Transcript Diapositiva 1

BEC Meeting, Trento, 2-3 May 2006
Ultracold Fermi gases
Sandro Stringari
University of Trento
INFM-CNR
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)
This talk
- HD expansion (aspect ratio and pair correlation function)
- collective oscillations and equation of state
- spin polarizability
EXPANSION OF
FERMI SUPERFLUID
Hydrodynamics predicts anisotropic
expansion of BEC gas
Hydrodynamics predicts
anisotropic expansion in
Fermi superfluids
(Menotti et al,2002)
Evidence for hydrodynamic
anisotropic expansion in
ultra cold Fermi gas
(O’Hara et al, 2003)
HD theory
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
g 2 (s)  1  exp[2 i
- Pair
si2
]
2
2 2
T (i t  1)
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: T it  t / mRi
QUESTION
Can we describe behaviour of pair correlation function during the
expansion 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)
unitarity
k F a  1/ 4
BEC limit
thermal bosons
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 ( k F a  1 ) 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)
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 kF s , expansion acts like a microscope
COLLECTIVE OSCILLATIONS
AND EQUATION OF STATE
COLLECTIVE OSCILLATIONS
IN SUPERFLUID PHASE (T=0)
- Surface modes: unaffected by equation of state
- Compression modes sensitive to equation of state.
-Theory of superfluids predicts
universal values when 1/a=0 :
rad  10 / 3
ax  12/ 5z
- In BEC regime one insetad finds
rad  2
ax  5 / 2z
Behaviour of equation of state through the crossover can
be inferred through the study of collective frequencies !
Radial compression mode
S. Stringari, Europhys. Lett. 65, 749 (2004)
10


3
Experiments on collective oscillations at
- Duke (Thomas et al..)
- Innsbruck (Grimm et al.)
Duke data agree with value 1.826 predicted at unitarity

unitarity
(mean field
BCS gap eq.)

Radial breathing mode at Innsbruck (2006)
(unpublished)
MC equation of state
BCS mean field
10/ 3  1.83
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 gas
A. Recati, I. Carusotto, C. Lobo and S.S., in preparation
Recent experiments and theoretical studies have focused on the
consequence of spin polarization ( P  ( N  N ) /( N  N ) ) on
the superfluid features of interacting Fermi gases
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.
 1
2
2
2
2
V (r )  m [ y  z  ( x  d ) ]
2
For non interacting gas, equilibrium corresponds to
rigid displacement of two spin clouds in opposite direction:
 1

n (r )  n0 (r  dxˆ )
2
This yields spin dipole moment (we assume N  N  N / 2 )

1


D(d )   dr x(n (r )  n (r ))  d
N
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
 1


1
VS (r )  (V (r )  V (r ))  m 2 [ x 2  y 2  z 2  d 2 ]
2
2
Competition between pairing effects and
external potential favouring spin polarization
 1
V (r )  m 2 [ y 2  z 2  ( x  d ) 2 ]
2
VS
V
V
At unitarity
SF
Equilibrium between superfluid and spin polarized phases
(Chevy 2005)
 S   F
  0.44


x  0 :  S (r )   (r )(2 )3 / 5 / 2


x  0 :  S (r )   (r )(2 )3 / 5 / 2


 S ( r )   0  VS ( r )


  (r )   0  V (r )
Spin dipole moment D(d)/d as a function of separation
distance d (in units of radius of the cloud)
ideal gas
  0.58
  0.44
Deep BEC
Further projects:
- Collective oscillations of spin polarized superfluid
- Rotational effects in spin polarized superfluids