1. Superfluid dynamics of BEC in a periodic potential Augusto Smerzi INFM-BEC & Department of Physics, Trento LANL, Theoretical Division, Los Alamos

2. Collaboration with: Chiara Menotti Chiara Menotti INFM-BEC & Department of Physics, Universita` di Trento INFM-BEC & Department of Physics, Universita` di Trento Andrea Trombettoni Andrea Trombettoni INFM & Department of physics, Universita` di Parma INFM & Department of physics, Universita` di Parma

4. BEC trapped in a periodic potential

5. BEC expanding in a 1D optical lattice No interaction --> a = 0 Density profile Momentum distribution Trapping potential

6. A. Trombettoni and A. Smerzi, PRL 86, 2353 (2000) BEC expanding in a 1D optical lattice

7. Interacting atoms --> a > 0 Density profile Trapping potential Momentum distribution

8. Preliminary experimental evidences of self-trapping B. Eiermann, M. Albiez, M. Taglieber, M. Oberthaler University of Konstanz

9. BEC expanding in a 1D optical lattice Interacting atoms --> a > 0 Array of weakly coupled BEC

10. Josephson oscillations Array of Josephson junctions driven by a harmonic external field 1.Atoms are condensed in the optical and magnetic fields. 2.The harmonic confinement is instantaneously shifted along the x direction.

11. Josephson oscillations The array is governed by a pendulum equation F.S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, M. Inguscio, Science 293, 843 (2001) Oscillations of the three peaks of the interferogram. Blue circles: no periodic potential

12. Small amplitude pendulum oscillations Triangles: GPE; stars: variational calculation of K Circles: experimental results Relation between the oscillation frequency and the tunneling rate

13. Breakdown of Josephson oscillations The interwell phase coherence breaks down for a large initial displacement of the BEC center of mass

14. Questions: 1) Why the interaction can break the inter-well phase coherence of a condensate at rest confined in a periodic potential ? 2) Why a large velocity of the BEC center of mass can break the inter-well phase coherence of a condensate confined in a periodic potential and driven by a harmonic field ? Which are the transport properties of BEC in periodic potentials ?

16. The discrete nonlinear equation (DNL)

17. Newtonian Dynamics of a wave-packet

18. Bloch energies & effective masses Effective masses depend on the height of the inter-well barriers and on the density See also M. Kramer, C. Menotti, L. Pitaevskii and S. Stringari, unpublished

19. Bogoliubov spectrum 1.Replace in the DNL 2.After linearization, retrieve the dispersion relation

20. Bogoliubov spectrum

21. Sound-wave & energetic instability Cfr. with B. Wu and Q. Niu, PRA64, 061603R (2001)

22. Dynamical instability New mechanism for the breakdown of superfluidity of a BEC in a periodic potential No dynamical instabilities The amplitude of the perturbation modes grows exponentially fast, dissipating the energy of the large amplitude wave-packet

23. Comparison between analytical and numerical dispersion relation Full line: analytical (DNL) Dots: numerical (GPE)

24. Breakdown of superfluidity for a BEC driven by a harmonic field A. Smerzi, A. Trombettoni, P.G. Kevrekidis, A.R. Bishop, PRL 89, 170402 (2002) Density at t=0,20,40 ms as a function of the Position. Initial displacements: 50, 120 sites Quasi-momentum vs. time for three different initial displacements: 40, 80, 90 sites

25. The Frontier Technological applications Technological applications Interferometry at the Heisenberg limit Interferometry at the Heisenberg limit Quantum information Quantum information Foundational problems Foundational problems Quantum – classical correspondence principle Quantum – classical correspondence principle Schroedinger cats, entanglement Schroedinger cats, entanglement Tools Tools Quantum many-body dynamical theory Quantum many-body dynamical theory