( ) Bose-Einstein condensates in fast rotation. Rotating condensates. Landau levels for a rotating gas. An interesting variant: the r 2 +r 4 potential

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1 Bose-instein condensates in fast otation Jean Dalibad Laboatoie Kastle Bossel cole nomale supéieue, Pais xp: Baptiste Battelie, Vincent Betin, Zoan Hadibabic, Sabine Stock Th: Amandine Aftalion, Xavie Blanc + many discussions with Yvan Castin, Sando Stingai, Goa Shlyapnikov Rotating condensates Rotation at angula fequency Ω Boulde: evapoative spin-up ; NS, MIT, Oxfod: stiing Abikosov lattice of votices nh vd. = Feynman, Onsage MIT M Rigid body otation v =Ω fo the coase-gain aveage of the velocity field if the votex suface density is: M Ω ρv = π Landau levels fo a otating gas An inteesting vaiant: the + 4 potential Isotopic hamonic tapping in the xy plane with fequency m=- Hamiltonian in the otating fame: m=- Ω= Ω < m= m= m= m= - - H ΩL When Ω~, one appoaches a situation with a macoscopically degeneate gound state fo the one-body hamiltonian Same physics as fo a chaged paticle in a magnetic field Ω Hamonic potential supeimposed with a small quatic tem (fomed using an additional lase beam popagating along the axis of otation): γ 4 = + in the xy plane ( = x + y ) V() m 4 Stiing fequency Ω of the gas: Ω Tapping + centifugal potential: ( ) γ m Ω Ω < Ω = Ω >

2 Coiolis vs. centifugal foces F = MΩ v Coiolis Fcentif. = MΩ p ( p A) H = + V( ) ΩL = + V( ) MΩ M M A = MΩ. Condensation tempeatue of an ideal otating gas Is the Coiolis foce significant fo fast otating gases?. Condensation tempeatue. quilibium shape of the condensate. Vibation modes of the condensate NO IT DPNDS YS two (little) miacles N = h Condensation theshold in the semi-classical appoximation d d p exp( (, p) / kt) ( p A) (, p) = + V ( ) MΩ M Semi-classical appoximation () Puely hamonic potential (Stingai 999): N =. ( kt / ) - Ω ( Ω ) expeimental check: Boulde Change of vaiables: ' p x = p x + MΩy ' p y = p y MΩx (Landau-Lifshit) Hamonic + quatic potential exactly at Ω = : γ V M M 4 4 () = + Ω γ 4 4 N = d d p' exp( (, p') / kt) p ' (, p') = + V ( ) MΩ M No effect of Coiolis foce Analogous to Boh- van Leeuven theoem: no classical magnetism 5/ M( kt) N =.9 γ Ae these esults compatible with a quantum desciption?

3 Calculation of T c with Landau levels (hamonic case) Calculation of T c with Landau levels ( + γ 4 /4) ' N = levels exp [( ) / kt] =Ω - - Ω 4 m If >> Ω, one can neglect the contibution of the levels with m < Same spectum as a D anisotopic oscillato: Ω,, kt >> ~ γ m 4M m Neglect the levels with m< Same spectum as a D system with: fee motion along diection with an effective mass M eff pop. to M /γ hamonic tap hamonic tap kt >> (class) N =. ( kt / ) ( Ω ) Ω Ω ( Ω) N = ( kt / ). ( Ω) 5/ (class) M( kt) N γ N 5/ Meff ( kt ) xpeimental esults in quadatic+quatic potential /π 65H = 5 atoms T = 5 nk. Ω: quilibium shape of a fast otating condensate Fo Ω=65-67 H, the votices become less and less visible: Not any moe a degeneate gas? T c = 6 nk Bending of the lines? Beakdown of mean-field appoximation? Phys. Rev. Lett. 9, 54 (4) D system: the motion is foen (gaussian of width a = /( m ) ) LLL physics, see also: Fische, Baym Watanabe, Baym, Pethick Komineas, Read, Coope xpeiments by the Boulde goup

