U1,t 0 e

i E2 t

0 U2,t

. (E.2)

U1,2,t are the kicked-particle evolution operators in the upper and in the lower atomic level, respectively. With the initial state 1 = 0 , 2 = , the state after the nal Ramsey pulse then reads 1,(t) = 2,(t) = 1 i E1 t (U1,t + e i Ramsey U2,t) e 2 1 i E2 t i Ramsey e (U1,te + U2,t) , 2 155

(E.3)

156

with the phase Ramsey (E2 E1)t . The momentum distribution in the lower state 1 is given by P1(p, t, Ramsey) = +

2 2 1 1 p|U1,t + p|U2,t 4 4 1 Re e i Ramsey U1,t|p p|U2,t 2

(E.4)

Integrating over p, the total probability in the lower level is obtained: P1 (t, Ramsey ) = 1 1 + Re e i Ramsey U1,t|U2,t 2 . (E.5)

We now consider the case when the initial pure state (of atomic motion) is replaced by a mixture which is described by the following statistical density operator = n |n n| .

n

The nal population in the lower level is then P 1 (t, Ramsey) = with G(t, Ramsey)

n

1 {1 + G(t, Ramsey)} , 2

(E.6)

where, at xed time t, diers from Ramsey , as well as from by a constant shift, and

2 2

F (t) =

n

2

=

n

is the delity, cf. equation (6.2). From equation (E.6) and subsequent ones, we see that the delity is equal to the dierence between the maximum and the minimum values taken by P 1 (t, ) while the original phase varies in [0, 2). We arrive at the conclusion that the delity is accessible in principle by an experimental setup as used in [142], cf. gure 6.1. Equation (E.8) coincides with the delity dened in (6.7) when identifying |n with the rotor states | (cf. sections 2.2.3 and 4.2.1).

Appendix F

The following formulas involving Bessel functions are used in part II of this thesis and previous appendices; these are taken from [157] or derived from formulas there (we give the corresponding numbers from [157] in []): 1 2

+2

2 n2 Jn (x) = n=

1 2 x 2

[9.1.76]

2 2 dx Jn (b sin(x)) = 0 0

cos(zx) dx 1 x2 1 1 z |n| Im(z) |Jn (z)| e n! 2 |z| 2 cos(z /4) J0 (z) z J0(z) = From [157], we further collect: sin(x) x0 1 [4.3.74] x m m! m (a + b) = ak bmk [3.1.1] k!(m k)! k=0 x n n x 1+ e [4.2.21] . n 157

158

In section 6.2.2 we used an integral for the Bessel function J0 adapted from [6.615] in [257]:

0

1 2 dy eay J0 (2b y) = J0 a

2b2 a

2b2 a

(F.12)

The following integral is used, for instance, when calculating the average energy in section 4.2.2: 2 sin2 (tx) dx = 2t . (F.13) sin2 (x) 0

Appendix G

Publications

Publications of the author related to the topics covered in this thesis: A classical scaling theory of quantum resonances Sandro Wimberger, Italo Guarneri, and Shmuel Fishman to appear in Phys. Rev. Lett. Decoherence as a probe of coherent quantum dynamics Michael B. dArcy, Rachel M. Godun, Gil S. Summy, Italo Guarneri, Sandro Wimberger, Shmuel Fishman, and Andreas Buchleitner to appear in Phys. Rev. E. Decay, interference, and chaos: How simple atoms mimic disorder Andreas Krug, Sandro Wimberger, and Andreas Buchleitner Eur. Phys. J. D, 26, 21 (2003). Quantum resonances and decoherence for -kicked atoms Sandro Wimberger, Italo Guarneri, and Shmuel Fishman Nonlinearity 16, 1381 (2003). Decay rates and survival probabilities in open quantum systems Sandro Wimberger, Andreas Krug, and Andreas Buchleitner Phys. Rev. Lett. 89, 263601 (2002). Signatures of Anderson localization in the ionization rates of periodically driven Rydberg states Sandro Wimberger and Andreas Buchleitner J. Phys. A 34, 7181 (2001).

