Correlations in Low-Dimensional Quantum Gases (eBook)

Supervisor’s Foreword 7
Abstract 8
Publications by the Author 9
References 10
Acknowledgements 11
Contents 13
1 Introduction: This Thesis 16
References 18
2 From 3D to 1D and Back to 2D 19
2.1 Introduction 19
2.2 Welcome to Lineland 20
2.2.1 Generalities on One-Dimensional Systems 20
2.2.2 Correlation Functions as a Universal Probe For Many-Body Quantum Systems 21
2.3 From 3D to 1D in Experiments 23
2.4 Analytical Methods to Solve 1D Quantum Models 27
2.4.1 Quantum Integrability and Bethe Ansatz Techniques 28
2.4.2 Exact Solution of the Tonks-Girardeau Model and Bose-Fermi Mapping 33
2.4.3 Bosonization and Tomonaga-Luttinger Liquids 38
2.4.4 Conformal Field Theory 46
2.5 From Lineland to Flatland: Multi-component Systems and Dimensional Crossovers 52
2.6 Summary of This Chapter 53
References 54
3 Ground-State Static Correlation Functions of the Lieb–Liniger Model 63
3.1 Introduction 63
3.2 Exact Ground-State Energy of the Lieb–Liniger Model 64
3.2.1 Ground-State Energy in the Finite-N Problem 64
3.2.2 Ground-State Energy in the Thermodynamic Limit 69
3.2.3 Weak-Coupling Regime 73
3.2.4 Strong- to Intermediate Coupling Regime 76
3.2.5 Illustrations 81
3.3 Local Correlation Functions 83
3.4 Non-local Correlation Functions, Notion of Connection 88
3.4.1 One-Body Non-local Correlation Function in the Tonks–Girardeau Regime 88
3.4.2 Large-Distance, One-Body Correlation Function at Finite Interaction Strength from the Tomonaga–Luttinger Liquid Formalism 90
3.4.3 Short-Distance, One-Body Correlation Function from Integrability, Notion of Connection 91
3.5 Momentum Distribution and Tan's Contact 95
3.6 Breakdown of Integrability, BALDA Formalism 96
3.6.1 Effect of a Harmonic Trap 96
3.6.2 Local-Density Approximation for the Density Profile in the Tonks–Girardeau Regime 97
3.6.3 From Local-Density Approximation to Bethe Ansatz LDA 98
3.6.4 Tan's Contact of a Trapped Bose Gas 99
3.7 Summary of This Chapter 104
3.8 Outlook of This Chapter 105
References 107
4 Dynamical Structure Factor of the Lieb–Liniger Model and Drag Force Due to a Potential Barrier 112
4.1 Introduction 112
4.2 Conceptual Problems Raised by Superfluidity, Lack of Universal Criterion 113
4.2.1 Experimental Facts, Properties of Superfluids 113
4.2.2 Landau's Criterion for Superfluidity 115
4.3 Drag Force as a Generalized Landau Criterion 117
4.4 Dynamical Structure Factor and Drag Force for a Tonks–Girardeau Gas 121
4.4.1 Dynamical Structure Factor 121
4.4.2 Drag Force Due to a Mobile, Point-Like Impurity in the Tonks–Girardeau Regime 126
4.4.3 Effect of a Finite Barrier Width on the Drag Force 127
4.5 Dynamical Structure Factor and Drag Force for a 1D Bose Gas … 129
4.5.1 State of the Art 129
4.5.2 Dynamical Structure Factor from the Tomonaga–Luttinger Liquid Theory 130
4.5.3 Drag Force from the Tomonaga–Luttinger Liquid Formalism 136
4.6 Exact Excitation Spectra from Integrability 138
4.7 Summary of This Chapter 143
4.8 Outlook of This Chapter 145
References 147
5 Dimensional Crossovers in a Gas of Noninteracting Spinless Fermions 152
5.1 Introduction 152
5.2 Energy-Momentum Space Dimensional Crossover in a Box Trap 153
5.3 Dimensional Crossovers in a Harmonic Trap 159
5.4 Low-Energy Approach for Fermions in a Box Trap, Cross-Dimensional … 162
5.5 Summary of This Chapter 165
5.6 Outlook of This Chapter 166
References 167
6 General Conclusion 169
A Complements to Chapter 2 174
A.1 Density Correlations of a Tomonaga-Luttinger Liquid in the Thermodynamic Limit at Zero Temperature 174
A.2 Density Correlations of a Tomonaga-Luttinger Liquid at Finite Temperature by Bosonization 177
A.2.1 First Contribution to the Density Correlation 177
A.2.2 Second Contribution to the Density Correlation Function 179
A.3 Density Correlations of a Tomonaga-Luttinger Liquid at Finite Size and Temperature by Bosonization 181
B Complements to Chapter 3 184
B.1 Exact Mapping from the Lieb-Liniger Model onto the Circular Plate Capacitor 184
B.2 Ristivojevic's Method of Orthogonal Polynomials 187
B.3 A General Method to Solve the Lieb Equation 189
B.4 Other Approaches to Local Correlation Functions 190
C Complements to Chapter 4 192
C.1 Around the Notion of Dynamical Structure Factor 192
C.2 Dynamical Structure Factor of the Tonks-Girardeau Gas at Zero Temperature 193
C.3 Dynamical Structure Factor of a Tomonaga-Luttinger Liquid in the Thermodynamic Limit 195
C.3.1 First Contribution: The Phonon-Like Spectrum Close to the Origin in Energy-Momentum Space 196
C.3.2 Second Contribution to the Dynamical Structure Factor: The Umklapp Region 197
C.4 Dynamical Structure Factor of a Tomonaga-Luttinger Liquid in the Thermodynamic Limit at Finite Temperature 199
C.4.1 First Contribution to the Dynamical Structure Factor of a Tomonaga-Luttinger Liquid at Finite Temperature 199
C.4.2 Second Contribution to the Dynamical Structure Factor at Finite Temperature 199
C.5 Drag Force Due to a Delta-Barrier in the Tomonaga-Luttinger Liquid Framework 202
References 203