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physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
, a Tonks–Girardeau gas is a
Bose gas An ideal Bose gas is a quantum-mechanical phase of matter, analogous to a classical ideal gas. It is composed of bosons, which have an integer value of spin and abide by Bose–Einstein statistics. The statistical mechanics of bosons were deve ...
in which the repulsive interactions between bosonic particles confined to one
dimension In physics and mathematics, the dimension of a mathematical 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 (1D) because only one coo ...
dominate the system's physics. It is named after physicists
Lewi Tonks Lewi Tonks (1897–July 30, 1971) was an American physicist who worked for General Electric on microwaves, plasma physics and nuclear reactors. Under Irving Langmuir, his work pioneered the study of plasma oscillations. He is also noted for the n ...
, who developed a classical model in 1936, and Marvin D. Girardeau who generalized it to the quantum regime. It is not a
Bose–Einstein condensate In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter that is typically formed when a gas of bosons at very low Density, densities is cooled to temperatures very close to absolute zero#Relation with Bose–Einste ...
as it does not demonstrate any of the necessary characteristics, such as off-diagonal long-range order or a unitary two-body
correlation function A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. If one considers the correlation function between random variables ...
, even in a
thermodynamic limit In statistical mechanics, the thermodynamic limit or macroscopic limit, of a system is the Limit (mathematics), limit for a large number of particles (e.g., atoms or molecules) where the volume is taken to grow in proportion with the number of ...
and as such cannot be described by a macroscopically occupied orbital (order parameter) in the Gross–Pitaevskii formulation. The Tonks–Girardeau gas is a particular case of the
Lieb–Liniger model In physics, the Lieb–Liniger model describes a gas of particles moving in one dimension and satisfying Bose–Einstein statistics. More specifically, it describes a one dimensional Bose gas with Dirac delta interactions. It is named after Elliott ...
.


Definition

A row of bosons all confined to a one-dimensional line cannot pass each other and therefore cannot exchange places. The resulting motion has been compared to a
traffic jam Traffic congestion is a condition in transport that is characterized by slower speeds, longer trip times, and increased vehicular queueing. Traffic congestion on urban road networks has increased substantially since the 1950s, resulting in m ...
: the motion of each boson is strongly correlated with that of its two neighbors. This can be thought of as the large-''c'' limit of the delta Bose gas. Because the particles cannot exchange places, their behavior might be expected to be
fermion In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...
ic, but their behavior 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 Fermi–Dirac statistics is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pauli exclusion principle. A result is the Fermi–Dirac distribution of part ...
. This is the phenomenon of bosonization which happens in 1+1 dimensions. 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 matches quantum
Nonlinear Schrödinger equation In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonli ...
for infinite repulsion, which can be efficiently analyzed by
quantum inverse scattering method In quantum physics, the quantum inverse scattering method (QISM), similar to the closely related algebraic Bethe ansatz, is a method for solving integrable models in 1+1 dimensions, introduced by Leon Takhtajan and L. D. Faddeev in 1979. It can ...
. This relation helps to study correlation functions. The correlation functions can be described by an
Integrable system In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
. In a simple case, it is a Painlevé transcendent. The quantum correlation functions of a Tonks–Girardeau gas can be described by means of classical, completely integrable, differential equations.
Thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
of Tonks–Girardeau gas was described by
Chen Ning Yang Yang Chen-Ning or Chen-Ning Yang (; born 1 October 1922), also known as C. N. Yang or by the English name Frank Yang, is a Chinese theoretical physicist who made significant contributions to statistical mechanics, integrable systems, gauge th ...
.


