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**Rotations and quantized vortices in Bose superfluids**

F.Dalfovo INFM-BEC Trento and Dipartimento di matematica e fisica, Università Cattolica, Brescia

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**Outline Irrotational velocity field and superfluidity**

Trento (past, present, future) Liquid Helium vs. trapped condensates

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**A superfluid has an irrotational velocity field**

Complex order parameter: n : density S : phase Velocity field : which implies:

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**Consequences: No circulation in a simply connected region**

Quantized circulation in toroidal geometry. Quantized vortices (n=0 on the vortex line). Vortex lattices

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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.

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**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 !!

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**Vortex-free configurations with angular momentum ℓ≠0**

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**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

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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)

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**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, …

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**Scissors mode below Tc :**

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

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Back to Helium

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

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**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)

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**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)

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**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) ↓

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Helium droplet + vortex + HCN ←

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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.

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DYNAMICS OF TRAPPED BOSE AND FERMI GASES Sandro Stringari University of Trento IHP Paris, June 2007 CNR-INFM Lecture 2.

DYNAMICS OF TRAPPED BOSE AND FERMI GASES Sandro Stringari University of Trento IHP Paris, June 2007 CNR-INFM Lecture 2.

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