Transcript Rotations and quantized vortices in Bose superfluids

Trento, June 4, 2009
Gentaro Watanabe, Franco Dalfovo, Giuliano Orso, Francesco Piazza, Lev P. Pitaevskii, and Sandro Stringari
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]
We were stimulated by:
6Li
Feshbach res. @ B=834G
N=106
weak lattice
vc is maximum
around unitarity
Superfluidity is robust at unitarity
What we have done first:
Calculation of
energy density
chemical potential
inverse compressibility
effective mass
sound velocity
Key quantities
for the
characterization
of the collective
properties of the
superfluid
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).
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, cutoff 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).
Results: compressibility and effective mass
Results: compressibility and effective mass
when EF << 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.
s=5
s=0
Two-body
results by
Orso et al.,
PRL 95, 060402
(2005)
Results: compressibility and effective mass
when EF << 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.
Two-body
results by
Orso et al.,
PRL 95, 060402
(2005)
The effective mass approaches the
solution of the two-body problem.
The effects of the lattice are larger
than for bosons!
Results: compressibility and effective mass
when EF >> ER :
Both quantities approach their
values for a uniform gas.
Analytic expansions in the
small parameter (sER/EF)
Results: compressibility and effective mass
when EF ~ ER :
Both quantities have a
maximum, caused by the band
structure of the quasiparticle
spectrum.
Sound velocity
Significant reduction of sound velocity the by lattice !
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
ħω/ER = 0.01
N=5105
s=5
!
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)]
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]
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
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)
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?
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.
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
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 j=n(x)v(x), the
Bernoulli equation for the stationary velocity field v(x) is
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.
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, cs[n(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 !
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 !
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:
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:
Bosons (BEC)
Unitary Fermions
α= g = 4πħ2as/m
α= (1+β)(3π2)2/3ħ2/2m
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:
Bosons (BEC)
Unitary Fermions
α= g = 4πħ2as/m
α= (1+β)(3π2)2/3ħ2/2m
Local sound velocity:
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:
Note: for bosons through a single barrier see also Hakim, and Pavloff et al.
Universal !!
Bosons and
Fermions in
any 1D
potential
LDA
Bosons through a barrier
Fermions through a barrier
fermions
bosons
Bosons in a lattice
Fermions in a lattice
LDA
The limit Vmax << 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.
LDA
The limit Vmax << 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
LDA
The limit Vmax << 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.
LDA
Question: when is LDA reliable?
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 >> ξ
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 >> ξ
For a single square barrier, L
is just its width.
For bosons with density n0, the
healing length is ξ=ħ/(2mgn0)1/2.
For an optical lattice, L is of
the order of the lattice spacing
(we choose L=d/2).
For fermions at unitarity, one has
ξ ≈ 1/kF, where kF = (3π2n0)1/3
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
- ξ 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 > 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.
LDA
Bosons through a barrier
Bosons in a lattice
Fermions through a barrier
Fermions in a lattice
LDA vs. GP/BdG
Fermions through a barrier
Bosons through a barrier
L/ξ=1
LkF=4 [Spuntarelli et al.]
5
10
bosons
Bosons in a lattice
fermions
Fermions in a lattice
L/ξ=0.5
LkF=0.5
1
0.89
1.57
3
5
10
1.11
1.92
2.5
1.57
Bosons (left) and Fermions (right) through single barrier
bosons
fermions
L/ξ=1
LkF=4
5
10
Bosons (left) and Fermions (right) through single barrier
bosons
fermions
L/ξ=1
LkF=4
5
10
L >> ξ : Hydrodynamic flow in LDA
Bosons (left) and Fermions (right) through single barrier
L < ξ : Macroscopic flow with quantum
effects beyond LDA
bosons
fermions
L/ξ=1
LkF=4
5
10
L >> ξ : Hydrodynamic flow in LDA
Bosons (left) and Fermions (right) through single barrier
Vmax> μ : quantum tunneling between
weakly coupled superfluids
(Josephson regime)
fermions
bosons
L/ξ=1
LkF=4
5
10
L >> ξ : Hydrodynamic flow in LDA
Bosons (left) and Fermions (right) in a periodic potential
The periodic potential gives results similar to the case of single barrier
bosons
fermions
L/ξ=0.5
LkF=0.5
1
0.89
1.57
3
5
10
1.11
1.92
2.5
1.57
Bosons (left) and Fermions (right) in a periodic potential
L < ξ : Macroscopic flow with
quantum effects beyond LDA
bosons
fermions
L/ξ=0.5
LkF=0.5
1
0.89
1.57
3
5
10
L >> ξ : Hydrodynamic flow in LDA
1.11
1.92
1.57
2.5
Vmax>> μ : quantum tunneling between
weakly coupled supefluids (Josephson
current regime)
Differences between single barrier and periodic potential (bosons)
Bosons through a barrier
L/ξ=1
5
10
Bosons in a lattice
L/ξ=0.5
1
1.57
3
5
10
Single barrier (bosons)
For a single barrier, μ and ξ 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/ ξ
or Vmax/ μ.
