Ultracold Fermi gases

University of Trento

BEC Meeting, Trento, 2-3 May 2006

INFM-CNR

Sandro Stringari

Atomic Fermi gases in traps

Ideal realization of non-interacting configuarations

with spin-polarized samples

- Bloch oscillations and sensors (Carusotto et al.),

- Quantum register (Viverit et al)

- Insulating-conducting crossover (Pezze’ et al.)

Role of interactions (superfluidity)

- HD expansion (aspect ratio and pair correlation function)

- collective oscillations and equation of state

- spin polarizability

This talk

EXPANSION OF FERMI SUPERFLUID

Hydrodynamics predicts anisotropicexpansion of BEC gas

Evidence for hydrodynamic anisotropic expansion in ultra cold Fermi gas (O’Hara et al, 2003)

HD theoryHydrodynamics predicts anisotropic expansion in Fermi superfluids (Menotti et al,2002)

normal collisionless

Pair correlations of an expanding superfluid Fermi gas C. Lobo, I. Carusotto, S. Giorgini, A. Recati, S. Stringari, cond-mat/0604282

Recent experiments on Hanbury-Brown Twiss effect with thermal bosons (Aspect, Esslinger, 2005) provide information on

- Pair correlation function measured after expansion - Time dependence calculated in free expansion approximation (no collisions)- Decays from 2 to uncorrelated value 1 (enhancement at short distances due Bose statistics).- For large times decay lengths approach anisotropic law: iiT mRtt /

])1(

2exp[1)(222

22

t

ssg

iT

i

i

Can we describe behaviour of pair correlation function during theexpansion in strongly interacting Fermi gases (eg. at unitarity) ?

- In situ correlation function calculated with MC approach (see Giorgini)

- Time dependence described working in HD approximation (local equilibrium assumption)

QUESTION

unitarity

BEC limit

thermal bosons

4/1akF

Pair spin up-down correlation function

Pair correlation function in interacting Fermi gas:

- Spin up-down correlation function strongly affected by interactions at short distances. Effect is much larger than for thermal bosons (Hanbury-Brown Twiss)

- In BEC regime ( ) pair correlation function approaches uncorrelated value 1 at distances of the order of scattering length (size of molecule)

- At unitarity pair correlation function approaches value 1 at distances of the order of interparticle distance (no other length scales available at unitarity)

1akF

Local equilibrium ansatz for expansion

- Dependence on s fixed by equilibrium result (calculated with local value of density)- Time dependence of density determined by HD equations.

Important consequences (cfr results for free expansion of thermal bosons)

- Pair correlation keeps isotropy during expansion- Measurement after expansion ‘measures’ equilibrium correlation function at local density- at unitarity, where correlation function depends on combination , expansion acts like a microscopeskF

COLLECTIVE OSCILLATIONS AND EQUATION OF STATE

- Surface modes: unaffected by equation of state - Compression modes sensitive to equation of state.-Theory of superfluids predicts universal values when 1/a=0 :

- In BEC regime one insetad finds

COLLECTIVE OSCILLATIONS IN SUPERFLUID PHASE (T=0)

3/10rad zax 5/12

2rad zax 2/5

Behaviour of equation of state through the crossover canbe inferred through the study of collective frequencies !

Radial compression mode

3

10

S. Stringari, Europhys. Lett. 65, 749 (2004)

Experiments on collective oscillations at

- Duke (Thomas et al..)

- Innsbruck (Grimm et al.)

unitarity

(mean fieldBCS gap eq.)

Duke data agree with value 1.826 predicted at unitarity

Radial breathing mode at Innsbruck (2006) (unpublished)

MC equation of state

BCS mean field83.13/10

Theory from Astrakharchik et al Phys. Rev. Lett. 95, 030405 (2005)

Crucial role of temperature:

- Beyond mean field (LHY) effects are easily washed out by thermal fluctuations finite T (Giorgini 2000) Conditions of Duke experiement

- Only lowering the temperature (new Innsbruck exp) one can see LHY effect

SPIN POLARIZABILITY

Spin Polarizability of a trapped superfluid Fermi gasA. Recati, I. Carusotto, C. Lobo and S.S., in preparation

Recent experiments and theoretical studies have focused on the consequence of spin polarization ( ) on the superfluid features of interacting Fermi gases

)/()( NNNNP

MIT, 2005

In situ density profiles for imbalanced configurations at unitarity(Rice, 2005)

Spin-up

Spin-down

difference

An effective magnetic field can be produced by separating rigidly the trapping potentials confining the two spin species.

])([2

1)( 2222 dxzymrV

For non interacting gas, equilibrium corresponds to rigid displacement of two spin clouds in opposite direction:

)ˆ(2

1)( 0 xdrnrn

This yields spin dipole moment (we assume )

drnrnxrdN

dD ))()((1

)(

2/NNN

We propose a complementary approach where we study the consequence of an effective magnetic field which can be tuned by properly modifying the trapping potentials.

Main motivation: Fermi superfluids cannot be polarized by external magnetic field unless it overcomes a critical value (needed to break pairs).

What happens in a trapped configuration? What happens at unitarity ?

In the superfluid phase atoms like to be paired. and feel the x-symmetric potential

][2

1))()((

2

1)( 22222 dzyxmrVrVrVS

Competition between pairing effects and external potential favouring spin polarization

])([2

1)( 2222 dxzymrV

VV

SV

SFAt unitarity

Equilibrium between superfluid and spin polarized phases (Chevy 2005)

2/)2)(()(:0

2/)2)(()(:05/3

5/3

rrx

rrx

S

S

44.0

FS

)()(

)()(

0

0

rVr

rVr SS

Spin dipole moment D(d)/d as a function of separation distance d (in units of radius of the cloud)

44.0

58.0

ideal gas

Deep BEC

Further projects:

- Collective oscillations of spin polarized superfluid- Rotational effects in spin polarized superfluids