Dissertation Of Andreas Wagner - COnnecting REpositories
Spinor Condensates in OpticalSuperlatticesInauguraldissertationzur Erlangung der Würde eines Doktors der Philosophievorgelegt der Philosophisch-Naturwissenschaftlichen Fakultätder Universität BaselvonAndreas Wagner,aus Konstanz, DeutschlandBasel, 2012
Genehmigt von der Philosophisch-Naturwissenschaftlichen Fakultät auf Antrag vonProf. Dr. Christoph BruderProf. Dr. Dieter JakschBasel, den 16. Oktober 2012,Prof. Dr. Jörg Schibler, Dekan
SummaryIn this thesis we study various aspects of spinor Bose-Einstein condensates in opticalsuperlattices using a Bose-Hubbard Hamiltonian that takes spin effects into account.We decouple the unit cells of the superlattice via a mean-field approach and takeinto account the dynamics within the unit cell exactly. In this way we derive theground-state phase diagram of spinor bosons in superlattices. The system supportsMott-insulating as well as superfluid phases. The transitions between these phasesare second-order for spinless bosons and second- or first-order for spin-1 bosons.Antiferromagnetic interactions energetically penalize high-spin configurations andelongate all Mott lobes, especially the ones corresponding to an even atom numberon each lattice site. We find that the quadratic Zeeman effect lifts the degeneracybetween different polar superfluid phases leading to additional metastable phasesand first-order phase transitions. A change of magnetic fields can drive quantumphase transitions in the same way as a change in the tunneling amplitude does.Furthermore we study the physics of spin-1 atoms in superlattices deep in theMott insulating phase when the superlattice decomposes into isolated double-wellpotentials. Assuming that a small number of spin-1 bosons is loaded in an opticaldouble-well potential, we study single-particle tunneling that occurs when one latticesite is ramped up relative to a neighboring site. Spin-dependent effects modify thetunneling events in a qualitative and quantitative way. Depending on the asymmetryof the double well different types of magnetic order occur, making the system ofspin-1 bosons in an optical superlattice a model for mesoscopic magnetism withan unprecedented control of the parameters. Homogeneous and inhomogeneousmagnetic fields are applied and the effects of the linear and the quadratic Zeemanshifts are examined. We generalize the concept of bosonic staircases to connecteddouble-well potentials. We show that an energy offset between the two sites of theunit cell in an extended superlattice induces a staircase of single-atom resonances inthe same way as in isolated double well. We also examine single-atom resonances inthe superfluid regime and find clear fingerprints of them in the superfluid density.We also investigate the bipartite entanglement between the sites and constructstates of maximal entanglement. The entanglement in our system is due to bothorbital and spin degrees of freedom. We calculate the contribution of orbital andspin entanglement and show that the sum of these two terms gives a lower boundfor the total entanglement.III
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KurzfassungIn der vorliegenden Dissertation werden verschiedene Aspekte von Bose-EinsteinKondensaten aus Spin-1-Atomen in optischen Supergittern studiert. Dazu wirdein Bose-Hubbard-Modell verwendet, das spinabhängige Wechselwirkungen berücksichtigt. Zunächst werden die Einheitszellen des Supergitters duch eine Molekularfeld-Näherung entkoppelt, wobei die Dynamik innerhalb der Einheitszelle exakt behandelt wird. Mit Hilfe dieser Näherung wird das Phasendiagramm von spinlosen,bosonischen Atomen und Spin-1-Atomen in Supergittern berechnet. Das System unterstützt Mott-isolierende sowie superfluide Phasen. Die Übergänge zwischen diesenPhasen sind zweiter Ordnung für spinlose Bosonen und zweiter oder erster Ordnungfür Spin-1-Bosonen.Antiferromagnetische