L/ξ=1
5
10
For L/ ξ >> 1: vc/cs  1 – const  (Vmax )1/2
For L/ ξ << 1: vc/cs  1 – const  (LVmax )2/3
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/ξ=0.5
1
1.57
3
5
10
Periodic potential (bosons)
Bloch band structure.
p = quasi-momentum
pB= Bragg quasi-momentum
ER= p2B/2m = recoil energy
Vmax=sER = lattice strength
Energy density vs. quasi-momentum
Lowest Bloch band for the same gn0=0.4ER and different s
Periodic potential (bosons)
Bloch band structure.
p = quasi-momentum
pB= Bragg quasi-momentum
ER= p2B/2m = recoil energy
Vmax=sER = lattice strength
Energy density vs. quasi-momentum
Periodic potential (bosons)
Bloch band structure.
p = quasi-momentum
pB= Bragg quasi-momentum
ER= p2B/2m = recoil energy
Vmax=sER = lattice strength
Energy density vs. quasi-momentum
The curvature at p=0 gives the
effective mass:
e = n0 p2/2m*
Periodic potential (bosons)
L/ξ=0.5
Bloch band structure.
1
p = quasi-momentum
pB= Bragg quasi-momentum
ER= p2B/2m = recoil energy
Vmax=sER = lattice strength
1.57
3
5
10
Energy density vs. quasi-momentum
Tight-binding limit (Vmax >> µ):
e = δJ [1-cos(πp/pB)]
(2ERn0/π2)(m/m*) = δJ = tunnelling energy
(Josephson current)
Critical p: pc=0.5pB
Critical velocity: vc = (2/π)(m/m*) ER/pB
m/m* proportional to e-2√s
Differences between single barrier and periodic potential (fermions)
Same qualitative behavior as for bosons
Fermions through a barrier
LkF=4
Fermions in a lattice
LkF=0.5
0.89
1.11
1.92
2.5
1.57
What about experiments?
BOSONS: Experiments at LENS-Florence
Weak lattice (energetic vs. dynamic instability)
L/ξ=0.5
1
3
5
10
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)
What about experiments?
BOSONS: Experiments at LENS-Florence
Weak lattice (energetic vs. dynamic instability)
L/ξ=0.5
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)
1
3
L/ξ ≈ 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)
What about experiments?
FERMIONS: Experiments at MIT
LkF=0.5
?
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)]
0.89
1.11
1.92
2.5
1.57
Problem: Which density n0? Which Vmax?
What about experiments?
FERMIONS: Experiments at MIT
LkF=0.5
?
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)]
0.89
1.11
1.92
1.57
Problem: Which density n0? Which Vmax?
2.5
n0
Depending on how one chooses n0 (and
hence EF) one gets
0.5 < EF/ER < 1
or
1 < LkF < 1.5
Vext
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)]
Expt
BdG theory
If EF is by the density at e-2
beam waist:
EF/ER ≈ 0.5 (LkF ≈ 1)
LDA
Significant discrepancies between
theory and MIT data.
Additional dissipation mechanisms?
Nonuniform nature of the gas/lattice ?
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)