- How can a scattering process have bound states?
We are familiar with bound states from our everyday experience. For example,
two hydrogen atoms interact through the Coulomb force. This leads to the
formation of a bound state, namely, the hydrogen molecule.
The most simple model of this situation is the square-well potential. This
potential has a finite strength $V_0$ for $r<R$, where $r$ is the distance
between the two atoms and $R$ is the size of the potential. For $r>R$, the
potential is zero.
It turns out that bound states (in vacuum) occur only for effectively-repulsive
square-well potentials. This connects nicely to the BEC/BCS problem in
ultracold atoms, as the repulsive side has molecules that condense (BEC),
whereas the attractive side has Cooper pairs with a slightly different story
(BCS).
- How can one physically understand this phenomenon of Feshbach resonance?
I will be simplistic in this explanation and will provide specific
references at the end of the answer. The basic ingredients are as follows.
a) Two different states of a pair of atoms. For example, the singlet
and the triplet state.
b) Only one of these states (let's say the singlet) supports a bound state
(the molecule).
c) The molecule has a different magnetic moment than the two atoms. This means
that the energy of the molecule (with respect to two free atoms) can be
shifted using the magnetic field.
d) It turns out that the energy of the molecule governs the scattering
of the two atoms. One way to imagine this is to think that
virtual molecules are formed, and depending on how easy it is to form
them, the scattering is stronger or weaker.
- Is it that one elastic scattering process is mapped to one bound state?
I am not completely sure that such a statement is precise.
I would rephrase it and say that due to the presence of a bound state
a cross-section of an elastic process can be controlled.
- How does this affect the s-wave scattering length?
- How is this process carried out in lab?
As explained in 2, the scattering length depends on the magnetic field.
In the simplest picture, the formula is
$$ a(B) = a_0 \left( 1 - \frac{\Delta B}{B-B_r} \right),$$
where $a_0$ is the scattering length far from the resonance in question,
$\Delta B$ is the so-called width of the resonance, and $B_r$ is the magnetic
field strength at which the resonance happens. Note that the resonance happens
when the energy of the two incoming atoms is equal to the binding energy
of the molecule.
Experimentally, this process is implemented by imposing a static magnetic
field on the ultracold gas sample. The concepts of open and closed channels
are useful here, since the thermal energy can be much less than the energy
shift due to magnetic field,
$$k_B T \ll \Delta \mu B_r,$$
which means that only one channel is open.
In order to understand how Feshbach resonances work in more detail, I found it
useful to get some scattering-theory basics (and look at the case of the square-well potential
in particular) first, and only then tackle the two-channel problem. One book
where both things are discussed is Ultracold quantum fields by Stoof.
From what I remember,
Bose-Einstein condensation in dilute gases by Pethick and Smith
also contains an explanation of the Feshbach resonance.
