In physics, a Tonks–Girardeau gas is a 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 be 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 fermionic, 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.
Realizing a TG gas
Until 2004, there were no known examples of TGs. However, in a paper in the 20 May 2004 edition of Nature, physicist Belén Paredes and coworkers present a technique of creating an array of such gases 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 waves along three orthogonal directions. This results in an array of optical dipole traps where atoms 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 collisional 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 many 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 there is a dispersion of temperatures and chemical potential between the different tubes which washes out many effects. For instance this configuration does not allow probing of fluctuations in the system. Thus it proved interesting to produce a single Tonks–Girardeau gas. In 2011 one team  managed to create a single one-dimensional Bose gas in this very peculiar regime by trapping rubidium atoms magnetically in the vicinity of a microstructure. Thibaut Jacqmin et al managed to measure density fluctuations in such a single strongly interacting gas. Those fluctuations proved to be sub-Poissonian, as expected for a Fermi gas.
- V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge University Press, 1993
- T. Jacqmin et al, Sub-Poissonian Fluctuations in a 1D Bose Gas: From the Quantum Quasicondensate to the Strongly Interacting Regime, Phys. Rev. Lett. 106, 230405 (2011)
- Tonks–Girardeau gas of ultracold atoms in an optical lattice, Nature 429, 277–281
- Ground-state properties of a one-dimensional system of hard-core bosons in a harmonic trap, Phys. Rev. A 63, 033601
- Sub-Poissonian Fluctuations in a 1D Bose Gas: From the Quantum Quasicondensate to the Strongly Interacting Regime, Phys. Rev. Lett. 106, 230405