Feshbach resonance
In physics, a Feshbach resonance can occur upon collision of two slow atoms when they temporarily stick together forming an unstable compound with short lifetime. It is a feature of many-body systems in which a bound state is achieved if the coupling between at least one internal degree of freedom and the reaction coordinates, which lead to dissociation, vanish. The opposite situation, when a bound state is not formed, is a shape resonance. It is named after Herman Feshbach, a physicist at MIT.
Feshbach resonances have become important in the study of cold atoms systems, including Fermi gases and Bose-Einstein condensates. In the context of scattering processes in many-body systems, the Feshbach resonance occurs when the energy of a bound state of an interatomic potential is equal to the kinetic energy of a colliding pair of atoms. In experimental settings, the Feshbach resonances provide a way to vary interaction strength between atoms in the cloud by changing scattering length, asc, of elastic collisions. For atomic species that possess these resonances, it is possible to vary the interaction strength by applying a uniform magnetic field. Among many uses, this tool has served to explore the transition from a BEC of fermionic molecules to weakly interacting fermion-pairs the BCS in Fermi clouds. For the BECs, Feshbach resonances have been used to study a spectrum of systems from the non-interacting ideal Bose gases to the unitary regime of interactions.
Introduction
Consider a general quantum scattering event between two particles. In this reaction, there are two reactant particles denoted by A and B, and two product particles denoted by A' and B' . For the case of a reaction, we may denote this scattering event byThe combination of the species and quantum states of the two reactant particles before or after the scattering event is referred to as a reaction channel. Specifically, the species and states of A and B constitute the entrance channel, while the types and states of A' and B' constitute the exit channel. An energetically accessible reaction channel is referred to as an open channel, whereas a reaction channel forbidden by energy conservation is referred to as a closed channel.
Consider the interaction of two particles A and B in an entrance channel C. The positions of these two particles are given by and, respectively. The interaction energy of the two particles will usually depend only on the magnitude of the separation, and this function, sometimes referred to as a potential energy curve, is denoted by. Often, this potential will have a pronounced minimum and thus admit bound states.
The total energy of the two particles in the entrance channel is
where denotes the total kinetic energy of the relative motion, is the contribution to the energy from couplings to external fields, and represents a vector of one or more parameters such as magnetic field or electric field. We consider now a second reaction channel, denoted by D, which is closed for large values of R. Let this potential curve admit a bound state with energy.
A Feshbach resonance occurs when
for some range of parameter vectors. When this condition is met, then any coupling between channel C and channel D can give rise to significant mixing between the two channels; this manifests itself as a drastic dependence of the outcome of the scattering event on the parameter or parameters that control the energy of the entrance channel. These couplings can arise from spin-exchange interactions or relativistic spin-dependent interactions.
Magnetic Feshbach resonance
In ultracold atomic experiments, the resonance is controlled via the magnetic field and we assume that the kinetic energy is approximately 0. Since the channels differ in internal degrees of freedom such as spin and angular momentum, their difference in energy is dependent on by the Zeeman effect. The scattering length is modified aswhere is the background scattering length, is the magnetic field strength where resonance occurs, and is the resonance width. This allows for manipulation of the scattering length to 0 or arbitrarily high values.
As the magnetic field is swept through the resonance, the states in the open and closed channel can also mix and a large number of atoms, sometimes near 100% efficiency, convert to Feshbach molecules. These molecules have high vibrational states, so they then need to be transitioned to lower, more stable states to prevent dissociation. This can be done through stimulated emissions or other optical techniques such as STIRAP. Other methods include inducing stimulated emission through an oscillating magnetic field and atom-molecule thermalization.