Ultra-cold Fermi gasesUltra-cold Fermi gases

F. ChevyF. Chevy

Laboratoire Kastler Brossel, ParisLaboratoire Kastler Brossel, Paris

Evaporative Cooling of ultra-cold atomsEvaporative Cooling of ultra-cold atoms

kTU0

position

energ

y

To maintain an efficient cooling, U0 decreases with T;

Collisions required

The Pauli Exclusion Principle and the

evaporative cooling of ultra-cold Fermi gases

The Pauli Exclusion Principle and the

evaporative cooling of ultra-cold Fermi gases

Collision between two atoms. Effective potential in the l-wave:

At low temperature, the atomes

cannot cross the centrifugalbarrier: only s-wave collisions.

Symmetrization for identicalparticles: even l-wave collisions

forbidden for polarized fermions.

2

eff 2

( 1)( ) ( )

2V r V r

mr

+= +

� � �

Interatomic potential

(long range~-1/r6)

centrifugal potential

l=0 l>0

Suppression of elastic collisions in

a spin polarized Fermi gas

Suppression of elastic collisions in

a spin polarized Fermi gas

Spin mixture (s-wave)

Spi

n po

lariz

ed(p

-wav

e)

Use spin mixtures or several atomic species (eg 6Li-7Li, K-Rb…)

B. DeMarco, J. L. Bohn, J.P. Burke, Jr., M. Holland, and D.S. Jin, Phys.

Rev. Lett. 82, 4208 (1999).

The non-interacting Fermi gas

Time of flight absorption imaging

The non-interacting Fermi gas

Time of flight absorption imaging

T/TF<0.05

Atom number~105

Fermi-Dirac

Gaussian Fit

Superconductivity

and superfluidity

Superconductivity

and superfluidity

Bose Einstein condensates Superconductivity and helium 3

Quantum fluidsQuantum fluids

Attractive fermions atzero temperature

Attractive fermions atzero temperature

Deep potential : 2-body bound state.Many body ground state : BEC of molecules

0V

0V

Shallow potential (V0<V0*):

no 2-body bound state. No superfluid?

Many body effects:BCS pairing atarbitrarily low V0

(3He, superconductors)

+k

-k

The BEC-BCS crossoverThe BEC-BCS crossover

Size of the molecules ~ aBinding energy 2 2/bE ma= −�

*

0 0 0V V a> ⇔ > *

0 0 0V V a< ⇔ <3| | ~ 1n a

??????

*

0 0~V V

At low energy, real potential replaced by

Binding energy1/ | |

~ Fk a

FE e−

BEC of molecules BCS state

2

pseudo

4( )

aV

m

πδ= r

�

Scattering length a~(V0-V0*)-1

The BEC-BCS « patchwork »The BEC-BCS « patchwork »

Alkali BEC Superfluid He-4

High Tc superconductors

Superfluid He-3

Normal superconductors

Ultra-cold fermions

Condensation tem

pera

ture

Interaction strength

(Fano) - Feshbach resonancesin cold atoms

(Fano) - Feshbach resonancesin cold atoms

0.0 0.5 1.0 1.5 2.0

-200

-100

0

100

200

scattering length

[nm

]

Magnetic field [kG]

6 Li

BF~834 G

6LiBosons near a Feshbach

resonance:Bose-Nova (C. Weiman).

Losses in 6Li (fermions)

Predicted position of

the resonance

Looking for the Feshbach resonanceof Lithium

Looking for the Feshbach resonanceof Lithium

Inhibition of inelastic losses

In fermionic gases

Inhibition of inelastic losses

In fermionic gases

1000,1

1

G [1

0^-

13

cm

^3/s

]

Scattering length [nm]

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

G 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.

JILA: 40K2

6Li2:Innsbruck

ENS6Li2

MIT6Li2

Also Rice, Duke, Melbourne 6Li2

Molecular BEC’saround the world

Molecular BEC’saround the world

Momentum distributionMomentum distribution

Broadening of the Fermi-Dirac distribution in the presence of

interactions (Viverit et al.PRA 69, 013607 (2004))

k/kF1

n(k)

1

a<0, T=0 ideal gas,width kF

a=¶, « unitary » gas

a>0, inner momentumdistribution, width~1/a

BCS (mean-field)

