Feshbach resonance

Feshbach resonances have become important in the study of cold atoms systems, including Fermi gases and Bose–Einstein condensates (BECs). 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, a_sc, of elastic collisions. For atomic species that possess these resonances (like K³⁹ and K⁴⁰), 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 (such as a nuclear reaction), we may denote this scattering event by

${\displaystyle A+B\to A^{\prime }+B^{\prime }}$ or ${\displaystyle A(B,B^{\prime })A^{\prime }}$.

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

${\displaystyle E=T+V_c(R)+\mathrm{\Delta }(\overrightarrow{P})}$,

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

A Feshbach resonance occurs when

${\displaystyle E_D\approx T+V_c(R)+\mathrm{\Delta }({\overrightarrow{P}}_0)}$

for some range of parameter vectors ${\displaystyle \{{\overrightarrow{P}}_0\}}$. 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.