1. Ultracold Fermi gases : the BEC-BCS crossover Roland Combescot Laboratoire de Physique Statistique, Ecole Normale Supérieure, Paris, France

2. Ultracold Fermi gases BCS Superfluidity Feshbach resonance The BEC-BCS transition Shift of the molecular threshold Collective oscillations in harmonic trap

3. - In practice alkali with even number of nucleons: 6 Li or 40 K C. Salomon group (ENS) - Trapping  parabolic potential  harmonic oscillator Ultracold Fermi gases First manifestation of statistics: - Gas cloud stays larger for fermions at low T due to Pauli repulsion (cf. white dwarfs) - No thermalisation possible  no evaporative cooling with fermions  "sympathetic" cooling with another atomic species (fermion or more often boson) - Ultracold :very low T  very low energy  s-wave scattering only - But prohibited for fermions by Pauli exclusion  no interaction ! (very good for atomic cloks) 7 Li 6 Li

4. - Pairing between atoms with attractive interaction  BCS superfluid BCS Superfluidity - s-wave  pairing between different atoms  different spins  two different (lowest) hyperfine states ( I = 1 for 6 Li) = "spins" - pb. equal population - Pauli  no s-wave interaction for s-wave scattering  single parameter scattering length a - Attraction  a < 0 - 6 Li spin triplet a = - 1140 Å (high field) ( almost bound state )  strong effective interaction identical atoms  p-wave (cf 3 He) (Kagan et al.  but very low T c ? 6 Li

5. - Allows to control effective interaction via magnetic field by changing scattering length a for energy ~ 0 - Scattering length a = ∞ if bound state with energy = 0 exists Feshbach resonance V(r) = (4  h 2 a/m)  (r) V r a < 0 no molecules a > 0 molecules a = ∞

6. 6 Li a < 0 no molecules a > 0 molecules -Actually atoms very near each other are not in the same spin configuration as when they are very far from each other ("closed channel" and "open channel")  sensitivity of bound state energy to magnetic field Feshbach resonance for 2 particles in vacuum - scattering amplitude Interaction tunable at will by magnetic field ! Allows a physical realization of the BEC-BCS transition

7. BEC-BCS transition - Leggett (80): Cooper pairs = giant molecules (  "molecular" physics found in Cooper pairs) Interest in looking at the continuous evolution (in particular  = 0) Keldysh and A. N. Kozlov(68) Eagles(69) - BCS Ansatz for ground state wavefunction: - Nozières - Schmitt-Rink (85): recover Bose-Einstein T c for molecules Accurate in weak-coupling(BCS) limit and strong-coupling(BEC) limit In between : physically quite reasonable "interpolation scheme" describes as well dilute gas of molecules, made of 2 fermions  only experiment can tell what happens in between !

8. - Sà de Melo, Randeria and Engelbrecht (93) - Use scattering length a known experimentally instead of BCS potential V (Leggett also) (goes back to Belaiev and Galitskii, see Fetter and Walecka) - Interest for high T c superconductivity  / E F 1/k F a -Gives  and  function of k F and a

9. Small a > 0 Molecules with T BEC / E F = 0.218 Excellent molecular stability - Lifetime ~ 1 s. around unitarity - Due to decrease of 3-body recombination process by Pauli repulsion

10. Also MIT and Innsbrück First Bose-Einstein condensates of molecules made of fermions ! Bose-Einstein condensates of molecules (2003-2004)

11. Vortices as evidence for superfluidity (2005) -Before : Anisotropic expansion (?) ~ BEC Collective mode damping (see below) 834 G

12. No singularity when a = ∞ is crossed OK with experiments : nothing special at « Feshbach resonance » - Everything obtained from scattering amplitude Shift of the molecular threshold ( normal state ) Fermi sea modifies molecular formation : - Favors Cooper pairs formation - Hinders molecular formation - Physical origin : Pauli exclusion ( dominant effect of Fermi sea )  " forbidden " states by Fermi sea  hinders molecular formation Effect depends on total momentum K - Strongest for K = 0, disappear for large K

13. Classical limit T   - Molecular instability shifted toward a > 0 ( instability for E b = ,  ) - BCS instability for a < 0 ( attraction ) normal superfluid Terminal point :  Pairs with Fermi sea (simple BCS) - Extends to a > 0 ( easier for molecules ! ) ( instability for E =  ) - Terminal point identical to BCS

14. - Continuity in the superfluid at  = 0 (compare with Leggett) - Would probably be different with " 3 HeA" (see Volovik) Self-consistent treatment (with X. Leyronas and M.Yu.Kagan) - Seen in vortex experiment ? - Should be seen in spectroscopic experiments

15. Collective oscillations in harmonic trap ( superfluid state ) Basic motivation: remarkable model system for normal and superfluid strongly interacting (and strongly correlated) Fermi systems - Hope for better understanding of High Tc superconductors - Equation of state = first (small) step - In situ experiments (no need for interpretation) - High experimental precision possible - Direct access to equation of state Cigar geometry  z <<  x,y

16. Equations of state a M = 0.6 a (PSS) a M = 2. a 0.44 0.59 - BCS equation of state - Monte-Carlo : should be reasonably accurate - Hydrodynamics -  Superfluid satisfies hydrodynamics, but  << E b (pair binding energy, i.e.  <<  on BCS side) - Radial geometry - Strong attenuation at 910 G  pair-breaking peak  r = 2  (T,B)  superfluid !

17. Axial geometry (with Astrakharchik, Leyronas and Stringari ) Experiment in better agreement with BCS ! No sign of LHY : Lee-Huang-Yang (57) Pitaevskii-Stringari (98) Experiment: Bartenstein et al. - BEC limit   = 2.5  2 - unitarity limit   = 2.4  2

18. Conclusion Extremely promising field !