Rotations and quantized vortices in Bose superfluids

1
Trento, June 4, 2009
Fermionic superfluidity in optical lattices

Gentaro Watanabe, Franco Dalfovo, Giuliano Orso,
Francesco Piazza, Lev P. Pitaevskii, and Sandro
Stringari
2
Summary

• Equation of state and effective mass of the
  unitary Fermi gas in a 1D periodic potential
  Phys. Rev. A 78, 063619 (2008)
• Critical velocity of superfluid flow work in
  progress

3
We were stimulated by
6Li Feshbach res. _at_ B834G N106
weak lattice
4
vc is maximum around unitarity
Superfluidity is robust at unitarity
5
What we have done first
Calculation of
energy density
chemical potential
Key quantities for the characterization of the
collective properties of the superfluid
inverse compressibility
effective mass
sound velocity
6
The theory that we have used
Bogoliubov de Gennes equation
Order parameter
Refs
A. J. Leggett, in Modern Trends in the Theory of
Condensed Matter, edited by A. Pekalski and R.
Przystawa (Springer-Verlag, Berlin, 1980)M.
Randeria, in Bose Einstein Condensation, edited
by A. Griffin, D. Snoke, and S. Stringari
(Cambridge University Press, Cambridge, England,
1995).
7
We numerically solve the BdG equations. We use a
Bloch wave decomposition.
From the solutions we get the energy density
Caveat a regularization procedure must be used
to cure ultraviolet divergence (pseudo-potential,
cut-off energy) as discussed by Randeria and
Leggett. See also G. Bruun, Y. Castin, R. Dum,
and K. Burnett, Eur. Phys. J D 7, 433 (1999) A.
Bulgac and Y. Yu, Phys. Rev. Lett. 88, 042504
(2002).
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Results compressibility and effective mass
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Results compressibility and effective mass
when EF ltlt ER The lattice favors the
formation of molecules (bosons). The
interparticle distance becomes larger than the
molecular size. In this limit, the BdG
equations describe a BEC of molecules. The
chemical potential becomes linear in density.
s5
s0
Two-body results by Orso et al., PRL 95,
060402 (2005)
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Results compressibility and effective mass
when EF ltlt ER The lattice favors the
formation of molecules (bosons). The
interparticle distance becomes larger than the
molecular size. In this limit, the BdG
equations describe a BEC of molecules. The
chemical potential becomes linear in density.
The system is highly compressible. The
effective mass approaches the solution of the
two-body problem. The effects of the lattice
are larger than for bosons!
Two-body results by Orso et al., PRL 95,
060402 (2005)
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Results compressibility and effective mass
when EF gtgt ER Both quantities approach their
values for a uniform gas. Analytic expansions in
the small parameter (sER/EF)
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Results compressibility and effective mass
when EF ER Both quantities have a maximum,
caused by the band structure of the quasiparticle
spectrum.
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Sound velocity
Significant reduction of sound velocity the by
lattice !
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Density profile of a trapped gas
From the results for µ(n) and using a local
density approximation, we find the density
profile of the gas in the harmonic trap 1D
lattice
Thomas-Fermi for fermions
!
Bose-like TF profile
aspect ratio 1 h?/ER 0.01 N5?105 s5
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Summary of the first part

• We have studied the behavior of a superfluid
  Fermi gas at unitarity in a 1D optical lattice by
  solving the BdG equations.
• The tendency of the lattice to favor the
  formation of molecules results in a significant
  increase of both the effective mass and the
  compressibility at low density, with a consequent
  large reduction of the sound velocity.
• For trapped gases, the lattice significantly
  changes
• the density profile
• the frequency of the collective oscillations

See Phys. Rev. A 78, 063619 (2008)
16

• Equation of state and effective mass of the
  unitary Fermi gas in a 1D periodic potential
  Phys. Rev. A 78, 063619 (2008)
• Critical velocity of superfluid flow work in
  progress

17

• The concept of critical current plays a
  fundamental role in the physics of superfluids.
• Examples
• Landau critical velocity for the breaking of
  superfluidity and the onset of dissipative
  effects. This is fixed by the nature of the
  excitation spectrum
• phonons, rotons, vortices in BEC superfluids
• single particle gap in BCS superfluids
• Critical current for dynamical instability
• (vortex nucleation in rotating BECs, disruption
• of superfluidity in optical lattices)
• Critical current in Josephson junctions.
• This is fixed by quantum tunneling

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• In ultracold atomic gases
• Motion of macroscopic impurities has revealed
  the onset of heating effect (MIT 2000)
• Quantum gases in rotating traps have revealed
  the occurrence of both energetic and dynamic
  instabilities (ENS 2001)
• Double well potentials are well suited to
  explore Josephson oscillations (Heidelberg 2004)
• Moving periodic potentials allow for the
  investigation of Landau critical velocity as well
  as for dynamic instability effects (Florence
  2004, MIT 2007)

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• Questions
• Can we obtain a unifying view of critical
  velocity phenomena driven by different external
  potentials and for different quantum statistics
  (Bose vs. Fermi)?
• Can we theoretically account for the observed
  values of the critical velocity?

