Rotations and quantized
vortices in Bose superfluids
F.Dalfovo
INFM-BEC Trento
and
Dipartimento di matematica e fisica,
Università Cattolica, Brescia
Outline
• Irrotational velocity field and superfluidity
• Work @ Trento (past, present, future)
• Liquid Helium vs. trapped condensates
A superfluid has an irrotational velocity field
Complex order parameter:
n e
1 / 2 iS
n : density
S : phase
Velocity field :
which implies:
v  ( / m)S
 v  0
Consequences:
• No circulation in a simply connected region
• Quantized circulation in toroidal geometry.
• Quantized vortices (n=0 on the vortex line).
• Vortex lattices
Vortices observed at:
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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.
A lot of physical questions:
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•
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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 !!
Vortex-free configurations with angular momentum ℓ≠0
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
The deformed condensate becomes
dynamically unstable
Complex dynamics with
nucleation of vortices at
the surface
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)
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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, 070405 (2002)
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A superfluid gyroscope with cold atomic gases
S. Stringari, Phys. Rev. Lett. 86, 4725 (2001)
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Shape deformations and angular momentum transfer in trapped Bose-Einstein condensates
F. Dalfovo and S. Stringari, Phys. Rev. A 63, 011601(R) (2001)
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Overcritical Rotation of a Trapped Bose-Einstein Condensate
A. Recati, F. Zambelli, and S. Stringari, Phys. Rev. Lett 86, 377 (2001)
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Moment of Inertia and Quadrupole Response Function of a Trapped Superfluid
F. Zambelli and S. Stringari, Phys. Rev. A 63, 033602 (2001)
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Free expansion of Bose-Einstein condensates with quantized vortices
F. Dalfovo and M. Modugno, Phys. Rev. A 61, 023605 (2000)
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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)
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Scissors mode and superfluidity of a trapped Bose-Einstein condensed gas
D. Guery-Odelin and S. Stringari, Phys. Rev. Lett 83, 4452 (1999)
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Phase diagram of quantized vortices in a trapped Bose-Einstein condensed gas
S. Stringari, Phys. Rev. Lett. 82, 4373 (1999)
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Quantized vortices and collective oscillations of a trapped Bose condensed gas
F. Zambelli and S. Stringari, Phys. Rev. Lett. 81, 1754 (1998)
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Moment of Inertia and Superfluidity of a Trapped Bose Gas ,
S. Stringari, Phys. Rev. Lett. 76, 1405 (1996)
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Bosons in anisotropic traps: ground state and vortices ,
F. Dalfovo and S. Stringari, Phys. Rev. A 53, 2477 (1996)
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, 021602 (2003)
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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/0211294
Present and next future:
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More about vortex lattices
Stationary configurations, Collective oscillations, elastic properties,
dynamics, …
Scissors mode below Tc :
the superfluid oscillates with frequency
(   )
2
x
2 1/ 2
y
Scissors mode above Tc :
the gas oscillates with frequencies
x   y
Back to Helium
Helium nanodroplets
From: “Superfluid Helium Droplets: An Ultracold Nanolaboratory”, J.P. Toennies, A.F. Vilesov, K.B. Whaley, Phys. Today 54 (2001)
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 0.15 - 0.4 K (evaporative cooling)
Density functional calculations
for helium nanodroplets:
Moment of inertia
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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
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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)
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Quantized Vortices in Mixed 3He-4He Drops, R. Mayol, M. Pi, and M. Barranco, and F. Dalfovo,
Phys. Rev. Lett. 87, 145301 (2001)
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)
↓
Helium droplet
+ vortex
+ HCN
←
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.