1. DYNAMICS OF TRAPPED BOSE AND FERMI GASES
IHP Paris, June 2007
DYNAMICS OF TRAPPED
BOSE AND FERMI GASES
Sandro Stringari
Lecture 1
University of Trento
CNR-INFM

2. Lecture 1:- Hydrodynamic theory of superfluid gases
- expansion
- collective oscillations and equation of state
- collective vs s.p. excitations :Landau critical velocity
- Fermi superfluids with unequal masses
- example of non HD behavior:
normal phase of spin polarized Fermi gas
Lecture 2:- Dynamics in rotating superfluid gases
- Scissors mode
- Expansion of rotating gas
- vortex lattices and rotational hydrodynamics
- Tkachenko oscillations

3. 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
More microscopic theories required to describe
other superfluid phenomena
(vortices, Landau critical velocity, pairing gap)

4. HYDRODYNAMIC THEORY OF SUPERFLUIDS
Basic assumptions:
Irrotationality constraint
(follows from the phase of order parameter)
Conservation laws
(equation of continuity, equation for the current)
Basic ingredient:
- Equation of state

5. HYDRODYNAMIC EQUATIONS AT ZERO TEMPERATURE
of superfluids (T=0)
Closed equations for
density and superfluid
velocity field
irrotationality
is atomic mass,
is superfluid density
is superfluid velocity,
is chemical potential (equation of state)
is external potential
T=0
Equation for velocity field involves gradient of chemical potential;
Includes smoothly varying external field
in local density approximation

6. 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
Depend on equation of state
(sensitive to quantum correlations, statistics, dimensionality, ...)
- Equilibrium solutions (v=0) consistent with LDA
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

7. 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
Expansion provides information on
release energy, sensitive to anisotropy
Collective frequencies are measurable with highest precision
and can provide accurate test of equation of state

8. Expansion from anysotropic trap
Initially the gas is confined in anisotropic trap
in situ density profile is anisotropic too
What happens after release of the trap ?
Expansion from anysotropic trap
Non interacting gas expands isotropically
- For large times density distribution become isotropic
Consequence of isotropy of momentum distribution
holds for ideal Fermi gas and ideal Bose gas above
Ideal BEC gas expands anysotropically because
momentum distribution of condensate is anysotropic

9. Superfluids expand anysotropically
Hydrodynamic equations can be solved
after switching off the external potential
For polytropic equation of state
(holding for unitary Fermi gas ( ) and in BEC gases ( )),
and harmonic trapping, scaling solutions are available in the form
(Castin and Dum 1996; Kagan et al. 1996)
with the scaling factors satisfying the equation
and
- Expansion inverts deformation of density distribution, being faster in
the direction of larger density gradients (radial direction in cigar traps)
- expansion transforms cigar into pancake, and viceversa.

10. Experiments probe HD nature of the
Expansion of BEC gases
ideal gas
ideal gas HD HD
Experiments probe HD nature of the
expansion with high accuracy. Aspect ratio

11. EXPANSION OF ULTRACOLD FERMI GAS
Hydrodynamics predicts anisotropic expansion in Fermi superfluids (Menotti et al,2002)
HD theory
First experimental evidence for hydrodynamic anisotropic expansion in ultra cold Fermi gas (O’Hara et al, 2002)
collisionless

12. Measured aspect ratio after expansion
along the BCS-BEC crossover of a Fermi gas
(R. Grimm et al., 2007)
prediction of HD
at unitarity
prediction of
HD for BEC

13. - Expansion follows HD behavior
on BEC side of the resonance and at unitarity.
- on BCS side it behaves more and more like
in non interacting gas (asymptotic isotropy)
Explanation:
- on BCS side superfluid gap becomes soon
exponentially small during the expansion
and superfluidity is lost.
At unitarity gap instead always remains of the
order of Fermi energy and hence pairs are
not easily broken during the expansion

14. 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

15. First measurement of thermal effect on the Casimir force
Recent example of high precision measurement
First measurement of thermal effect on the Casimir force
(JILA, Obrecht et al. PRL 2007)
Lifschitz result
(thermal equilibrium)
Asymptotic behavior of the force out of thermal equilibrium
(Antezza et al. PRL , 2005)

16. Results for the frequency shift of the center of mass
oscillation due to the Casimir-Polder force produced
by a substrate at different temperatures
theory
Relative
frequency shift

17. For unitary Fermi gas ( )
Propagation of sound in elongated traps
In uniform medium HD theory gives trivial sound wave solution
with
If wave length is larger than radial size of
elongated trapped gas sound has 1D character
where and n is determined by TF eq.
one finds
For BEC gas ( )
(Zaremba, 1998)
For unitary Fermi gas ( )
(Capuzzi et al, 2006)

18. function of central density
Sound wave packets propagating in a BEC (Mit 97)
velocity of sound as a
function of central density

19. Sound wave packets propagating in an
Interacting Fermi gas (Duke, 2006)
behavior along the crossover
BCS mean field
QMC
Difference bewteen BCS and QMC reflects:
at unitarity: different value of in eq. of state
On BEC side different dimer-dimer scattering length

20. Collective oscillations in harmonic trap
When wavelength is of the order of the size of the atomic cloud sound
is no longer a useful concept. Solve linearized 3D HD equations
where is non uniform
equilibrium Thomas Fermi profile
well understood in
atomic BEC Stringari,1996
here we focus
on Fermi gases
Solutions of HD equations predict
surface and compression modes
Surface modes
Surface modes are unaffected by equation of state
For isotropic trap one finds where is angular momentum
surface mode is driven by external potential, not by surface tension
Dispersion law differs from ideal gas value (interaction effect)

