THE ATTRACTIVE FERMI GAS

THE WEAKLY INTERACTING BEC

1. The BEC-BCS crossover

Bose Einstein condensates Superconductivity and helium 3

Quantum Fluids

DEEP POTENTIAL : stable 2-body bound state with binding energy Eb=-2/ma2. Many-body ground state: BEC of molecules (a>0)

SHALLOW POTENTIAL (V0<V0*, a<0):

No 2-body bound state, but Cooper pairing stabilized by Pauli blocking.

V0

V0

BOSE ENSTEIN CONDENSATION AND BCS THEORY APPEAR AS TWO LIMITING CASES OF A UNIFYING THEORY: THE BEC-BCS CROSSOVER

Fermionic superfluidity

Alkali BEC

Superfluid helium

High Tc superconductors

Superfluid 3He

Metal superconductors

Ultra-cold fermions

Cri

tica

l tem

per

atu

re

Interactions

EXPERIMENTAL REALIZATION IN COLD ATOMS (2003) : Innsbruck, JILA, MIT, ENS, Rice, Duke, Melbourne, Tokyo…

The experimental BEC-BCS Patchwork

Losses in 6Li (fermions)

Predicted position of the resonance

Looking for Feshbach Resonances in Lithium

1000,1

1

G [10^-

13 c

m^3

/s]

Scattering length [nm]

Scaling law G ~ a -2.0 +/-0.8 (theory Petrov et al. ) 2.55G a

2 body (dimers) losses mainly : decay towards deeply bound states)

N G n N

3/4 atom loss requires 2 atoms of same spin close to each other.

Inhibition of losses in Fermi gases

2003-2013: ten years of strongly correlated Fermi gases

Boulder

Innsbruck

MIT

ENS

Is the Fermi superfluid a superfluid?

Vortices at MIT

Ato

mic

BEC

M

ole

cula

r

BEC

B

CS

sup

erfl

uid

Equation of State in the BEC-BCS crossover

Goal: calculate the equation of state of the gas as a function of 1/kFa

Dimensionally: 3/ )

2(1

5

Fb F

NENEE k a

22 2

2/3

2

2

(3 )2

( )

F

b

FF

kE k n

m

E ama

Asymptotic behavior:

1/kFa0-(BCS Limit): weakly attractive Fermi gas, 1 1/kFa0+(BEC Limit): repulsive Bose-Einstein condensate of dimers, 0

Asymptotic expansion of the Equation of State

Lee-Yang expansion:

Lee-Huang-Yang expansion:

2

2

3 10 4(11 2ln 2)1 ( )

5 9 21...F

F FENE

k a k a

Lee & Yang Phys. Rev. 105, 1119 (1957) (repulsive fermions) Diener et al. Phys. Rev. A 77, 023626 (2008). (attractive fermions)

23mmmm

1281 ...

2 2 15b

aNEE N n na

m

Lee, Huang, Yang, Phys. Rev. 106, 1135 (1957) X. Leyronas & R. Combescot PRL 99, 170402 (2007) (Composite bosons)

Lee-Huang-Yang corrections and Few body physics

mm 0.6a a Petrov et al. PRL 93, 090404 (2004) (Schrödinger equation) Brodsky et al. PRA 73, 032724 (2006) (Diagrammatic)

2

3 3 3mmmm mm mm

128 81 4 3 3 ln( ) ...

2 2 315b

aNE N n na n naE B

m

How far can one describe the dimers as point-like bosons?

Non-universal term: depends on the internal structure of the boson.

Higher order expansion (Wu 1957, Braaten et al. PRL 2002)

The Unitary Fermi gas

3(0)

5

FNEE ~ Ideal Fermi gas. (0) = « Bertsch parameter »

•BCS mean-field theory: (0)<0.6 •DFT (McNeil Forbes et al., PRL 2011): (0)<0.38(1) •Current experimental estimate (ENS-MIT): 0.37(1)

At the surface of a neutron star:

010; / 10Fk a a r

Density profile of the unitary Fermi gas.

Local density approximation: 2 2

1/3(0) (3 )2

m RN