Aharonov–Casher effect

The Aharonov–Casher effect is a quantum mechanical phenomenon predicted in 1984 by Yakir Aharonov and Aharon Casher,[1] in which a traveling magnetic dipole is affected by an electric field. It is dual to the Aharonov–Bohm effect, in which the quantum phase of a charged particle depends upon which side of a magnetic flux tube it comes through. In the Aharonov–Casher effect, the particle has a magnetic moment and the tubes are charged instead. It was observed in a gravitational neutron interferometer in 1989[2] and later by fluxon interference of magnetic vortices in Josephson junctions.[3] It has also been seen with electrons and atoms.

In both effects the particle acquires a phase shift (${\displaystyle \varphi }$) while traveling along some path P. In the Aharonov–Bohm effect it is

${\displaystyle \varphi _{\rm {AB}}={\frac {q}{\hbar }}\int _{P}\mathbf {A} \cdot d\mathbf {x} }$

While for the Aharonov–Casher effect it is

${\displaystyle \varphi _{\rm {AC}}={\frac {1}{\hbar c^{2}}}\int _{P}(\mathbf {E} \times {\boldsymbol {\mu }})\cdot d\mathbf {x} }$

where ${\displaystyle q}$ is its charge and ${\displaystyle {\boldsymbol {\mu }}}$ is the magnetic moment. The effects have been observed together.[4]