Corinna Kollath (Ecole Polytechnique)

Quantum dynamics, cold atoms and numerical methods (9 h)

1. Introduction to ultracold atoms

Bose-Einstein condensation.
Optical lattices.
Chosen experiments.

2. Quantum dynamics in ultracold atoms

Collapse and revival of a wave function.
Time-resolved control of super-exchange interaction.

3. Dynamics after a sudden change of parameters: quantum quenches

Experiments in ultracold gases.
Short time dynamics.
Long time dynamics.

4. Numerical methods for quantum dynamics I: 'Static' density matrix renormalization group method (DMRG)

Schmidt decomposition.
Algorithm.
Convergence analysis.

5. Numerical methods for quantum dynamics II

Numerical methods for time-evolution.
Trotter-Suzuki algorithm.
Runge-Kutta algorithm, and more specialized algorithms.

6. Numerical methods for quantum dynamics III: Generalization of the DMRG to time-dependent (finite temperature) situations

Adaptation of the effective Hilbert space.
Algorithm.
Convergence analysis.