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). 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

$A + B \rightarrow A' + B'$ 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.

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.^[1] 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'"^[2] and may be further contrasted to a shape resonance depending on the angular momentum.^[3] Because of their transient existence, they can require special techniques for analysis and measurement, for example.^[4][5][6][7]

References

1. On the Dynamics of Single-Electron Tunneling in Semiconductor Quantum Dots under Microwave Radiation Dissertation Physics Department of Ludwig-Maximilians-Universitat Munchen by Hua Qin from Wujin, China 30 July 2001, Munchen
2. Schulz George Resonances in Electron Impact on Atoms and Diatomic Molecules Reviews of Modern Physics vol 45 no 3 pp378-486 July 1973
3. Donald C. Lorents, Walter Ernst Meyerhof, James R. Peterson Electronic and atomic collisions: invited papers of the XIV International Conference on the Physics of Electronic and Atomic Collisions, Palo Alto, California, 24-30 July, 1985 North-Holland, 1986 ISBN 0-444-86998-0, ISBN 978-0-444-86998-2 page 800
4. D. Field1 *, N. C. Jones1, S. L. Lunt1, and J.-P. Ziesel2 Experimental evidence for a virtual state in a cold collision: Electrons and carbon dioxide Phys. Rev. A 64, 022708 (2001) 10.1103/PhysRevA.64.022708
5. B. A. Girard and M. G. Fuda Virtual state of the three nucleon system Phys. Rev. C 19, 579 - 582 (1979) 10.1103/PhysRevC.19.579
6. Tamio Nishimura * and Franco A. Gianturco Virtual-State Formation in Positron Scattering from Vibrating Molecules: A Gateway to Annihilation Enhancement Phys. Rev. Lett. Volume 90Issue 18 Phys. Rev. Lett. 90, 183201 (2003) 10.1103/PhysRevLett.90.183201
7. Kurokawa, Chie; Masui, Hiroshi; Myo, Takayuki; Kato, Kiyoshi Study of the virtual state in νc10Li with the Jost function method American Physical Society, First Joint Meeting of the Nuclear Physicists of the American and Japanese Physical Societies October 17 - 20, 2001 Maui, Hawaii Meeting ID: HAW01, abstract #DE.004

• R.J. Fletcher, A.L. Gaunt, N. Navon, R. Smith, Z. Hadzibabic (2013). "Stability of a Unitary Bose Gas". Phys. Rev. Lett. 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. 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. doi:10.1063/1.1724312.
• Claude Bloch (1958). "Sur la théorie des perturbations des états liés". Nucl. Phys. 6: 329. doi:10.1016/0029-5582(58)90116-0.