Tonks-Girardeau gas
Encyclopedia
In physics
, a Tonks–Girardeau gas
is a quantum degenerate Bose Gas in which the repulsive interactions between bosonic particles
confined to one dimension
dominate the physics of the system. It is named after physicists Marvin D. Girardeau
and Lewi Tonks
. Strictly speaking this is not a Bose–Einstein condensate
as it does not demonstrate any of the characteristics, such as off diagonal long range order or a unitary two body correlation function, even in a thermodynamic limit and as such cannot by described by a macroscopically occupied orbital (order parameter) in the Gross Pitaevskii formulation.
Consider a row of bosons all confined to a one-dimensional line. They cannot pass each other and therefore cannot exchange places. The resulting motion has been compared to a traffic jam: the motion of each boson would be strongly correlated with that of its two neighbours.
Because the particles cannot exchange places, one might expect their behaviour to be fermion
ic, but it turns out that their behaviour differs from that of fermions in several important ways: the particles can all occupy the same momentum state which corresponds to neither Bose–Einstein nor Fermi–Dirac statistics.
The fermionic exchange rule
implies more than the exclusion of two particles from the same point: in addition, the momentum of two identical fermions can never be the same, wherever they are located. Mathematically, there is an exact one-to-one mapping
of impenetrable bosons (in a one-dimensional system) onto a system of fermions that do not interact at all.
In the case of a Tonks–Girardeau gas (TG), so many properties of this one-dimensional string of bosons would be sufficiently fermion-like that the situation is often referred to as the 'fermionization' of bosons.Tonks–Girardeau gas coincide with quantum Nonlinear Schrödinger equation
for infinite repulsion, which can be efficiently analyzed by Quantum inverse scattering method
. This relation help to study Correlation function (statistical mechanics)
. The correlation functions
can be described by Integrable system
. In a simple case it is Painlevé transcendents
. A textbook explains in detail the description of quantum correlation functions of Tonks–Girardeau gas by means of classical completely integrable differential equations. Thermodynamics
of Tonks–Girardeau gas was described by Chen Ning Yang.
, physicist Belén Paredes and coworkers present a technique of creating such a gas using an optical lattice
.
The optical lattice is formed by six intersecting laser
beams, which generate an interference pattern. The beams are arranged as standing wave
s along three orthogonal directions. This results in an array of optical dipole traps
where atom
s are stored in the intensity
maxima of the interference pattern.
The researchers first loaded ultracold rubidium
atoms into one-dimensional tubes formed by a two-dimensional lattice (the third standing wave is off for the moment). This lattice is very strong, so that the atoms do not have enough energy to tunnel between neighbouring tubes. On the other hand, the density is still too low for the transition to the TG regime. For that, the third axis
of the lattice is used. It is set to a lower intensity than the other two axes, so that tunneling in this direction stays possible. For increasing intensity of the third lattice, atoms in the same lattice well are more and more tightly trapped, which increases the collision
al energy. When the collisional energy becomes much bigger than the tunneling energy, the atoms can still tunnel into empty lattice wells, but not into or across occupied ones.
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 Tonks–Girardeau gas
Gas
Gas is one of the three classical states of matter . Near absolute zero, a substance exists as a solid. As heat is added to this substance it melts into a liquid at its melting point , boils into a gas at its boiling point, and if heated high enough would enter a plasma state in which the electrons...
is a quantum degenerate Bose Gas in which the repulsive interactions between bosonic particles
Boson
In particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....
confined to one dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
dominate the physics of the system. It is named after physicists Marvin D. Girardeau
Marvin D. Girardeau
Marvin D. Girardeau is a quantum physicist, currently a research professor at the University of Arizona. One of Girardeau's achievements was to predict the existence of the Tonks-Girardeau gas in 1960...
and Lewi Tonks
Lewi Tonks
Lewi Tonks was an American quantum physicist noted for his discovery of the Tonks-Girardeau gas.Tonks was employed by the General Electric for most of his working life, researching microwaves and ferromagnetism...
. Strictly speaking this is not a Bose–Einstein 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...
as it does not demonstrate any of the characteristics, such as off diagonal long range order or a unitary two body correlation function, even in a thermodynamic limit and as such cannot by described by a macroscopically occupied orbital (order parameter) in the Gross Pitaevskii formulation.
Consider a row of bosons all confined to a one-dimensional line. They cannot pass each other and therefore cannot exchange places. The resulting motion has been compared to a traffic jam: the motion of each boson would be strongly correlated with that of its two neighbours.
Because the particles cannot exchange places, one might expect their behaviour to be fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
ic, but it turns out that their behaviour differs from that of fermions in several important ways: the particles can all occupy the same momentum state which corresponds to neither Bose–Einstein nor Fermi–Dirac statistics.
