# Feshbach resonance in simple terms

I was reading up Feshbach resonances in cold atoms and I was unable to grasp the concept. I will tell you what I have understood. We consider two body scattering processes elastic as well as inelastic. There are entry channels, i.e. the incoming particles and exit channel viz., the scattered particles. Then there are open channels which are the energetically allowed exit channels (depending on energy of the entry channel) and closed channels, which are forbidden energetically. Now, I am unable to grasp the statement "Elastic scattering in one channel can be altered dramatically if there is a low-energy bound state in a second channel which is closed."

1. How can a scattering process have bound states?

2. How can one physically understand this phenomenon of Feshbach resonance?

3. Is it that one elastic scattering process is mapped to one bound state?

4. How does this affect the s-wave scattering length?

5. How is this process carried out in lab?

It would be helpful if somebody could answer what they have understood and not just give references.

• Good question, I will try to explain if I have time later this week. – ffc Apr 21 '15 at 10:18

1. 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).

1. 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.

1. 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.

1. How does this affect the s-wave scattering length?
2. 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.

• Regard to your first point: "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)." <<<1>>> Why the bound states can (only) occur in the effective repulsive potential? It's counter-intuitive. <<<2>>> And, according to your argument, you seems to say that cooper pairs is not bound states? – 喵喵是我的猫猫 Jan 30 '16 at 2:43
• Can you briefly explain the word "Resonance". The magnetic field changes the energy of the atom, so the atom resonant with what? – 喵喵是我的猫猫 Jan 30 '16 at 2:51