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 neighbors. This can be thought of as the large-c limit of the delta Bose gas. Because the particles cannot exchange places, one might expect their behavior to be fermionic, but it turns out that 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. This is the phenomenon of bosonization which happens in 1+1 dimensions. In the case of a Tonks–Girardeau gas, 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 coincides 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. 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 integrabledifferential equations. Thermodynamics of Tonks–Girardeau gas was described by Chen Ning Yang.
Realizing a TG gas
There were no known examples of TGs until 2004 when Paredes and coworkers presented a technique of creating an array of such gases using an optical lattice. In a different experiment, Kinoshita and coworkers also succeeded in observation of a strongly correlated 1D Tonks–Girardeau gas. 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. This lattice is very strong so that the atoms do not have enough energy to tunnel between neighboring tubes. On the other hand, the interaction 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 and shorter time 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 wash 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.