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Acknowledgements

This thesis in the frame of a binational dissertation would not have been accomplished without the support, advice, and help of many people whom I am extremely grateful to: many thanks to Andreas Buchleitner who spent a lot of his precious time on bureaucratic issues, making the binational doctorate possible. Seine nichtendende Geduld mit allen Problemchen gab mir die ntige Sttze in mancher komplexen o u Situation. Italo Guarneri for accepting the oer of the rst binational doctorate programme with the Universit` degli Studi dell Insubria. His advice on physia cal problems in general as well as on detailed questions was often given with outstanding humour. La ringrazio molto! Herrn Dr. Herbert Sthr gebhrt besonderer Dank fr sein Engagement, welches o u u die vorliegende binationale Promotion an der Fakultt fr Physik der Ludwiga u Maximilians-Universitt Mnchen mglich machte. a u o professore Giulio Casati and his group at the International Centre for the Study of Dynamical Systems in Como. Many bureaucratic issues could be solved by the ingenious eorts of the professores Giorgio Mantica and Roberto Artuso. Mille, mille grazie! many thanks also to my collaborator Prof. Shmuel Fishman. His ideas, accompanied by continuously imposed pressure of exactness, led to fruitful results. Dr. Andreas Krug for computing 3D data for the driven hydrogen problem, and his very useful visit in Como. Wir vermissen unseren Rechenmeister! dott. Marcello Terraneo for the collaboration on uctuations, and for many advises what concerns quantum maps. Dr. Gregor Veble whose help with a physicists everyday-life problems, his cooking skills, together with Mojcas charm made my life pleasant in Como. Dr. Gil Summy, Dr. Michael dArcy for their courtesy to use their experimental data in this thesis as well as for the generous invitation to Oxford which, actually, I had missed a lot. Our discussions fostered the work presented in this thesis. Many thanks also to Dr. Rachel Godun, Dr. Simon Gardiner, Frau Dipl.-Phys. Sophie Schlunk, and the alumnus at the Oxford quantum-chaos experiments, Herrn Prof. Dr. Markus Oberthaler. dott. Giuliano Benenti for discussions on delity.

den Mitgliedern und Ehemaligen meiner Dresdner Arbeitsgruppe: den Herren Peter Schlagheck, Klaus Hornberger, Cord Mller, sowie den Herren Thomas u Wellens, Javier Madroero, Florian Minter. Besondern Dank den lehrreichen n Diskussion mit Herrn Boris Fine, Herrn Andr R. R. Carvalho und Herrn Vye acheslav Shatokhin. Dr. Christian Miniatura and Prof. Ken Taylor for discussions on spontaneous emission processes and scattering theory. dott. Fabrizio Lillo e professore Rosario Mantegna per lospitalit` a Palermo. a ai membri del gruppo Optical Nonlinear Processes, Dr. Ottavia Jedrkiewicz, Dr. Jos` Trull e professore Paolo Di Trapani per alcune discussioni a Como. e ringrazio le nostre secretarie a Como per tutto laiuto: Barbara Arcari, Roberta Meroni, Lorenza Paolucci e Barbara Span`. In particolare, ringrazio Alessandra o Parassole per essere la mia mamma italiana, e Silvia Ceccarelli per le gite in Questura. Mille grazie anche ad Annalisa Bardelli per tutti i libri che mi ha aiutato a trovare. vielen herzlichen Dank an Frau Claudia Lantsch, Frau Gabriele Makolies und Frau Ilona Auguszt fr die Meisterung organisatorischer Probleme in Dresu den. Unserer Bibliothekarin Frau Heidi Nther fr die vielen Buch- und Ara u tikelbestellungen. ai miei amici, Damiano Monticelli, Giampoalo Cristadoro, Gloria Tabacchi, Alice Barale, e Franz Josef Kaiser. Mille grazie anche a Daniela Fanetti per le corse vicino alla nostra Via Pannilani. my friends, Mario Brandhorst & Kelley Wilder, Bettina Hartz, Stephanie Hutzler, Chien-Nan Liu, Lieselot Vandevenne, and Simone Zerer for their visits in Como. Ivo Hring and Nilfer Baba for many stimuli beyond the ivory tower of physics. a u The Finite Systems Group, and the football teams of the MPI-PKS and the MPI-CPFS for the international atmosphere in Dresden. e mille baci ad Andreana che mi fa piacere lItalia come nessun altro.

Alles Vergngliche a ist nur ein Gleichnis; das Unzulngliche a Hier wirds Ereignis; Das Unbeschreibliche, Hier ist es getan, Das Ewig-Weibliche Zieht uns hinan. J. v. Goethe, Faust, 2. Teil

Lebenslauf

Name Geburtsdatum/ort Staatsangehrigkeit o Adresse Sandro Marcel Wimberger 22. Oktober 1974 in Passau deutsch Im Tnental 28 a D-94051 Hauzenberg

Schulbildung

1981 - 1985 1985 - 1994 1994 Grundschule Hauzenberg Gymnasium Leopoldinum Passau (humanistischer Zweig) Abitur (Note: 1.0)

1994 - 2000 1997 - 1998 1999 - 2000 14. November 2000 Ludwig-Maximilians-Universitt Mnchen a u Visiting student an der University of Oxford (England) Visiting student am Weizmann Institute of Science (Israel) Physik Diplom (mit Auszeichnung) Diplomarbeit: Der Leitwert von Atomen, Betreuer: PD Dr. Andreas Buchleitner, angefertigt am Max-Planck-Institut fr u Physik komplexer Systeme, Dresden

2000 - 2003 an der Fakultt fr Physik der LMU Mnchen a u u und an der Universit` degli Studi dell Insubria a Betreuer: PD Dr. Andreas Buchleitner (Mnchen/Dresden) u und Prof. Italo Guarneri (Como)

Stipendien

1994 - 2000 1997 - 2000 2001 - 2003 Stipendium nach dem Bayerischen Begabtenfrderungsgesetz o Studienstiftung des deutschen Volkes Contratto di Collaborazione des Istituto Nazionale per la Fisica della Materia (Italien)