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
-wave scattering length
, and hence the strength of atom-atom
interactions, to essentially any value including zero and
.
Degenerate quantum gases near a Feshbach resonance have recently received a great
deal of interest both experimentally and theoretically. At resonance
(
) the 3D scattering cross-section
is fixed by the
unitary condition,
, where
is the relative wave vector of the
two atoms. In this regime it is predicted that, if the effective range
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
[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
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
.
In quasi-one dimensional geometries a new length scale becomes relevant, namely,
the oscillator length
of the tightly confined
transverse motion, where
is the mass of the atoms and
is the
angular frequency of the harmonic trapping potential. For
, the
gas is expected to exhibit a universal behavior if the effective range
of the
atom-atom interaction potential is much smaller than
and the mean
interparticle distance is much larger than
. 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
.
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
scattering length
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
, 2) reaches the regime of a TG gas for a
critical positive value of the 3D scattering length
, 3) enters a unitary
regime for large values of
, that is, for
, where the properties of the quasi-one dimensional Bose gas become
independent of the actual value of
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
.
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 bosons under
quasi-one dimensional confinement. Section 4.4 presents
our MC results for
and
atoms in highly-elongated harmonic traps over a
wide range of values of the 3D scattering length
. A comparison of the
energetics of the lowest-lying gas-like state for the 3D and the 1D Hamiltonian is
carried out.
In the
case, we additionally compare with the essentially exact
results presented in Sec. 4.2. In the
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
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.