Lecture IV
Bose-Einstein condensate
Superfluidity
New trends
Theoretical description of the condensate
pi2
H 
 V ( ri )  W (ri  rj )
i 1 2m
i j
N
The Hamiltonian:
Confining
potential
Interactions
between atoms
At low temperature, we can replace the real potential W ( ri  rj ) by :
W ( ri  rj )
g  (ri  rj )
Hartree approximation:
4 2 a , a : scattering legnth
g
m
( r1 , r2 ,..., rN )   ( r1 ) ( r2 )...  ( rN )
Gross-Pitaevski equation (or non-linear Schrödinger’s equation) :
2

2
  V (r )  Ng  (r )  (r )   ( r )

 2m

Different regime of interactions
The scattering length can be modified: a ( B ) Feshbach’s resonances
a > 0 : Repulsive interactions
a = 0 : Ideal gas
a < 0 : Attractive interaction
a=0
a>0
a < 0, 3D
a < 0, 1D
N < Nc
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Gaussian
Parabolic
Soliton
Experimental realization
2 ms
1,0
int. opt. dens [arb. units]
3 ms
4 ms
5 ms
6 ms
7 ms
8 ms
8 ms
7 ms
0,8
6 ms
0,6
2 ms
0,4
0,2
0,0
-0,5
0,0
0,5
150
150
125
125
100
100
 [m]
 [m]
axial direction [mm]
75
50
50
25
25
0
75
0
0
1
2
3
4
5
6
Temps [ms]
7
8
9
0
1
2
3
4
5
6
7
8
9
Temps [ms]
Science 296, 1290 (2002)
Time-dependent Gross-Pitaevski equation
Hydrodynamic equations Review of Modern Physics 71, 463 (1999)
with the normalization
Phase-modulus formulation
evolve according to a set of hydrodynamic equations (exact formulation):
continuity
euler
Thomas Fermi approximation in a trap
Appl. Phys. B 69, 257 (1999)
Thomas Fermi energy point of view
Kinetic energy
Potential energy
Interaction energy
87
Rb : a = 5 nm
N = 105
R = 1 m
Scaling solutions
Scaling parameters
Time dependent
Scaling ansatz
Normalization
Equation of continuity
Euler equation
Scaling solutions: Applications
Monopole mode
Quadrupole mode
• Coupling between monopole and quadrupole
mode in anisotropic harmonic traps
• Time-of-fligth: microscope effect
1 m
100 m
Bogoliubov spectrum
Equilibrium state
in a box
uniform
Linearization of
the hydrodynamic
equations
We obtain
speed of
sound
Landau argument for superfluidity
At low momentum, the collective
excitations have a linear dispersion
relation:
E(P*)
Microscopic probe-particle:
before collision and
after collision
P*
A solution can exist
if and only if
Conclusion : For
the probe cannot deposit energy
in the fluid. Superfluidity is a consequence of
interactions.
For a macroscopic probe: it also exists a threshold velocity, PRL 91, 090407 (2003)
HD equations: Rotating Frame, Thomas Fermi
regime
velocity in the laboratory frame
position in the rotating frame
Stationnary solution
Introducing the irrotational ansatz
We find a shape which is the inverse of a parabola
But with modified frequencies
PRL 86, 377 (2001)
Determination of a
Equation of continuity gives
From which we deduce the equation for a
We introduce the anisotropy parameter
Determination of a
Center of mass unstable
Solutions which break the symmetry of the hamiltonian
It is caused by two-body interactions
dashed line: non-interacting gas
Velocity field: condensate versus classical
Condensate
Classical gas
Moment of inertia
The expression for the angular momentum is
It gives the value of the moment of inertia, we find
Strong dependence
with anisotropy !
where
PRL 76, 1405 (1996)
Scissors Mode
PRL 83, 4452 (1999)
Scissors Mode: Qualitative picture (1)
Kinetic energy for rotation
Moment
of
Inertia
Extra potential energy due to anisotropy
For classical gas
For condensate
Scissors Mode: Qualitative picture (2)
classical
condensate
We infer the existence of a low frequency mode for
the classical gas, but not for the Bose-Einstein
condensate
Scissors Mode: Quantitative analysis
Classical gas: Moment method for <XY>
Two modes
and
One mode
Bose-Einstein condensate in the Thomas-Fermi regime
Linearization of HD equations
One mode
Experiment (Oxford)
Experimentl proof of reduced moment of inertia
associated as a superfluid behaviour
PRL 84, 2056 (2001)
Vortices in a rotating quantum fluid

iS ( r )
In a condensate  ( r )   ( r ) e
the velocity v 
m
S is such that
nh
 v . dr  m
incompatible with rigid body rotation v    r
Liquid superfluid helium
Below a critical rotation c, no motion at all
Above c, apparition of singular lines on which the density is zero
and around which the circulation of the velocity is quantized
Onsager - Feynman
Preparation of a condensate with vortices
1. Preparation of a quasi-pure condensate (20 seconds)
Laser+evaporative cooling of
87Rb
atoms in a magnetic trap
105 to 4 105 atoms
1
1
2
2
2
m   x  y   m z2 z 2
2
2
6 m
  / 2  200 Hz
T < 100 nK
120 m
 z / 2  10 Hz
2. Stirring using a laser beam (0.5 seconds)
Y
16 m
t
X
1
 U( r )  m 2  X X 2   Y Y 2 
2
controlled with
acousto-optic
modulators
X=0.03 , Y=0.09
From single to multiple vortices
PRL 84, 806 (2000)
Just below
the critical
frequency
Just above
the critical
frequency
Notably above
the critical
frequency
It is a real quantum vortex: angular momentum h
For large numbers
of atoms:
Abrikosov lattice
PRL 85, 2223 (2000)
also at MIT, Boulder, Oxford
Dynamics of nucleation
PRL 86, 4443 (2001)
Dynamically
unstable
branch
Stable
branch