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Generalized Quantum Mean-Field Systems and Their Application to Ultracold Atoms

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Generalized Quantum Mean-Field Systems and Their Application to Ultracold Atoms Generalized Quantum Mean-Field Systems and their Application to Ultracold Atoms Von der Fakult¨atf¨urMathematik und Physik der Gottfried Wilhelm Leibniz Universit¨atHannover zur Erlangung des Grades Doktorin der Naturwissenschaften Dr. rer. nat. genehmigte Dissertation von Dipl.-Phys. Friederike Annemarie Trimborn-Witthaut geboren am 15.08.1983 in Ludwigshafen am Rhein 2011 ii Referent: Prof. Dr. R. F. Werner Koreferent: apl. Prof. Dr. H. J. Korsch Tag der Promotion: 18.11.2011 Abstract Strongly interacting many-body systems consisting of a large number of indistinguish- able particles play an important role in many areas of physics. Though such systems are hard to deal with theoretically since the dimension of the respective Hilbert space increases exponentially both in the particle number and in the number of system modes. Therefore, approximations are of considerable interest. The mean-field approximation describes the behaviour in the macroscopic limit N , which leads to an effective nonlinear single-particle problem. Although this approximation! 1 is widely used, rigor- ous results on the applicability and especially on finite size corrections are extremely rare. One prominent example of strongly interacting many-body systems are ultracold atoms in optical lattices, which are a major subject of this thesis. Typically these systems consist of a large but well-defined number of particles, such that corrections to the mean-field limit can be systematically studied. This thesis is divided into two parts: In the first part we study generalized quantum mean-field systems in a C∗-algebraic framework. These systems are characterized by their intrinsic permutation symmetry. In the limit of infinite system size, N , the intensive observables converge to the commutative algebra of weak∗-continuous! 1 functions on the single particle state space. To quantify the deviations from the mean- field prediction for large but finite N, we establish a differential calculus for state space functions and provide a generalized Taylor expansion around the mean-field limit. Furthermore, we introduce the algebra of macroscopic fluctuations around the mean-field limit and prove a quantum version of the central limit theorem. On the basis of these results, we give a detailed study of the finite size corrections to the ground state energy and establish bounds, for both the quantum and the classical case. Finally, we restrict ourselves to the subspace of Bose-symmetric states and discuss their representation by quantum phase space distributions in terms of generalized coherent states. In particular, this allows for an explicit calculation of the evolution equations and bounds for the ground state energy. In the second part of this thesis we analyse the dynamics of ultracold atoms in optical lattices described by the Bose-Hubbard Hamiltonian, which provide an important ex- ample of the generalized quantum mean-field systems treated in the first part. In the mean-field limit the dynamics is described by the (discrete) Gross-Pitaevskii equation. We give a detailed analysis of the interplay between dissipation and strong interactions in different dynamical settings, where we especially focus on the relation between the mean-field description and the full many-particle dynamics given by a master equation. Keywords: Ultracold Atoms; Quantum Mean-Field Systems; Dissipative Quantum Sys- tems; Finite Size Corrections iv Zusammenfassung Stark wechselwirkende Vielteilchensysteme spielen in vielen Gebieten der Physik eine herausragende Rolle. Die theoretische Beschreibung ist jedoch sehr aufw¨andig,da die Dimension des Hilbertraums exponentiell mit der Anzahl der Teilchen und der An- zahl der Moden skaliert. Daher sind N¨aherungenvon großem Interesse. Die mean-field N¨aherungbeschreibt das System im Grenzwert N als effektives nichtlineares Ein- teilchensystem. Auch wenn diese N¨aherungoft angewendet! 1 wird, gibt es nur wenige mathematisch rigorose Ergebnisse. Ein wichtiges Beispiel f¨urein solches System sind ultrakalte Atome in optischen Gittern, die in dieser Arbeit untersucht werden. Typ- ischerweise bestehen diese Systeme aus einer großen, aber wohlbestimmten Anzahl an Atomen, so dass Abweichungen von mean-field Grenzwert systematisch analysiert werden k¨onnen. Diese Arbeit gliedert sich in zwei Teile: Im ersten Teil befassen wir uns mit der Beschreibung von verallgemeinerten mean-field Systemen mithilfe von C∗-Algebren, die durch ihre Permutationssymmetrie charakterisiert sind. Im makroskopischen Gren- zwert N konvergieren intensive Observablen gegen die kommutative Algebra der schwach!