Feshbach Resonances

Part of the Theoretical and Mathematical Physics book series (TMP)


In Sect. 10.3.3 we discussed the possibility of a shape resonance in the scattering length of two atoms. Such a shape resonance occurs when the interatomic potential has a bound state that is very weakly bound and, therefore, has an energy that lies very close to the continuum threshold of the atoms. Because this bound state occurs in the same potential with which the atoms interact, it is difficult to experimentally control such a resonance, and thereby the scattering properties of atoms. To do so would require changing the actual shape of the interatomic potential, which is impossible by a static magnetic bias field. Put differently, for a shape resonance the magnetic moment of the weakly-bound molecule is exactly the same as the magnetic moment of the colliding atoms. As a result, an external magnetic field affects the energy of the bound state in exactly the same manner as the energy of the colliding atoms. One thus cannot change the energy difference between the bound state and the atomic continuum, which would drastically affect the outcome of the scattering process and therefore the effective interaction strength between the atoms. For completeness, we should mention that in principle time-dependent electric fields can be used to influence the scattering length of the atoms in this case, but the physics then actually turns out to be very similar to the physics of Feshbach resonances that we discuss next.

The crucial difference between a shape resonance and a Feshbach resonance is that in the latter case, the molecular state responsible for the resonance has a magnetic moment that is different from the magnetic moment of the colliding atoms [17, 183]. Therefore, the energy difference between the bound state and the atomic continuum can now be easily controlled experimentally by an external magnetic bias field. Theoretically this implies that the scattering process is not described by a single-channel Schrödinger equation, as in the case of a shape resonance, but by a multi-channel or matrix Schrödinger equation, because the scattering wavefunction has now a nonzero amplitude in a number of different spin states.


Molecular State Shape Resonance Feshbach Resonance Closed Channel Ladder Diagram 
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