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Universality in Driven-Dissipative Quantum Many-Body Systems

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Universality in Driven-Dissipative Quantum Many-Body Systems Universality in Driven-Dissipative Quantum Many-Body Systems DISSERTATION zur Erlangung des akademischen Grades “Doctor of Philosophy” an der Fakultat¨ fur¨ Mathematik, Informatik und Physik der Leopold-Franzens-Universitat¨ Innsbruck vorgelegt von Mag. rer. nat. Lukas M. Sieberer M¨arz2015 Doktorvater/Thesis Advisor: Prof. Dr. Sebastian Diehl, Dresden Zusammenfassung In jungerer¨ Zeit durchgefuhrte¨ experimentelle Studien von Kondensationsphanomenen¨ in getrie- ben-dissipativen Quantenvielteilchensystemen bringen die Frage auf, welche Art von neuem univer- sellem Verhalten unter Nicht-Gleichgewichtsverhaltnissen¨ auftreten kann. Wir untersuchen verschie- dene Aspekte von Universalitat¨ in diesem Kontext. Unsere Resultate haben Relevanz fur¨ zahlreiche offene Quantensysteme, von Exziton-Polariton-Kondensaten bis zu kalten atomaren Gasen. In Teil I charakterisieren wir das dynamische kritische Verhalten am Bose-Einstein-Kondensati- ons-Phasenubergang¨ in getriebenen offenen Quantensystemen in drei raumlichen¨ Dimensionen. Ob- wohl sich thermodynamische Gleichgewichtsverhaltnisse¨ bei niedrigen Frequenzen einstellen, wird die Annaherung¨ an dieses thermalisierte Regime niedriger Frequenzen durch einen kritischen Ex- ponenten beschrieben, der kennzeichnend fur¨ den Nicht-Gleichgewichts-Ubergang¨ ist und diesen außerhalb der Standardklassifikation von dynamischem kritischem Verhalten im Gleichgewicht ein- stuft. Unser theoretischer Ansatz beruht auf der funktionalen Renormierungsgruppe im Rahmen der Keldysh-Nicht-Gleichgewichts-Feldtheorie, welche aquivalent¨ zu einer mikroskopischen Beschrei- bung der Dynamik des offenen Systems in Form einer Quantenvielteilchen-Mastergleichung ist. Universelles Verhalten der Koharenzeigenschaften¨ von getrieben-dissipativen Kondensaten in reduzierten Dimensionen wird in Teil II untersucht. Wir zeigen, dass getriebene zweidimensiona- le Bose-Systeme keine algebraische Ordnung wie im thermodynamischen Gleichgewicht aufweisen konnen,¨ sofern sie nicht hinreichend anisotrop sind. Dennoch finden wir Hinweise, dass sogar isotro- pe Systeme einen endlichen superfluiden Anteil haben konnen.¨ In eindimensionalen Systemen sind Nicht-Gleichgewichtsbedingungen im Verhalten der Autokorrelationsfunktion nachweisbar. Wir er- zielen diese Resultate durch eine Abbildung der Dynamik des Kondensats auf langen Wellenlangen¨ auf die Kardar-Parisi-Zhang-Gleichung. In Teil III zeigen wir, dass Systeme in thermodynamischem Gleichgewicht eine besondere Sym- metrie haben, welche sie von allgemeinen getriebenen offenen Systemen unterscheidet. Das neuartige universelle Verhalten, welches in den Teilen II und III beschrieben wird, kann auf das Fehlen dieser Symmetrie außerhalb des Gleichgewichts zuruckgef¨ uhrt¨ werden. Von einem praktischen Standpunkt aus bietet diese Symmetrie einen effizienten Test fur¨ das Vorliegen von thermodynamischen Gleich- gewichtsbedingungen und macht die explizite Berechnung von Fluktuations-Dissipations-Relationen unnotig.¨ Im klassischen Grenzfall finden wir die bekannte Gleichgewichtssymmetrie der stochasti- schen Dynamik klassischer, an thermische Bader¨ gekoppelter Systeme wieder. Abstract Recent experimental investigations of condensation phenomena in driven-dissipative quantum many-body systems raise the question of what kind of novel universal behavior can emerge under non-equilibrium conditions. We explore various aspects of universality in this context. Our results are of relevance for a variety of open quantum systems on the interface of quantum optics and condensed matter physics, ranging from exciton-polariton condensates to cold atomic gases. In Part I we characterize the dynamical critical behavior at the Bose-Einstein condensation phase transition in driven open quantum systems in three spatial dimensions. Although thermodynamic equilibrium conditions are emergent at low frequencies, the approach to this thermalized low-frequency regime is described by a critical exponent which is specific to the non-equilibrium transition, and places the latter beyond the standard classification of equilibrium dynamical critical behavior. Our theoretical approach is based on the functional renormalization group within the framework of Keldysh non-equilibrium field theory, which is equivalent to a microscopic description of the open system dy- namics in terms of a many-body quantum master equation. Universal behavior in the coherence properties of driven-dissipative condensates in reduced di- mensions is investigated in Part II. We show that driven two-dimensional Bose systems cannot ex- hibit algebraic order as in thermodynamic equilibrium, unless they are sufficiently anisotropic. How- ever, we find evidence