  Using a Luttinger-liquid approach we study the quantum fluctuations of a
Bose-Josephson junction, consisting of a Bose gas confined to a quasi
one-dimensional ring trap which contains a localized repulsive potential
barrier. For an infinite barrier we study the one-particle and two-particle
static correlation functions. For the one-body density-matrix we obtain
different power-law decays depending on the location of the probe points with
respect to the position of the barrier. This quasi-long range order can be
experimentally probed in principle using an interference measurement. The
corresponding momentum distribution at small momenta is also shown to be
affected by the presence of the barrier and to display the universal power-law
behavior expected for an interacting 1D fluid. We also evaluate the particle
density profile, and by comparing with the exact results in the Tonks-Girardeau
limit we fix the nonuniversal parameters of the Luttinger-liquid theory. Once
the parameters are determined from one-body properties, we evaluate the
density-density correlation function, finding a remarkable agreement between
the Luttinger liquid predictions and the exact result in the Tonks-Girardeau
limit, even at the length scale of the Friedel-like oscillations which
characterize the behavior of the density-density correlation function at
intermediate distance. Finally, for a large but finite barrier we use the
one-body correlation function to estimate the effect of quantum fluctuations on
the renormalization of the barrier height, finding a reduction of the effective
Josephson coupling energy, which depends on the length of the ring and on the
interaction strength.
