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Lecture IV Bose-Einstein condensate Superfluidity New trends. Theoretical description of the condensate. Hartree approximation:. Gross-Pitaevski equation (or non-linear Schrödinger’s equation) :. The Hamiltonian:. Interactions between atoms. Confining potential. - PowerPoint PPT Presentation
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Lecture IV
Bose-Einstein condensateSuperfluidityNew trends
The Hamiltonian:2
1
( ) ( )2
Ni
i i ji i j
pH V r W r r
m
Confiningpotential
Interactionsbetween atoms
At low temperature, we can replace the real potential by :( )i jW r r
( )i jW r r
( )i jg r r
, a : scattering legnth
Hartree approximation: 1 2 1 2( , ,..., ) ( ) ( )... ( )N Nr r r r r r
Gross-Pitaevski equation (or non-linear Schrödinger’s equation) :2
2( ) ( ) ( ) ( )
2V r Ng r r r
m
24 ag
m
Theoretical description of the condensate
The scattering length can be modified: a ( B ) Feshbach’s resonances
a > 0 : Repulsive interactionsa = 0 : Ideal gasa < 0 : Attractive interaction
a = 0 a > 0
Gaussian Parabolic
a < 0, 3D
N < Nc
« Collapse »
a < 0, 1D
Soliton
Different regime of interactions
-0,5 0,0 0,5
0,0
0,2
0,4
0,6
0,8
1,0
int.
opt.
dens
[arb
.uni
ts]
axial direction [mm]
2 m s
3 m s
4 m s
5 m s
6 m s
7 m s
8 m s
8 ms
7 ms
6 ms2 ms
0 1 2 3 4 5 6 7 8 90
25
50
75
100
125
150
[
m]
Temps [ms]0 1 2 3 4 5 6 7 8 9
0
25
50
75
100
125
150
[
m]
Temps [ms]
Experimental realization
Science 296, 1290 (2002)
Time-dependent Gross-Pitaevski equationHydrodynamic 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 nmN = 105
R = 1 m
Scaling solutions
Equation of continuity
Scaling ansatzScaling parametersTime dependent
Normalization
Euler equation
Scaling solutions: Applications
Quadrupole modeMonopole mode
• Time-of-fligth: microscope effect
• Coupling between monopole and quadrupole mode in anisotropic harmonic traps
1 m
100 m
Bogoliubov spectrum
Equilibrium statein a box
uniform
Linearization ofthe hydrodynamicequations
We obtain
speed of sound
At low momentum, the collective excitations have a linear dispersion relation:
P*
E(P*)
Microscopic probe-particle:
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)
Landau argument for superfluidity
before collision and after collision
A solution can exist if and only if
HD equations: Rotating Frame, Thomas Fermi regime
velocity in the laboratory frame
position in the rotating frame
Stationnary solution
We find a shape which is the inverse of a parabola
But with modified frequencies
Introducing the irrotational ansatz
PRL 86, 377 (2001)
Determination of
Equation of continuity gives
From which we deduce the equation for
We introduce the anisotropy parameter
Determination of
dashed line: non-interacting gas
Solutions which break the symmetry of the hamiltonianIt is caused by two-body interactions
Center of mass unstable
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
where
Strong dependence with anisotropy !
PRL 76, 1405 (1996)
Scissors Mode
PRL 83, 4452 (1999)
Scissors Mode: Qualitative picture (1)
Kinetic energy for rotation
For classical gas
For condensate
Extra potential energy due to anisotropy
MomentofInertia
classical
condensate
We infer the existence of a low frequency mode for the classical gas, but not for the Bose-Einstein condensate
Scissors Mode: Qualitative picture (2)
Scissors Mode: Quantitative analysis
Classical gas: Moment method for <XY>
Two modes and
One mode
Bose-Einstein condensate in the Thomas-Fermi regime
One modeLinearization of HD equations
Experiment (Oxford)
PRL 84, 2056 (2001)
Experimentl proof of reduced moment of inertiaassociated as a superfluid behaviour
Vortices in a rotating quantum fluid
In a condensate
the velocity is such that
( )( ) ( ) iS rr r e
v Sm
.nh
v drm
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
incompatible with rigid body rotation v r
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
2 2 2 2 21 1
2 2 zm x y m z
/ 2 200 Hz
/ 2 10 Hzz
105 to 4 105 atoms
T < 100 nK
120 m
6 m
2. Stirring using a laser beam (0.5 seconds)
16 m
Y t
X
2 2 21U( )
2 X Yr m X Y X=0.03 , Y=0.09
controlled with acousto-optic modulators
From single to multiple vortices
Just belowthe critical frequency
Just abovethe critical frequency
Notably abovethe critical frequency
For large numbersof atoms:Abrikosov lattice
PRL 84, 806 (2000)
It is a real quantum vortexangular momentum h PRL 85, 2223 (2000)
also at MIT, Boulder, Oxford
Stable branch
Dynamicallyunstable branch
Dynamics of nucleation
PRL 86, 4443 (2001)
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