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, asc, 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.


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

or .

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 and , respectively. The interaction energy of the two particles will usually depend only on the magnitude of the separation , and this function, sometimes referred to as 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


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 by D, which is closed for large values of R. 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 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]


  1. ^ Chin, Cheng; Grimm, Rudolf; Julienne, Paul; Tiesinga, Eite (2010-04-29). "Feshbach resonances in ultracold gases". Reviews of Modern Physics. 82 (2): 1225–1286. arXiv:0812.1496. Bibcode:2010RvMP...82.1225C. doi:10.1103/RevModPhys.82.1225.
  2. ^ 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
  3. ^ Schulz George Resonances in Electron Impact on Atoms and Diatomic Molecules Reviews of Modern Physics vol 45 no 3 pp378-486 July 1973
  4. ^ 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
  5. ^ 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
  6. ^ 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
  7. ^ 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
  8. ^ 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
BCS theory

BCS theory or Bardeen–Cooper–Schrieffer theory (named after John Bardeen, Leon Cooper, and John Robert Schrieffer) is the first microscopic theory of superconductivity since Heike Kamerlingh Onnes's 1911 discovery. The theory describes superconductivity as a microscopic effect caused by a condensation of Cooper pairs into a boson-like state. The theory is also used in nuclear physics to describe the pairing interaction between nucleons in an atomic nucleus.

It was proposed by Bardeen, Cooper, and Schrieffer in 1957; they received the Nobel Prize in Physics for this theory in 1972.


A bosenova or bose supernova is a very small, supernova-like explosion, which can be induced in a Bose–Einstein condensate (BEC) by changing the external magnetic field, so that the "self-scattering" interaction transitions from repulsive to attractive due to the Feshbach resonance, causing the BEC to "collapse and bounce" or "rebound."Although the total energy of the explosion is very small, the "collapse and bounce" scenario qualitatively resembles a condensed matter version of a core-collapse supernova, hence the term bosenova. The nomenclature is a play of words on the Brazilian music style, bossa nova.

Bose–Einstein condensate

A Bose–Einstein condensate (BEC) is a state of matter of a dilute gas of bosons cooled to temperatures very close to absolute zero (-273.15 °C). Under such conditions, a large fraction of bosons occupy the lowest quantum state, at which point microscopic quantum phenomena, particularly wavefunction interference, become apparent macroscopically. A BEC is formed by cooling a gas of extremely low density, about one-hundred-thousandth the density of normal air, to ultra-low temperatures.

This state was first predicted, generally, in 1924–1925 by Satyendra Nath Bose and Albert Einstein.

Core-excited shape resonance

A core-excited shape resonance is a shape resonance in a system with more than one degree of freedom where, after fragmentation, one of the fragments is in an excited state. It is sometimes very difficult to distinguish a core-excited shape resonance from a Feshbach resonance.

Hendricus Stoof

Hendricus Theodorus Christiaan "Henk" Stoof (born 1962) is a professor in theoretical physics at Utrecht University in the Netherlands. His main interests are atomic physics, condensed matter physics and many-body physics. He is a Fellow of the American Physical Society.

During the last ten years, the group of H.T.C. Stoof has been involved in the study of various aspects of the physics of ultracold atomic gases. In addition, they performed research on skyrmion lattices in the quantum Hall effect and collective modes in supersolid 4He. Below the results obtained from the study of degenerate Fermi gases are briefly summarized.

Already in 1996 they predicted that an atomic gas of 6Li (a fermionic isotope of lithium) becomes a Bardeen-Cooper-Schrieffer (BCS) superfluid at experimentally obtainable temperatures. They have also performed a detailed study of the superfluid behaviour of this gas below the critical temperature. Motivated by this work, at least six experimental groups from around the world, including the groups of R. Grimm, R.G. Hulet, D.S. Jin, and W. Ketterle, started trying to achieve the necessary conditions for the BCS transition in 6Li.

In the last seven years the study of superfluidity in Fermi gases has been at the center of attention of the ultracold atoms community. It is fair to say that the very successful experiments, that ultimately have led to the creation of the superconductor with, as a fraction of the Fermi energy, the highest critical temperature ever, have only been possible due to the use of so-called Feshbach resonances. These resonances were theoretically co-discovered by H.T.C. Stoof in the alkalis in 1993.

At that time, the full potential of a Feshbach resonance for studying the crossover from a BCS superconductor of Bose-Einstein condensed Cooper pairs to a Bose-Einstein condensate (BEC) of molecules was not realized yet, but this crossover is now well understood due to the strong connection between experiment and ab initio theory that is possible in this field. The group has made important contributions to the present understanding of how many-body physics affects the BEC-BCS crossover, and how to incorporate the two-body physics of the Feshbach resonance exactly into the many-body theory. Henk Stoof was elected as an APS Fellow for these contributions.

