1. Superfluidity of ultracold
GSI-Darmstadt
September 2008
Superfluidity of ultracold
atomic Fermi gases
Sandro Stringari
University of Trento
CNR-INFM

2. Important manifestations of superfluidity
- Absence of viscosity (Landau’s criterion)
Hydrodynamic behavior at T=0 (irrotationality)
Quenching of moment of inertia
Occurrence of quantized vortices
Josephson oscillations
- Second sound
……..
Furthermore, in Fermi gases
Pairing gap (single particle excitations)
and (at unitarity)
- Phase separation in the presence of
- Spin polarization
Adiabatic rotation

3. of atomic Fermi gases interesting?
Why is superfluidity
of atomic Fermi gases interesting?

4. Important knobs available in atomic Fermi gases
Temperature
Scattering length
Polarization
Intensity of periodic potential
Rotation

5. BEC-BCS Crossover in Fermi gases
unitarity
In the presence of Feshbach
resonance different superfluid
regimes can be explored by
tuning the magnetic field
BCS
When scattering length is positive and small ( )
bound molecules of size a and binding energy are formed.
At small T they condense giving rise to BEC superfluidity
In the opposite regime of negative values of a the interaction
gives rise to Cooper pairs and BCS superfluidity
- At resonance the many body system exhibits strong correlations:
unitary superfluidity (analog of high Tc superconductivity)

6. Unitary Fermi gas: main features
diluteness
(interparticle distance >> range of inetraction)
strong interactions
(scattering length >> interparticle distance)
universality
(no dependence on interaction parameters)
high Tc (of the order of Fermi temperature)
robust superfluidity (high critical velocity)

7. Understanding superfluid features
requires theory for transport phenomena
(crucial interplay between dynamics and superfluidity)
Macroscopic dynamic phenomena in superfluids
(expansion, collective oscillations, moment of inertia)
are described by theory of irrotational hydrodynamics
(see also Thomas’talk)
More microscopic theories required to describe
other superfluid phenomena
(vortices, Landau critical velocity, pairing gap)

8. HYDRODYNAMIC THEORY OF SUPERFLUIDS AT T=0
Basic assumption:
Irrotationality
(follows from the phase of order parameter)
Basic ingredient:
Equation of state
(sensitive to quantum correlations, statistics, dimensionality, ...)

9. What do we mean by macroscopic, low energy phenomena ?
KEY FEATURES OF HD EQUATIONS OF SUPERFLUIDS
Have classical form (do not depend on Planck constant)
Velocity field is irrotational
(should be distinguished from rotational hydrodynamics)
- Applicable to low energy, macroscopic, phenomena
Hold for both Bose and Fermi superfluids
What do we mean by macroscopic, low energy phenomena ?
size of
Cooper pairs
BEC superfluids
BCS Fermi superfluids
healing
length
more restrictive
than in BEC
superfluid gap

10. WHAT ARE THE HYDRODYNAMIC
EQUATIONS USEFUL FOR ?
They provide quantitative predictions for
Expansion of the gas follwowing sudden release of the trap
- Collective oscillations excited by modulating harmonic trap
Quantities of highest interest from both
theoretical and experimental point of view

11. Collective oscillations in trapped gases
Collective oscillations: unique tool to explore consequence
of superfluidity and test the equation of state of
interacting quantum gases (both Bose and Fermi)
Experimental data for collective frequencies
are available with high precision
Solutions of HD equations in harmonic trap
predict both surface and compression modes
(first investigated in dilute BEC gases, Stringari 96)

12. Minimum damping near unitarity
HD theory predicts
Ideal gas model
Surface modes
Quadrupole mode
(Altmeyer et al. 2007)
Ideal gas value
HD prediction
Enhancement
of damping
Minimum damping near unitarity

13. Experiments on surface oscillations (quadrupole, scissors)
confirm HD behavior on BEC side of resonance
(including unitarity !)
on BCS side superfluidity is broken
for relatively small values of
(when gap is of the order of radial oscillator frequency)
Deeper in BCS regime frequency takes collisionless value
Damping is minimum near resonance
At unitarity hydrodynamic behavior is observed also
at high T (in normal phase). Consequence of collisions

