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

CHAPTER 2. THEORY
by Fano [54], and is known in cold atoms as a Feshbach resonance.
The bound state in the singlet potential exists in different flavors. The
total nuclear spin I leads to an energy shift, splitting the singlet potential
into two, resulting in two Feshbach resonances. The same bound state
exists also in the p-wave scattering potential, where it is shifted because
of the centrifugal barrier mentioned above. As p-wave scattering is
allowed even for fermions in the same internal state, they couple to all
possible combinations of two atoms in the shown states, as shown in the
inset of figure 2.4.
The different couplings call for a rather complicated coupled-channel
treatment, as found in reference [47]. Here we choose a more simplified
model potential [55, 56], which we can actually calculate analytically. For
the open channel, we choose a spherical potential well of depth ¯h 2 q2 o/m
with an interaction range r0, while we use a spherical box for the closed
channel, where the bottom of the box is at ¯h 2 q2 c/m as, depicted in figure
2.5. We assume a coupling term Ω between the two potentials. This is
much weaker than the short range potential between the atoms, so we
can suppose that Ω ≪ q2 o/c . This gives the Schrödinger equation
¯h 2
m (−∇2 + V)|ψ〉 = E|ψ〉 (2.12)
⎧
�
q2 c Ω
Ω q2 o
⎪⎨ −
with V = �
0
⎪⎩
0
�
0
∞
�
for r < r0
for r > r0
(2.13)
In the limit of zero scattering energy E = 0, and setting ψ(r) = χ(r)
r
as above, we can write the general solution to this problem as
|χ〉 = (r − a)|o〉 for r > r0 (2.14a)
|χ〉 = cos φ sin(q+r)|+〉 + sin φ sin(q−r)|−〉 for r < r0 (2.14b)
a is the scattering length, q± are determined by inserting equations (2.14)
into the Schrödinger equation. φ is fixed by the boundary conditions.
|o〉 denotes the open channel, as |c〉 will denote the closed one. |±〉 are
defined by
|+〉 = cos θ|o〉 + sin θ|c〉
|−〉 = − sin θ|o〉 + cos θ|c〉
24
(2.15)