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 . The behavior of confined Bose gases strongly depends on
the ratio of the harmonic oscillator length in the tight transverse direction,
, to the interaction range
and to the average interparticle distance
, where
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 -wave scattering length
through
application of an external magnetic field in the proximity of a Feshbach resonance.
For
, the scattering length
determines to a good
approximation the effective 1D scattering length
and the effective 1D
coupling constant
, which can be, just as the 3D coupling constant, tuned
to essentially any value including zero and
. By exploiting Feshbach
resonance techniques, one should be able to achieve strongly-correlated quasi-one
dimensional systems. The strong coupling regime is achieved for
, 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
. In the unitary regime, the gas is dilute, that is,
, and at
the same time strongly-correlated, that is,
.
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 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 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
. Our numerical results suggest that the stability condition can be
expressed as
. 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.