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 the cold atoms systems, both the Fermi gases as well as the Bose–Einstein condensates (BECs).^[1] 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, which have hyperfine structure coupled via Coulomb or exchange interactions. 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 region of the BEC (of fermionic molecules) to the BCS (of weakly interacting fermion-pairs) transition 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

A+B\rightarrow A'+B'

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.

Unstable State

A virtual state, or unstable state is a bound or transient state which can decay into a free state or relax at some finite rate.^[2] This state may be the metastable state of a certain class of Feshbach resonance, "A special case of a Feshbach-type resonance occurs when the energy level lies near the very top of the potential well. Such a state is called 'virtual'"^[3] and may be further contrasted to a shape resonance depending on the angular momentum.^[4] Because of their transient existence, they can require special techniques for analysis and measurement, for example.^[5]^[6]^[7]^[8]

References

R.J. Fletcher; A.L. Gaunt; N. Navon; R. Smith; Z. Hadzibabic (2013). "Stability of a Unitary Bose Gas". Phys. Rev. Lett. Bibcode:2013PhRvL.111l5303F. arXiv:1307.3193 . doi:10.1103/PhysRevLett.111.125303.
Pethick; Smith (2002). Bose–Einstein Condensation in Dilute Gases. Cambridge. ISBN 0-521-66580-9.
Herman Feshbach (1958). "Unified theory of nuclear reactions". Annals of Physics. 5: 357. Bibcode:1958AnPhy...5..357F. doi:10.1016/0003-4916(58)90007-1.
Ugo Fano: Nuovo Cimento 156, 12 (1935)
Ugo Fano: Phys. Rev. 124, 1866 (1961) doi:10.1103/PhysRev.124.1866
Per-Olov Löwdin (1962). "Studies in Perturbation Theory. IV. Solution of Eigenvalue Problem by Projection Operator Formalism". J. Math. Phys. 3,: 969–982. Bibcode:1962JMP.....3..969L. doi:10.1063/1.1724312.
Claude Bloch (1958). "Sur la théorie des perturbations des états liés". Nucl. Phys. 6: 329. Bibcode:1958NucPh...6..329B. doi:10.1016/0029-5582(58)90116-0.

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