The fermionic exchange rule
Identical particles
Identical particles, or indistinguishable particles, are particles that cannot be distinguished from one another, even in principle. Species of identical particles include elementary particles such as electrons, and, with some clauses, composite particles such as atoms and molecules.There are two...
implies more than the exclusion of two particles from the same point: in addition, the momentum of two identical fermions can never be the same, wherever they are located. Mathematically, there is an exact one-to-one mapping
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
of impenetrable bosons (in a one-dimensional system) onto a system of fermions that do not interact at all.
In the case of a Tonks–Girardeau gas (TG), so many properties of this one-dimensional string of bosons would be sufficiently fermion-like that the situation is often referred to as the 'fermionization' of bosons.Tonks–Girardeau gas coincide with quantum Nonlinear Schrödinger equation
Nonlinear Schrödinger equation
In theoretical physics, the nonlinear Schrödinger equation is a nonlinear version of Schrödinger's equation. It is a classical field equation with applications to optics and water waves. Unlike the Schrödinger equation, it never describes the time evolution of a quantum state...
for infinite repulsion, which can be efficiently analyzed by Quantum inverse scattering method
Quantum inverse scattering method
Quantum inverse scattering method relates two different approaches:1) Inverse scattering transform is a method of solving classical integrable differential equations of evolutionary type.Important concept is Lax representation....
. This relation help to study Correlation function (statistical mechanics)
Correlation function (statistical mechanics)
In statistical mechanics, the correlation function is a measure of the order in a system, as characterized by a mathematical correlation function, and describes how microscopic variables at different positions are correlated....
. The correlation functions
can be described by Integrable system
Integrable system
In mathematics and physics, there are various distinct notions that are referred to under the name of integrable systems.In the general theory of differential systems, there is Frobenius integrability, which refers to overdetermined systems. In the classical theory of Hamiltonian dynamical...
. In a simple case it is Painlevé transcendents
Painlevé transcendents
In mathematics, Painlevé transcendents are solutions to certain nonlinear second-order ordinary differential equations in the complex plane with the Painlevé property , but which are not generally solvable in terms of elementary functions...
. A textbook explains in detail the description of quantum correlation functions of Tonks–Girardeau gas by means of classical completely integrable differential equations. Thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...
of Tonks–Girardeau gas was described by Chen Ning Yang.
Realizing a TG gas
Until 2004, there were no known examples of TGs. However, in a paper in the 20 May 2004 edition of NatureNature (journal)
Nature, first published on 4 November 1869, is ranked the world's most cited interdisciplinary scientific journal by the Science Edition of the 2010 Journal Citation Reports...
, physicist Belén Paredes and coworkers present a technique of creating such a gas using an optical lattice
Optical lattice
An optical lattice is formed by the interference of counter-propagating laser beams, creating a spatially periodic polarization pattern. The resulting periodic potential may trap neutral atoms via the Stark shift. Atoms are cooled and congregate in the locations of potential minima...
.
The optical lattice is formed by six intersecting laser
Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...
beams, which generate an interference pattern. The beams are arranged as standing wave
Standing wave
In physics, a standing wave – also known as a stationary wave – is a wave that remains in a constant position.This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling...
s along three orthogonal directions. This results in an array of optical dipole traps
Optical tweezers
Optical tweezers are scientific instruments that use a highly focused laser beam to provide an attractive or repulsive force , depending on the refractive index mismatch to physically hold and move microscopic dielectric objects...
where atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...
s are stored in the intensity
Intensity (physics)
In physics, intensity is a measure of the energy flux, averaged over the period of the wave. The word "intensity" here is not synonymous with "strength", "amplitude", or "level", as it sometimes is in colloquial speech...
maxima of the interference pattern.
The researchers first loaded ultracold rubidium
Rubidium
Rubidium is a chemical element with the symbol Rb and atomic number 37. Rubidium is a soft, silvery-white metallic element of the alkali metal group. Its atomic mass is 85.4678. Elemental rubidium is highly reactive, with properties similar to those of other elements in group 1, such as very rapid...
atoms into one-dimensional tubes formed by a two-dimensional lattice (the third standing wave is off for the moment). This lattice is very strong, so that the atoms do not have enough energy to tunnel between neighbouring tubes. On the other hand, the density is still too low for the transition to the TG regime. For that, the third axis
Cartesian coordinate system
A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...
of the lattice is used. It is set to a lower intensity than the other two axes, so that tunneling in this direction stays possible. For increasing intensity of the third lattice, atoms in the same lattice well are more and more tightly trapped, which increases the collision
Collision
A collision is an isolated event which two or more moving bodies exert forces on each other for a relatively short time.Although the most common colloquial use of the word "collision" refers to accidents in which two or more objects collide, the scientific use of the word "collision" implies...
al energy. When the collisional energy becomes much bigger than the tunneling energy, the atoms can still tunnel into empty lattice wells, but not into or across occupied ones.