Integrable quantum many-body systems traditionally belong to the domain of mathematical physics, with little or no connection to experiments. However, the experiments on confined quantum-degenerate gases has recently yielded faithful AMO realizations of a number of integrable systems. The new phenomenological relevance of integrable models, unique to atomic, molecular, and optical physics, opens up the possibility of a new research direction that is the focus of this proposal: the experimental manifestations of integrability in cold dilute quantum gases. It is shown that the presence of nontrivial conserved quantities in a quantum system leads to dramatic, initial-state-dependent discrepancy between the equilibrium state of the system and the predictions of thermodynamics. A new thermodynamic ensemble is suggested and successfully numerically tested: there all integrals of motion participate on an equal footing. Furthermore, the relaxation dynamics in an integrable system is conjectured to be very different from generic: it cannot be reduced to the analysis of few-body collisions, but is rather a substantially many-body effect. Overall, it is argued that the kinetic and thermodynamic properties of integrable quantum gases are so different from the usual, that they well-qualify for a new state of quantum matter. The objects of study chosen include bosons in one-dimensional optical lattices in the deep Mott regime, spin-0 Bose and spin-1/2 Fermi gases confined to waveguides, and two-dimensional harmonically trapped Bose condensates, all of which have been experimentally realized already. Momentum distribution and chemical composition are suggested as the simplest experimental observables sensitive to the effects of integrability.
