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Ultracold Atomic Gases in Artificial Magnetic Fields Ultracold Atomic Gases in Artificial Magnetic Fields Von der Fakultat¨ fur¨ Mathematik und Physik der Gottfried Wilhelm Leibniz Universitat¨ Hannover zur Erlangung des Grades Doktor der Naturwissenschaften — Doctor rerum naturalium — (Dr. rer. nat.) genehmigte Dissertation von Dipl. Phys. Klaus Osterloh geboren am 15. April 1977 in Herford in Nordrhein-Westfalen 2006 Referent : Prof. Dr. Maciej Lewenstein Korreferent : Prof. Dr. Holger Frahm Tag der Disputation : 07.12.2006 ePrint URL : http://edok01.tib.uni-hannover.de/edoks/e01dh06/521137799.pdf To Anna-Katharina “A thing of beauty is a joy forever, its loveliness increases, it will never pass into nothingness.” John Keats Abstract A phenomenon can hardly be found that accompanied physical paradigms and theoretical con- cepts in a more reflecting way than magnetism. From the beginnings of metaphysics and the first classical approaches to magnetic poles and streamlines of the field, it has inspired modern physics on its way to the classical field description of electrodynamics, and further to the quan- tum mechanical description of internal degrees of freedom of elementary particles. Meanwhile, magnetic manifestations have posed and still do pose complex and often controversially debated questions. This regards so various and utterly distinct topics as quantum spin systems and the grand unification theory. This may be foremost caused by the fact that all of these effects are based on correlated structures, which are induced by the interplay of dynamics and elementary interactions. It is strongly correlated systems that certainly represent one of the most fascinat- ing and universal fields of research. In particular, low dimensional systems are in the focus of interest, as they reveal strongly pronounced correlations of counterintuitive nature. As regards this framework, the quantum Hall effect must be seen as one of the most intriguing and complex problems of modern solid state physics. Even after two decades and the same number of Nobel prizes, it still keeps researchers of nearly all fields of physics occupied. In spite of seminal progress, its inherent correlated order still lacks understanding on a microscopic level. Despite this, it is obvious that the phenomenon is thoroughly fundamental of nature. To resolve some puzzles of this nature is a key topic of this thesis. The chosen approach might be, at first sight, confusing, as ultracold atomic gases are electrically neutral objects, and their quantum mechan- ical characteristics only pave the way for successful cooling in optical and magnetic traps. Yet, it is because of the rapid development of quantum optical experimental techniques during the last decade that there are methods at hand to grant access to various interesting models and problems of the theory of condensed matter. Exactly the amalgamation of different physical disciplines, here atomic and molecular physics and quantum optics, have contributed to this decisively. Not only is it possible to trap atoms in nearly perfect periodic crystals of light, in which their interactions can be determined in strength and type quasi at will, but also to subject them to artificial magnetic fields by phase imprinting and rotation-induced centrifugal poten- tials. In such systems, the trapped particles react as if they were charged objects and are subject to effective vector potentials. The dimensionality of the trap systems can also be controlled. Paradigms of strongly correlated systems, such as the Mott insulator and the one-dimensional Tonks Girardeau gas, have recently been experimentally achieved and detected. Experiments with rotating Bose gases have continuously drawn nearer to the quantum Hall regime. Great effort is being made to achieve a stable rotation of fermions soon, and to venture into the low- est strongly correlated Landau level in microscopic atomic probes. A main focus of this thesis is devoted to the ground state structures appearing in this regime and their elementary excita- tions on the crossover from weakly interacting states to strongly correlated Laughlin liquids. The possibilities of quantum-mechanical manipulation of ultracold atomic gases have, as re- gards the above phenomena, yet by far not been exhausted. Completely novel physical systems, which cannot be found in nature in this form, can be achieved by mixing atomic species or by simultaneously controlling the internal atomic degrees of freedom. The latter is facilitated by the realization of effective isospins, which can be exposed to state-selective vector potentials in optical lattices. These gauge potentials act on the space of atomic isospins and can be ac- cordingly modeled as operators of symmetry groups. Thereby, non-Abelian vector potentials can be