Trapping of neutral particles

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Lecture IVBose-Einstein condensate Superfluidit
y New trends
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Theoretical description of the condensate
The Hamiltonian
Interactions between atoms
Confining potential
At low temperature, we can replace the real
potential by
, a scattering legnth
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Different regime of interactions
The scattering length can be modified a ( B )
Feshbachs resonances
a 0
a gt 0
a lt 0, 3D
a lt 0, 1D
N lt Nc  Collapse 
Gaussian
Parabolic
Soliton
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Experimental realization
Science 296, 1290 (2002)
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Time-dependent Gross-Pitaevski equation Hydrodynam
ic 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
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Thomas Fermi approximation in a trap
Appl. Phys. B 69, 257 (1999)
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Thomas Fermi energy point of view
Kinetic energy Potential energy
Interaction energy
87 Rb a 5 nm N 105 R 1 mm
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Scaling solutions
Equation of continuity
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Scaling solutions Applications
Quadrupole mode
Monopole mode

• Coupling between monopole and quadrupole
• mode in anisotropic harmonic traps

• Time-of-fligth microscope effect

1 mm
100 mm
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Bogoliubov spectrum
uniform
Equilibrium state in a box
Linearization of the hydrodynamic equations
We obtain
speed of sound
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Landau argument for superfluidity
At low momentum, the collective excitations have
a linear dispersion relation
E(P)
Microscopic probe-particle
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)
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HD equations Rotating Frame, Thomas Fermi regime
velocity in the laboratory frame
position in the rotating frame
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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)
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Determination of a
Equation of continuity gives
From which we deduce the equation for a
We introduce the anisotropy parameter
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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
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Velocity field condensate versus classical
Condensate
Classical gas
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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)
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Scissors Mode
PRL 83, 4452 (1999)
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Scissors Mode Qualitative picture (1)
Kinetic energy for rotation
For classical gas
Moment of Inertia
For condensate
Extra potential energy due to anisotropy
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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
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Scissors Mode Quantitative analysis
Classical gas Moment method for ltXYgt
Two modes
and
One mode
Bose-Einstein condensate in the Thomas-Fermi
regime
Linearization of HD equations
One mode
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Experiment (Oxford)
Experimentl proof of reduced moment of
inertia associated as a superfluid behaviour
PRL 84, 2056 (2001)
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Vortices in a rotating quantum fluid
In a condensate
the velocity is such that
incompatible with rigid body rotation
Liquid superfluid helium
Below a critical rotation Wc, no motion at all
Above Wc, apparition of singular lines on which
the density is zero and around which the
circulation of the velocity is quantized
Onsager - Feynman
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Preparation of a condensate with vortices
1. Preparation of a quasi-pure condensate (20
seconds)
Laserevaporative cooling of 87Rb atoms in a
magnetic trap
105 to 4 105 atoms
T lt 100 nK
6 mm
120 mm
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From single to multiple vortices
PRL 84, 806 (2000)
Just below the critical frequency
Just above the critical frequency
Notably above the critical frequency
For large numbers of atoms Abrikosov lattice
It is a real quantum vortex angular momentum h
PRL 85, 2223 (2000)
also at MIT, Boulder, Oxford
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Dynamics of nucleation
PRL 86, 4443 (2001)
Dynamically unstable branch
Stable branch