Feshbach resonance

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In quantum mechanics, a Feshbach resonance, named after Herman Feshbach, in contrast to a shape resonance, is a resonance of a system with more than one degree of freedom, which would turn into a bound state if the coupling between some degrees of freedom and the degrees of freedom associated to the fragmentation (reaction coordinates) were set to zero.

1 Restriction
2 Examples
3 Historical remark
4 External link

[edit] Restriction

This definition makes sense only if the separable model, which supposes the two groups of degrees of freedom uncoupled, is a meaningful approximation.

[edit] Examples

In the case of atomic and molecular electronic structure problems, it is well known that the self-consistent field (SCF) approximation is relevant at least as a starting point of more elaborated methods. The Slater determinants built from SCF orbitals (atomic or molecular orbitals) are Feshbach resonances if more than one electronic transition are required to emit one electron.

For example, if the ground state of an electronic system can be labeled by $a 2 p 0$ with $a$ a doubly occupied orbital and $p$ a virtual orbital, and if $f E$ corresponds to an incident electron of kinetic energy $E$ , then

$a^2p^0 + f_E \to a^2p^1 \Rightarrow$ shape resonance
$a^2p^0 + f_E \to a^1p^2 [$ with $a^1p^2 \to a^1p^1 + f_{E'>0}$ not allowed $] \Rightarrow$ Feshbach resonance
$a^2p^0 + f_E \to a^1p^2 [$ with $a^1p^2 \to a^1p^1 + f_{E'>0}$ allowed $] \Rightarrow$ core-excited shape resonance

In case (1), $a 2 p 1$ is a shape resonance because the $p \to f_E$ transition is the only one required for the emission of an electron. In case (2), $a 1 p 2$ is a Feshbach resonance because both transitions $p \to f_E$ and $p \to a$ are required to emit one electron. In case (3), $a 1 p 2$ is a shape resonance because only the transition $p \to f_E$ is required to emit one electron. This is a core-excited shape resonance because one of the fragment is left in the excited state $a 1 p 1$ .

A well-known example of vibrational Feshbach resonance is a weakly bound system formed by a rare gas atom and a vibrationally excited diatomic molecule. When the rare gas atom is far from the molecule, it sees a weakly attractive potential (usually Van der Waals interaction). During the collision it may excite the molecule in an excited vibrational state, lose some energy and "fall" into the well of the weakly attractive potential. It would stay trapped into this Feshbach resonance if the coupling between the degrees of freedom corresponding to the movement of the rare gas atom in the neighbourhood of the diatomic and the degree of freedom of the diatomic molecule were zero. This coupling switches this bound state into a Feshbach resonance and is responsible for its finite lifetime.

Feshbach resonances have become important in the study of fermi gases, as these resonances allow for the creation of Bose-Einstein condensates. In the context of Bose-Einstein condensates, 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. This bound state is a bound state of an interatomic Born-Oppenheimer potential, which connects asymptotically to a state with a different hyperfine structure than the colliding pair of atoms does. The magnetic moment of this bound state is different from that of the colliding atoms, because the Born-Oppenheimer potential which "contains" the bound state "has a different hyperfine state" than the colliding pair of atoms. This condition can be satisfied for ultracold (some $μ$ K) alkali atoms. In this case, the energy position of the bound state can be controlled by an external magnetic field which is adjusted in such a way that the ultracold collisions becomes resonant. Recent experiments allows one to use an electric field instead of a magnetic field to create cold molecules via Feshbach resonance. The major difficulty in creating Bose-Einstein condensates of molecules is to cool down their vibrational temperature.

[edit] Historical remark

The name "Feshbach resonance" comes from the seminal paper of Herman Feshbach (Ann. Phys. (N.Y.) 5 (1958) 357) which introduces a method for computing resonant cross sections. This method had previously been introduced by Ugo Fano in Nuovo Cimento 12 (1935) 156 and further developed by Fano in Physical Review 124 (1961) 1866. This method is called the Feshbach–Fano partitioning method. Though this method is particularly well adapted for Feshbach resonances it is very general and can be also applied to shape resonances. This method is closely related to Löwdin's partitioning method (J. Math. Phys. 3 (1962) 969) and Bloch's perturbation theory (Nucl. Phys. 6 (1958) 329).