Martin Teichmann Atomes de lithium-6 ultra froids dans la ... - TEL

5.3. HYDRODYNAMIC EXPANSION
and take the ellipticity inversion as a signature for superfluidity [108].
For the collisional regime, however, it was pointed out by Jackson et al.
[109] 2 that at very strong interactions the cloud will also show ellipticity
inversion, but this effect is much smaller than the ellipticity resulting
from a superfluid cloud. Therefore we calculate the ellipticity that we
expect from a hydrodynamic theory and compare to the experiment,
and take the gas to be superfluid once the ellipticity inversion is close to
the predicted value.
5.3.2 The scaling ansatz
We now want to calculate the ellipticity of an expanding superfluid. In a
superfluid, all particles have condensed into one macroscopic wavefunction
ψ. This wavefunction follows the continuity equation of quantum
mechanics,
∂|ψ| 2
+ ∇ · j = 0 (5.9)
∂t
where the probability current j is defined by 2mj = ψ∇ψ∗ − ψ∗∇ψ. We write the wave function as ψ = |ψ| exp iφ. Then the density is
n = |ψ| 2 and the probability current becomes j = (¯h/m)n∇φ. After
having defined a velocity v = j/n, we are left with the continuity equation
of classical fluid mechanics:
∂n
∂t
+ ∇ · (nv) = 0 (5.10)
A superfluid is a liquid with vanishing viscosity. Such a liquid is described
by the Euler equation, which reads, after having used the Gibbs-Duhem
relation [40], as
m ∂v
∂t
�
+ ∇ µ(n) + Vext(r) + 1
2 mv2
�
= 0 (5.11)
Now we are supposed to plug in the results from BCS theory, see
section 2.4, for the equation of state µ(n). Unfortunately, this leads to
coupled differential equations which are only numerically solvable. A
good approximation is to use a polytropic equation of state, µ(n) ∝ n γ ,
where γ is the polytropic exponent [108].
Recently, a large amount of literature was published regarding this
exponent since it is crucial in describing the Innsbruck [29] and Duke
2 This paper uses the term “hydrodynamic” only for the superfluid state as we do,
opposed to Gehm et al., see last footnote
97