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• NOTE The mean-field calculations by Spuntarelli
  et al. PRL 99, 040401 (2007) for fermions
  through a single barrier show a similar
  dependence of the critical velocity on the
  barrier height.

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Our goal establishing an appropriate framework
in which general results can be found in order to
compare different situations (bosons vs. fermions
and single barrier vs. optical lattice) and
extract useful indications for available and/or
feasible experiments.
L
Vmax
d
Vmax
22
The simplest approach Hydrodynamics in Local
Density Approximation (LDA)
Assumption the system behaves locally as a
uniform gas of density n, with energy density
e(n) and local chemical potential, µ(n). The
density profile of the gas at rest in the
presence of an external potential is given by
the Thomas-Fermi relation
If the gas is flowing with a constant current
density jn(x)v(x), the Bernoulli equation for
the stationary velocity field v(x) is
23
The simplest approach Hydrodynamics in Local
Density Approximation (LDA)
This equation gives the density profile, n(x),
for any given current j, once the equation of
state µ(n) of the uniform gas of density n is
known.
24
The simplest approach Hydrodynamics in Local
Density Approximation (LDA)
The system becomes energetically unstable when
the local velocity, v(x), at some point x becomes
equal to the local sound velocity, csn(x).
For a given current j, this condition is first
reached at the point of minimum density, where
v(x) is maximum and cs(x) is minimum.
here the density has a minimum and the local
velocity has a maximum !
25
The simplest approach Hydrodynamics in Local
Density Approximation (LDA)
The same happens in a periodic potential
here the density has a minimum and the local
velocity has a maximum !
26
The simplest approach Hydrodynamics in Local
Density Approximation (LDA)
To make calculations, one needs the equation of
state µ(n) of the uniform gas!
We use a polytropic equation of state
27
The simplest approach Hydrodynamics in Local
Density Approximation (LDA)
To make calculations, one needs the equation of
state µ(n) of the uniform gas!
We use a polytropic equation of state
Unitary Fermions
Bosons (BEC)
a (1ß)(3p2)2/3h2/2m
a g 4ph2as/m
28
The simplest approach Hydrodynamics in Local
Density Approximation (LDA)
To make calculations, one needs the equation of
state µ(n) of the uniform gas!
We use a polytropic equation of state
Unitary Fermions
Bosons (BEC)
a (1ß)(3p2)2/3h2/2m
a g 4ph2as/m
Local sound velocity
29
The simplest approach Hydrodynamics in Local
Density Approximation (LDA)
Inserting the critical condition
into the Bernoulli equation
one gets an implicit relation for the critical
current
Universal !! Bosons and Fermions in any 1D
potential
Note for bosons through a single barrier see
also Hakim, and Pavloff et al.
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LDA
Fermions through a barrier
Bosons through a barrier
fermions
bosons
Fermions in a lattice
Bosons in a lattice
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LDA
The limit Vmax ltlt µ corresponds to the usual
Landau criterion for a uniform superfluid flow in
the presence of a small external perturbation,
i.e., a critical velocity equal to the sound
velocity of the gas.
32
LDA
The limit Vmax ltlt µ corresponds to the usual
Landau criterion for a uniform superfluid flow in
the presence of a small external perturbation,
i.e., a critical velocity equal to the sound
velocity of the gas.
the critical velocity decreases because the
density has a local depletion and the velocity
has a corresponding local maximum
33
LDA
The limit Vmax ltlt µ corresponds to the usual
Landau criterion for a uniform superfluid flow in
the presence of a small external perturbation,
i.e., a critical velocity equal to the sound
velocity of the gas.
the critical velocity decreases because the
density has a local depletion and the velocity
has a corresponding local maximum
When Vmax µ the density vanishes and the
critical velocity too.
34
LDA
Question when is LDA reliable?
35
LDA
Question when is LDA reliable? Answer the
external potential must vary on a spatial
scale much larger than the healing length of the
superfluid.
L gtgt ?
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LDA
Question when is LDA reliable? Answer the
external potential must vary on a spatial
scale much larger than the healing length of the
superfluid.
L gtgt ?
For a single square barrier, L is just its
width. For an optical lattice, L is of the
order of the lattice spacing (we choose Ld/2).
For bosons with density n0, the healing length
is ?h/(2mgn0)1/2. For fermions at unitarity,
one has ? 1/kF, where kF (3p2n0)1/3
37
LDA
Question when is LDA reliable? Answer the
external potential must vary on a spatial
scale much larger than the healing length of the
superfluid.

