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Lecture IV Bose-Einstein condensate Superfluidity New trends

Lecture IV Bose-Einstein condensate Superfluidity New trends. Theoretical description of the condensate. Hartree approximation:. Gross-Pitaevski equation (or non-linear Schrödinger’s equation) :. The Hamiltonian:. Interactions between atoms. Confining potential.

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Lecture IV Bose-Einstein condensate Superfluidity New trends

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  1. Lecture IVBose-Einstein condensate Superfluidity New trends

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

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

  4. Experimental realization 8 ms 7 ms 6 ms 2 ms 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 = 105 R = 1 mm

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

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

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

  11. Landau argument for superfluidity before collision and after collision At low momentum, the collective excitations have a linear dispersion relation: E(P*) Microscopic probe-particle: P* A solution can exist if and only if 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)

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

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

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

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

  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 Strong dependence with anisotropy ! where PRL 76, 1405 (1996)

  18. Scissors Mode PRL 83, 4452 (1999)

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

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

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

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

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

  24. 2. Stirring using a laser beam (0.5 seconds) controlled with acousto-optic modulators eX=0.03 , eY=0.09 Preparation of a condensate with vortices 1. Preparation of a quasi-pure condensate (20 seconds) Laser+evaporative cooling of 87Rb atoms in a magnetic trap 105 to 4 105 atoms T < 100 nK 6 mm 120 mm

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

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

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