21. Quadrupole mode measured on ultracold Fermi gas
along the crossover (Altmeyer et al. 2007)
Ideal gas value
HD prediction
Enhancement
of damping
Minimum damping near unitarity

22. - Experiments on collective oscillations show that
on the BCS side of the resonance superfluidity is
broken for relatively small values of
(where gap is of the order of radial oscillator frequency)
Deeper in BCS regime frequency takes
collisionless value
- Damping is minimum near resonance

23. 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

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

25. On BEC side of the resonance QMC confirms result
for the dimer-dimer scattering
length (Petrov et al., 2005)
and points out occurrence of
Lee-Huang-Yang correction
in equation of state
LHY w
In harmonic trap LHY effect results in enhancement of radial compression
frequency according to:
Pitaevskii and Stringari (1998)
Braaten and Pearson (1999)

26. compression mode in cigar trap
Frequency of radial
compression mode in cigar trap
(Stringari 2004)
BEC
beyond mean field
unitarity
Collective frequency has non monotonic behaviour
(effect missed by BCS theory, accounted for by MC)
can be evaluated numerically in whole regime
starting from knowledge of equation of state

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

28. Main conclusions concerning the m=0 radial compression mode
Accurate confirmation of the universal HD value
predicted at unitarity.
Accurate confirmation of QMC equation of state on the BEC side of
the resonance.
First evidence for Lee Huang Lee effect.
LHY effect (and in general enhancement of compression frequency
above the BEC value) very sensitive to thermal effects (thermal
fluctuations prevail on quantum fluctuations except at very low temperature)

29. Landau’s critical velocity
While in BEC gas sound velocity provides critical velocity,
in a Fermi BCS superfluid critical velocity is fixed
by pair breaking mechanisms (role of the gap)

30. 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)
Hydrodynamic theory cannot predict value of critical velocity
Needed full description of dispersion law (beyond phonon regime)
- 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)

31. Dispersion law along BCS-BEC crossover of Fermi gas
gap
gap
gap
unitarity
BEC
Critical velocity
(Combescot, Kagan and Stringari
2006)
Sound velocity B
resonance
Landau’s critical velocity is
highest near unitarity !!
Healing length
is smallest !!
(Marini et al 1997)
BCS
BEC

32. Lecture 1:- Hydrodynamic theory of superfluid gases
- expansion
- collective oscillations and equation of state
- collective vs s.p. excitations :Landau critical velocity
- Fermi superfluids with unequal masses
- example of non HD behavior:
normal phase of spin polarized Fermi gas

33. Fermi superfluids with unequal masses ( )
(G.Orso, L.Pitaevskii, S.S, in preparation)
Hydrodynamic eqs. keep standard form:
- Relevant frequencies for
HD oscillations
If oscillator lengths are equal
(natural condition
to cool down into superfluid phase)
one has
with

34. What happens if one suddenly switches off ?
- Sudden change in trapping conditions:
- In the absence of interactions atoms down will fly away
- Superfluidity keeps component down confined even if
- obvious in molecular regime
- non trivial many-body effect at unitarity
- weak effect in BCS regime
- Condition for energetic stability of superfluid with respect to separation
of two components
- Gas starts oscillating with frequency
(unitarity)
easily satisfied if

35. Example: mixture (aspect ratio 0.2) Oscillation of radial size
if one removes K-trapping
Equilibrium of superfluid
in new trapping conditions
Oscillation of radial size
if one removes Li-trapping
Will superfluid survive in conditions
of strong energetic instability?

36. Collective oscillations in spin polarized Fermi gas (normal phase)
Lobo, Recati, Giorgini and S.S
Recent experiments on spin polarized Fermi gas show phase separation
between superfluid and spin polarized normal gas if
Density profiles at unitarity well agree with theory (Lobo et al. 2006)
based on QMC equation of state of superfluid and normal phases
Equation of state in normal phase well approximated by expansion
holding for small
concentrations
QMC simulation yields A = 0.97, m/m* = 1.04

37. normal
Axial densities
in spin polarized
Fermi gas
P=0.80
Exp. Shin et al. 2006
Theory
ideal gas
Lobo et al. 2006
normal plus
superfluid
P=0.58
Critical value for
normal-superfluid
phase separation
P = 0.75
Clogston-Chandrasekhar
limit at unitarity (and in trap)

38. At zero temperature oscillations in normal spin polarized phase
are not governed by hydrodynamics
(like in superfluid),
but by Landau’s theory of Fermi liquids
(collisionless regime + mean field)
Spin down feel potential
generated by spin up
resulting in renormalization
of oscillator frequency

39. QUESTION: Will the spin mode be strongly damped?
If spin down atoms
oscillate with frequencies
dipole
quadrupole
Spin dipole mode difficult to excite because shift of harmonic trap results
in excitation of center of mass motion (Kohn’s theorem)
Spin quadrupole is instead excited by quadrupole modulation of trap.
In general one expect to excite two modes (sum rule approach)
Quadrupole
modes
Exciting Q one excites
also spin mode !!
QUESTION: Will the spin mode be strongly damped?
Pauli principle is expected to quench damping at small T
(collaboration with Ch. Pethick)

40. Main conclusions
- Collective oscillations in trapped quantum gases
can be measured with high precision.
They provide unique information on the
consequences of superfluidity and accurate test
of equation of state in Fermi superfluids
Can their measurement be considered a proof of superfluidity?
Lecture 2:- Dynamics in rotating superfluid gases
- Scissors mode
- Expansion of rotating gas
- vortex lattices and rotational hydrodynamics
- Tkachenko oscillations