Consider a two-component atomic Fermi gas confined in a highly-elongated trap. The
fermionic atoms are assumed to belong to the same atomic species, that is, to have
the same mass , but to be trapped in different hyperfine states
, where
represents a generalized spin or angular momentum,
or
. The trapping potential is assumed to be harmonic and axially
symmetric,
If the Fermi gas is kinematically in 1D, it can be described by an effective 1D
Hamiltonian with contact interactions,
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function of the 3d -wave scattering length
, which can be varied
continuously by application of an external field. The effective 1D interaction is
repulsive,
, for
(
), and
attractive,
, for
and for
.
By varying
, it is possible to go adiabatically from the weakly-interacting
regime (
) to the strongly-interacting repulsive regime (
or
), as well as from the weakly-interacting
regime to the strongly-interacting attractive regime (
or
)8.2
For two fermions with different spin the Hamiltonian ,
Eq. 8.3, supports one bound state with binding energy
and spatial extent
for
, and no bound state for
, that is, the molecular state becomes
exceedingly weakly-bound and spatially-delocalized as
[Ols98,BMO03]. In the following we investigate the properties of a
gas with
fermions,
, for both effectively
attractive and repulsive 1D interactions with and without longitudinal
confinement.