Lecture IV BoseEinstein condensate Superfluidity New trends Theoretical

Lecture IV Bose-Einstein condensate Superfluidity New trends

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

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

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

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

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

Thomas Fermi energy point of view Kinetic energy Potential energy Interaction energy 87 Rb : a = 5 nm N = 105 R = 1 mm

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

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

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

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

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

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)

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

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

Velocity field: condensate versus classical Condensate Classical gas

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)

Scissors Mode PRL 83, 4452 (1999)

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

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

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

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

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

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 105 to 4 105 atoms 6 mm T < 100 n. K 120 mm 2. Stirring using a laser beam (0. 5 seconds) controlled with acousto-optic modulators e. X=0. 03 , e. Y=0. 09

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

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