Ultracold Fermi gases :

the

BEC-BCS crossover

Roland Combescot

Laboratoire de Physique Statistique, Ecole Normale Supérieure,

Paris, France

• Ultracold Fermi gases

• BCS Superfluidity

• Feshbach resonance

• The BEC-BCS transition

• Shift of the molecular threshold

• Collective oscillations in harmonic trap

- In practice alkali with even number of nucleons: 6Li or 40K

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)

7Li

6Li

- 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 6Li) = "spins" - pb. equal population

- Pauli no s-wave interaction for

• s-wave scattering single parameter scattering length a- Attraction a < 0- 6Li spin triplet a = - 1140 Å (high field) ( almost bound state )

strong effective interaction

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identical atoms p-wave (cf 3He) (Kagan et al.but very low Tc ?

6Li

- 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) = (4h2a/m) (r)

V

ra < 0 no molecules

a > 0 moleculesa = ∞

0,0 0,5 1,0 1,5 2,0

-200

-100

0

100

200

scattering length [nm]

Magnetic field [kG]

6Li

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

f ( ω ) = - 1

a- 1

+ i k

ε

b =

1

m a2

• Interaction tunable at will by magnetic field !Allows a physical realization of the BEC-BCS transition

• 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 Tc 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 !

- 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 Tc superconductivity

/ EF

1/kFa-Gives and function of kF and a

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Small a > 0Molecules withTBEC / EF = 0.218

• Excellent molecular stability- Lifetime ~ 1 s. around unitarity- Due to decrease of 3-body recombination process by Pauli repulsion

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• Also MIT and Innsbrück

• First Bose-Einstein condensates of molecules made of fermions !

• Bose-Einstein condensates of molecules (2003-2004)

• Vortices as evidence for superfluidity (2005)

-Before : Anisotropic expansion (?) ~ BEC Collective mode damping (see below)

0,0 0,5 1,0 1,5 2,0

-200

-100

0

100

200

scattering length [nm]

Magnetic field [kG]

834 G

• 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

1

a

= 4 n ΛT

2

= kF

8

3

EF

T

n =

kF

3

6 2

EF

=

kF

2

2 m

• Classical limit T

- Molecular instability shifted toward a > 0( instability for Eb = , )

T0

EF

= 0 , 9 9

1

kF

a0

= 0 , 6 8

- 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

- Continuity in the superfluid at = 0 (compare with Leggett)- Would probably be different with "3HeA" (see Volovik)

• Self-consistent treatment (with X. Leyronas and M.Yu.Kagan)

- Seen in vortex experiment ?- Should be seen in spectroscopic experiments

• 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 ωz << ωx,y

• Equations of state

aM = 0.6 a(PSS)

aM = 2. a

0.44

0.59- BCS equation of state

- Monte-Carlo : should be reasonably accurate

- Hydrodynamics

-Superfluid satisfies hydrodynamics, but ω << Eb (pair binding energy, i.e. ω << on BCS side)

- Radial geometry

- Strong attenuation at 910 G pair-breaking peak r = 2 (T,B)

superfluid !

• 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 ω2 = 2.5 2

- unitarity limit ω2 = 2.4 2

• Conclusion

Extremely promising field !