14. Compression modes Sensitive to the equation of state
analytic solutions for collective frequencies
available for polytropic equation of state
Example: radial compression mode in cigar trap
At unitarity ( ) one predicts universal value
For a BEC gas one finds
Non monotonic behavior along the crossover (Stringari 2004)

15. Equation of state along BCS-BEC crossover
- Fixed Node Diffusion MC (Astrakharchick et al., 2004)
Comparison with mean field BCS theory ( )

16. Radial breathing mode at Innsbruck (Altmeyer et al., 2007)
MC equation of state (Astrakharchick et al., 2005)
includes quantum
correlations
does not includes
quantum correlations
BCS eq. of state
(Hu et al., 2004)
universal value
at unitarity
Measurement of collective frequencies
provides accurate test of equation of state !!

17. Landau’s critical velocity
Dispersion law of
elementary excitations
- Landau’s criterion for superfluidity (metastability):
fluid moving with velocity smaller than critical velocity cannot decay
(persistent current)
- Ideal Bose gas and ideal Fermi gas one has
In interacting Fermi gas one predicts two limiting cases:
BEC (Bogoliubov dispersion)
BCS (role of the gap)
(sound velocity)

18. Dispersion law along BCS-BEC crossover
gap
gap
unitarity
gap
BEC
(R. Combescot, M. Kagan and S. Stringari 2006)

19. Results for Landau’s critical velocity
theory
experiment
Sound velocity B
resonance
(Combescot et al, 2006)
(Miller et al, 2007)
Landau’s critical velocity is
highest near unitarity !!

20. Measurement of Landau’s critical velocity (proof of superfluidity)
Above critical value dissipative effects are observed
(Miller et al, 2007)

21. Phase separation between superfluid
and normal components
Differently from BEC’s
phase separation is not easily observed
by imaging in situ profiles of Fermi gas
(bimodal distribution is absent
at unitarity as well as in BCS
Phase separation can be nevertheless
produced by playing with
- Spin imbalance
- Adiabatic rotation

22. Assumption: phase separation below
Spin imbalance:
what happens by increasing P ?
If P=0 system is superfluid
If P=1 system is ideal Fermi gas (no interaction in s-channel)
For intermediate P different scenarios are possible:
- uniform spin polarized superfluid
- periodic modulations (FFLO)
- phase separation between superfluid and normal gas
Clogston-Chandrasekhar limit :
critical value of P above which the system becomes normal
Assumption: phase separation below
compare eq. of state of superfluid and normal phases
determine equilibrium conditions (pressure and chem. potential)

23. (phase contrast imaging, MIT 2006)
Occurrence of phase separation
well supported experimentally at unitarity (see also Rice exp)
Density difference
(phase contrast imaging, MIT 2006)
In superfluid phase
In polarized normal phase

24. BEC (after ramping) and in situ density difference
unitarity
Critical polarization
Pc=0.75
BEC (after ramping) and in situ density difference
in strongly interacting Fermi gas at MIT

25. Phase separation at unitarity:
unpolarized superfluid vs polarized normal gas at T=0
C. Lobo, A. Recati, S. Giorgini, S. Stringari, Phys. Rev. Lett. 97, (2006)
Recati, C. Lobo and S. Stringari, PRA, cond-mat/
Equilibrium between the two phases in uniform matter ensured by
equal pressure :
and equal chemical potential
Look for equation of state of superfluid and normal phases

26. Equation of state of superfluid and normal phases at untarity
Lobo et al. (PRL 2006)
Unpolarized superfluid
universal interaction parameter
Polarized normal gas
relative concentration of spin down and spin up
Fermi motion of spin down particles
binding energy of spin down particles
Parameters and originate from interaction
between spin up and spin down (Giorgini’s MC predictions, see also
Chevy, Prokofeev, Recati, Combescot….). A=0 in BCS mean field eqs.

