Quantum vortex

In physics, a quantum vortex is a topological defect exhibited in superfluids and superconductors. The existence of these quantum vortices were independently predicted by Richard Feynman and Alexei Alexeyevich Abrikosov in the 1950s. They were later observed experimentally in Type-II superconductors, liquid helium, and atomic gases (see Bose-Einstein condensate).

A quantum vortex in a superfluid is different than one in a superconductor. The key similarity is that they are both topological defects, and they are both quantized. In addition, the make up of each quantum vortex is neither superfluid nor superconductor, for each system. In a superfluid, a quantum vortex "carries" the angular momentum, thus allowing the superfluid to rotate; in a superconductor, the vortex carries the magnetic flux.

Vortex in a superfluid

In a superfluid, a quantum vortex is a hole with the superfluid circulating around the vortex; the inside of the vortex may contain excited particles, air, vacuum, etc. The thickness of the vortex depends upon the chemical make-up of the superfluid; in liquid helium, the thickness is on the order of a few Angstroms.

A superfluid has the special property of having phase, given by the wavefunction, and the velocity of the superfluid is proportional to the gradient of the phase. The circulation around any closed loop in the superfluid is zero, if the region enclosed is simply connected. The superfluid is deemed irrotational. However, if the enclosed region actually contains a smaller region that is an absence of superfluid, for example a rod through the superfluid or a vortex, then the circulation is,

:$oint\_\{C\}\; mathbf\{v\}cdot,dmathbf\{l\}\; =\; frac\{hbar\}\{m\}oint\_\{C\}\; ablaphicdot,dmathbf\{l\}\; =\; frac\{hbar\}\{m\}Deltaphi,$

where $hbar$ is Planck's constant divided by $2pi$, m is the mass of the superfluid particle, and $Deltaphi$ is the phase difference around the vortex. Because the wavefunction must return to its same value after an integral number of turns around the vortex (similar to what is described in the Bohr model), then $Deltaphi\; =\; 2pi\; n$, where n is an integer. Thus, we find that the circulation is quantized:

:$oint\_\{C\}\; mathbf\{v\}cdot,dmathbf\{l\}\; =\; frac\{2pihbar\}\{m\}n.$

Vortex in a superconductor

A principal property of superconductors is that they expel magnetic fields; this is called the Meissner effect. If the magnetic field becomes sufficiently strong, one scenario is for the superconductive state to be "quenched". However, in some cases, it may be energertically favorable for the superconductor to form a quantum vortex, which carries a quantized amount of magnetic flux through the superconductor. Meanwhile, the superconductive state prevails in the regions around the vortex. A superconductor that is capable of carrying a vortex is called a type-II superconductor.

Over some enclosed area S, the magnetic flux is

:$Phi\; =\; oint\_Smathbf\{B\}cdotmathbf\{hat\{n,d^2x\; =\; oint\_\{partial\; S\}mathbf\{A\}cdot\; dmathbf\{l\}$.

Substituting a result of London's second equation: $mathbf\{j\}\_s\; =\; -frac\{n\_se\_s^2\}\{m\}mathbf\{A\}\; -\; frac\{n\_se\_shbar\}\{m\}mathbf\{\; abla\}phi$, we find

:$Phi\; =-frac\{m\}\{n\_s\; e^2\}oint\_\{partial\; S\}mathbf\{j\}\_scdot\; dmathbf\{l\}\; +frac\{hbar\}\{e\_s\}oint\_\{partial\; S\}mathbf\{\; abla\}phicdot\; dmathbf\{l\}$,

where "n_s", "m", and "e_s" are the number density, mass and charge of the Cooper pairs.

If the region, S, is large enough so that $mathbf\{j\}\_s\; =\; 0$ along $partial\; S$, then

:$Phi\; =\; frac\{hbar\}\{e\_s\}oint\_\{partial\; S\}mathbf\{\; abla\}phicdot\; dmathbf\{l\}\; =\; frac\{hbar\}\{e\_s\}Deltaphi\; =\; frac\{2pihbar\}\{e\_s\}n$.

tatistical Mechanics of Vortex Lines

If the temperature is raised in a superfluid or a superconductor, the vortex loopsundergo a second-order phase transition. This happens when the configurational entropy overcomes the
Boltzmann factor which suppresses the thermal generation of vortex lines.The lines form a condensate. Since the center of the lines, the vertex cores, are normal liquid or normal conductors, respectively, the condensation transforms the superfluid or superconductor into the normal state.The ensembles of vortex lines and their phase transitions can be described efficiently by a gauge theory.The gauge theory has great similarity with the gauge theory of electrons and photons, the famous quantum electrodynamics (QED)and is therefore called "quantum vortex dynamics" (QVD).

ee also

* vortex
* superfluid
* superfluid film
* Bose-Einstein condensate
* superconductor
* Abrikosov vortex
* Josephson vortex
* Fractional vortices

References

*cite journal |last=Feynman |first=R. P. |authorlink= |coauthors= |year=1955 |month= |title=Application of quantum mechanics to liquid helium |journal=Progress in Low Temperature Physics |volume=1 |issue= |pages=17–53 |issn=00796417 |url= |quote=
* Hagen Kleinert (1985), "Towards a Quantum Field Theory of Defects and Stresses--Quantum Vortex Dynamics in a Film of Superfluid Helium", Int. J. Engng. Sci. 23, 927 (1985), online readable [http://www.physik.fu-berlin.de/~kleinert/121/121.pdf here] .

Books

* Kleinert, H.: "Gauge Fields in Condensed Matter", Vol. I, " SUPERFLOW AND VORTEX LINES", pp. 1--742, [http://www.worldscibooks.com/physics/0356.htm World Scientific (Singapore, 1989)] ; Paperback ISBN 9971-5-0210-0 " (also available online: [http://www.physik.fu-berlin.de/~kleinert/kleiner_reb1/contents1.html Vol. I] . Read pp. 618--688).
* H. Kleinert, "Multivalued Fields in in Condensed Matter, Electrodynamics, and Gravitation", [http://www.worldscibooks.com/physics/6742.html World Scientific (Singapore, 2008)] (also available [http://www.physik.fu-berlin.de/~kleinert/re.html#B9 online] ).