A Mesh-free Method for three-dimensional Nonlinear Schrödinger Equation
Thomas C.L. Yue
[email protected] Feb 09, 2011
Overview • Physical motivation of the problem – Gross-Pitaevskii equation (GPE)
• Radial basis functions (RBF) • Meshfree solution of cubic Nonlinear Schrodinger Equation – Numerical experiments and validation
Physical Motivation
Physical Motivation History of Bose Einstein Condensation (BEC) [1,2] • First predicted by Bose & Einstein (1924) • Experimentally observed in University of Colorado JILA lab (1995)
What is BEC? [1,2] • A phase of matter where all particles occupy the same quantum state • Occurs when diulated bosons (integer spin particles) gas are cooled to extremely low temperature (10-9K) • Individual particle wave functions behave as a single wave function
Physical Motivation
1. High temperature particle behaviour dominated
3. T=Tcrit Bose Einstein Condensate
2. Low temperature λdB α T -0.5
4. T=0 Giant Matter Wave
Fig1.A visual description of how a gas of bosonic-atoms behave at various temperatures (T). [1]
Experimental Results of BEC JILA (95’,Rb,5,000)
ETH (02’,Rb, 300,000)
Gross–Pitaevskii equation • Hartree–Fock approximation [1,2] – The many-body wavefunction is written as productsof individual wave functions of each bosons [1,2]
• The Hamiltonian
• The conserved quantities
Gross–Pitaevskii equation • At temperature T