generated, inducing complex correlated atom dynamics, which can be solely observed in this form in elementary field theories. The resulting opportunities for generalized quantum Hall systems and simulation of lattice gauge theories conclude the investigations of this thesis. Keywords: strongly correlated systems, quantum Hall effect, non-Abelian gauge theories Zusammenfassung Kaum ein Ph¨anomenhat die Entwicklung physikalischer Gedankenmodelle und theoretischer Konzepte reflektierender begleitet als der Magnetismus. Von den Anf¨angender Metaphysik und ersten klassischen Beschreibungen magnetischer Pole und Feldlinien inspirierte er die mo- derne Physik auf ihrem Weg zur klassischen Feldbeschreibung der Elektrodynamik bis hin zur quantenmechanischen Beschreibung innerer Freiheitsgrade der Elementarteilchen. Magnetische Erscheinungsformen werfen hierbei bis zum heutigen Tage komplexe und oftmals kontrovers diskutierte Fragestellungen auf, und dies in so vielf¨altigenund scheinbar grundverschiedenen Gebieten wie Quantenspinsystemen und der großen Vereinheitlichungstheorie. Dies mag vor allem darin begr¨undetsein, dass diesen Effekten korrelierte Strukturen zugrunde liegen, die durch das Wechselspiel von Dynamik und elementaren Wechselwirkungen induziert werden. Stark korrelierte Systeme stellen wohl eines der spannendsten und universellsten Forschungs- gebiete der modernen Physik. Insbesondere niederdimensionale Systeme sind hierbei von be- sonderem Interesse, da in diesen Korrelationen besonders ausgepr¨agtund von kontraintuitiver Natur sind. Der Quanten-Hall Effekt ist in diesem Rahmen sicher eines der fesselndsten und komplexesten Probleme moderner Festk¨orperphysikund besch¨aftigtselbst nach zwei Jahrzehn- ten und der gleichen Zahl ausgelobter Nobelpreise Forscher aus nahezu allen Teilbereichen der Physik. Die ihm inh¨arentemagnetisch korrelierte Ordnung ist trotz wegweisender Fortschritte mikroskopisch weiterhin nur unvollst¨andigverstanden. Offenkundig ist hingegen, dass dieses Ph¨anomenvon ausgesprochen fundamentaler Natur ist. Dieser ein wenig n¨aherzu kommen, ist ein Kernthema dieser Arbeit. Der Zugang scheint hierbei zun¨achstverwirrend, da ultrakalte atomare Gase elektrisch neutrale Objekte sind und ihre quantenmagnetischen Charakteristika lediglich den Weg zu ihrer erfolgreichen K¨uhlungin optischen und magnetischen Fallen bah- nen. Aufgrund der rapiden Entwicklung quantenoptischer Experimentiertechniken im letzten Jahrzehnt stehen jedoch mittlerweile Methoden zur Verf¨ugung,die einen Zugang zu vielf¨altigen interessanten Modellen und Problemen der Theorie kondensierter Materie gew¨ahrleisten.Ge- rade die Verschmelzung verschiedener physikalischer Disziplinen wie die der Atom- und Mo- lek¨ulphysikmit der Quantenoptik hat hierzu entscheidend beigetragen. Nicht nur lassen sich heutzutage Atome in nahezu perfekten periodischen Potentialstrukturen fangen, in denen ihre Wechselwirkung quasi nach Belieben in St¨arke und Art durchstimmbar ist, sondern dar¨uber hinaus durch Phasenaufpr¨agungund rotationsinduzierte Zentrifugalpotentiale dem Effekt artifi- zieller Magnetfelder unterwerfen. In solchen Systemen verhalten sich die gefangenen Teilchen als w¨arensie geladene Objekte und stehen in Wechselwirkung mit effektiven Vektorpotentia- len. Die Dimensionalit¨atder Fallensysteme kann hierbei ebenfalls kontrolliert werden. Paradig- men stark wechselwirkender Systeme wie der Mott Isolator und das eindimensionale Tonks- Girardeau Gas sind k¨urzlichexperimentell realisiert und detektiert worden. Experimente mit rotierenden Bose Gasen n¨ahernsich kontinuierlich dem Quanten-Hall Regime. Große Anstren- gungen werden unternommen, bald auch Fermionen stabil rotieren zu k¨onnenund gerade in mikroskopischen atomaren Proben ins stark korrelierte niedrigste Landau Niveau vorzustoßen. Den in diesem Regime auftretenden Grundzustandsstrukturen und ihren elementaren Anregun- gen im Ubergang¨ von schwach wechselwirkenden Zust¨andenzu stark korrelierten Laughlin Fl¨ussigkeiten ist ein Schwerpunkt dieser Arbeit gewidmet. Die Analyse behandelt hierbei kurz- reichweitig wechselwirkende Bose und dipolare Fermi Gase. Im Fokus der Untersuchungen stehen hierbei bosonische Vortexstrukturen und eine kritische Gegen¨uberstellungder verschie- denen Wechselwirkungen. Die M¨oglichkeiten quantenmechanischer Manipulation ultrakalter atomarer Gase sind im Rahmen der obigen Ph¨anomenejedoch bei weitem nicht ersch¨opft.Voll- kommen neuartige Quantensysteme, die in dieser Form in der Natur nicht vorkommen, lassen sich durch Mischung atomarer Spezies oder die simultane Kontrolle innerer atomarer Freiheits- grade verwirklichen. Letzteres erm¨oglichtinsbesondere die Realisierung effektiver Isospins, die zustandsselektiven Vektorpotentialen in optischen Gittern ausgesetzt werden k¨onnen.Diese
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