Wechselwirkungen verursachen eine Verlängerung der MottInseln hin zu grösseren Tunnelamplituden und bevorzugen allgemein niedrige SpinKonfigurationen. Die Mott-Inseln, die einer geraden Anzahl von Atomen pro Gitterplatz entsprechen, werden besonders vergrössert, da eine gerade Anzahl von Spin1-Atomen immer Spin-Singletts bilden können. Es werden verschiedene superfluidePhasen beschrieben und herausgestellt, dass durch schwache magnetische Felder dieEntartung der verschiedenen polaren superfluiden Phasen aufgehoben wird, waszu zusätzlichen metastabilen Phasen führt. Phasenübergange lassen sich durchVeränderung des Magnetfeldes ebenso wie durch eine Veränderung der Wechselwirkungsstärke verursachen.Weiterhin wird die Physik von Spin-1-Atomen in Supergittern tief in der MottPhase studiert, wenn das Supergitter in isolierte Doppelmuldenpotentiale zerfällt.Es folgt eine Untersuchung der Besetzungwahrscheinlichkeit in asymmetrischen Doppelmulden für eine geringe Anzahl von Atomen. Für diese Systeme können Einteilchen-Resonanzen festgestellt werden. Diese Einteilchen-Resonanzen werden durchspinabhängige Wechselwirkungen qualitativ und quantitativ verändert. Abhängigvon der Asymmetrie der Doppelmulde treten verschiedene magnetische Ordnungen auf; dadurch wird das System von Spin-1-Atomen in optischen Supergittern zueinem Modell für mesoskopischen Magnetismus, wobei in diesem Modell alle Parameter mit einem sehr hohen Grad der Kontrolle verändert werden können. Es wirddie Wirkung von homogenen und inhomogenen Magnetfeldern untersucht, wobeider lineare und quadratische Zeeman Effekt berücksichtigt wird. Weiter wird dasKonzept der Einteilchen-Resonanzen auf Supergitter verallgemeinert und gezeigt,dass eine Asymmetrie in den Einheitszellen des Supergitters ebenso EinteilchenResonanzen verursacht. Im Anschluss werden Einteilchen-Resonanzen in dem suV
perfluiden Regime untersucht und festgestellt, dass diese auch in der superfluidenDichte sichtbar sind.Im letzten Kapitel dieser Dissertation werden Verschränkungseigenschaften zwischen Gitterplätzen untersucht und maximal verschränkte Zustände konstruiert. DieVerschränkung in dem System von Spin-1-Atomen resultiert aus orbitalen und SpinFreiheitsgraden. Es werden die Beiträge beider untersucht und argumentiert, dassdie Summe beider eine untere Grenze für die gesamte Verschränkung ist.VI
AcknowledgmentsFirst, I want to express my gratitude to my advisor, Christoph Bruder. It hasbeen great to be part of his group. I really enjoyed working under his supervision,he was always available for discussions and I benefited a lot from his friendliness,knowledge and experience. I also would like to thank Andreas Nunnenkamp forhis energy, his ability to motivate me and highly valuable input. Of course, I alsoneed to thank all current and former members of the theory group in Basel: Myoffice-mates Thomas Schmidt, Ying-Dan Wang, Samuel Aldana, Grégory Strübi andPatrick Hofer, but also every other member of the group, especially Stefano Chesi,Daniel Becker, Mathias Duckheim, Jan Fischer, Verena Koerting, Roman Riwar,Beat Röthlisberger, Manuel Schmidt, Dimitrije Stepanenko, Mircea Trif, OleksandrTsyplyatyev, Kevin van Hoogdalem, Robert Zak, Gerson Ferreira, Diego Rainis, Peter Stano, Vladimir M. Stojanovic, Rakesh Tiwari, James Wootton, Adrian Hutter,Jelena Klinovaja, Christoph Klöffel, Viktoriia Kornich, Franziska Maier, ChristophOrth, Fabio Pedrocchi, Luka Trifunovic and Robert Zielke.Throughout my doctoral studies, I also benefited a lot from the discussions withother scientists. Specifically, I would like to thank Eugene Demler, Rosario Fazioand Daniel Burgarth.All this support would count nothing without my family. I want to thank mywife Beata for her love, encouragement and unconditional backing. I also want tothank Julius. He did not really help, he was even a source of permanent distraction.Nevertheless, he was my main source of motivation.VII