that even isotropic systems may have a finite superfluidity fraction. In one- dimensional systems, non-equilibrium conditions are traceable in the behavior of the autocorrelation function. We obtain these results by mapping the long-wavelength condensate dynamics onto the Kardar-Parisi-Zhang equation. In Part III we show that systems in thermodynamic equilibrium have a specific symmetry, which makes them distinct from generic driven open systems. The novel universal behavior described in Parts I and II can be traced back to the absence of this symmetry out of equilibrium. From a practi- cal viewpoint, this symmetry provides an efficient check for thermodynamic equilibrium conditions, making the explicit calculation of fluctuation-dissipation relations unnecessary. In the classical limit we recover the known equilibrium symmetry of the stochastic dynamics of classical systems coupled to thermal baths. Eidesstattliche Erklarung¨ Ich erklare¨ hiermit an Eides statt durch meine eigenhandige¨ Unterschrift, dass ich die vorliegende Arbeit selbstandig¨ verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel verwendet habe. Alle Stellen, die wortlich¨ oder inhaltlich den angegebenen Quellen entnommen wurden, sind als solche kenntlich gemacht. Die vorliegende Arbeit wurde bisher in gleicher oder ahnlicher¨ Form noch nicht als Dissertation eingereicht. Contents Zusammenfassung v Abstract vii Eidesstattliche Erklarung¨ ix Contents xi Chapter 1. Introduction 1 1.1 Quantum Many-Body Systems out of Equilibrium . 1 1.2 Universality in Non-Equilibrium Stationary States . 6 1.3 Non-Equilibrium Field Theory and the Functional Renormalization Group . 8 1.4 Outline of this Thesis . 9 I Dynamical Critical Phenomena in Driven-Dissipative Systems 19 Chapter 2. Publication: Dynamical Critical Phenomena in Driven-Dissipative Systems 21 2.A Open system dynamics . 27 2.B Functional Renormalization group equation . 28 2.C The second variational derivative . 29 2.D The regulator function . 30 2.E Flow of the effective potential . 30 2.F Flow of the inverse propagator . 31 2.G Rescaled flow equations . 32 2.H Critical properties . 33 2.I Ginzburg Criterion . 34 xii Contents Chapter 3. Publication: Nonequilibrium functional renormalization for driven-dissipative Bose-Einstein condensation 39 3.1 Introduction . 40 3.2 Key Results and physical picture . 42 3.3 The Model . 47 3.3.1 Quantum Master equation . 47 3.3.2 Keldysh functional integral . 49 3.3.3 Effective action . 52 3.4 Preparatory Analysis . 53 3.4.1 Mean-field theory . 53 3.4.2 Infrared divergences near criticality . 54 3.4.3 Canonical power counting . 56 3.4.4 Equilibrium symmetry . 57 3.5 Functional Renormalization Group . 58 3.5.1 FRG approach for the Keldysh effective action . 58 3.5.2 Truncation . 59 3.6 Relation to equilibrium dynamical models . 64 3.6.1 Model A with N = 2 and reversible mode couplings (MAR) . 65 3.6.2 Truncation for MAR . 66 3.6.3 Fluctuation-dissipation theorem . 66 3.6.4 Geometric interpretation of the equilibrium symmetry . 67 3.7 Non-Equilibrium FRG flow equations . 68 3.8 Scaling solutions . 72 3.8.1 Scaling form of the flow equations . 72 3.8.2 Gaussian fixed point . 73 3.8.3 Wilson-Fisher fixed point: critical behavior . 73 3.9 Numerical integration of flow equations . 78 3.10 Conclusions . 83 3.A Markovian dissipative action . 84 3.A.1 Translation table: Master equation vs. Keldysh functional integral . 84 3.A.2 Derivation in the Keldysh setting . 85 3.B Symmetry constraints on the action and truncation for MAR . 89 3.C Non-Equilibrium FRG flow equations . 89 Contents xiii 3.C.1 Expansion in fluctuations . 90 3.C.2 Flow equation for the effective potential . 92 3.C.3 Flow equation for the inverse propagator . 95 3.C.4 Computation of gradient coefficient anomalous dimensions . 96 II Universality in Driven-Dissipative Systems in Reduced Dimensions 105 Chapter 4. Publication: Two-dimensional superfluidity of exciton-polaritons requires strong anisotropy 107 4.1 Introduction . 107 4.2 Model . 109 4.3 Mapping to a KPZ equation . 110 4.4 Isotropic systems . 112 4.5 Strong anisotropy . 115 4.6 Outlook . 117 4.7 Acknowledgements . 117 4.A Mapping to the Kardar-Parisi-Zhang equation . 118 4.B Polariton condensate model with reservoir . 118 Chapter 5. Additional Material: Superfluidity in Driven-Dissipative Systems 125 5.1 Introduction . 125 5.2 Superfluidity in systems without particle number conservation . 127 5.3 Model . 130 5.3.1 Quantum master equation and Keldysh action . 130 5.3.2 Current in open systems . 132 5.3.3 Density-phase representation of the Keldysh action . 138 5.4 Superfluid density of a driven-dissipative condensate . 141 5.5 The influence of topological defects . 149 5.A Equilibrium symmetry in the density-phase representation . 151 5.B Retarded response functions in the Keldysh formalism . 154 5.C Density-phase representation for a weakly
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