Research Collection. Excitations and transport in strongly correlated bosonic matter. Doctoral Thesis. ETH Library. Author(s): Huber, Sebastian David
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1 Research Collection Doctoral Thesis Excitations and transport in strongly correlated bosonic matter Author(s): Huber, Sebastian David Publication Date: 2008 Permanent Link: Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library
2 DISS. ETH No EXCITATIONS AND TRANSPORT IN STRONGLY CORRELATED BOSONIC MATTER A dissertation submitted to ETH ZURICH for the degree of Doctor of Sciences presented by Sebastian David Huber Dipl. Phys. ETH born citizen of Unterlunkhofen accepted on the recommendation of Prof. Dr. J. W. Blatter, examiner Prof. Dr. M. Lewenstein, co-examiner Prof. Dr. M. Sigrist, co-examiner 2008
3 Abstract In this thesis we study cold bosonic atoms trapped in a lattice potential. The recent progress in cooling alkali atoms and manipulating their interactions allows for the investigation of properties beyond the ground state, namely excitations and non-equilibrium physics. When confined to an array of lattice sites, bosonic atoms undergo a zero temperature phase transition between a superfluid and an insulating state. The excitations in a system of such confined lattice bosons strongly depend on the lattice potential and the interaction strength. In the insulating state a particle and a hole-type mode are responsible for the low-energy physics. For the superfluid phase, where a macroscopic number of particles occupy a single quantum state, a gapless sound and a gapped amplitude mode determine the low-energy properties. The content of the present work is to derive these excitation and to investigate their influence on the low-energy physics. To understand the physics of a Bose Einstein condensate, a state of matter where a quantum mechanical wave function becomes observable at a macroscopic length scale, one can apply a description in terms of a nonlinear Schrödinger equation. While such an approach accounts for all relevant properties of harmonically trapped atoms, in particular the gapless sound mode, it fails to properly explain the physics of a condensate subject to a strong lattice potential. The broken translation invariance, together with the effect of strong interactions gives rise to a different low-energy theory, a nonlinear Klein Gordon equation. Such a relativistic theory enables an additional type of excitation, a gapped amplitude modulation of the condensate fraction. Here, we analyze how such a mode disappears when the parameters are changed from the strongly interacting regime towards the domain where the nonlinear Schrödinger equation is valid. We find that the gapped mode, named Higgs mode after its similarity with the Higgs particle, exists throughout a large portion of the phase diagram and only disappears in a high energy continuum once the lattice can be fully absorbed in a renormalized mass. We discuss its evolution through the domain of intermediate interaction and show how it can be observed in an experiment. In the insulating phase, where interactions force the atoms in the ground state to be localized to single sites, the system carries two type of excitations: v
4 A particle moving on top of the regular array or a hole moving through the otherwise static background. How these excitations couple to experimental probes available is subject of the second chapter. To answer this question we develop a method capable of incorporating fluctuations on top of a meanfield ground state in the insulating phase as well as in the close by superfluid. Doing so, we provide a signature for the zero temperature phase transition in terms of measurable properties of the excitation spectrum. Furthermore, our scheme allows for a unified approach, valid for both thermodynamic phases, and provides the connection to the discussion of the collective modes in the first part of this thesis. The recent progress in preparing quantum gases in optical lattices and obtaining in-situ information which goes beyond an averaged momentum distribution allows for the study of effects related to the finite spatial extension of the atom cloud. Due to the inhomogeneous nature of the laboratory systems, a heterostructure of alternating superfluid and insulating shells develops in the harmonic trap. We discuss mesoscopic aspects of such a structure by applying a first quantized formalism similar to the description of spatially nonuniform superconductors. By calculating the heat conductivity we obtain results relevant to the recent discussion on the heating problem in optical lattices close to the insulating regime. The field of cold atoms develops currently at a pace where more and more quantum many-body effects can be simulated in a controlled environment. In addition, the high degree of isolation and control over these systems offer pathways to new interesting physics. With the present thesis we hope to provide some of the understanding demanded by this newly established field located between traditional condensed matter physics and atomic and molecular physics. vi
