Lecture III Trapped gases in the classical regime

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Lecture IIITrapped gases in the classical
regime
Bilbao 2004
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Outline
I-Boltzmann equation
II-Method of averages
III-Scaling factors method
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I-Boltzmann equation
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Trapped gases in the dilute regime
d interparticle length l de Broglie
wavelength
l ltlt d collisions dominate (irreversibility) l
gtgt d mean field dominate
To describe the gas The Boltzmann equation
Confinement term
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Mean field and dimensionality
où
For a  pure condensate it remains only the
contribution of the mean field
Gross - Pitaevskii
PRA 66 033613 (2002)
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Stationary solution of the BE in a box
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Exact solutions of the BE in a box
Class of solutions
Normalization
Tail
Gaussian
One can work out explicitly
M. Krook and T. T. Wu, PRL 36 1107 (1976)
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Exact solutions of the BE in an isotropic
harmonic potential
Stationary solution
L. Boltzmann, in Wissenschaftliche Abhandlungen,
edited by F. Hasenorl (Barth, Leipzig, 1909),
Vol. II, p. 83.
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Two   classical types of experiments thermal
gas versus BEC
Time of flight
time
Excitation modes
monopole
quadrupole
time
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II-Method of averages
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Averages
BE
with
and
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Collisional invariants
This is still valid for the quantum Boltzmann
equation
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Monopole mode
Harmonic and isotropic confinement
We obtain a closed set of linear equations
(1)
Linear only for harmonic
confinement
(2)
(3)
We readily obtain the conservation of energy Eq.
(1) Eq. (3)
Valid for bosons or fermions.
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Quadrupolar mode
Linear set of equations for the averages
To solve we need further approximations 1_ One
relaxation time 2_ Gaussian ansatz similar to
the previous approach, but gives also an estimate
for the relaxation time
Test the accuracy by means of a molecular
dynamics (Bird)
Only term affected by collisions
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Quadrupolar modes (results experiments)
Exp ENS
PRA 60 4851 (1999).
Theory
Acta Physica Polonica B 33 p 2213 (2002).
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Quadrupolar mode BEC / thermal cloud in the
hydrodynamic limit
Disk shape
Cigar shape
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Application spinning up a classical gas
Average methods combined with time relaxation
aproach well suited to quadratic potential
Angular momentum can be transferred only
throught elastic collisions. What is the
typical time scale to transfer angular mometum
to the gas ?
PRA 62 033607 (2000).
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Spinning up a classical gas (results)
Angular momentum (rotating anisotropy)
Collisionless regime
Dissipation of angular momentum (static
anisotropy)
with
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Why it could be interested to spin up the thermal
gas
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III-Scaling factors method
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Collisionless gas in 1D
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Equilibrium solution such that
We search for a solution of Eq. 1 of the form
with


Can be easily integrated We find an exact
solution of Eq. 1.
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Collisionless gas in 1D (results)
Modes
By linearizing, oscillation frequency ,
i.e. monopole mode.
time of flight
Lost the information on the initial state We
probe the velocity distribution, it permits to
measure the temperature.
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Time of flight of a collisionless gas in 2D and
3D
Equations
Ellipticity
Ellipticity
reflects the isotropy of the velocity
distribution
temps
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The opposite limit hydrodynamic regime
We search for a solution of the form
Continuity equation
Euler Equation adiabaticity
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Time of flight in the hydrodynamic regime
Inversion of ellipticity at long times i.e.
similar behaviour as for superfluid phases !
Necessity of a quantitative theorie which links
the elastic collision rate to the evolution of
ellipticity.
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Time of flight from an anisotropic trap
Evolution of ellipticity as a function of time
for different collision rate
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Scaling ansatz and approximations
BE with mean field in the time relaxation
approach
Scaling ansatz
Scaling form for the relaxation time
PRA 68 043608 (2003)
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Equations for the scaling parameters
This approach permits to find all the known
results in the collisionless or hydrodynamic
regime, it gives an interpolation from the
collisionless regime to the hydrodynamic
regime. Consistent with numerical
simulations. Recently generalized to include
Fermi statistics
EuroPhys. Lett. 67, 534 (2004)
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Equations for the scaling parameters
Circle experimental points Solid line theory of
scaling parameters with no adjustable parameter
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How to link t0 and the collision rate ?
Ellipticity as a function of time (result of
simulation) fitted with the scaling laws with
only one parameter t0
Deviation from the gaussian anstaz in the
hydrodynamic regime
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Quadrupolar mode (2D)
One can also compare modes and time of flight