Physical realization

The first example of TGs came in 2004 when Paredes and coworkers created an array of such gases using an
optical lattice An optical lattice is formed by the Interference (wave propagation), interference of counter-propagating laser beams, creating a spatially periodic intensity pattern. The resulting periodic scalar potential, potential may trap neutral atoms via ...
. In a different experiment, Kinoshita and coworkers observed a strongly correlated 1D Tonks–Girardeau gas. The optical lattice is formed by six intersecting
laser A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word ''laser'' originated as an acronym for light amplification by stimulated emission of radi ...
beams, which generate an
interference Interference is the act of interfering, invading, or poaching. Interference may also refer to: Communications * Interference (communication), anything which alters, modifies, or disrupts a message * Adjacent-channel interference, caused by extra ...
pattern. The beams are arranged as
standing wave In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect t ...
s along three
orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...
directions. This results in an array of
optical dipole traps Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of optical instruments, instruments that use or Photodetector, detect it. Optics usually describes t ...
where
atom Atoms are the basic particles of the chemical elements. An atom consists of a atomic nucleus, nucleus of protons and generally neutrons, surrounded by an electromagnetically bound swarm of electrons. The chemical elements are distinguished fr ...
s are stored in the
intensity Intensity may refer to: In colloquial use * Strength (disambiguation) *Amplitude * Level (disambiguation) * Magnitude (disambiguation) In physical sciences Physics *Intensity (physics), power per unit area (W/m2) *Field strength of electric, m ...
maxima of the interference pattern. The researchers loaded ultracold
rubidium Rubidium is a chemical element; it has Symbol (chemistry), symbol Rb and atomic number 37. It is a very soft, whitish-grey solid in the alkali metal group, similar to potassium and caesium. Rubidium is the first alkali metal in the group to have ...
atoms into one-dimensional tubes formed by a two-dimensional lattice (the third standing wave is initially off). This lattice is strong so that the atoms have insufficient energy to
tunnel A tunnel is an underground or undersea passageway. It is dug through surrounding soil, earth or rock, or laid under water, and is usually completely enclosed except for the two portals common at each end, though there may be access and ve ...
between neighboring tubes. The interaction is too low for the transition to the TG regime. For that, the third
axis An axis (: axes) may refer to: Mathematics *A specific line (often a directed line) that plays an important role in some contexts. In particular: ** Coordinate axis of a coordinate system *** ''x''-axis, ''y''-axis, ''z''-axis, common names ...
of the lattice is used. It is set to a lower intensity and shorter time than the other two, so that tunneling in this direction is possible. For increasing intensity of the third lattice, atoms in the same lattice well are more and more tightly trapped, which increases the
collision In physics, a collision is any event in which two or more bodies exert forces on each other in a relatively short time. Although the most common use of the word ''collision'' refers to incidents in which two or more objects collide with great for ...
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. This technique has been used by other researchers to obtain an array of one-dimensional Bose gases in the Tonks-Girardeau regime. However, the fact that an array of gases is observed only allows the measurement of averaged quantities. Moreover, the temperatures and
chemical potential In thermodynamics, the chemical potential of a Chemical specie, species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potent ...
between the different tubes are dispersed, which wash out many effects. For instance, this configuration does not allow probing of system fluctuations. Thus it proved interesting to produce a single Tonks–Girardeau gas. In 2011 one team created a single one-dimensional TG gas by trapping rubidium atoms magnetically in the vicinity of a microstructure. Thibaut Jacqmin ''et al.'' measured density fluctuations in that single strongly interacting gas. Those fluctuations proved to be
sub-Poissonian In mathematics, a super-Poissonian distribution is a probability distribution that has a larger variance than a Poisson distribution with the same mean. Conversely, a sub-Poissonian distribution has a smaller variance. An example of super-Poissonia ...
, as expected for a
Fermi gas A Fermi gas is an idealized model, an ensemble of many non-interacting fermions. Fermions are particles that obey Fermi–Dirac statistics, like electrons, protons, and neutrons, and, in general, particles with half-integer spin. These statis ...
.


See also

*
BCS theory In physics, the Bardeen–Cooper–Schrieffer (BCS) theory (named after John Bardeen, Leon Cooper, and John Robert Schrieffer) is the first microscopic theory of superconductivity since Heike Kamerlingh Onnes's 1911 discovery. The theory descr ...
*
Quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
*
Super Tonks–Girardeau gas Super may refer to: Computing * SUPER (computer program), or Simplified Universal Player Encoder & Renderer, a video converter/player * Super (computer science), a keyword in object-oriented programming languages * Super key (keyboard button) ...


References


External links

* * * * * * {{DEFAULTSORT:Tonks-Girardeau gas Condensed matter physics