4 Physics in the lowest Landau level Gound state in the LLL appoximation Ω /a e / ( x e a n + /a ( x+ e Geneal one-paticle state in the LLL: /a e P( x+ Polynomial o analytic function LLL If the chemical potential is much smalle than, the physics is esticted to the lowest Landau level /a = x + y a = / n j= m e ( u u ) u = x+ iy u j : votices j Minimiation of = + ho + ot + int = ψ int In the LLL: = Ω ψ ψ * ho = ψ ot [ L ] 4 = ψ g = 8 π as / a a s : scatteing length Minimiation of: = ho = + ot Λ =Ω+ ( Ω ) ψ + ψ only one paamete: 4 Ω = M = = LLL vs. centifugal foce appoximation ( = ) Minimiation of Thomas-Femi appoximation: Validity: ( Ω) Λ 4 ψ + ψ atom R ρ Ω R Λ /4 5 Centifugal foce appoximation: Ω D >> : Thomas-Femi egime, simila to LLL ψ + ψ + ψ ψ = µψ ( ) /4 << : Gaussian egime, with a width Ω >> R vey diffeent fom the LLL pediction

5 Physics in the lowest Landau level Gound state in the LLL appoximation Ω /a e / ( x e a n + /a ( x+ e Geneal one-paticle state in the LLL: /a e P( x+ Polynomial o analytic function LLL If the chemical potential is much smalle than, the physics is esticted to the lowest Landau level /a = x + y a = / n j= m e ( u u ) u = x+ iy u j : votices j Minimiation of = + ho + ot + int = ψ int In the LLL: = Ω ψ ψ * ho = ψ ot [ L ] 4 = ψ g = 8 π as / a a s : scatteing length Minimiation of: = ho = + ot Λ =Ω+ ( Ω ) ψ + ψ only one paamete: 4 Ω = M = = LLL vs. centifugal foce appoximation ( = ) LLL solution fo Minimiation of 4 ψ + ψ Thomas-Femi appoximation: Λ atom R ρ Ω R Λ /4 n /a j= ψ ( xy, ) = e ( u u) u = x+ iy j 5 Validity: ( Ω) Centifugal foce appoximation: Ω ψ + ψ + ψ ψ = µψ -5 D >> : Thomas-Femi egime, simila to LLL ( ) /4 << : Gaussian egime, with a width Ω >> R vey diffeent fom the LLL pediction atomic distibution Rb atoms, Ω=.99 / (π) = 5 H votex distibution stong distotion on the edges igid body otation (T.L. Ho)!

6 The tansvese monopole oscillation Fo a non otating gas in a hamonic tap, this "beathing mode" coespond to a scaling tansfom of the steady-state distibution. Beathing mode of a otating condensate monopole = Chevy et al, PRL 88, 54 () Guilleumas & Pitaevskii, Jackson et al, Kagan et al Pediction fo a otating gas in a hamonic tap (Coini & Stingai) = and not = Ω monopole xpeiment in a quadatic+ quatic tap.5..5 monopole monopole S. Stock et al, uophys. Lett. 65, 594 (4) Stiing fequency (H) The stuctue of the monopole mode Fo elatively low otation fequencies, usual beathing mode: 6 H Ω /.9 6 ms xcitation of the beathing mode of the fast otating condensate One pictue evey ms Ω /.4 Fo otation fequencies above the tap fequency, stange stuctue of the beathing mode Th: Stingai, Fette, Jackson

7 Conclusions The Coiolis foce plays no ole fo the value of the citical tempeatue of a otating gas, if kt but it is essential to calculate the shape and the modes of the BC Fo fast otations in the mean-field egime, the LLL basis povides a convenient fomalism: deviation fom igid-body otation + 4 : Mexican hat potentials and pesistent cuents Pespectives: Layeed otating supefluid calculation by I. Danaila Reach N votex ~N atomes : coelated states and Factional Quantum Hall effect