Feshbach resonance - Wikipedia, the free encyclopedia.pdf

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Feshbach resonanceFrom Wikipedia, the free encyclopedia

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 BoseEinstein 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 var

interaction strength between atoms in the cloud by changing scattering length, asc, of elastic collisions. For

atomic species that possess these resonances (like K39and 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 regio

of the BEC (of fermionic molecules) to the BCS (of weakly interacting fermion-pairs) transition in Fermiclouds. For the BECs, Feshbach resonances have been used to study a spectrum of systems from the non-

interacting ideal Bose gases to the unitaryregime of interactions.

Introduction

Consider a general quantum scattering event between two particles. In this reaction, there are two reactant

particles denoted byAandB, and two product particles denoted byA'andB'. For the case of a reaction

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

or .

The combination of the species and quantum states of the two reactant particles before or after the

scattering eventis referred to as a reaction channel. Specifically, the species and states ofAandBconstitu

the entrance channel, while the types and states ofA'andB'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 particlesAandBin an entrance channel C. Thepositionsof 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 tas 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

,
https://en.wikipedia.org/wiki/Bound_stateshttps://en.wikipedia.org/wiki/Energy_conservationhttps://en.wikipedia.org/wiki/Nuclear_reactionhttps://en.wikipedia.org/w/index.php?title=Reaction_product&action=edit&redlink=1https://en.wikipedia.org/wiki/BEChttps://en.wikipedia.org/wiki/BCS_theoryhttps://en.wikipedia.org/wiki/Kinetic_energyhttps://en.wikipedia.org/wiki/Hyperfine_structurehttps://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_condensatehttps://en.wikipedia.org/w/index.php?title=Cold_atoms&action=edit&redlink=1https://en.wikipedia.org/w/index.php?title=Cold_atoms&action=edit&redlink=1https://en.wikipedia.org/w/index.php?title=Cold_atoms&action=edit&redlink=1https://en.wikipedia.org/wiki/Fermi_gashttps://en.wikipedia.org/wiki/Fermi_gashttps://en.wikipedia.org/wiki/Physicshttps://en.wikipedia.org/wiki/Herman_Feshbachhttps://en.wikipedia.org/wiki/Many-bodyhttps://en.wikipedia.org/wiki/Bound_stateshttps://en.wikipedia.org/wiki/Potential_energy_surfacehttps://en.wikipedia.org/wiki/Energy_conservationhttps://en.wikipedia.org/w/index.php?title=Reaction_channel&action=edit&redlink=1https://en.wikipedia.org/wiki/Nuclear_reactionhttps://en.wikipedia.org/w/index.php?title=Reaction_product&action=edit&redlink=1https://en.wikipedia.org/wiki/Reactanthttps://en.wikipedia.org/wiki/Scatteringhttps://en.wikipedia.org/wiki/Quantumhttps://en.wikipedia.org/wiki/BCS_theoryhttps://en.wikipedia.org/wiki/BEChttps://en.wikipedia.org/wiki/Coulombhttps://en.wikipedia.org/wiki/Hyperfine_structurehttps://en.wikipedia.org/wiki/Kinetic_energyhttps://en.wikipedia.org/wiki/Interatomic_potentialhttps://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_condensatehttps://en.wikipedia.org/wiki/Fermi_gashttps://en.wikipedia.org/w/index.php?title=Cold_atoms&action=edit&redlink=1https://en.wikipedia.org/wiki/Shape_resonancehttps://en.wikipedia.org/wiki/Reaction_coordinatehttps://en.wikipedia.org/wiki/Many-bodyhttps://en.wikipedia.org/wiki/Herman_Feshbachhttps://en.wikipedia.org/wiki/Physics


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where denotes the total kinetic energy of the relative motion (center-of-mass motion plays no role in the

two-body interaction), 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 byD, which is closed for large values ofR. 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 channe

Cand channelDcan 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 (http://edoc.ub.uni-muenchen.de/413/1/Qin_Hua.pdf)

2. Schulz George Resonances in Electron Impact on Atoms and Diatomic Molecules Reviews of Modern Physics

vol 45 no 3 pp378-486 July 1973 (http://www.nist.gov/data/nsrds/NSRDS-NBS50.pdf)

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

(http://books.google.com/books?

id=uqcsAAAAYAAJ&q=%22virtual+state%22+%22feshbach+resonance%22&dq=%22virtual+state%22+%22f

hbach+resonance%22&lr=&ei=IuaRSpvoBJG-ywS7rOy3Bw) 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

(http://prola.aps.org/abstract/PRA/v64/i2/e022708)

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 (http://prola.aps.org/abstract/PRC/v19/i3/p579_1)

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 (http://prola.aps.org/abstract/PRL/v90/i18/e183201)

7. Kurokawa, Chie Masui, Hiroshi Myo, Takayuki Kato, Kiyoshi Study of the virtual state in c10Li with the Jo

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
http://adsabs.harvard.edu/abs/2001APS..HAW.DE004Khttp://prola.aps.org/abstract/PRL/v90/i18/e183201http://prola.aps.org/abstract/PRC/v19/i3/p579_1http://prola.aps.org/abstract/PRA/v64/i2/e022708http://books.google.com/books?id=uqcsAAAAYAAJ&q=%22virtual+state%22+%22feshbach+resonance%22&dq=%22virtual+state%22+%22feshbach+resonance%22&lr=&ei=IuaRSpvoBJG-ywS7rOy3Bwhttp://www.nist.gov/data/nsrds/NSRDS-NBS50.pdfhttp://edoc.ub.uni-muenchen.de/413/1/Qin_Hua.pdfhttp://-/?-http://-/?-http://-/?-http://-/?-http://-/?-https://en.wikipedia.org/wiki/Shape_resonancehttp://-/?-http://-/?-https://en.wikipedia.org/wiki/Electric_fieldhttps://en.wikipedia.org/wiki/Magnetic_field


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(http://adsabs.harvard.edu/abs/2001APS..HAW.DE004K)

R.J. Fletcher, A.L. Gaunt, N. Navon, R. Smith, Z. Hadzibabic (2013). "Stability of a Unitary BoseGas".Phys. Rev. Lett. doi:10.1103/PhysRevLett.111.125303.Pethick Smith (2002).BoseEinstein Condensation in Dilute Gases. Cambridge. ISBN 0-521-66589.Herman Feshbach (1958). "Unified theory of nuclear reactions".Ann. Phys. (N.Y.)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(https://dx.doi.org/10.1103%2FPhysRev.124.1866)Per-Olov Lwdin (1962). "Studies in Perturbation Theory. IV. Solution of Eigenvalue Problem byProjection Operator Formalism".J. Math. Phys.3,: 969-982. doi:10.1063/1.1724312.Claude Bloch (1958). "Sur la thorie des perturbations des tats lis".Nucl. Phys.6: 329.doi:10.1016/0029-5582(58)90116-0.

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