# Feshbach resonance

In the field of physics, a Feshbach resonance, named after Herman Feshbach, is a feature of many-body systems in which a bound state is achieved if the coupling(s) 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.

Feshbach resonances have become important in the study of Fermi gases, as these resonances allow for the creation of Bose–Einstein condensates (BECs). In the context of a BEC, 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, which have hyperfine structure coupled via Coulomb or exchange interactions. This condition is rare in our local conditions, but can be satisfied in ultracold alkali atoms.

## 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 (such a as nuclear reaction), we may denote this scattering event by

$A + B \rightarrow A' + B'$ or $A(B,B')A'$.

The 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 $\vec{r}_A$ and $\vec{r}_B$, respectively. The interaction energy of the two particles will usually depend only on the magnitude of the separation $R \equiv |\vec{r}_A - \vec{r}_B|$, and this function, sometimes referred to as a potential energy curve, is denoted by $V_c(R)$. 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

$E = T + V_C(R) + \Delta(\vec{P})$,

where $T$ denotes the total kinetic energy of the relative motion (center-of-mass motion plays no role in the two-body interaction), $\Delta$ is the contribution to the energy from couplings to external fields, and $\vec{P}$ 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 $V_D(R)$ admit a bound state with energy $E_D.$.

A Feshbach resonance occurs when

$E_D \approx T + V_C(R) + \Delta(\vec{P}_0)$

for some range of parameter vectors $\lbrace\vec{P}_0\rbrace$. 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.