  We study zero-temperature quantum fluctuations in harmonically trapped
one-dimensional interacting Bose gases, using the self-consistent
multiconfigurational time-dependent Hartree method. We define $phase$
$fluctuations$ from the full single-particle density matrix by the spatial
decay exponent of off-diagonal long-range order. In a regime of mesoscopic
particle numbers and moderate contact couplings, we derive the spatial
dependence of the amplitude of phase fluctuations, determined from the {\em
self-consistently} derived shape of the field operator orbitals and Fock space
orbital occupation amplitudes. It is shown that the phase fluctuations display
a peak, which in turn corresponds to a dip of the first-order correlations in
position space, akin to what has previously been obtained in the
Tonks-Girardeau limit of very large interactions and low densities.