Wave function

Experiment vs theoryExperiment vs theory

-1 0 1-0,2

0,0

0,2

0,4

0,6

0,8

1,0

1,2In

teg

rate

dd

en

siit

y[a

.u]

k/kF

BCS theory

Thomas-Fermi for ideal gas

Ideal gas

Strongly correllated Fermi gas

0.46(10)JILA

0.46(5)Rice

0.41(15)ENSExperiment

0.42(1)Carlson

0.455Perali

0.42(1)Astrakharchick

0.59BCSTheory

Measurement of the zero temperature

universal equation of state of

the strongly interacting Fermi gas

Measurement of the zero temperature

universal equation of state of

the strongly interacting Fermi gas

( )2

2/326

2F

n Em

µ ξ π ξ↑

= =�

Determination of x

For a=¶, f(1/na3)=f(0), independent on density

( ) )/1(62

33/222

nafnm

πµ�

=↑Dimensional analysis:

Equation of state at finite temperatureEquation of state at finite temperature

Goal: measure the

thermodynamical equation ofstate S(U,N)

•Prepare an ideal gas andmeasure T/TF~S

• Ramp slowly the magnetic

field to a=¶•Measure the potential energy

E=mw2<x2>/2=U/6 (Virialtheorem)

J. T. Stewart, J. P. Gaebler, C. A. Regal, and D. S. Jin, Phys. Rev. Lett. 97, 220406

(2006).

L. Luo, B. Clancy, J. Joseph, J. Kinast, and J. E. Thomas, Phys. Rev. Lett. 98, 080402

(2007)

Superfluidity: vortex lattices (MIT)Superfluidity: vortex lattices (MIT)

Vortices at MIT

Ato

mic

BE

CM

ole

cula

r

BE

C

BC

S

su

pe

rflu

id

Fermions in

optical lattices

Fermions in

optical lattices

Ultra cold atoms in Optical LatticesUltra cold atoms in Optical Lattices

Optical lattices: periodic potential created by the interference of several laser

beams

Imaging the first Brillouin ZoneImaging the first Brillouin Zone

Momentum distribution after time of flight for various

filling factors.

M. Köhl, H. Moritz, T. Stöferle, K. Günter, T. Esslinger

Phys. Rev. Lett. 94, 080403 (2005).

Repulsive fermions in an optical latticeRepulsive fermions in an optical lattice

For bosons see M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch and I. Bloch

Nature 415, 39-44 (2002)

Fermi-Hubbard hamiltonian

Small U/J: conductor Large U/J: insulator

Mott

transitio

n

† † †

,

i j i i i ii j i

H J c c U c c c cσ σ

σ

↑ ↑ ↓ ↓= +∑ ∑

Mott transition in superfluid Fermi gasesMott transition in superfluid Fermi gases

R. Jördens, N. Strohmaier, K. Günter, H. Moritz and Tilman Esslinger, Nature 455,

204-207 (11 September 2008)

U. Schneider, L. Hackermuller, S. Will, Th. Best, I. Bloch, T. A. Costi, R. W. Helmes,

D. Rasch, A. Rosch, Science 322,1520 (2008)

Double occupancy spectroscopic

measurement

2 atoms persite

Isolated atom

Interaction

energy

hn hn’

High resolution imaging in optical latticesHigh resolution imaging in optical lattices

A. Itah, H. Veksler, O. Lahav, A. Blumkin, C. Moreno, C.

Gordon, J. Steinhauer, arXiv:0903.3282

Polarized Fermi gasesPolarized Fermi gases

Fermionic superfluid with population imbalanceFermionic superfluid with population imbalance

Polarized phase : One spin species (Carlson, PRL 95, 060401 (2005))

Chandrasekar and Clogston: robustness of the paired state :

Paired state stable for

And beyond?

µ µ↑ ↓>

µ µ↑ ↓

− < ∆

FFLO Phase (Fulde Ferrell Larkin Ovshinikov) : pairing in (C. Mora et R. Combescot, PRB 71, 214504 (2005))

0↓↑ − ≠k k

Sarma phase (internal gap) : pairing in opening of a gap in the Fermi sea of majorityspecies. (Liu, PRL 90, 047002 (2003))

0↓↑ − =k k

Experimental results at Feshbach resonanceExperimental results at Feshbach resonance

MIT: 3 phases- Fully paired superfluid core

-Intermediate mixture

-Fully polarized rim

Rice: 2 phasesFully paired superfluid core

Fully polarized rim

M.W. Zwierlein, et al., Science, 311

(2006) 492.

