  We introduce the thermal-difference states (TDS), a three-parameter family of
single-mode non-Gaussian bosonic states whose density operator is a weighted
difference of two thermal states. We show that the states of "heralded photons"
generated via parametric down-conversion (PDC) are precisely those among the
TDS that are nonclassical, meaning they have a negative $P$-function. The three
parameters correspond in that context to the initial brightness of PDC and the
transmittances, characterizing the linear loss in the signal and the idler
channels. At low initial brightness and unit transmittances, the heralded
photon state is known to be a single-photon state. We explore the influence of
brightness and linear loss on the heralded state of the signal mode. In
particular, we analyze the influence of the initial brightness and the loss on
the state nonclassicality by computing several measures of nonclassicality,
such as the negative volume of the Wigner function, the sum of quantum Fisher
information for two quadratures, and the ordering sensitivity, introduced
recently by us [Phys. Rev. Lett. 122, 080402 (2019)]. We argue finally that the
TDS provide new benchmark states for the analysis of a variety of properties of
single-mode bosonic states.