ContentsSummaryIIIKurzfassungVAcknowledgmentsVII1 Introduction12 Ultracold Atoms in Optical Lattices2.1 Optical Dipole Traps . . . . . . . . .2.2 Interactions between Ultracold Atoms2.3 Optical Lattices . . . . . . . . . . . .2.4 The Bose-Hubbard Model . . . . . .2.5 Spinor Condensates . . . . . . . . . .2.6 Probing Ultracold Atoms . . . . . . .3 Quantum Phase Transitions in the BHM3.1 The Mott-Superfluid Quantum Phase Transition3.2 Ultracold Spin-1 Atoms in Optical Lattices . . .3.3 Spinless Bosons in Superlattices . . . . . . . . .3.4 Spin-1 Bosons in Superlattices . . . . . . . . . .4 Bosonic Staircases4.1 Spinless Bosons . . . . . . . . . . . . . .4.2 Spin-1 Atoms . . . . . . . . . . . . . . .4.3 Bosonic Staircases for Spinor Atoms . . .4.4 Beyond Ground-State Analysis . . . . .4.5 Effects of Magnetic Fields . . . . . . . .4.6 Single Atom Resonances in Superlattices5 Entanglement in Superlattices5.1 Entanglement in Double-Well Potentials5.2 Two Spin-1 Bosons . . . . . . . . . . . .5.3 Three Spin-1 Bosons . . . . . . . . . . .5.4 Arbitrary Number of Bosons . . . . . . .IX.9101314212332.3738536574.838386939597101.107. 111. 112. 115. 118.
CONTENTS6 Conclusions125A Mean-Field Calculations129A.1 Mathematica Code for Mean-Field Calculations . . . . . . . . . . . . 129A.2 Matlab Code for Mean Field Calculations . . . . . . . . . . . . . . . . 131B Calculations for Bosonic Staircases139B.1 Spinless Atoms in a Double-Well Potential . . . . . . . . . . . . . . . 139B.2 Spin-1 Atoms in a Double-Well Potential . . . . . . . . . . . . . . . . 140Bibliography147C Curriculum Vitae163X
Chapter 1IntroductionIn 1995, atomic gases were cooled down to such low temperatures that a largenumber of atoms occupied a single quantum state and formed a Bose-Einstein condensate.1 That was the first experimental realization of this novel state of matterpredicted by Albert Einstein2 following the quantum statistics of bosons suggestedby Satyendranath Bose.3 A Bose-Einstein condensate in an atomic cloud is formedwhen the de Broglie wavelength of the (bosonic) atoms is of the order of the meaninter-atomic distance; at such low temperatures the atoms are called ultracold. Ultracold atomic gases are quantum liquids in which macroscopic characteristics of theliquid derive directly from quantum coherences. Thus, ultracold atomic gases offerthe possibility to observe quantum effects on a macroscopic scale. In this thesis wewill discuss ultracold spinless and spin-1 atoms in optical superlattices.For atoms trapped in a magneto-optical trap the spin degree of freedom is frozenand the atoms become effectively spinless. If, however, the quantum gas is trappedby purely optical means, the atoms keep their spin degree of freedom and the orderparameter describing the superfluid phase becomes a spinor. The spinor degree offreedom in optically trapped alkaline gases corresponds to the manifold of degenerate Zeeman hyperfine levels. Spinor Bose-Einstein condensates possess an internaldegree of freedom, similar to quantum liquids such as d-wave and p-wave superconductors or superfluid 3 He. It is therefore tempting to use spinor condensates as aquantum simulator for these quantum liquids which still lack a thorough theoreticalunderstanding. This idea dates back to a proposal of R. Feynman to create quantumsimulators which are controllable quantum systems that can model the behavior ofmore complicated systems.4Nevertheless, spinor condensates are also interesting in their own right. Theinteraction between the external and internal degrees of freedom leads to a numberof phenomena unfamiliar from studies of scalar quantum liquids. The experimentalexamination of spinor condensates started in 1998 with experiments on ultra-cold1234[Anderson et al.(1995), Bradley et al.(1995), Davis et al.(1995)], see also an(1982), Buluta and Nori(2009)]1
