  We analyze the phase diffusion, quantum fluctuations and their spectral
features of an one-dimensional Bose-Josephson junction (BJJ) coupled to a
bosonic heat bath. We show the dependence of the phase diffusion coefficient on
the on-site interaction parameter $U$ and the temperature in zero-phase and
$\pi$-phase modes. We find that in the $\pi$-phase mode, the phase diffusion
co-efficient as a function of $U$ decreases so long as $U$ is below a critical
value while it increases above the critical value. This criticality of on-site
interaction reflects a transition between Josephson oscillation and macroscopic
quantum self-trapping (MQST) regime. Based on the thermal canonical Wigner
distribution, we calculate the coherence factor to understand its dependence on
temperature and on-site interaction energy in Josephson oscillation and MQST
regime. Furthermore, we discuss coherent and incoherent spectral properties in
connection with the fluctuations of the relative phase and the population
imbalance in both zero and $\pi$-phase modes from weak to strong dissipation
regime.
