Diapositiva 1

1
Trento PhD course on Ultracold atomic Fermi
gases (February-March 2008)
Lecture 7
Superfluidity and hydrodynamic behavior
Sandro Stringari
University of Trento
CNR-INFM
2
Piano del corso
Lecture 2. BEC gases and the role of the
interaction The Gross-Pitaevskii equation
Lecture 3. Bose-Einstein condensates in harmonic
traps
Lecture 4. Ideal Fermi gas and the role of the
interaction
Lecture 5. BEC-BCS crossover and the Bogoliubov
de Gennes equations
Lecture 6. Interacting Fermi gases in harmonic
traps
Lecture 7. Superfluidity and hydrodynamic
behavior
Lecture 8. Rotating Fermi superfluids
Lecture 9. Spin polarized Fermi gases
Lecture 10. Fermi gases in periodic potentials
3
Manifestations of superfluidity

• - Absence of viscosity (Landaus criterion)
• Hydrodynamic behavior at T0 (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

4
Understanding superfluid features requires
theory for transport phenomena (crucial
interplay between dynamics and superfluidity)
Macroscopic dynamic phenomena (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, gap,
thermodynamics)
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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
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HYDRODYNAMIC EQUATIONS AT ZERO TEMPERATURE
Hydrodynamic equations of superfluids
(T0) Closed equations for density and
superfluid velocity field
irrotationality
is atomic mass,
is superfluid density (density of each spin
spiecies) is superfluid velocity,
is atomic chemical potential, is
potential felt by each spin species
T0

• Equation for velocity field involves gradient of
  chemical potential
• Includes smoothly varying external field
• in local density approximation

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KEY FEATURES OF HD EQUATIONS OF SUPERFLUIDS

• Equations have classical form (do not depend on
  Planck constant)
• Eqs. involve density, not order parameter
• Velocity field is irrotational (gradient of
  phase of order parameter)
• Should be distinguished from rotational
  hydrodynamics.
• - Applicable to low energy, macroscopic,
  phenomena
• HD equations hold for both Bose and Fermi
  superfluids
• HD equations depend on equation of state
  
• (sensitive to quantum correlations, statistics,
  dimensionality, ...)
• - Equilibrium solutions (v0) consistent with LDA
  

What do we mean by low energy, macroscopic
phenomena ?
size of pairs
BEC superfluids
BCS superfluids
healing length
more restrictive than in BEC
superfluid gap
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WHAT ARE THE HYDRODYNAMIC EQUATIONS USEFUL FOR ?

• They provide quantitative predictions for
• Expansion of the gas follwoing 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

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• 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
• BEC ideal gas expands anysotropically because
• momentum distribution of condensate is
  anysotropic !!

10
Superfluids expand anysotropically
Hydrodynamic equations can be solved after
switching off the external potential
For polytropic equation of state (holding for
example at unitary ( ) and in BEC
limit ( )), 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
disc, and viceversa.
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Expansion of BEC gases
ideal gas
ideal gas
HD
HD
Experiments probe HD nature of the expansion
with high accuracy. Aspect ratio
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EXPANSION OF ULTRACOLD FERMI GAS
HD theory
Hydrodynamics predicts anisotropic expansion
inFermi superfluids (Menotti et al,2002)
First experimental evidence for hydrodynamic
anisotropic expansion inultra cold Fermi gas
(OHara et al, 2002)
normal collisionless
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Fermions
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
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• - 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

15
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
16
Propagation of sound in trapped gases
In uniform medium HD theory gives sound wave
solution with
In trapped gases sound waves can propagate if
wave length is smaller than axial size of the
condensate. Condition is easily satisfied in
elongated condensates.
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Propagation of sound in elongated traps

• 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)
(Capuzzi et al, 2006)
For unitary Fermi gas ( )

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Bosons
Sound wave packets propagating in a BEC (Mit 97)
velocity of sound as a function of central
density
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Fermions
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 molecule-molecule
  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
Solutions of HD equations in harmonic
trap predict both surface and compression
modes (first investigated in dilute BEC gases
(Stringari 96)
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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)

Surface modes in BECs, Mit 2000
Bosons
m2
m4
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l2 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
23
Fermions

• - 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

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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

25
Radial breathing mode at Innsbruck (2006)
universal value at unitarity
Measurement of compression mode provides
accurate test of unitary behavior !!
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Compression modes on the BEC side

• By expanding equation of state from BEC side
• one can also evaluate beyond mean field
  corrections to the
• collective frequencies (Pitaevskii, Stringari
  1998 Braaten, Pearson 1999).

Lee,Huang,Yang correction
mean field
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Molecule-molecule interaction in the BEC side

• Molecular scattering length determines
  properties of the
• molecular condensate (size of the condensate,
  release energy)
• Molecular scattering lenght calculated by solving
  exactly the low
• energy scattering problem with 4 atoms
• (Petrov et al 2003, see also Pieri and Strinati,
  2000).
• (BCS theory predicts )
• Correct value of molecular
• scattering length confirmed by
• Monte Carlo calculation of
• the equation of state
• (in BEC limit eq. of state
• corresponds to dilute Molecular gas)
• - experimental data on release
• energy and on in situ radial size

28
Application to radial compression mode in cigar
geometry
( , and
)
beyond mean field effect?
ENS experiment on BEC gas (2002)

Theory predicts Compared to mean field
value



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Behavior of compression mode between BEC and
Unitarity (Stringari 2004)
beyond mean field
unitarity
BEC
0

• 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)

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Radial breathing mode at Innsbruck (2006)
MC equation of state (Manini and Salasnich,
2005
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 !!
31
Fermions

• - 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

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Hydrodynamics and superfluidity

• Is the measurement of anisotropic expansion and
  
• collective frequencies a proof of
  superfluidity?
• These measurement actually probe validity of
  hydrodynamic theory
• and predictions for equation of state
• More direct proofs of superfluidity concern
• - absence of viscosity (Landau critical
  velocity)
• - rotations (response to transverse field)

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Landaus critical velocity
34
Landaus critical velocity
Dispersion law of elementary excitations

• - Landaus 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

BCS (role of the gap)
BEC (Bogoliubov dispersion)
(sound velocity)
35
Dispersion law along BCS-BEC crossover
BCS
BEC
gap
gap
gap
unitarity
BEC
(R. Combescot, M. Kagan and S. Stringari 2006)
36
Results for Landaus critical velocity
theory
experiment
Sound velocity
B
resonance
(Combescot et al, 2006)
(Miller et al, 2007)
Landaus critical velocity is highest near
unitarity !!
37
Measurement of Landaus critical velocity (proof
of superfluidity)
Above critical value dissipative effects are
observed
(Miller et al, 2007)