We discuss stability and phase coherence of 1D trapped Bose gases and find that inelastic decay processes, such as three-body recombination, are suppressed in the strongly interacting (TonksGirardeau) and intermediate regimes. This is promising for achieving these regimes with a large number of particles. "Fermionization" of the system reduces the phase coherence length, and at T = 0 the gas is fully phase coherent only deeply in the weakly interacting (Gross-Pitaevskii) regime.PACS numbers: 03.75Fi, 05.30JpRecent success in creating quantum degenerate 1D trapped atomic gases [1,2,3] has stimulated an interest in correlation properties of these systems. The 1D regime is reached by tightly confining the radial motion of atoms in a cylindrical trap to zero point oscillations. The difference between such a kinematically 1D gas and a purely 1D gas is related only to the value of the interparticle interaction, which depends on the radial confinement.The 1D Bose gas with repulsive short-range interactions characterized by the coupling constant g > 0 exhibits remarkable properties. Counterintuitively, it becomes more non-ideal with decreasing 1D density n [4,5]. The equation of state and correlation functions depend crucially on the parameterwhere m is the atom mass. For comparatively large n, the parameter γ ≪ 1 and √ γ represents the ratio of the mean interparticle separation 1/n to the correlation length l c =h/ √ mng. In this case the gas is in the weakly interacting or Gross-Pitaevskii (GP) regime, and the amplitude of the boson field obeys the familiar Gross-Pitaevskii equation. For sufficiently low densities (or large g), one has γ ≫ 1 and the strongly interacting or Tonks-Girardeau (TG) regime is reached. In this regime the 1D gas acquires fermionic properties in the sense that the wave function strongly decreases as particles approach each other (see [4,5] and cf. [6]). For γ = ∞ any correlation function of the density can be calculated straightforwardly [7] by using the exact mapping onto the system of free fermions found by Girardeau [4].A uniform 1D system of bosons interacting via a deltafunctional potential is integrable by using the Bethe ansatz and has been a subject of extensive theoretical studies. Lieb and Liniger [5] have calculated the ground state energy and excitation spectrum for any value of the parameter γ. Yang and Yang [8] have studied thermodynamic properties of this system at finite temperatures and found no phase transition for T > 0. The absence of long-range order and true Bose-Einstein condensate in finite-temperature 1D Bose gases follows from the Bogoliubov k −2 theorem as has been expounded in [9]. The long-range order is destroyed by long-wave fluctuations of the phase leading to an exponential decay of the oneparticle density matrix at large distances [10,11]. A similar picture is found at T = 0 [12], where the density matrix undergoes a power-law decay [11,13,14]. In the past few decades, a general approach has been developed for exact calculations of one-and two-particle correla...

We calculate the two-particle local correlation for an interacting 1D Bose gas at finite temperature and classify various physical regimes. We present the exact numerical solution by using the Yang-Yang equations and Hellmann-Feynman theorem and develop analytical approaches. Our results draw prospects for identifying the regimes of coherent output of an atom laser, and of finitetemperature "fermionization" through the measurement of the rates of two-body inelastic processes, such as photo-association. The knowledge of the exact solutions to the 1D models allows us to go far beyond the mean-field Bogoliubov approximation. In the current stage of studies of experimentally feasible 1D Bose gases, one of the most important issues that requires such an approach is understanding the correlation properties in the various regimes at finite temperatures.In this Letter we give an exact calculation of the finitetemperature two-particle local correlation for an interacting uniform 1D Bose gas,is the field operator and n = Ψ † (x)Ψ(x) is the linear (1D) density. As a result, we identify and classify various finite-temperature regimes of the 1D Bose gas. Aside from this, the pair correlations are responsible for the rates of inelastic collisional processes [9], and are of particular importance for the studies of coherence properties of atom "lasers" produced in 1D waveguides.At T = 0, the local two-and three-particle correlations of a uniform 1D Bose gas have recently been calculated in Ref. [10]. Here one has two well-known and physically distinct regimes of quantum degeneracy. For weak couplings or high densities, the gas is in a coherent or Gross-Pitaevskii (GP) regime with g (2) → 1. In this regime, long-range order is destroyed by longwavelength phase fluctuations [11] and the equilibrium state is a quasi-condensate characterized by suppressed density fluctuations. For strong couplings or low densities, the gas reaches the strongly interacting or TonksGirardeau (TG) regime and undergoes "fermionization" [3,4]: the wave function strongly decreases as particles approach each other, and g (2) → 0.At the non-zero temperatures studied here, one has to further extend the classification of different regimes. For strong enough couplings or low densities, we obtain the TG regime with g (2) → 0 not only at low temperatures, but also at high temperatures. In addition, and in contrast to previous T = 0 results, we find a weak-coupling or high-density regime in which fluctuations are enhanced. Asymptotically, they reach the non-interacting Bose gas level of g (2) → 2 (rather than g (2) → 1), for any finite temperature T .The emergence of this behavior at low temperatures implies that one can identify three physically distinct regimes of quantum degeneracy: the strong-coupling TG regime of "fermionization" with g (2) → 0, a coherent GP regime with g (2) ≃ 1 at intermediate coupling strength, and a fully decoherent quantum (DQ) regime with g (2) ≃ 2 at very weak couplings. In the GP regime, where the density fluctuations are suppresse...

Recent developments in the physics of ultracold gases provide wide possibilities for reducing the dimensionality of space for magnetically or optically trapped atoms. The goal of these lectures is to show that regimes of quantum degeneracy in two-dimensional (2D) and one-dimensional (1D) trapped gases are drastically different from those in three dimensions and to stimulate an interest in low-dimensional systems. Attention is focused on the new physics appearing in currently studied low-dimensional trapped gases and related to finite-size and finite-temperature effects.