1. System and definitions In harmonic trap (ideal): er

2. Dilute interacting Bosons Single particle field operators: Macroscopic occupation assumption: Homogeneous result:

3. Dilute interacting Bosons Inhomogeneous (time and space): Single particle density matrix formalism: Scattering theory (see ahead): Time evolution of operator in Heisenberg Rep. Time-Dependent Gross- Pitaevskii equation (TDGPE) Mean-field assumption – discard fluctuating part Not an operator! an operator!

4. A short review of scat. theory Fourier Trans. Born Approx. Indistinguishable particles… Low k limit (“s-wave”) Eigenvalue scattering problem: Effective potential!

5. GPE – ground state properties Interaction energy: Variational derivation + Energy functional Smallness parameter: Kinetic energy: Weak interactions ≠ ideal gas behavior! (still small depletion, but strongly non-ideal)

6. GPE – ground state properties TDGPE: Ansatz + normalization: TIGPE: Note: energy is not a good quantum number (nonlinear problem!)

7. Numerical solution of TDGPE Imaginary time evolution: Interacting ground-state Non-interacting ground-state (Mean-field repulsion causes increase in Size)

8. Thomas-Fermi approx. Neglect kinetic term: Relaxed T.F.

9. Excitations – Bogoliubov equations Ansatz (plug Into TDGPE): Bogoliubov equations (“linearized GPE”): Homogeneous system (u(r) and v(r) are plane waves): Neglect terms of order u 2, v 2 and uv

10. Homogeneous Bogoliubov spectrum “healing length” Interaction vs. Quantum Pressure k(    E( 

11. Bragg Spectroscopy M. Kozuma, et. al., PRL 82, 871 (1999). J. Stenger, et. al., PRL 82, 4569 (1999).

12. The Measured Excitation Spectrum (using Bragg spectroscopy) Liquid Helium (scaled for comparison)

13. Phonon Region

14. Superfluidity! Landau criteria: Superfluid velocity A few mm/sec in experimental systems! Interactions – lead to superfluidity!

15. Many body theory (homogeneous) Assume macroscopic occupation of S.P. Ground state: Put in assumption + keep terms of orderand The number operator is conserved – can be placed in

16. Many body theory (homogeneous) Neglected: Bogoliubov Transform: Atomic commutation relations give:

17. Many body theory (homogeneous) Eliminate off-diagonal third line: Convenient representation: Solution of quasi-particle amplitudes:

18. Diagonalized Hamiltonian Energy spectrum: (again) Ground state is a highly non-trivial Superposition of all momentum states: Ground state energy:

19. Quasi-particle physics Inverse transformation: Particle creation Particle Annihilation Quasi-particle factors for repulsive condensates Low k limit High k limit

20. Quasi-particle physics ??? Don’t Forget Bosonic Enhancement!

21. Quantum depletion of S.P. ground state Evaluate the non-single-particle component of the ground state at T=0 About 1% for “standard” experiments

22. Attractive collapse! Complex energy – unstable to excitation! Finite size can save us (cutoff in Low k’s) Experimental values: A few thousand atoms!

23. Structure factor and Feynman relation Static structure factor (Fourier transform of the density-density correlation function) T=0

24. Static Structure Factor Measure of: Response at k Fluctuations with wave-number k Feynman Relation

25. Excitation Spectrum of Superfluid 4 He D. G. Henshaw and A. D. B. Woods, Phys. Rev. 121, 1266 (1961). Feynman Relation D. G. Henshaw, Phys. Rev. 119, 9 (1960).

26. Higher order – Beliaev and Landau damping Beliaev k k-q q Landau k k-q q A kq The many-body suppression factor:

27. Damping rate Fermi golden rule: The  function can be turned into a geometrical condition:

28. Damping rate Impurities Excitations

29. Points not covered - Inhomogeneous Bogoliubov theory - Beyond T=0 - Coherent collisions of excitations (FWM) - Hydrodynamic representation of GPE - Na 3 ~ 1 – theory and experiment