Ultracold Fermi gases : the BEC-BCS crossover

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 •Feshbachresonance •The BEC-BCS transition •Shift of the molecular threshold •Collective oscillations in harmonic trap

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

4. •BCS Superfluidity 6Li • Pairingbetweenatoms • with attractive interaction BCS superfluid • Paulino s-wave interaction for identical atomsp-wave(cf3He)(Kagan et al.) but very low Tc ? - s-wavepairingbetweendifferentatoms different spins  two different(lowest)hyperfine states( I = 1 for 6Li) ="spins" - pb.equal population • s-wave scattering  single parameter scattering lengtha - Attraction a < 0 - 6Li spin triplet a = - 1140 Å (high field) ( almost bound state ) strong effective interaction

5. •Feshbachresonance • Allows to control effectiveinteraction via magnetic field • by changing scattering length a V(r) = (4ph2a/m) d(r) for energy ~ 0 - Scattering length a= ∞ if bound state withenergy = 0 exists V a < 0 no molecules r a = ∞ a > 0 molecules

6. 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 a > 0 molecules 6Li a < 0 no molecules • Interaction tunable at will by magnetic field ! Allows a physical realization of the BEC-BCS transition

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

8. -Interest for high Tc superconductivity - Sà de Melo, Randeria and Engelbrecht (93) - Use scattering length aknown experimentally instead of BCS potential V (Leggett also) (goes back to Belaiev and Galitskii, see Fetter and Walecka) m / EF 1/kFa • Gives D and m function • of kF and a

9. Small a > 0 Molecules with TBEC / EF = 0.218 •Excellentmolecular stability - Lifetime ~ 1 s. aroundunitarity - Dueto decrease of 3-body recombination process by Pauli repulsion

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

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

12. •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 Fermisea  hinders molecular formation •Effect depends on total momentum K - Strongest forK=0 , disappear for large K - Everything obtained from scattering amplitude •No singularity when a = ∞ is crossed •OK with experiments : nothing special at «  Feshbach resonance  »

13. •PairswithFermi sea(simple BCS) - BCSinstability for a < 0 ( attraction ) - Extends to a > 0 ( easier for molecules ! ) ( instability for E = m > 0 ) normal • Terminal point : superfluid -Molecularinstabilityshifted toward a > 0 ( instability for Eb = 0 , m < 0) - Terminal point identical to BCS •Classical limit T  

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

15. •Collective oscillations in harmonic trap ( superfluid state ) Cigar geometry wz << wx,y - In situ experiments (no need for interpretation) - High experimental precision possible -Direct access to equation of 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

16. •Equations of state -Monte-Carlo : should be reasonably accurate -BCS equation of state 0.59 0.44 -Hydrodynamics aM = 0.6 a(PSS) aM = 2. a - T0Superfluid satisfies hydrodynamics, but w << Eb (pair binding energy, i.e. w << Don BCS side) - Radial geometry - Strong attenuation at 910 G  pair-breaking peak Wr = 2 D (T,B)  superfluid!

17. •Axial geometry (with Astrakharchik, Leyronas and Stringari ) - BEC limit w2 = 2.5W2 - unitarity limit w2 = 2.4W2 • Experiment in better agreement with BCS ! Experiment: Bartenstein et al. • No sign of LHY : Lee-Huang-Yang (57) Pitaevskii-Stringari (98)

18. •Conclusion Extremely promising field !