Additional problem (AC Stark shift)
2 lectures on basic Bose-Hubbard physics.
Lecture 3: Scattering, pseudopotentials and the microscopic model.
1 research talk.
1) Derivation of the Bose-Hubbard model
- Single particle in a periodic potential, Bloch functions.
- Wannier functions
- Statement and explanation of the second-quantised field operator Hamiltonian
- Simple derivation of the Bose-Hubbard model, with quantitative justification for the various approximations (single-band, nearest neighbour tunnelling, etc.)
2) Introduction to the Basic Physics of the Bose-Hubbard model
- Hopping term in momentum space, relation to tight-binding model, -2J cos(ka) band shape
- Overview of the Phase diagram
- States in limit of large and small U/J
- Introduction of the Single-particle density matrix, long range order, condensate mode
- relationship to momentum distributions.
- Examples of the Superfluid and MI states in 1D, with Single-particle density matrices and off-diagonal behaviour
- Local density approximation and emergence of the layer structure in a Harmonic trap (This should connect to what Simon will present later)
3) Scattering and Pseudopotentials
- Two-Body scattering processes
- Use of the delta-function pseudopotential
- justification of the microscopic second-quantised Hamiltonian (the starting point for the Bose-Hubbard model).
4) Research seminar (topic t.b.d.) e.g. Adiabatic potentials for creating addressible, sub-wavelength lattices. Transport, especially Andreev reflections, but also potentially new ideas to measure currents in lattice systems. Atomic lattice excitons, and the study of excited many-body states on lattices more generally.