Introduction

Quasi-1D Bose gases have been realized in highly-elongated traps by tightly confining the transverse motion of the atoms to their zero point oscillations [GVL+01,SKC+01,GBM+01]. As compared to the 3D case, the role of quantum fluctuations is enhanced in 1D and these systems are predicted to exhibit peculiar properties, which cannot be described using traditional mean-field theories, but require more advanced many-body approaches. Particularly intriguing is the strong coupling regime, where, due to repulsion between particles, the quasi-one dimensional Bose gas behaves as if it consisted of fictitious spinless fermions (Tonks-Girardeau gas[Ton36,Gir60,Ols98,PSW00]). This regime has not been achieved yet, but is one of the main focus areas of present experimental investigations in this field [MSKE03,RT03]. An interesting possibility to approach the strongly correlated TG regime is provided by magnetic field induced atom-atom Feshbach resonances [IAS+98,CCR+00b,LRT+02,OHG+02,BCK+03]. By utilizing this technique one can tune the $3D$ $s$-wave scattering length $a_{3D}$, and hence the strength of atom-atom interactions, to essentially any value including zero and $\pm\infty$.

Degenerate quantum gases near a Feshbach resonance have recently received a great deal of interest both experimentally and theoretically. At resonance ( $\vert a_{3D}\vert\to\infty$) the 3D scattering cross-section $\sigma$ is fixed by the unitary condition, $\sigma=4\pi/k^2$, where $\k$ is the relative wave vector of the two atoms. In this regime it is predicted that, if the effective range $R$ of the atom-atom interaction potential is much smaller than the average interparticle distance, the behavior of the gas is universal, i.e., independent of the details of the interatomic potential and independent of the actual value of $a_{3D}$ [Hei01,CHM+02]. This is known as the unitary regime [HM04]. In the case of 3D Bose gases, this unitary regime can most likely not be realized in experiments since three-body recombination is expected to set in when $a_{3D}$ becomes comparable to the average interparticle distance. Three-body recombination leads to cluster formation and hence makes the gas-like state unstable. The situation is different for Fermi gases, for which the unitary regime has already been reached experimentally [OHG+02,BCK+03]. In this case, the Fermi pressure stabilizes the system even for large $\vert a_{3D}\vert$.

In quasi-one dimensional geometries a new length scale becomes relevant, namely, the oscillator length $a_\perp=\sqrt{\hbar/(m\omega_\perp)}$ of the tightly confined transverse motion, where $m$ is the mass of the atoms and $\omega _\perp$ is the angular frequency of the harmonic trapping potential. For $\vert a_{3D}\vert\gg a_\perp$, the gas is expected to exhibit a universal behavior if the effective range $R$ of the atom-atom interaction potential is much smaller than $a_\perp$ and the mean interparticle distance is much larger than $a_\perp$. It has been predicted that three-body recombination processes are suppressed for strongly interacting 1D Bose gases (see Eq. 5.12). These studies raise the question whether the unitary regime can be reached in Bose gases confined in highly-elongated traps, that is, whether the quasi-one dimensional bosonic gas-like state is stable against cluster formation as $a_{3D}\to\pm\infty$.

This chapter is devoted to the investigation of the properties of a quasi-one dimensional Bose gas at zero temperature over a wide range of values of the $3D$ scattering length $a_{3D}$ using Quantum Monte Carlo techniques (see Chapter 2). We find that the system 1) is well described by a 1D model Hamiltonian with contact interactions and renormalized coupling constant [Ols98] for any value of $a_{3D}$, 2) reaches the regime of a TG gas for a critical positive value of the 3D scattering length $a_{3D}$, 3) enters a unitary regime for large values of $\vert a_{3D}\vert$, that is, for $\vert a_{3D}\vert \rightarrow \infty$, where the properties of the quasi-one dimensional Bose gas become independent of the actual value of $a_{3D}$ and are similar to those of a 1D gas of hard-rods and 4) becomes unstable against cluster formation for a critical value of the 1D gas parameter, or equivalently, for a critical negative value of the 3D scattering length $a_{3D}$.

The structure of this Chapter is as follows. Section 4.2 discusses the energetics of two bosons in quasi-one dimensional harmonic traps. to a 1D model Hamiltonian with contact interactions and renormalized coupling constant [Ols98]. The eigenenergies of the system are calculated by exact diagonalization of both the 3D and the 1D Hamiltonian. We use these results for two particles to benchmark our quantum MC calculations presented in Sec. 4.4. Section 4.3 discusses the relation between the 3D and the 1D Hamiltonian for $N$ bosons under quasi-one dimensional confinement. Section 4.4 presents our MC results for $N=2$ and $N=10$ atoms in highly-elongated harmonic traps over a wide range of values of the 3D scattering length $a_{3D}$. A comparison of the energetics of the lowest-lying gas-like state for the 3D and the 1D Hamiltonian is carried out. In the $N=2$ case, we additionally compare with the essentially exact results presented in Sec. 4.2. In the $N=10$ case, we additionally compare with the energy of the lowest-lying gas-like state of the 1D Hamiltonian calculated using the local density approximation (LDA). Section 4.5 discusses the stability of the lowest-lying gas-like state against cluster formation when $a_{3D}$ is negative using the variational Monte Carlo (VMC) method. We provide a quantitative estimate of the criticality condition. Finally, Sec. 4.6 draws our conclusions.