5 Kurzfassung In dieser Dissertation werden kalte Atome, die einem Gitterpotential ausgesetzt sind, diskutiert. Fortschritte in der Kühlung von Alkaliatomen und der Kontrolle ihrer Wechselwirkung erlauben die Untersuchung von Eigenschaften solcher Systeme, die über den Grundzustand hinausgehen. Bosonische Atome, restringiert auf einzelne Gitterplätze, durchlaufen einen Nulltemperaturphasenübergang zwischen einem supraflüssigen und einem isolierenden Zustand. Die Anregungen in einem derart eingeschlossenen System hängen wesentlich von der Tiefe des Gitterpotentiales und der Stärke der Wechselwirkung ab. Während im isolierenden Zustand eine Teilchen- und eine Lochanregung für die Niederenergiephysik verantwortlich zeichnen, sind es in der supraflüssigen Phase eine weiche Dichteanregung und eine massive Amplitudenmode. Der Inhalt der vorliegenden Arbeit ist die Herleitung dieser Anregungen und die Untersuchung ihrer Auswirkungen auf die Niederenergiephysik. Das Verständnis der Physik eines Bose-Einstein-Kondensates, eines Materiezustandes bei welchem eine quantenmechanische Wellenfunktion auf einer makroskopischen Längenskala beobachtbar wird, basiert auf dem Studium der nichtlinearen Schrödingergleichung. Während ein solcher Zugang die Berechnung vieler relevanten Grössen erlaubt, namentlich die Geschwindigkeit von Dichtewellen, versagt er in der Beschreibung von Kondensaten, welche tiefen Gitterpotentialen ausgesetzt werden. Die Aufhebung der Translationsinvarianz, zusammen mit dem Effekt starker Wechselwirkung, lassen eine andere Niederenergietheorie in Form einer nichtlinearen Klein-Gordon- Gleichung erscheinen. Solch eine relativistische Beschreibung erlaubt einen neuen Typus von Anregungen: eine Amplitudenmodulation des Kondensatanteils, welche durch eine Anregungslücke charakterisiert wird. Hier wollen wir analysieren, wie eine solche Mode gegen den Bereich schwacher Wechselwirkung, wo die nichtlineare Schrödingergleichung Gültigkeit hat, verschwindet. Wir finden, dass diese Higgsmode, benannt nach ihrer Ähnlichkeit mit dem Higgsteilchen aus der Teilchenphysik, über einen weiten Bereich des Phasendiagrams erhalten bleibt und erst im Grenzfall schwacher Wechselwirkung, wo das Gitter in einer renormierten Masse absorbiert werden kann, vii
6 in einem Hochenergiekontinuum verschwindet. Wir diskutieren ihre Evolution durch diesen intermediären Wechselwirkungsbereich und zeigen, wie man eine solche Anregung im Experiment nachweisen kann. Im isolierenden Zustand, in welchem die Atome auf den einzelnen Gitterplätzen lokalisiert sind, trägt das System zwei Typen von Anregungen: Es kann sich ein zusätzliches Teilchen über den lokalisierten Hintergrund bewegen oder eine Fehlstelle (ein Loch) propagiert durch die sonst statische Anordnung von Atomen. Der Inhalt des zweiten Kapitels ist die Beschreibung, wie diese Anregungen an experimentelle Proben koppeln. Der Einbezug von Fluktuationen über einem Molekularfeldgrundzustand, welcher beide Phasen, den Isolator und das Superfluidum, beschreibt, erlaubt die Beantwortung dieser Frage. Wir zeigen auf, wie man die Signatur des Nulltemperaturphasenübergangs im Experiment erkennen kann. Des weiteren lässt uns die entwickelte Methode eine Brücke zu den kollektiven Anregungen des ersten Kapitels schlagen. Der neueste Fortschritt in der Präparation von Quantengasen in optischen Gittern und die Möglichkeit, in-situ Informationen zu erhalten, die über die gemittelte Impulsverteilung hinausgehen, erlaubt das Studium von Effekten, die von der endlichen räumlichen Ausdehnung der Atomwolke herrühren. Die inhomogene Falle führt zu einer Heterostruktur von alternierenden supraflüssigen und isolierenden Schalen welche man üblicherweise den Hochzeitskuchen nennt. Wir diskutieren mesoskopische Aspekte solcher Strukturen mittels einer erstquantisierten Sprache ähnlich der Beschreibung räumlich inhomogener Supraleiter. Die Berechnung der Wärmeleitfähigkeit erlaubt uns, Aussagen zu machen, welche für die Diskussion von Temperatureffekten in optischen Gittern relevant sind. Das Feld der kalten Atome entwickelt sich im Moment mit einer Geschwindigkeit, welche mehr und mehr Quantenvielteilcheneffekte einer kontrollierten Simulation zugänglich machen. Zusätzlich werden durch den hohen Grad an Isolation und Kontrolle neue Aspekte solcher Vielteilchensysteme sichtbar. Mit der vorliegenden Dissertation hoffen wir Einsichten zur Verfügung zu stellen, welche dieses neu etablierte Feld zwischen der traditionellen Festkörperphysik und der Quantenoptik verlangt. viii
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