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Superfluidity in atomic Fermi gases

Superfluidity in atomic Fermi gases. Luciano Viverit. University of Milan and CRS-BEC INFM Trento. CRS-BEC inauguration meeting and Celebration of Lev Pitaevskii’s 70th birthday. Outline. Why superfluidity in atomic Fermi gases? Some ways to attain superfluidity

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Superfluidity in atomic Fermi gases

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  1. Superfluidity in atomic Fermi gases Luciano Viverit University of Milan and CRS-BEC INFM Trento CRS-BEC inauguration meeting and Celebration of Lev Pitaevskii’s 70th birthday

  2. Outline • Why superfluidity in atomic Fermi gases? • Some ways to attain superfluidity • How to detect superfluidity and current • experimental developments • Vortices

  3. Why superfluidity in atomic Fermi gases? Test ground for various theories: • 1) Superfluidity in dilute gases • Gorkov and Melik-Barkhudarov, JETP 13,1018 (1961) • Stoof, Houbiers, Sackett and Hulet, PRL 76, 10 (1996) • Papenbrock and Bertsch, PRC 59, 2052 (1999) • Heiselberg, Pethick, Smith and LV, PRL 85, 2418 (2000) a < 0 ; kF |a| <<1

  4. Why superfluidity in atomic Fermi gases? • 2) Detailed study of effective interactions in medium and • consequences on pairing • Berk and Schrieffer, PRL 17, 433 (1966) (superconductors) • Schulze et al., Phys. Lett. B375, 1 (1996) (neutron stars) • Barranco et al., PRL 83, 2147 (1999) (nuclei) • Combescot, PRL 83, 3766 (1999) (atomic gases) • LV, Barranco, Vigezzi and Broglia, work in progress a < 0 ; kF |a| ~1

  5. + Why superfluidity in atomic Fermi gases? • 3) Boson enhanced pairing in Bose-Fermi mixtures • Bardeen, Baym and Pines, PRL 17, 372 (1966) (3He-4He) • Heiselberg, Pethick, Smith and LV, PRL 85, 2418 (2000) • Bijlsma, Heringa and Stoof, PRA 61, 053601 (2000) • LV, PRA 66, 023605 (2002)

  6. + Why superfluidity in atomic Fermi gases? • 3) Boson enhanced pairing in Bose-Fermi mixtures • Bardeen, Baym and Pines, PRL 17, 372 (1966) (3He-4He) • Heiselberg, Pethick, Smith and LV, PRL 85, 2418 (2000) • Bijlsma, Heringa and Stoof, PRA 61, 053601 (2000) • LV, PRA 66, 023605 (2002)

  7. Why superfluidity in atomic Fermi gases? • 4) BCS-BEC crossover • Leggett (1980) • Nozières and Schmitt-Rink, JLTP 59, 195 (1985) • Sà de Melo, Randeria and Engelbrecht, PRL 71, • 3202 (1993) • Pieri and Strinati, PRB 61, 15370 (2000) 1/ kFa  -∞1/ kFa+∞

  8. Why superfluidity in atomic Fermi gases? • 4a) Resonance superfluidity • Holland, Kokkelmans, Chiofalo and Walser • PRL 87, 120406 (2001) • Ohashi and Griffin, PRL 89, 130402 (2002)

  9. Why superfluidity in atomic Fermi gases? • 4a) Resonance superfluidity • Holland, Kokkelmans, Chiofalo and Walser • PRL 87, 120406 (2001) • Ohashi and Griffin, PRL 89, 130402 (2002)

  10. Why superfluidity in atomic Fermi gases? • 5) Superfluid-insulator transition in (optical) lattices • Micnas, Ranninger and Robaszkiewicz RMP 62, • 113 (1990) (High Tc) • Hofstetter, Cirac, Zoller, Demler and Lukin • PRL 89, 220407 (2002)

  11. Why superfluidity in atomic Fermi gases? • 5) Superfluid-insulator transition in (optical) lattices • Micnas, Ranninger and Robaszkiewicz RMP 62, • 113 (1990) (High Tc) • Hofstetter, Cirac, Zoller, Demler and Lukin • PRL 89, 220407 (2002)

  12. Gap equation at Tc,0: Number equation at Tc,0: Ways to attain superfluidity 1)BCS in a uniform dilute gas (a<0, kF|a|<<1) where .