In the last three years, the group of R.G. Hulet at Rice and the group of W. Ketterle at MIT have pioneered the experimental study of spin imbalance on the superfluid state. These experiments have especially concentrated on the strongly interacting or unitarity limit exactly at resonance where the attraction between the atoms is as large as quantum mechanics allows. Again, the group of H.T.C. Stoof made important contributions to this topic. For example, they were first to predict the topology of the universal phase diagram of this unitarity gas, that is now confirmed by the experiments and that contains a tricritical temperature below which the gas phase separates between an (almost) equal density superfluid and a polarized normal gas. Making use of renormalization group techniques, they are up to now the only theoretical group that has been able to accurately calculate this strong-coupling tricritical temperature from first principles. They believe that an important reason for the success in this area of physics is that they have a background in both the microscopic atomic physics and in the macroscopic condensed-matter physics. It is only through a combination of this knowledge that one can arrive at sophisticated many-body theories that can be directly compared with experiment without any fitting parameters.

Herman Feshbach

Herman Feshbach (February 2, 1917, in New York City – 22 December 2000, in Cambridge, Massachusetts) was an American physicist. He was an Institute Professor Emeritus of physics at MIT. Feshbach is best known for Feshbach resonance and for writing, with Philip M. Morse, Methods of Theoretical Physics.

Landau–Zener formula

The Landau–Zener formula is an analytic solution to the equations of motion governing the transition dynamics of a two-state quantum system, with a time-dependent Hamiltonian varying such that the energy separation of the two states is a linear function of time. The formula, giving the probability of a diabatic (not adiabatic) transition between the two energy states, was published separately by Lev Landau, Clarence Zener, Ernst Stueckelberg, and Ettore Majorana, in 1932.

If the system starts, in the infinite past, in the lower energy eigenstate, we wish to calculate the probability of finding the system in the upper energy eigenstate in the infinite future (a so-called Landau–Zener transition). For infinitely slow variation of the energy difference (that is, a Landau–Zener velocity of zero), the adiabatic theorem tells us that no such transition will take place, as the system will always be in an instantaneous eigenstate of the Hamiltonian at that moment in time. At non-zero velocities, transitions occur with probability as described by the Landau–Zener formula.

Negative temperature

In quantum thermodynamics, certain systems can achieve negative temperature; that is, their temperature can be expressed as a negative quantity on the Kelvin or Rankine scales.

A system with a truly negative temperature on the Kelvin scale is hotter than any system with a positive temperature. If a negative-temperature system and a positive-temperature system come in contact, heat will flow from the negative- to the positive-temperature system. A standard example of such a system is population inversion in laser physics.

Temperature is loosely interpreted as the average kinetic energy of the system's particles. The existence of negative temperature, let alone negative temperature representing "hotter" systems than positive temperature, would seem paradoxical in this interpretation. The paradox is resolved by considering the more rigorous definition of thermodynamic temperature as the tradeoff between internal energy and entropy contained in the system, with "coldness", the reciprocal of temperature, being the more fundamental quantity. Systems with a positive temperature will increase in entropy as one adds energy to the system, while systems with a negative temperature will decrease in entropy as one adds energy to the system.Classical thermodynamic systems cannot achieve negative temperatures: adding heat always increases their entropy. The possibility of a decrease in entropy as energy increases requires the system to "saturate" in entropy. This is only possible if the number of high energy states is limited. In classical Boltzmann statistics, the number of high energy states is unlimited (particle speeds can in principle be increased indefinitely). Systems bounded by a maximum amount of energy are generally forbidden in classical mechanics, and the phenomenon of negative temperature is strictly a

quantum mechanical phenomenon. Some systems, however (see the examples below), have a maximum amount of energy that they can hold, and as they approach that maximum energy their entropy actually begins to decrease.


A polaron is a quasiparticle used in condensed matter physics to understand the interactions between electrons and atoms in a solid material. The polaron concept was first proposed by Lev Landau in 1933 to describe an electron moving in a dielectric crystal where the atoms move from their equilibrium positions to effectively screen the charge of an electron, known as a phonon cloud. This lowers the electron mobility and increases the electron's effective mass.

The general concept of a polaron has been extended to describe other interactions between the electrons and ions in metals that result in a bound state, or a lowering of energy compared to the non-interacting system. Major theoretical work has focused on solving Fröhlich and Holstein Hamiltonians. This is still an active field of research to find exact numerical solutions to the case of one or two electrons in a large crystal lattice, and to study the case of many interacting electrons.

