PhD thesis in English

magnetic field ⃗ B = 2M ⃗ Ω.3. Rotating ideal BECHowever, once harmonically trapped Bose-Einstein condensate is rotated critically,i.e. the rotation frequency becomes so large that it fully compensates theradially confining harmonic trapping, the system turns out to be radially no longerconfined. In the absence of additional potential terms, the condensate would start toexpand in directions perpendicular to the rotation axis. For an overcritical rotation,this expansion would even be accelerated by the presence of a residual centrifugalforce. This poses a severe problem to the experimental achievement of strongly correlatedbosonic states. In order to reach experimentally this delicate regime, Fettersuggested in Ref. [74] a small quartic term to be added to the harmonic trap potential,which would provide a confinement in the radial direction in the case of acritical and overcritical rotation.The proposal has been experimentally realized in Paris by the Dalibard group fora BEC of 87 Rb atoms [12, 7], by superimposing to the magnetic trap an additionalGaussian laser beam propagating in the z-direction,U(x, y) = U 0 e −2(x2 +y 2 )w 2 . (3.2)In the previous equation, w denotes the laser beam waist and U 0 is the intensity ofthe beam. Within the laser beam waist, the potential can be approximated by:U(x, y) ≈ U 0 − 2 U 0x 2 + y 2w 2 + 2 U 0(x 2 + y 2 ) 2w 4 , (3.3)and this is how the quartic term is brought about. An additional laser beam thatcreates a small anisotropic potential in the x − y plane is used for stirring the pureBEC at the rotation frequency Ω. After the angular momentum has been introducedinto the system, the rotation is stopped and the condensate is allowed to equilibrate.Taking into account Eqs. (3.1) and (3.3), the resulting axially-symmetric trap witha small quartic anharmonicity in the x − y plane, seen by individual atoms, has theformV BEC = M 2 (ω2 − Ω 2 )(x 2 + y 2 ) + M 2 ω2 z z2 + κ 4 (x2 + y 2 ) 2 , (3.4)with the trap frequencies ω = 2π × 64.8 Hz, ω z = 2π × 11.0 Hz, and the trapanharmonicity κ = κ BEC = 2.6 × 10 −11 Jm −4 . The rotation frequency Ω, measuredin units of ω and expressed by the ratio η = Ω/ω, represents the tunable controlparameter, which could be experimentally varied in the range between 0 and 1.04.58