∗ 1-stetigen Funktionen auf dem Einteilchenzustandsraum. Um die Abwe- ichungen von einer solchen Beschreibung f¨urgroße, aber endliche Systemgr¨oßenzu quantifizieren f¨uhren wir einen Differentialkalk¨ulf¨urFunktionen auf dem Zustand- sraum ein, mit dessen Hilfe wir eine verallgemeinerte Taylorentwicklung um den mean- field Extremwert definieren k¨onnen. Dar¨uber hinaus diskutieren wir die Algebra der Fluktuationen um den Grenzwert und beweisen einen nicht-kommutativen zentralen Grenzwertsatz. Mithilfe dieser Ergebnisse analysieren wir die Korrekturen aufgrund der endlichen Systemgr¨oßef¨urdas Grundzustandsproblem und ermitteln Schranken f¨urden klassischen und den quantenmechanischen Fall. Im letzten Kapitel des ersten Teils betrachten wir Bose-symmetrische Zust¨andeund deren Phasenraumdarstellungen mithilfe verallgemeinerter koh¨arenter Zust¨ande.Diese Darstellung erm¨oglicht es unter anderem die Bewegungsgleichungen und Schranken f¨urdie Grundzustandsenergie zu bestimmen. Im zweiten Teil dieser Arbeit besch¨aftigenwir uns mit der Dynamik von ultrakalten Atomen in optischen Gittern, welche durch das Bose-Hubbard-Modell beschrieben wer- den und eine wichtige Anwendung der verallgemeinerten mean-field Systeme darstellen. Im mean-field Grenzwert wird die Dynamik durch die Gross-Pitaevskii-Gleichung be- schrieben. Wir stellen eine detaillierte Analyse des Zusammenspiels von Dissipation und starker Wechselwirkung zwischen den Teilchen vor. Insbesondere befassen wir uns mit der Beziehung zwischen den Vorhersagen der mean-field N¨aherungund der vollen Vielteilchendynamik, welche durch eine Mastergleichung beschrieben wird. Stichw¨orter: Ultrakalte Atome; Quanten Mean-Field Systeme; Dissipative Quanten- systeme; Korrekturen aufgrund endlicher Systemgr¨oße vi Contents Abstract iii Zusammenfassung v I Generalized Mean-Field Systems 1 1 Mathematical preliminaries 3 1.1 C∗-Algebras . .3 1.1.1 Composed systems . .5 1.1.2 States on C∗-algebras . .5 1.2 CCR-algebra and quasi-free states . .7 1.3 Tools from functional analysis . .8 2 Introduction to mean-field systems 9 2.1 Mean-field observables: Basic concepts . .9 2.2 Strictly symmetric sequences . 10 2.3 Approximately symmetric sequences . 12 2.4 The limiting algebra . 14 2.5 Symmetric states . 18 2.5.1 Størmer’s theorem and the mean-field limit . 19 2.5.2 Finite quantum de Finetti theorem . 19 3 Mean-field differential calculus 23 3.1 Introduction to the calculus of state space functions . 23 3.2 Higher order derivatives . 25 3.3 Extremal problems . 28 3.4 Taylor expansion . 31 viii CONTENTS 4 The algebra of fluctuations 35 4.1 Quantum fluctuations . 35 4.2 Central limit theorem . 38 4.3 Expansion in fluctuation operators . 41 5 The ground state problem 47 5.1 Introduction . 47 5.2 Some illustrative examples from statistical mechanics . 49 5.2.1 The mean-field Heisenberg Model . 49 5.2.2 Mean-field Ising Model with transverse field . 51 5.3 Relations to other problems . 52 5.3.1 Finite de Finetti theorems . 53 5.3.2 Spin squeezing inequalities and entanglement detection . 54 5.4 Inner bound in terms of fluctuations . 59 5.4.1 Minimization of the fluctuation Hamiltonian . 62 5.4.2 Determination of the mean-field limit of the corrections . 63 5.4.3 Illustrative example . 66 5.5 Classical ground state problem . 67 5.6 Conclusion and outlook . 73 6 Quantum phase space 75 6.1 Generalized coherent states . 76 6.1.1 Glauber states . 77 6.1.2 SU(M)-coherent states . 78 6.2 Differential algebra . 79 6.2.1 Flatland . 80 6.2.2 From the plane to the sphere . 81 6.3 Quantum dynamics in phase space . 82 6.3.1 Quantum phase space distributions . 83 6.3.2 Expectation values . 83 6.3.3 Evolution equations . 85 6.3.4 Numerical simulation using phase space ensembles . 87 7 Relation to the general theory 89 7.1 The Bose-Hubbard model as a quantum mean-field system . 89 7.1.1 Formulation in symmetric operators . 90 7.1.2 The ground state problem . 91 7.2 Mean-field dynamical semigroups . 94 CONTENTS ix 7.2.1 Hamiltonian flows . 94 7.2.2 Limiting dynamics for the Bose-Hubbard Hamiltonian . 97 7.2.3 Symplectic structure . 100 7.2.4 Some comments on dissipative systems . 102 7.3 Applications of the phase space approach . 104 7.3.1 A quantum de Finetti theorem starting from differential operators104 7.3.2 A quantum de Finetti theorem in phase space . 108 7.3.3 Discussion of the bounds . 114 7.3.4 A note on quantum phase space entropies . 115 7.3.5 The ground-state variational problem . 116 II Dynamics of Ultracold Atoms in Optical Lattices 121 8 Ultracold atoms in optical lattices { models and methods 123 8.1 Bose-Einstein condensation . 123 8.2 Experimental overview . 127 8.2.1 From laser cooling to the creation of a BEC . 127 8.2.2 Trapping by light . 129 8.3 The Bose-Hubbard model . 130 8.4 The mean-field limit and beyond . 133 8.4.1 The time-independent Gross-Pitaevskii equation . 133 8.4.2 Bogoliubov theory . 135 8.4.3 The time-dependent Gross-Pitaevskii equation . 137 8.4.4 Dynamics beyond mean-field . 139 8.5 Noise and dissipation in a trapped Bose-Einstein condensate . 141 8.5.1 Master equation . 142 8.5.2 Monte Carlo wave function method . 144 9 Dynamics of an open two-mode Bose-Einstein condensate 147 9.1 Ultracold atoms in a double-well trap . 147 9.2 Dissipative mean-field dynamics . 151 9.2.1 Relation to the non-hermitian Gross-Pitaevskii equation . 151 9.2.2 Analysis of fixed points .
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