Quantum vortex
Encyclopedia
In physics
, a quantum vortex
is a topological defect
exhibited in superfluids and superconductors. Superfluids and superconductors are states of matter without friction. They exist only at very low temperatures. The existence of these quantum vortices was 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 from one in a superconductor. The key similarity is that they are both topological defects, or surface 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
.
, 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,
where
is Planck's constant divided by 
, m is the mass of the superfluid particle, and 
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
, where n is an integer
. Thus, we find that the circulation is quantized:
. If the magnetic field becomes sufficiently strong, one scenario is for the superconductive state to be "quenched". However, in some cases, it may be energetically 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
.
Substituting a result of London's second equation:
, we find
,
where ns, m, and es are the number density, mass and charge of the Cooper pairs.
If the region, S, is large enough so that
along 
, then
.
The flow of current can cause vortices in a superconductor to move, it causes the electric field due to the phenomenon of electromagnetic induction
. In some circumstances, this leads to energy dissipation and causes the material to display a small amount of electrical resistance
while in the superconducting state.
or a superconductor, the vortex loops
undergo a second-order phase transition. This happens when the configurational entropy
overcomes the
Boltzmann factor
which suppresses the thermal or heat generation of vortex lines.
The lines form a condensate
. Since the center of the lines, the vortex 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).
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, a quantum vortex
Vortex
A vortex is a spinning, often turbulent,flow of fluid. Any spiral motion with closed streamlines is vortex flow. The motion of the fluid swirling rapidly around a center is called a vortex...
is a topological defect
Topological defect
In mathematics and physics, a topological soliton or a topological defect is a solution of a system of partial differential equations or of a quantum field theory homotopically distinct from the vacuum solution; it can be proven to exist because the boundary conditions entail the existence of...
exhibited in superfluids and superconductors. Superfluids and superconductors are states of matter without friction. They exist only at very low temperatures. The existence of these quantum vortices was independently predicted by Richard Feynman
Richard Feynman
Richard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...
and Alexei Alexeyevich Abrikosov
Alexei Alexeyevich Abrikosov
Alexei Alexeyevich Abrikosov is a Soviet and Russian theoretical physicist whose main contributions are in the field of condensed matter physics. He was awarded the Nobel Prize in Physics in 2003.- Biography :...
in the 1950s. They were later observed experimentally in Type-II superconductors, liquid helium
Helium
Helium is the chemical element with atomic number 2 and an atomic weight of 4.002602, which is represented by the symbol He. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas that heads the noble gas group in the periodic table...
, and atomic gases (see Bose-Einstein condensate).
A quantum vortex in a superfluid is different from one in a superconductor. The key similarity is that they are both topological defects, or surface 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
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
, thus allowing the superfluid to rotate; in a superconductor, the vortex carries the magnetic flux
Magnetic flux
Magnetic flux , is a measure of the amount of magnetic B field passing through a given surface . The SI unit of magnetic flux is the weber...
.
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 heliumHelium
Helium is the chemical element with atomic number 2 and an atomic weight of 4.002602, which is represented by the symbol He. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas that heads the noble gas group in the periodic table...
, the thickness is on the order of a few Angstroms.
A superfluid
Superfluid
Superfluidity is a state of matter in which the matter behaves like a fluid without viscosity and with extremely high thermal conductivity. The substance, which appears to be a normal liquid, will flow without friction past any surface, which allows it to continue to circulate over obstructions and...
has the special property of having phase, given by the wavefunction
Wavefunction
Not to be confused with the related concept of the Wave equationA wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of...
, and the velocity of the superfluid is proportional to the gradient
Gradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
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,
where
is Planck's constant divided by , m is the mass of the superfluid particle, and 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 modelBohr model
In atomic physics, the Bohr model, introduced by Niels Bohr in 1913, depicts the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits around the nucleus—similar in structure to the solar system, but with electrostatic forces providing attraction,...
), then
, where n is an integerInteger
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
. Thus, we find that the circulation is quantized:
Vortex in a superconductor
A principal property of superconductors is that they expel magnetic fields; this is called the Meissner effectMeissner effect
The Meissner effect is the expulsion of a magnetic field from a superconductor during its transition to the superconducting state. The German physicists Walther Meissner and Robert Ochsenfeld discovered the phenomenon in 1933 by measuring the magnetic field distribution outside superconducting tin...
. If the magnetic field becomes sufficiently strong, one scenario is for the superconductive state to be "quenched". However, in some cases, it may be energetically 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
Magnetic flux
Magnetic flux , is a measure of the amount of magnetic B field passing through a given surface . The SI unit of magnetic flux is the weber...
is
.Substituting a result of London's second equation:
, we find,where ns, m, and es are the number density, mass and charge of the Cooper pairs.
If the region, S, is large enough so that
along , then.The flow of current can cause vortices in a superconductor to move, it causes the electric field due to the phenomenon of electromagnetic induction
Electromagnetic induction
Electromagnetic induction is the production of an electric current across a conductor moving through a magnetic field. It underlies the operation of generators, transformers, induction motors, electric motors, synchronous motors, and solenoids....
. In some circumstances, this leads to energy dissipation and causes the material to display a small amount of electrical resistance
Electrical resistance
The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...
while in the superconducting state.
Statistical Mechanics of Vortex Lines
If the temperature is raised in a superfluidSuperfluid
Superfluidity is a state of matter in which the matter behaves like a fluid without viscosity and with extremely high thermal conductivity. The substance, which appears to be a normal liquid, will flow without friction past any surface, which allows it to continue to circulate over obstructions and...
or a superconductor, the vortex loops
undergo a second-order phase transition. This happens when the configurational entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...
overcomes the
Boltzmann factor
Boltzmann factor
In physics, the Boltzmann factor is a weighting factor that determines the relative probability of a particle to be in a state i in a multi-state system in thermodynamic equilibrium at temperature T...
which suppresses the thermal or heat generation of vortex lines.
The lines form a condensate
Bose–Einstein condensate
A Bose–Einstein condensate is a state of matter of a dilute gas of weakly interacting bosons confined in an external potential and cooled to temperatures very near absolute zero . Under such conditions, a large fraction of the bosons occupy the lowest quantum state of the external potential, at...
. Since the center of the lines, the vortex cores, are normal liquid or
normal conductors, respectively, the condensation transforms the superfluid
Superfluid
Superfluidity is a state of matter in which the matter behaves like a fluid without viscosity and with extremely high thermal conductivity. The substance, which appears to be a normal liquid, will flow without friction past any surface, which allows it to continue to circulate over obstructions and...
or superconductor into the normal state.
The ensembles of vortex lines and their phase transitions can be described efficiently by a gauge theory
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
.
The gauge theory has great similarity with the gauge theory of electrons and photons, the famous quantum electrodynamics
Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...
(QED),
and is therefore called "quantum vortex dynamics" (QVD).