27. superfluid and (partially) polarized phase
Equilibrium between
superfluid and (partially) polarized phase
best fit
to MC
fully polarized
Fermi gas
small x
approximation
-Ax
energy of
superfluid
critical concentration
- for x<0.44 gas is in normal phase
- for x>0.44 equilibrium between normal and superfluid phases

28. By using LDA one derives equations
Consequence of phase separation in harmonic trap
By using LDA one derives equations
for the profiles of superfluid, spin-up
and spin-down components
(surface effects ignored, see Rice, Stoof)

29. N S Shell structure in trapped spin polarized Fermi gas
(Lobo et al. 2006) N S
Predicted value for critical polarization
(Chandrasekhar-Clogston limit)
in good agreement with MIT result (0.75)
(mean field BdeG eq. predicts 0.97)

30. Column density of polarized Fermi gas
Exp: MIT
Theory: Trento

31. (In situ) density jump at the interface
Spin up density practically
continuous at the interface
Spin down density exhibits
jump at the interface
Exp: MIT
Theory: Trento

32. Axial density difference above and below normal P=0.80
Exp. Shin et al. 2006
Theory
ideal gas
Lobo et al. 2006
normal plus
superfluid
P=0.58

33. Main conclusion: Some open questions:
Interactions play a crucial role in both the
superfluid and normal polarized phase
- MC simulation + LDA provide excellent
description of CC limit and density profiles
(inadequacy of mf to describe CC limit at unitarity)
Some open questions:
Dynamic behavior of phase separated configuration
Motion of single impurities (polaron vs molecule)
Phase separation when

34. Phase separation induced
by adiabatic rotation

35. Spin polarization is not the only way
to produces phase separation at T=0
Phase separation induced by adiabatic rotation
At small angular velocity superfluid is unable to rotate
(consequence of irrotationality of moment of inertia)
By increasing the system has two possibilities:
Lines of singular vorticity are created if syste
is allowed to jump into lowest energy configuration
- Superfluid keeps irrotationality without vortices
(metastability, should be increased adiabatically)

36. Proof of superfluidity
Quantized vortices in Fermi gases
observed along the BEC-BCS crossover
(MIT, Nature June 2005, Zwierlein et al.)
Scattering length is suddenly ramped to small and positive values
in order to increase visibility of vortex lines

37. Adiabatic rotation of Fermi superfluid at unitarity
I. Bausmerth, A.Recati and S.S., PRL, 77, (2008)
Most relevant features
- Fermi superfluidity is fragile near the border (density is small),
superfluid energy gain being vanishingly small at unitarity
System prefers to loose superfluidity
gaining centrifugal energy
Rotation produces phase separation between a non rotating
superfluid core and an external rigidly rotating normal component.
Phase separation is characterized by density discontinuity
At higher angular velocity system exhibits
spontaneous rotational symmetry breaking, due to instability
of quadrupole surface modes (similar to BEC’s).

38. Equation of state in moving frame at T=0
Superfluid
universal interaction parameter
Normal gas (metastable in the absence of rotation)
Parameters available from Monte Carlo
simulation (Giorgini, Carlson).
In the absence of rotation (v=0) superfluid is energetically favourable.
However the rotation can make the normal component favourable

39. Hard sphere Fermi gas
MC for normal phase
BCS mean field
Nnnn
MC for superfluid phase

40. Effect of the rotation
on the density profiles
(LDA predictions)
Density jump at
the interface
knee in
column density
(directly measured
quantity)

41. Emergence of spontaneous symmetry breaking
(Bausmerth et al.
PRL 2008)
BEC
Rigidly rotating
normal gas
Unitary
Fermi gas

42. Availability of normal phase of
unpolarized T=0 Fermi gas at unitarity
new opportunities for investigating
Viscosity at small temperature
Landau parameters and spin zero sound
(different from liquid He3
similar to neutron matter )

43. Collaborators in Trento
Lev Pitaevskii
Stefano Giorgini
Alessio Recati
Ingrid Bausmerth
Theory of ultracold atomic Fermi gases
S.Giorgini, L.Pitaevskii and S. Stringari
Rev.Mod.Phys, in press (2008)
cond-mat/