G.B. Partridge, W. Li , R.I. Kamar, Y.-A.

Liao, R.G. Hulet, Science, 311 (2006)

503.

Rice : 2 phase model(F. Chevy, PRL 96, 130401 (2006))

Rice : 2 phase model(F. Chevy, PRL 96, 130401 (2006))

Rice experiment: fully described by a 2 phase model, without any adjustable parameter.

External radius

Superfluid inner radius

(N�-N�)/(N�+N�)

Surface tension

(T. N. De Silva and E. Mueller, PRL., 97 070402 (2006))

Surface tension

(T. N. De Silva and E. Mueller, PRL., 97 070402 (2006))

Local density approximation:

int erfacesF Sγ=

2 4/3

2

n

mγ ε=

�

Fit of data:

With surface tension:

310ε −≈

(1st order transition)

What about other phases.What about other phases.

Fully polarized

ideal gas

?

αηβη

Fully paired

Superfluid

/η µ µ↓ ↑=

1-1

0/P P

3/5(2 ) 1

0.099

cη ξ= −

≈ −

Grand potential W=-PVflground state has the highest pressure.

Theoretical evidence for an intermediate

phase

Theoretical evidence for an intermediate

phase

Step 1: calculate the energy E of a single impurity atomimmersed in a Fermi sea (E=m�(n�=0+)).

Step 2: dP/dmσ=ns>0

General properties of a mixed branch?

/η µ µ↓ ↑=cη

P/P0cβη η<

cβη η<

: the new branch isstable

: the new branch isunstable

cβη η>

cβη η>

See A. Bulgac and M.

McNeil Forbes,

PRA 75, 031605 (2007)

A spin down particle immersed in a Fermi sea of spin

up atoms:

The Fermi-polaron problem

A spin down particle immersed in a Fermi sea of spin

up atoms:

The Fermi-polaron problem

(Dressed atom, Higgs mechanism, Landau’s Fermi liquid theory…)

Variational upper bound for hb:the Fermi swimmer

Variational upper bound for hb:the Fermi swimmer

One impurity: restrict the effect of interactions to the formation ofa single particle-hole pair.

0 ,

,

0 ,ϕ ϕΨ = +∑ k q

k q

k q

=kq

k

q

q-k0 =

For a=¶, E=-0.606 EF�flηβ<-0.606<ηc~-0.1

Comparison with exact results :hb=-0.58(1) (C. Lobo et al. PRL. 97, 200403 (2006)); hb=-0.62 (Prokof’ev and Svistunov, Phys. Rev. B 77, 020408 (2008))

Systematic expansion R. Combescot and S. Giraud, Phys. Rev. Lett. 101, 050404 (2008)

Structure of the intermediate phase (Combescot et al.PRL 98, 180402 (2007))

Structure of the intermediate phase (Combescot et al.PRL 98, 180402 (2007))

Ideal gas of fermionic impurities dressed by particle-hole pairs

2 *( ) / 2F

E p E p mβη= +

0.6βη = − * ~ 1.15m mVariational calculation:

MC

Ideal gas approx.

Comparison with experiment:

Shell radii and critical polarisation

Comparison with experiment:

Shell radii and critical polarisation

If polarisation (N�-N�)/(N�+N�) is to large : superfluid core vanishes

Rb

R1

Ra

-0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,80,2

0,4

0,6

0,8

1,0

Cri

tical p

ola

risa

tion

1/kFa

No adjustableparameter

See also Martin Zwierlein’s talk

Comparison with experiments:

Radio-frequency experiments(A. Schirotzek et al. Phys. Rev. Lett. 102, 230402 (2009))

Comparison with experiments:

Radio-frequency experiments(A. Schirotzek et al. Phys. Rev. Lett. 102, 230402 (2009))

Short term issues

Conclusion et future directionsConclusion et future directions

Ultra-cold atomic gases are wonderful tools to simulate

condensate matter systems:BEC-BCS crossover

Mott transitionClogston-Chandrasekhar limit

Dynamical studies, measurement of the effective mass of theFermi polaron (Nascimbène et al. arXiv:0907.3032)

l>0 superfluids (p-wave?)

Magnetic properties in optical lattices

What was demonstrated?

The ENS ultra-cold Fermi groupThe ENS ultra-cold Fermi group

The lithium group The lithium-potassium group

Theory support: R. Combescot, C. Mora (ENS), S. Stringari, C. Lobo, A. Recati

(Trento)