  13. If kF|a|<<1 solutions: • Sà de Melo, Randeria and Engelbrecht, PRL 71, 3202 • (1993) • Stoof, Houbiers, Sackett and Hulet, PRL 76, 10 (1996)

  14. Now include the effects of induced interactions to second order in a (important also in the dilute limit)

  15. Now include the effects of induced interactions to second order in a (important also in the dilute limit) 0 0 =0 =0 a a a a =0 ~ c (kFa)2; c>0 0 a

  16. Since kF|a|<<1 then By carrying out detailed calculations one finds and thus

  17. Gorkov and Melik-Barkhudarov, JETP 13,1018 (1961) • Heiselberg, Pethick, Smith and LV, PRL 85, 2418 (2000) Formula ~ valid also in trap if

  18. Practical problem: If kF|a|<<1 then Best experimental performances with present techniques • Gehm, Hemmer, Granade, O’Hara and Thomas, • e-print cond-mat/0212499 • Regal and Jin, e-print cond-mat/0302246 Not enough if the gas is dilute!

  19. Idea A: let kF|a| approach 1 (but stillkF|a|<1) • Combescot, PRL 83, 3766 (1999) (=2kF a /)

  20. WHY?? • Exchange of density and spin collective • modes (higher orders in kFa than previously • considered) and • Fragmentation of single particle levels • both strongly influence Tc!

  21. So what? • Answer difficult, no completely reliable theory • Answer interesting for several physical systems • LV, Barranco, Vigezzi and Broglia, work in progress • We wait for experiments ...

  22. Idea B: BCS-BEC crossover Back to BCS equations. Gap equation at Tc,0: Number equation at Tc,0:

  23. Solution in1/ kFa  -∞

  24. Including gaussian fluctuations in  about the saddle-point: • Sà de Melo, Randeria and Engelbrecht, PRL 71, 3202 • (1993) BEC critical temperature

  25. Superfluid transition in unitarity limit (kFa) predicted at BUT • Exchange of density and spin modes, and • Fragmentation of single particle levels • not included in the theory.Then:

  26. ? Strong interaction between theory and experiments needed.

  27. What is happening with experiments? • O’Hara et al., Science 298, 2179 (2002) (Duke) • Regal and Jin, e-print cond-mat/0302246 (Boulder) • Bourdel et al., eprint cond-mat/0303079 (Paris) • Modugno et al., Science 297, 2240 (2002) (Firenze) • Dieckmann et al., PRL 89, 203201 (2002) (MIT) Two component Fermi gas at T ~ 0.1 TF in unitarity conditions (kFa ±∞).

  28. According to theory the gas could be superfluid. But is it? Problem: How do we detect superfluidity? No change in density profile (at least in w.c. limit) Suggestion 1: Look at expansion. • Menotti, Pedri and Stringari, PRL 89, 250402 (2002)

  29. Theory Ei / Eho=0 Ei / Eho=-0.4 Experiment Ei / Eho>0 Ei / Eho=0 Ei / Eho<0

  30. Problem: If the gas is in the hydrodynamic regime then expansion of normal gas = expansion of superfluid. Suggestion 1 cannot distinguish. Suggestion 2: Rotate the gas to see quantization of angular momentum.

  31. Normal hydrodynamic gas can sustain rigid body • rotation • Superfluid gas can rotate only by forming vortices • (because of irrotationality)

  32. Superfluid vortex structure. Simple model Vortex velocity field Kinetic energy (per unit volume) Condensation energy (per unit volume)

  33. By imposing one finds: where

  34. Vortex energyin a cylindical bucket of radius Rc Rc Vortex core normal matter

  35. Vortex energyin a cylindical bucket of radius Rc

  36. Vortex energyin a cylindical bucket of radius Rc Factor 1.36 model dependent. Let then • Bruun and LV, PRA 64, 063606 (2001)

  37. From microscopic calculation ... • Nygaard, Bruun, Clark and Feder, e-print cond-mat/0210526

  38. Above formula for v with D=2.5 D=2.5 kFa=-0.43 kFa=-0.59

  39. The critical frequency for formation of first vortex is thus since - ~ h per particle.

  40. In unitarity limit one expects: and thus Very recent microscopic result ...

  41. D Density • Bulgac and Yu, e-print cond-mat/0303235

  42. In traps Vortex forms if In dilute limit this means which is fulfilled if In unitarity limit it reads

  43. Rough estimate for c1in unitarity limit in trap(C=1) In the case of Duke experiment one finds

  44. No angular momentum transfer to the gas for stirring frequencies below c1 if the gas is superfluid! Example with bosons: • Chevy, Madison, Dalibard, PRL 85, 2223 (2000)

  45. How can one do the experiment? e.g. Lift of degeneracy of quadrupolar mode Normal hydrodynamic for arbitrarily small stirring frequency . Superfluid - only if <lz>  h/2, ( > c1) and zero otherwise.

  46. Splitting for a single vortex For fermions then

  47. Again in the case of Duke experiment one finds

  48. Conclusions • I showed various reasons why superfluidity in atomic gases is very interesting and important • I illustrated recent experimental developments • I showed how superfluidity can be detected by means of the rotational properties of the gas (vortices) • I pointed out several open questions which have to be addressed in the next future

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