1. Lecture IV Bose-Einstein condensate Superfluidity New trends

2. The Hamiltonian: Confining potential Interactions between atoms At low temperature, we can replace the real potential by :, a : scattering legnth Hartree approximation: Gross-Pitaevski equation (or non-linear Schrödinger’s equation) : Theoretical description of the condensate

3. The scattering length can be modified: a ( B ) Feshbach’s resonances a > 0 : Repulsive interactions a = 0 : Ideal gas a < 0 : Attractive interaction a = 0a > 0 GaussianParabolic a < 0, 3D N < N c « Collapse » a < 0, 1D Soliton Different regime of interactions

4. 8 ms 7 ms 6 ms 2 ms Experimental realization Science 296, 1290 (2002)

5. Time-dependent Gross-Pitaevski equation Hydrodynamic equations Review of Modern Physics 71, 463 (1999) with the normalization Phase-modulus formulation evolve according to a set of hydrodynamic equations (exact formulation): continuity euler

6. Thomas Fermi approximation in a trap Appl. Phys. B 69, 257 (1999)

7. Thomas Fermi energy point of view Kinetic energy Potential energy Interaction energy 87 Rb : a = 5 nm N = 10 5 R = 1  m

8. Scaling solutions Equation of continuity Scaling ansatz Scaling parameters Time dependent Normalization Euler equation

9. Scaling solutions: Applications Quadrupole modeMonopole mode Time-of-fligth: microscope effect Coupling between monopole and quadrupole mode in anisotropic harmonic traps 1  m 100  m

10. Bogoliubov spectrum Equilibrium state in a box uniform Linearization of the hydrodynamic equations We obtain speed of sound

11. At low momentum, the collective excitations have a linear dispersion relation: P*P* E(P * ) Microscopic probe-particle: Conclusion : For the probe cannot deposit energy in the fluid. Superfluidity is a consequence of interactions. For a macroscopic probe: it also exists a threshold velocity, PRL 91, 090407 (2003) Landau argument for superfluidity before collision and after collision A solution can exist if and only if

12. HD equations: Rotating Frame, Thomas Fermi regime velocity in the laboratory frame position in the rotating frame

13. Stationnary solution We find a shape which is the inverse of a parabola But with modified frequencies Introducing the irrotational ansatz PRL 86, 377 (2001)

14. Determination of  Equation of continuity gives From which we deduce the equation for  We introduce the anisotropy parameter

15. Determination of  dashed line: non-interacting gas Solutions which break the symmetry of the hamiltonian It is caused by two-body interactions Center of mass unstable

16. Velocity field: condensate versus classical Condensate Classical gas

17. Moment of inertia The expression for the angular momentum is It gives the value of the moment of inertia, we find where Strong dependence with anisotropy ! PRL 76, 1405 (1996)

18. Scissors Mode PRL 83, 4452 (1999)

19. Scissors Mode: Qualitative picture (1) Kinetic energy for rotation For classical gas For condensate Extra potential energy due to anisotropy Moment of Inertia

20. classical condensate We infer the existence of a low frequency mode for the classical gas, but not for the Bose-Einstein condensate Scissors Mode: Qualitative picture (2)

21. Scissors Mode: Quantitative analysis Classical gas: Moment method for Two modesand One mode Bose-Einstein condensate in the Thomas-Fermi regime One mode Linearization of HD equations

22. Experiment (Oxford) PRL 84, 2056 (2001) Experimentl proof of reduced moment of inertia associated as a superfluid behaviour

23. Vortices in a rotating quantum fluid In a condensate the velocity is such that Liquid superfluid helium Below a critical rotation  c, no motion at all Above  c, apparition of singular lines on which the density is zero and around which the circulation of the velocity is quantized Onsager - Feynman incompatible with rigid body rotation

24. Preparation of a condensate with vortices 1. Preparation of a quasi-pure condensate (20 seconds) Laser+evaporative cooling of 87 Rb atoms in a magnetic trap 10 5 to 4 10 5 atoms T < 100 nK 120  m 6  m 2. Stirring using a laser beam (0.5 seconds)  X =0.03,  Y =0.09 controlled with acousto-optic modulators

25. From single to multiple vortices Just below the critical frequency Just above the critical frequency Notably above the critical frequency For large numbers of atoms: Abrikosov lattice PRL 84, 806 (2000) It is a real quantum vortex  angular momentum h PRL 85, 2223 (2000) also at MIT, Boulder, Oxford

26. Stable branch Dynamically unstable branch Dynamics of nucleation PRL 86, 4443 (2001)