This paper presents a thorough study of correlation properties of a one-dimensional
gas of bosons at zero temperature. In a homogeneous system the behavior is fixed by
the product of linear density and one-dimensional scattering length
. In the strongly interacting regime
the bosonic system
behaves effectively as a system of non interacting fermions. In this limit the
energy, pair distribution function
, static structure factor
are known
explicitly and are same as the ones of the corresponding fermionic system. For
arbitrary value of the gas parameter no complete description was known so far.
Switching on an external harmonic potential leads to modification in properties as
new length, the oscillator length
is introduced.
Quasi one-dimensional systems have been already realized in a number of experiments
with elongated traps. Many new experiments with condensates in a same geometry, in a
waveguide or on a chip are expected to appear. The characteristic parameter
can be varied by changing number of atoms in the condensate, trapping
frequencies or by adjusting the scattering length using the Feshbach resonance.
Momentum distribution is accessible from ballistic expansion and static factor can
be measured by the Bragg scattering.
We find for the first time full description of the correlation functions in a wide
range of the characteristic parameter starting from Tonks-Girardeau
regime and up to Gross-Pitaevskii regime. We benchmark our Diffusion Monte Carlo
calculations by recovering the ground state energy known from solution of the
Lieb-Liniger integral equations. We completely recover all properties of the
Tonks-Girardeau gas and known asymptotic behavior of the momentum distribution and
correlation functions. We obtain the one-body density matrix
and pair
distribution function
for all densities. In particular we have the
description of the most nontrivial regime
which is relevant
for current experiments.
We study the dependence of the value at zero of the three-body correlation function
on the density
.
This function is of a great interest as it governs the three-body recombination
rate, which leads to loss of the atoms out of the condensate. The data of an
experimental measurement of
is available [TOH+04] and is compared
with the predictions of the Lieb-Liniger theory. An agreement between theory and
experiment is found.
By the means of Fourier transformation we extract the momentum distribution
and static structure factor
. Low momentum part is described by phonon
hydrodynamic theory which is expected to be applicable at distances
larger
than the healing length
. We judge that
shows phononic power-law
divergence for values of
considerably smaller than
.
Finally we discuss how the presence of a harmonic trapping modifies the correlation
functions. We plot the pair distribution function in typical experimental
configurations. We discuss possibility of finding in traces of divergent
behavior, which is characteristic for a one-dimensional infinite system, in a
trapped system.