Center of Mass (CoM) Motion

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Table of contents

Introduction 
I. Theory 
1. Two-Particle Problem 
1.1. Scattering
1.1.1. Center of Mass (CoM) Motion
1.1.2. Radial Schrödinger Equation
1.1.3. Scattering Potential
1.1.4. Spherical Waves
1.1.5. Elastic Scattering
1.1.6. Scattering Amplitude
1.1.7. Scattering Cross Section
1.1.8. The Unitary Limit
1.1.9. Low-Temperature Limit: Bosons versus Fermions
1.1.10. Scattering Length
1.2. Feshbach Resonances – Tuning the Scattering Length
1.2.1. Two-Channel Model
1.2.2. Determining the Position and Width
1.3. Summary
2. Three-Particle Scattering 
2.1. Elastic Scattering
2.1.1. Three-Particle Hamiltonian
2.1.2. Hyperangular Problem
2.1.3. Scattering Regimes
2.2. Unitary Interactions – Efimov’s Ansatz
2.2.1. Hyperspherical Waves
2.2.2. Short-Distance Scattering – R<Rm
2.2.2.1. Elastic Scattering
2.2.2.2. Efimov Bound States
2.2.2.3. Zero-Range Model
2.2.3. Long-Distance Scattering
2.2.3.1. Long-Range Wavefunction
2.2.3.2. Coupling of the Long-Range to the Short-Range
2.3. Finite-a – Hyperspherical Channels
2.3.1. Long-Distance Scattering
2.3.1.1. Long-Range Wavefunction
2.3.1.2. Coupling of the Long-Range to the Short-Range
2.3.1.3. Effective Two-Channel System
2.4. Inelastic Three-Particle Processus
2.4.1. Elastic versus Inelastic Scattering
2.4.2. Short-Range
2.4.2.1. Elastic → Inelastic Scattering
2.4.2.2. Inelastic Zero-Range Model (ZRM)
2.4.3. Long-Range
2.4.3.1. Resonant Interactions: Efimov Physics
2.4.3.2. Finite Interactions
2.4.4. Flux and Recombination
2.4.5. Temperature Average
2.4.6. Optical Resonator Analogy
2.4.7. Oscillations of L3(T) at Unitarity
2.4.8. Numerical Analysis of L3(T, a)
2.5. Three-Particle Losses on the Positive-a Side
2.5.1. Weakly Bound Dimer
2.5.2. Weakly Bound Dimers and the Efimov Channel
2.5.3. Atom-Dimer Decay with Chemical Equilibrium
2.6. Summary
3. Three-Particle Recombination in a Harmonic Trap 
3.1. Three-Particle Losses in a Trap
3.1.1. Trapping Potential
3.1.2. Weakly and Deeply Bound Dimers in a Trap
3.1.3. Number Decay
3.2. Heating Effects
3.2.1. Weakly-Interaction Limit
3.2.2. Extending the Model to Include Strong Interactions
3.3. Evaporation
3.3.1. A Simple Evaporation Model
3.3.2. More Advanced Model of Evaporation Effects
3.3.3. “Magic” η
3.4. Summary
II. Experiments 
4. The Road to Strongly Interacting Bose Gases 
4.1. Experimental sequence
4.1.1. Lithium-7
4.1.2. Laser System
4.1.3. Zeeman slower
4.1.4. Magneto-Optical Trap (MOT)
4.1.5. Optical Pumping
4.1.6. Magnetic Trapping and Evaporation
4.1.7. Optical Dipole Trap (ODT)
4.1.8. Radio-Frequency (RF) Transitions
4.1.9. Imaging
4.2. Feshbach Resonance in 7Li
4.3. Summary
5. Lifetime of the Resonant Bose Gas 
5.1. Recombination Rate Measurements and Assumptions
5.1.1. Quasi-Thermal Equilibrium Condition
5.1.2. Separation of Time Scales
5.1.3. Starting Point for the Measurements
5.1.4. Number Calibration
5.1.4.1. Pressure calibration
5.1.4.2. Recombination and Temperature calibration
5.1.5. Constant Temperature
5.1.6. Data Analysis
5.2. Results – Unitary Interactions
5.2.1. Temperature Dependence of L3 at Unitarity
5.2.2. Reanalysis using the Advanced Evaporation Model
5.3. Results – Finite Interactions
5.3.1. Saturation of L3 for Resonant Interactions
5.3.2. Comparison with Previous Data – 133Cs
5.3.2.1. The First Efimov Resonance
5.3.2.2. Resonance Position
5.3.3. Temperature Behavior of L3 – 39K
5.3.3.1. Validating the 1/T 2 Law for L3(T)
5.3.3.2. Excess Heat Measurements
5.4. Summary
Concluding remarks 
Perspectives 
Acknowledgements
Appendix A. Technical Details – Theory
A.1. Jacobian and Hyperspherical Coordinates
A.1.1. Jacobian Coordinates
A.1.2. Jacobian → Hyperspherical Coordinates
A.1.3. Jacobian → Hyperspherical Hamiltonian
A.1.4. Hyperradial and Hyperangular Schrödinger Equations
A.2. Incoming and Outgoing waves
A.3. Saddle Point Method
A.4. Efimov’s Ansatz
A.5. The s-matrix at Unitarity
Appendix B. Peer-reviewed papers
B.1. Dynamics and Thermodynamics of the Low-Temperature Strongly
Interacting Bose Gas
B.2. Lifetime of the Bose Gas with Resonant Interactions
B.3. -enhanced sub-Doppler cooling of lithium atoms in D1 gray molasses
Appendix C. Data Loss Measurements
C.1. Unitary Interactions
C.2. Finite-a Interactions
Appendix D. Efimov resonances
D.1. Caesium-133
D.1.1. Universality of the Efimov resonances
D.2. Lithium-7
D.2.1. L3 vs. a
D.3. Rubidium-85
D.4. Potassium-39
D.4.1. Efimov resonance
D.4.2. Universality of the Efimov resonances in 39K
D.5. How to determine the Efimov parameters
D.6. Summary
Bibliography 
Abstract
Résumé