CHAPTER 1. INTRODUCTIONsodium1 and rubidium.2 Seminal theoretical work on the ground-state properties ofspinor Bose-Einstein condensates in single traps has been done soon afterwards byT. Ho and Ohmi et al.3 Experimentally, long-lived alkali spinor gases have beenexplored in the F 1 manifold both of 23 Na (by D. Stamper-Kurn et al.(1998)4 )and 87 Rb (by M. Barrett et al.(2001)5 ) and the higher energy F 2 manifold of87Rb.6 Further experiments on spinor condensates in harmonic traps highlightingspin dynamics, spin textures and properties of the superfluid order parameter havebeen performed in the following.7Atoms can be trapped via the ac-Stark effect in optical lattices, which are created by counter-propagating laser-beams; in case there are only a few atoms persite, they build up so-called “optical crystals” or “artificial solids”. In a typical natural solid, electrons are moving in a lattice generated by the periodic array of atomcores. This can be simulated with ultracold neutral atoms moving in an opticallattice.8 Ultracold atoms in optical latices offer the unique opportunity to studyquantum many-body effects in an extremely clean and well-controlled environment.In contrast to most condensed matter systems they are characterized by the absence of disorder and other imperfections. Ultracold atoms in optical lattices offerrobust quantum coherence, a unique controllability and powerful read-out tools liketime-of-flight measurements9 or in situ imaging.10 Experiments with cold atoms inoptical lattices were done already at the beginning of the 1990’s in the micro-kelvinrange.11 But not until realization of Bose-Einstein Condensates, when much coldertemperatures became possible, the field started to become such an interesting andlively field of research.One of the most prominent examples illustrating how cold atoms in optical lattices can be used to study genuine many-body phenomena is the quantum phasetransition between a
enFakultätaufAntragvon Prof. Dr. ChristophBruder Prof. Dr. DieterJaksch Basel,den16. Oktober2012, Prof. Dr .
Andreas Werner The Mermin-Wagner Theorem. How symmetry breaking occurs in principle Actors Proof of the Mermin-Wagner Theorem Discussion The Bogoliubov inequality The Mermin-Wagner Theorem 2 The linearity follows directly from the linearity of the matrix element 3 It is also obvious that (A;A) 0 4 From A 0 it naturally follows that (A;A) 0. The converse is not necessarily true In .
opponent affective reaction. Wagner’s SOP theory (Brandon & Wagner, 1991; Brandon, Vogel, & Wagner, 2002; Wagner & Brandon, 1989, 2001) is an extension of opponent-process theory that can explain why the CR sometimes seems the same as and sometimes different from the UCR. According to Wagner, the UCS elicits two unconditioned
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its original connecting rod before another connecting rod bearing cap is removed. 5. Remove the connecting rod bearing cap with the connecting rod bearing. 6. Inspect the connecting rod bearing for damage. If the connecting rod bearing is damaged, replace all main and connecting rod bearings. a. Acceptable bearing wear (1). b.
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Andreas Wagner, Karlsruhe Institute of Technology, Germany MIT Symposium - May 6, 2013 Andreas Wagner with acknowledgement to all contributions of researchers from the different universities and research institutions involved in the research programs to be presented here . Content German research programs on building energy efficiency Innovative building technologies and performance of .
Andreas Wagner { Integrated Electricity Spot and Forward Model 16/25. MotivationFrameworkModel and ResultsConclusions Volatility of supply-functional This observation motivates the following volatility structure (as in Boerger et al. [2009]) Volatility structure ( ;t) e (t ) 1; 2(t) ; where 1 is the (additional) short-term volatility, is a positive constant controlling the in .