• Quantum effects beyond LDA become important when
• - ? is of the same order or larger than L they
  cause a smoothing of both density and velocity
  distributions, as well as the emergence of
  solitonic excitations (and vortices in 3D).
• Vmax gt µ in this case LDA predicts a
  vanishing current, while quantum tunneling
  effects yield Josephson current.
• Quantitative estimates of the deviations from
  the predictions of LDA can be obtained by using
  quantum many-body theories, like Gross-Pitaevskii
  theory for dilute bosons and Bogoliubov-de Gennes
  equations for fermions.

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LDA
Fermions through a barrier
Bosons through a barrier
Bosons in a lattice
Fermions in a lattice
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LDA vs. GP/BdG
Fermions through a barrier
Bosons through a barrier
L/?1
LkF4 Spuntarelli et al.
5
10
bosons
fermions
Bosons in a lattice
Fermions in a lattice
L/?0.5
LkF0.5
1
0.89
1.57
1.11
1.92
3
1.57
5
2.5
10
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Bosons (left) and Fermions (right) through single
barrier
bosons
fermions
L/?1
LkF4
5
10
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Bosons (left) and Fermions (right) through single
barrier
bosons
fermions
L/?1
LkF4
5
10
L gtgt ? Hydrodynamic flow in LDA
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Bosons (left) and Fermions (right) through single
barrier
L lt ? Macroscopic flow with quantum effects
beyond LDA
bosons
fermions
L/?1
LkF4
5
10
L gtgt ? Hydrodynamic flow in LDA
43
Bosons (left) and Fermions (right) through single
barrier
Vmaxgt µ quantum tunneling between weakly
coupled superfluids (Josephson regime)
bosons
fermions
L/?1
LkF4
5
10
L gtgt ? Hydrodynamic flow in LDA
44
Bosons (left) and Fermions (right) in a periodic
potential
The periodic potential gives results similar to
the case of single barrier
bosons
fermions
L/?0.5
LkF0.5
1
0.89
1.57
1.11
1.92
3
1.57
5
2.5
10
45
Bosons (left) and Fermions (right) in a periodic
potential
L lt ? Macroscopic flow with quantum effects
beyond LDA
bosons
fermions
L/?0.5
LkF0.5
1
0.89
1.57
1.11
1.92
3
1.57
5
2.5
10
Vmaxgtgt µ quantum tunneling between weakly
coupled supefluids (Josephson current regime)
L gtgt ? Hydrodynamic flow in LDA
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Differences between single barrier and periodic
potential (bosons)
Bosons through a barrier
L/?1
5
10
Bosons in a lattice
L/?0.5
1
1.57
3
5
10
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Single barrier (bosons)
For a single barrier, µ and ? are fixed by the
asymptotic density n0 only. They are unaffected
by the barrier. All quantities behave smoothly
when plotted as a function of L/ ? or Vmax/
µ. For L/ ? gtgt 1 vc/cs ? 1 const ? (Vmax
)1/2 For L/ ? ltlt 1 vc/cs ? 1 const ? (LVmax
)2/3
L/?1
5
10
48
Periodic potential (bosons)
In a periodic potential the barriers are
separated by distance d. The energy density, e,
and the chemical potential, µ, are not fixed by
the average density n0 only, but they depend also
on Vmax. They exhibit a Bloch band structure.
L/?0.5
1
1.57
3
5
10
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Periodic potential (bosons)
Bloch band structure. p quasi-momentum pB
Bragg quasi-momentum ER p2B/2m recoil
energy VmaxsER lattice strength
Energy density vs. quasi-momentum
Lowest Bloch band for the same gn00.4ER and
different s
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Periodic potential (bosons)
Bloch band structure. p quasi-momentum pB
Bragg quasi-momentum ER p2B/2m recoil
energy VmaxsER lattice strength
Energy density vs. quasi-momentum
