1. Rotations and quantized vortices in Bose superfluids
F.Dalfovo
INFM-BEC Trento
and
Dipartimento di matematica e fisica,
Università Cattolica, Brescia

2. Outline Irrotational velocity field and superfluidity
Trento (past, present, future)
Liquid Helium vs. trapped condensates

3. A superfluid has an irrotational velocity field
Complex order parameter:
n : density
S : phase
Velocity field :
which implies:

4. Consequences: No circulation in a simply connected region
Quantized circulation in toroidal geometry.
Quantized vortices (n=0 on the vortex line).
Vortex lattices

9. Vortices observed at:
JILA-Boulder
ENS-Paris
MIT
Oxford
Produced with different techniques:
Phase imprinting, rotating laser spoon, rotating magnetic trap, rotating thermal cloud, selective evaporation, decay of solitons, etc.

10. A lot of physical questions:
Nucleation mechanisms.
Observation of density and phase.
Stability, decay, precession.
Shape and dynamics of a single vortex.
Formation and dynamics of vortex lattices.
Fast rotating condensates and giant vortices.
Coreless vortices and textures in spinor condensates.
Interaction with thermal atoms, solitons, surface modes.
Vortex rings, vortex-antivortex pairs, etc.
A lot of theoretical papers !!

11. Vortex-free configurations with angular momentum ℓ≠0

19. Possible route to vortex nucleation
Almost spherical condensate
in a rotating trap with Ω close to ω┴/√2
New stable configuration, spherical, with vortices
Many quadrupole shape deformations are excited
Vortices enter the condensate
Highly deformed condensate with irrotational field
Complex dynamics with nucleation of vortices at the surface
The deformed condensate becomes dynamically unstable

24. Work done in Trento
Vortex nucleation and quadrupole deformation of a rotating Bose-Einstein condensate M. Kraemer, L. Pitaevskii, S. Stringari, F. Zambelli, Laser Physics 12, 113 (2002) 
Consequence of superfluidity on the expansion of a rotating Bose-Einstein condensate M. Edwards, C. W. Clark, P. Pedri, L. Pitaevskii, and S. Stringari, Phys. Rev. Lett. 88, (2002) 
A superfluid gyroscope with cold atomic gases S. Stringari, Phys. Rev. Lett. 86, 4725 (2001)   
Shape deformations and angular momentum transfer in trapped Bose-Einstein condensates F. Dalfovo and S. Stringari, Phys. Rev. A 63, (R) (2001) 
Overcritical Rotation of a Trapped Bose-Einstein Condensate A. Recati, F. Zambelli, and S. Stringari, Phys. Rev. Lett 86, 377 (2001) 
Moment of Inertia and Quadrupole Response Function of a Trapped Superfluid F. Zambelli and S. Stringari, Phys. Rev. A 63, (2001) 
Free expansion of Bose-Einstein condensates with quantized vortices F. Dalfovo and M. Modugno, Phys. Rev. A 61, (2000) 
Pinning of quantized vortices in helium drops by dopant atoms and molecules F. Dalfovo, R. Mayol, M. Pi, and M. Barranco, Phys. Rev. Lett. 85, 1028 (2000) 
Scissors mode and superfluidity of a trapped Bose-Einstein condensed gas D. Guery-Odelin and S. Stringari, Phys. Rev. Lett 83, 4452 (1999)  
Phase diagram of quantized vortices in a trapped Bose-Einstein condensed gas S. Stringari, Phys. Rev. Lett. 82, 4373 (1999)  
Quantized vortices and collective oscillations of a trapped Bose condensed gas F. Zambelli and S. Stringari, Phys. Rev. Lett. 81, 1754 (1998)  
Moment of Inertia and Superfluidity of a Trapped Bose Gas , S. Stringari, Phys. Rev. Lett. 76, 1405 (1996)  
Bosons in anisotropic traps: ground state and vortices , F. Dalfovo and S. Stringari, Phys. Rev. A 53, 2477 (1996)

25. Present and next future:
Most recent activity:
Scissors mode in rotating condensates
Scissors mode of a rotating Bose-Einstein condensate,
M.Cozzini, S. Stringari, V. Bretin, P. Rosenbusch, J. Dalibard, PRA 67, (2003)
Macroscopic dynamics of vortex lattices
Macroscopic dynamics of a Bose-Einstein condensate containing a vortex lattice,
Marco Cozzini and Sandro Stringari, e-print cond-mat/
Present and next future:
More about vortex lattices
Stationary configurations, Collective oscillations, elastic properties, dynamics, …

27. Scissors mode below Tc :
the superfluid oscillates with frequency
Scissors mode above Tc :
the gas oscillates with frequencies

39. Back to Helium

40. Helium nanodroplets
From: “Superfluid Helium Droplets: An Ultracold Nanolaboratory”, J.P. Toennies, A.F. Vilesov, K.B. Whaley, Phys. Today 54 (2001)

41. Helium droplet ↔ trapped BEC
Helium is dense
Condensate fraction is 10% in bulk at T=0
Superfluid fraction is 100% in bulk at T=0
Helium droplets are self bound (no confinement)
Temperature of droplets is about K (evaporative cooling)

43. Density functional calculations for helium nanodroplets:
Moment of inertia
A superfluid hydrodynamic model for the enhanced moments of inertia of molecules in liquid 4He, C. Callegari, A. Conjusteau, I. Reinhard, K. K. Lehmann, G. Scoles, F. Dalfovo Phys. Rev. Lett. 83, 5058 (1999)
Quantized vortices
Pinning of quantized vortices in helium drops by dopant atoms and molecules , F. Dalfovo, R. Mayol, M. Pi, and M. Barranco, Phys. Rev. Lett. 85, 1028 (2000)
Quantized Vortices in Mixed 3He-4He Drops, R. Mayol, M. Pi, and M. Barranco, and F. Dalfovo, Phys. Rev. Lett. 87, (2001)

45. Trapped BEC with a vortex
Helium droplet with a vortex
F. D., R. Mayol, M. Pi, and M. Barranco, Phys. Rev. Lett. 85, 1028 (2000) ←
Trapped BEC with a vortex
F. D. and S. Stringari,
Phys. Rev. A 53, 2477 (1996) ↓

46. Helium droplet
+ vortex
+ HCN ←

47. Conclusions
Rotational properties and quantized vorticity are intimately connected to superfluidity.
Dilute condensates in traps represent a wonderful testing ground for theories on quantum fluids.
Dilute condensates and liquid helium are good friends. They look different, but they speak the same language.