  We investigate the loss of nonclassicality and non-Gaussianity of a
single-mode state of the radiation field in contact with a thermal reservoir.
The damped density matrix for a Fock-diagonal input is written using the Weyl
expansion of the density operator. Analysis of the evolution of the
quasiprobability densities reveals the existence of two successive
characteristic times of the reservoir which are sufficient to assure the
positivity of the Wigner function and, respectively, of the $P$ representation.
We examine the time evolution of non-Gaussianity using three recently
introduced distance-type measures. They are based on the Hilbert-Schmidt
metric, the relative entropy, and the Bures metric. Specifically, for an
$M$-photon-added thermal state, we obtain a compact analytic formula of the
time-dependent density matrix that is used to evaluate and compare the three
non-Gaussianity measures. We find a good consistency of these measures on the
sets of damped states. The explicit damped quasiprobability densities are shown
to support our general findings regarding the loss of negativities of Wigner
and $P$ functions during decoherence. Finally, we point out that Gaussification
of the attenuated field mode is accompanied by a nonmonotonic evolution of the
von Neumann entropy of its state conditioned by the initial value of the mean
photon number.