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Periodic potential (bosons)
Bloch band structure. p quasi-momentum pB
Bragg quasi-momentum ER p2B/2m recoil
energy VmaxsER lattice strength
Energy density vs. quasi-momentum
The curvature at p0 gives the effective mass
e n0 p2/2m
52
Periodic potential (bosons)
L/?0.5
Bloch band structure. p quasi-momentum pB
Bragg quasi-momentum ER p2B/2m recoil
energy VmaxsER lattice strength
1
1.57
3
5
10
Energy density vs. quasi-momentum
Tight-binding limit (Vmax gtgt µ) e dJ
1-cos(pp/pB) (2ERn0/p2)(m/m) dJ
tunnelling energy (Josephson current) Critical
p pc0.5pB Critical velocity vc (2/p)(m/m)
ER/pB m/m proportional to e-2vs
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Differences between single barrier and periodic
potential (fermions)
Same qualitative behavior as for bosons
Fermions through a barrier
LkF4
Fermions in a lattice
LkF0.5
0.89
1.11
1.92
1.57
2.5
54
What about experiments?
BOSONS Experiments at LENS-Florence
Weak lattice (energetic vs. dynamic
instability) L. De Sarlo, L. Fallani, J. E. Lye,
M. Modugno, R. Saers, C. Fort, M. Inguscio,
Unstable regimes for a Bose-Einstein condensate
in an optical lattice Phys. Rev. A 72, 013603
(2005)
L/?0.5
1
3
5
10
55
What about experiments?
BOSONS Experiments at LENS-Florence
Weak lattice (energetic vs. dynamic
instability) L. De Sarlo, L. Fallani, J. E. Lye,
M. Modugno, R. Saers, C. Fort, M. Inguscio,
Unstable regimes for a Bose-Einstein condensate
in an optical lattice Phys. Rev. A 72, 013603
(2005)
L/?0.5
1
3
L/? 0.7 and Vmax/µ 10 - 25
5
10
Strong lattice (Josephson current regime) F.
S. Cataliotti, S. Burger, C. Fort, P. Maddaloni,
F. Minardi, A. Trombettoni, A. Smerzi, M.
Inguscio Josephson Junction arrays with
Bose-Einstein Condensates Science 293, 843 (2001)
56
What about experiments?
FERMIONS Experiments at MIT
D. E. Miller, J. K. Chin, C. A. Stan, Y. Liu, W.
Setiawan, C. Sanner, W. Ketterle Critical
velocity for superfluid flow across the BEC-BCS
crossover PRL 99, 070402 (2007)
LkF0.5
?
0.89
1.11
1.92
Problem Which density n0? Which Vmax?
1.57
2.5
57
What about experiments?
FERMIONS Experiments at MIT
D. E. Miller, J. K. Chin, C. A. Stan, Y. Liu, W.
Setiawan, C. Sanner, W. Ketterle Critical
velocity for superfluid flow across the BEC-BCS
crossover PRL 99, 070402 (2007)
LkF0.5
?
0.89
1.11
1.92
Problem Which density n0? Which Vmax?
1.57
2.5
n0
Depending on how one chooses n0 (and hence EF)
one gets 0.5 lt EF/ER lt 1 or 1 lt LkF lt
1.5
Vext
58
What about experiments?
FERMIONS Experiments at MIT
D. E. Miller, J. K. Chin, C. A. Stan, Y. Liu, W.
Setiawan, C. Sanner, W. Ketterle Critical
velocity for superfluid flow across the BEC-BCS
crossover PRL 99, 070402 (2007)
If EF is by the density at e-2 beam waist
EF/ER 0.5 (LkF 1)
BdG theory
Expt
LDA
Significant discrepancies between theory and
MIT data. Additional dissipation mechanisms?
Nonuniform nature of the gas/lattice ?
59
Summary of the second part

• Unifying theoretical picture of critical
  velocity phenomena in 1D geometries (including
  single barrier and periodic potentials as well
  as Bose and Fermi statistics).
• Different scenarios considered, including LDA
  hydrodynamics and Josephson regime
• Comparison with experiments reveals that the
  onset of energetic instability is affected by
  nonuniform nature of the gas (also in recent
  MIT experiment).
• Need for more suitable geometrical
  configurations. For example toroidal geometry
  with rotating barrier would provide cleaner and
  more systematic insight on criticality of
  superfluid phenomena (including role of quantum
  vorticity)