1. Interazioni e transizione superfluido-Mott

2. Bose-Hubbard model for interacting bosons in a lattice: Interacting bosons in a lattice SUPERFLUID Long-range phase coherence Poissonian number fluctuations Gapless excitation spectrum Compressible MOTT INSULATOR No phase coherence No number fluctuations (Fock states) Gap in the excitation spectrum Not compressible

3. Superfluid to Mott Insulator transition first experimental demonstration in M. Greiner et al., Nature 415, 39 (2002) momentum distribution of the 87 Rb atomic sample after expansion (LENS, 2006) measuring coherence via matter-wave interference

4. Time-of-flight imaging P. Pedri et al., Phys. Rev. Lett. 87, 220401 (2001)

5. Time-of-flight imaging M. Greiner et al., Nature 415, 39 (2002)

6. Superfluid-Mott transition M. Greiner et al., Nature 415, 39 (2002) measuring coherence after time-of-flight: superfluid-Mott transition

7. Superfluid-Mott transition M. Greiner et al., Nature 415, 39 (2002) reversible quantum phase transition Restoring phase coherence from a Mott insulator

8. Collapse & Revival M. Greiner et al., Nature 419, 51 (2002)

9. Excitation spectrum Example: weakly-interacting BEC Bogoliubov spectrum Small momentum k  collective excitations, phonons Large momentum k  single particle spectrum

10. BEC Excitation spectrum Excitation spectrum of a weakly interacting BEC in harmonic trap J. Steinhauer et al., PRL 88, 120407 (2002)

11. Excitation spectrum U 2U Mott Insulator spectrum --- measurement of excitation spectrum via lattice modulation (T. Stöferle et al., PRL 92, 130403 (2004))

12. Excitation spectrum M. Greiner et al., Nature 415, 39 (2002) application of a magnetic field gradient

13. In-situ density measurement in-situ imaging of the density distribution incompressible central region with 1 atom/site N. Gemelke et al., Nature 460, 995 (2009).

14. Quantum information

15. Quantum bit (qubit) Qubits coherent superposition

16. Entangled states Entanglement

17. Quantum gates via ultracold collisions D. Hayes, P. Julienne, I. Deutsch, PRL 98, 070501 (2007) Quantum Logic via the Exchange Blockade in Ultracold Collisions Example of implementation of a quantum gate for identical fermionic particles: Symmetrization of the wavefunction:  collisional phase-shift  no collisional phase-shift Implementation of the gate with fidelity ~ 1

18. Quantum gates via ultracold collisions M. Anderlini et al., Nature 448, 452 (2007) Controlled exchange interactions between pairs of neutral atoms in OL Experimental realization of the exchange phase-shift in a lattice of double wells:

19. Detecting correlations

20. Hanbury-Brown & Twiss effect interference between quantum-mechanical paths of identical particles correlations between joint probability at detector positions U. Fano, Am. J. Phys. 29, 539 (1961) 2

21. approx. 50000 detectors in a single image! absorption image of a Mott Insulator state HBT interferometry in quantum gases d

22. Noise correlations

23. Noise interferometry: bunching for bosons (Mott) S. Foelling et al., Nature 434, 481 (2005)

24. Noise interferometry: antibunching for fermions T. Rom et al., Nature 444, 733 (2006)

25. Detecting correlations M. Schellekens et al., Science 310, 648 (2005) Time-resolved time-of-flight detection of metastable He atoms

26. Bunching for bosons (thermal) M. Schellekens et al., Science 310, 648 (2005)

27. Antibunching for fermions He-4 He-3 T. Jeltes et al., Nature 445, 402 (2007)