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

CHAPTER 2. THEORY
a shape resonance. The treatment is very generic and holds for many
scattering problems.
If there is a bound state in the scattering potential very close to the
dissociation limit, the scattering length can take on very large values.
This is called a shape resonance and the situation is sketched in figure
2.2.2. For large distances r we can neglect the potential V. In this case,
a bound state wave function takes the form
χ(r ≫ r0) = Ae −r√ 2m|ε|/¯h
(2.8)
where ε is the binding energy. At small distances and for cold atoms,
we can neglect the total energy E in the Schrödinger equation χ ′′ −
2m
¯h 2 U(r)χ = Eχ, since the potential energy is much larger. As we have
neglected the total energy, all wave functions of low scattering energies
are the same, including the bound state. The solutions to this equation
must be connected to the solution for large distances at some point,
and since this solution does not change much as r � 0, we can do this
formally at r = 0. We match the logarithmic boundary condition χ ′ /χ at
r = 0 for the two limiting cases, equation 2.6 and 2.8 and get
�
cot ka = − |ε|/E (2.9)
where E is the energy of the incoming particles, E = ¯h 2 k 2 /2µ.
Here we see that the scattering length can become very large once
the binding energy ε gets small. Our approximation performed in
equation (2.7) does not hold anymore, it would lead to infinite cross
sections, which has no physical meaning. Instead we can write that [46]
a
f = −
1 − 1
2r∗k 2a + ika
(2.10)
Here r ∗ is called the effective range of the potential. The effective range
term does not follow from equation (2.7) but it is introduced to explain
better the behavior at high momentum k. We can use it to explore the
limits of our approximations: the effective range is very small in our
experiments, and there is a natural cut-off for the momenta, the Fermi
momentum kF, such that the effective range term is normally negligible
[47].
However, there are two noteworthy exceptions to this rule. Firstly,
the group at Rice University studied the formation of molecules in the
narrow Feshbach resonance [48], see the next section, where the effective
range is large and negative, which complicated the interpretation of
their data.
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