Superfluid dynamics of BEC in a periodic potential

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 INFM-BEC & Department of Physics, Universita` di Trento Andrea Trombettoni INFM & Department of physics, Universita` di Parma

3. Condensed matter Quantum optics BEC Atomic physics Solid state

4. BEC trapped in a periodic potential

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

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

7. BEC expanding in a 1D optical lattice Interacting atoms --> a > 0 Density profile Trappingpotential 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 • Atoms are condensed in the optical and magnetic fields. • The harmonic confinement is instantaneously shifted along the x direction. Array of Josephson junctions driven by a harmonic external field

11. Josephson oscillations Oscillations of the three peaks of the interferogram. Blue circles: no periodic potential 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)

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

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

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