Conclusions

This paper presents a thorough study of the properties of inhomogeneous, harmonically-confined quasi-one dimensional Bose gases as a function of the 3D scattering length $a_{3D}$. The behavior of confined Bose gases strongly depends on the ratio of the harmonic oscillator length in the tight transverse direction, $a_\perp$, to the interaction range $R$ and to the average interparticle distance $1/n^{1/3}$, where $n$ denotes the 3D central density.

Quasi-1D bosonic gases have been realized experimentally in highly-elongated harmonic traps. The strength of atom-atom interactions can be varied over a wide range by tuning the value of the 3D $s$-wave scattering length $a_{3D}$ through application of an external magnetic field in the proximity of a Feshbach resonance. For $R\ll a_\perp$, the scattering length $a_{3D}$ determines to a good approximation the effective 1D scattering length $a_{1D}$ and the effective 1D coupling constant $g_{1D}$, which can be, just as the 3D coupling constant, tuned to essentially any value including zero and $\pm\infty$. By exploiting Feshbach resonance techniques, one should be able to achieve strongly-correlated quasi-one dimensional systems. The strong coupling regime is achieved for $1/n^{1/3}\gg a_\perp$, it includes the TG gas, where a system of interacting bosons behaves as if it consisted of non-interacting spinless fermions, and the so-called unitary regime, where the properties of the gas become independent of the actual value of $a_{3D}$. In the unitary regime, the gas is dilute, that is, $nR^3\ll 1$, and at the same time strongly-correlated, that is, $n\vert a_{3D}\vert^3\gg 1$.

The present analysis is carried out within various theoretical frameworks. We obtain the 3D energetics of the lowest-lying gas-like state of the system using a microscopic FN-MC approach, which accounts for all degrees of freedom explicitly. The resulting energetics are then used to benchmark our 1D calculations. Full microscopic 1D calculations for contact interactions with renormalized coupling constant $g_{1D}$ result in energies that are in excellent agreement with the full 3D energies. This agreement implies that a properly chosen many-body 1D Hamiltonian describes quasi-one dimensional Bose gases well.

We also consider the LL and the hard-rod equation of state of a 1D system treated within the LDA. These approaches provide a good description of the energy of the lowest-lying gas-like state for as few as five or ten particles. Finite size effects are to a good approximation negligible. Our detailed microscopic studies suggest that these LDA treatments provide a good description of quasi-one dimensional Bose gases. In particular, we suggest a simple treatment of 1D systems with negative $g_{1D}$ using the hard-rod equation of state.

Finally, we address the question of whether the lowest-lying gas-like state of inhomogeneous quasi-one dimensional Bose gases is actually stable. We find, utilizing a variational 1D many-body framework, that the lowest-lying gas-like state is stable for negative coupling constants, up to a minimum critical value of $\vert g_{1D}\vert$. Our numerical results suggest that the stability condition can be expressed as $n_{1D} a_{1D} \simeq0.35$. Since our conclusions are derived from variational 1D calculations, more thorough microscopic calculations are needed to confirm our findings. We believe, however, that our findings will hold even in a 3D framework or when three-body recombination effects are included explicitly.

While our study was performed for inhomogeneous quasi-one dimensional Bose gases, many findings also apply to homogeneous quasi-one dimensional Bose gases. Furthermore, the Fermi-Bose mapping [Gir60,GB04,CS99,GO03], which allows one to map an interacting 1D gas of spin-polarized fermions to an interacting 1D gas of spin-polarized bosons, suggests that many of the results presented here for quasi-one dimensional Bose gases may directly apply to quasi-one dimensional Fermi gases.