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? Superfluidity in dilute gases 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) Test ground for various theories: a < 0 ; k F |a| <<1 a < 0 ; k F | a| <<1

4. Why superfluidity in atomic Fermi gases? Detailed study of effective interactions in medium and 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 ; k F |a| ~1 a < 0 ; k F | a| ~ 1

5. Why superfluidity in atomic Fermi gases? Boson enhanced pairing in Bose-Fermi mixtures 3) Boson enhanced pairing in Bose-Fermi mixtures Bardeen, Baym and Pines, PRL 17, 372 (1966) ( 3 He- 4 He) 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? Boson enhanced pairing in Bose-Fermi mixtures 3) Boson enhanced pairing in Bose-Fermi mixtures Bardeen, Baym and Pines, PRL 17, 372 (1966) ( 3 He- 4 He) 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? BCS-BEC crossover 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/ k F a  -∞1/ k F a  +∞ 1/ k F a  -∞ 1/ k F a  +∞

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

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

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

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

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

15. a 0 a a a a0 0 =0 ~ c (k F a) 2 ; c>0

16. Since k F |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 k F |a|<<1 then Best experimental performances with present techniques Not enough if the gas is dilute! Gehm, Hemmer, Granade, O’Hara and Thomas, e-print cond-mat/0212499 Regal and Jin, e-print cond-mat/0302246

19. Idea A: Idea A: let k F |a| approach 1 (but still k F |a|<1) Combescot, PRL 83, 3766 (1999) ( =2k F a /  )

20. WHY?? Exchange of density and spin collective modes (higher orders in than previously modes (higher orders in k F a than previously considered) and considered) and Fragmentation of single particle levels both strongly influence T c !

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 Gap equation at T c,0 Gap equation at T c,0 : Number equation at T c,0 Number equation at T c,0 : Back to BCS equations.

23. Solution in1/ k F a  -∞ Solution in 1/ k F a  -∞

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 (k F a  ) predicted at BUT Exchange of density and spin modes, and Fragmentation of single particle levels not included in the theory.Then: 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 T F in unitarity conditions (k F a  ±∞).

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

29. E i / E ho =0 E i / E ho =-0.4 Theory Experiment E i / E ho >0 E i / E ho =0 E i / E ho <0

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

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

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

34. By imposing one finds: where

35. Vortex energy in a cylindical bucket of radius R c Vortex core normal matter RcRc

36. Vortex energy in a cylindical bucket of radius R c

37. Factor 1.36 model dependent. Let then Bruun and LV, PRA 64, 063606 (2001)

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

39. k F a=-0.43 k F a=-0.59 D=2.5 Above formula for  v with D=2.5

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

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

42.  Density Bulgac and Yu, e-print cond-mat/0303235

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

44. Rough estimate for  c1 in unitarity limit in trap(C=1) Rough estimate for  c1 in unitarity limit in trap (C=1) In the case of Duke experiment one finds

45. 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)

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

47. Splitting for a single vortex For fermions then

48. Again in the case of Duke experiment one finds

49. 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