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This graduate level textbook provides a numerical and analytical perspective of the Bose Einstein Condensates, starting the most basic level. The authors will trace the origin of BECs from scratch (in almost a layman’s perspective). They will then sketch the algorithmic approach needed to generate BECs in different integrable models analytically while the numerical codes developed will give a flavour of the dynamics of BECs in nonintegrable models. End-of-chapter summaries, worked examples, and end of chapter problems will be included. Numerical codes in Fortran/C++ and ML algorithms will be provided.
Key features:
- Introduces the subject at the very basic level
- Includes the analytical and theoretical methods
- Contains numerical methods
- Provides code in Fortfran/C++
- Provides worked examples, end-of-chapter summaries, and homework problems
Chapter 1. Introduction
1.1. History of Bose-Einstein Condensates (BECs)
1.2. Theoretical Perspective of BECs
1.3. Experimental Realization
1.4. Mean Field Description of BECs and Gros-Pitaevskii (GP) Equation
1.5. Beyond Mean Field Description
1.6. Ultracold Fermi Gases
1.7. Overview
1.8. References
Chapter 2. Analytical Methods
2.1. Inverse Scattering Transform
2.2. Gauge Transformation Approach
2.3. Darboux transformation Approach
2.4. Hirota (Direct) Method
2.5. Approximation Methods
2.6. Problems
2.7. References
Chapter 3. Numerical Techniques
3.1. Finite Difference Method
3.2. Finite Element Method
3.3. Machine Learning Approach
3.4. Problems 3.5. References
Chapter 4. Bose Einstein Condensates and Optical Solitons
4.1. Analogy Between GP and Nonlinear Schrodinger (NLS) Equation
4.2. Switching of Energy (Intensity)
4.3. New Signatures of Manakov Model (Coupled NLS Equation)
4.4. Coherently Coupled NLS Equation
4.5. Generalized Coupled NLs Equation with Four Wave Mixing (FWM)
4.6. Problems
4.7 References
Chapter 5. Dynamics of scalar BECs with Short Range Interactions
5.1. Quasi One dimensional BECs in a Time Independent Harmonic Trap
5.2. Impact of Transient Trap on BECs
5.3. Matter Wave Interference Pattern in the Collision of Solitons
5.4. Condensates with Three Body (attractive and repulsive) Interactions
5.5. Problems
5.6. References
Chapter 6. Vectorial Condensates
6.1. Model of Vector BECs and Feshbach Resonance Management
6.2. Impact of Temporal Rabi Coupling.
6.3. Spatially Coupled BECs
6.4. Taming of Rogue Waves in Vector BECs
6.5. Designer Quasi Particle Condensates 6.6. Electromagnetically Induced Transparency (EIT) and its Signatures
6.7. Spinor BECs
6.8. Problems
6.9. References
Chapter 7. Spinor and Spin Orbit Coupled BECs
7.1. Synthetic Spin-Orbit (SO) and Rabi Coupling in BECs
7.2. Coupled Gross-Pitaevskii Equations
7.3. Plane Wave and Stripe Patterns
7.4. Dynamics of SO coupled BECs
7.5. Problems
7.6. References
Chapter 8. BECs with Long Range Interactions
8.1. Dipolar BECs
8.2. Stability of Trapped Dipolar BECs in One-, Two- and Three-Dimensions
8.3. Dynamics of Dipolar BECs
8.4. Problems 8.5. References
Chapter 9. Collisionally Inhomogenous BECs
9.1. Collisionally Inhomogenous BECs with Short Range Interactions
9.2. Faraday and Resonant Waves in Scalar BECs
9.3. Faraday and Resonant Waves in Binary Condensates
9.4. Solitons Under Spatially Localized Cubic-Quintic-Septimal Nonlinearities
9.5. Problems
9.6. References
Chapter 10. Vortices in BECs
10.1. Rotating BECs
10.2. Vortices and Vortex Lattice
10.3. Analytical Analysis
10.4. Numerical Simulations of Vortices
10.5. Problems
10.6. References
Chapter 11. Nascent Outlook of Polariton Condensates
Chapter 12. Glimpses of Roadmap Ahead
Appendix A1. Fortran Programs and Computer Simulation tools