  We aim at constraining the stellar population properties of quiescent
galaxies. These properties reveal how these galaxies evolved and assembled
since $z\sim1$ up to the present time. Combining the ALHAMBRA multi-filter
photo-spectra with the SED-fitting code MUFFIT, we build a complete catalogue
of quiescent galaxies via the dust-corrected stellar mass vs colour diagram.
This catalogue includes stellar population properties, such as age,
metallicity, extinction, stellar mass and photometric redshift, retrieved from
the analysis of composited populations based on two independent sets of SSP
models. We develop and apply a novel methodology to provide, for the first
time, the analytic probability distribution functions (PDFs) of mass-weighted
age, metallicity, and extinction of quiescent galaxies as a function of
redshift and stellar mass. We adopt different star formation histories to
discard potential systematics in the analysis. The number density of quiescent
galaxies is found to increase since $z\sim1$, with a more substantial variation
at lower mass. Quiescent galaxies feature extinction $A_V<0.6$, with median
values in the range $A_V = 0.15\mathrm{-}0.3$. At increasing stellar mass,
quiescent galaxies are older and more metal rich since $z\sim1$. A detailed
analysis of the PDFs reveals that the evolution of quiescent galaxies is not
compatible with passive evolution and a slight decrease is hinted at median
metallicity $0.1\mathrm{-}0.2$~dex. The intrinsic dispersion of the age and
metallicity PDFs show a dependence with stellar mass and/or redshift. These
results are consistent with both sets of SSP models and the alternative SFH
assumptions explored. Consequently, the quiescent population must undergo an
evolutive pathway including mergers and/or remnants of star formation to
reconcile the observed trends, where the `progenitor' bias should also be taken
into account.