Experimentally, polarons are important to the understanding of a wide variety of materials. The electron mobility in semiconductors can be greatly decreased by the formation of polarons. Organic semiconductors are also sensitive to polaronic effects, which is particularly relevant in the design of organic solar cells that effectively transport charge. The electron phonon interaction that forms Cooper pairs in low-Tc superconductors (type-I superconductors) can also be modeled as a polaron, and two opposite spin electrons may form a bipolaron sharing a phonon cloud. This has been suggested as a mechanism for Cooper pair formation in high-Tc superconductors (type-II superconductors). Polarons are also important for interpreting the optical conductivity of these types of materials.

The polaron, a fermionic quasiparticle, should not be confused with the polariton, a bosonic quasiparticle analogous to a hybridized state between a photon and an optical phonon.


In mathematics, a q-analog of a theorem, identity or expression is a generalization involving a new parameter q that returns the original theorem, identity or expression in the limit as q → 1. Typically, mathematicians are interested in q-analogs that arise naturally, rather than in arbitrarily contriving q-analogs of known results. The earliest q-analog studied in detail is the basic hypergeometric series, which was introduced in the 19th century.q-analogues are most frequently studied in the mathematical fields of combinatorics and special functions. In these settings, the limit q → 1 is often formal, as q is often discrete-valued (for example, it may represent a prime power).

q-analogs find applications in a number of areas, including the study of fractals and multi-fractal measures, and expressions for the entropy of chaotic dynamical systems. The relationship to fractals and dynamical systems results from the fact that many fractal patterns have the symmetries of Fuchsian groups in general (see, for example Indra's pearls and the Apollonian gasket) and the modular group in particular. The connection passes through hyperbolic geometry and ergodic theory, where the elliptic integrals and modular forms play a prominent role; the q-series themselves are closely related to elliptic integrals.

q-analogs also appear in the study of quantum groups and in q-deformed superalgebras. The connection here is similar, in that much of string theory is set in the language of Riemann surfaces, resulting in connections to elliptic curves, which in turn relate to q-series.

Q-exponential distribution

The q-exponential distribution is a probability distribution arising from the maximization of the Tsallis entropy under appropriate constraints, including constraining the domain to be positive. It is one example of a Tsallis distribution. The q-exponential is a generalization of the exponential distribution in the same way that Tsallis entropy is a generalization of standard Boltzmann–Gibbs entropy or Shannon entropy. The exponential distribution is recovered as

Originally proposed by the statisticians George Box and David Cox in 1964, and known as the reverse Box–Cox transformation for a particular case of power transform in statistics.

Scientific phenomena named after people

This is a list of scientific phenomena and concepts named after people (eponymous phenomena). For other lists of eponyms, see eponym.

Shape resonance

A shape resonance is a metastable state in which an electron is trapped due the shape of a potential barrier.

Altunata describes a state as being a shape resonance if, "the internal state of the system remains unchanged upon disintegration of the quasi-bound level."

A more general discussion of resonances and their taxonomies in molecular system can be found in the review article by Schulz,; for the discovery of the Fano resonance line-shape and for the Majorana pioneering work in this field by Antonio Bianconi; and for

a mathematical review by Combes et al.


Superstripes is a generic name for a phase with spatial broken symmetry that favors the onset of superconducting or superfluid

quantum order. This scenario emerged in the 1990s when no-homogeneous metallic heterostructures at the atomic limit with a broken spatial symmetry have been found to favor superconductivity. Before a broken spatial symmetry was expected to compete and suppress the superconducting order. The driving mechanism for the amplification of the superconductivity critical temperature in superstripes matter has been proposed to be the shape resonance in the energy gap parameters ∆n that is a type of Fano resonance for coexisting condensates.The superstripes show multigap superconductivity near a 2.5 Lifshitz transition where the renormalization of chemical potential at the metal-to-superconductor transition is not negligeable and the self-consistent solution of the gaps equation is required. The superstripes lattice scenario is made of puddles of multigap superstripes matter forming a superconducting network where different gaps are not only different in different portions of the k-space but also in different portions of the real space with a complex scale free distribution of Josephson junctions.

Virtual state

In quantum physics, a virtual state is a very short-lived, unobservable quantum state.In many quantum processes a virtual state is an intermediate state, sometimes described as "imaginary" in a multi-step process that mediates otherwise forbidden transitions. Since virtual states are not eigenfunctions of anything, normal parameters such as occupation, energy and lifetime need to be qualified. No measurement of a system will show one to be occupied, but they still have lifetimes derived from uncertainty relations. While each virtual state has an associated energy, no direct measurement of its energy is possible but various approaches have been used to make some measurements (for example see and related work on virtual state spectroscopy) or extract other parameters using measurement techniques that depend upon the virtual state's lifetime. The concept is quite general and can be used to predict and describe experimental results in many areas including Raman spectroscopy, non-linear optics generally, various types of photochemistry, and/or nuclear processes.

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