  The effect of unstable quasiparticles in the out-of-equilibrium dynamics of
certain integrable systems has been the subject of several recent studies. In
this paper we focus on the stationary value of the entanglement entropy
density, its growth rate, and related functions, after a quantum quench. We
consider several quenches, each of which is characterised by a corresponding
squeezed coherent state. In the quench action approach, the coherent state
amplitudes $K(\theta)$ become input data that fully characterise the large-time
stationary state, thus also the corresponding Yang-Yang entropy. We find that,
as function of the mass of the unstable particle, the entropy growth rate has a
global minimum signalling the depletion of entropy that accompanies a slowdown
of stable quasiparticles at the threshold for the formation of an unstable
excitation. We also observe a separation of scales governed by the interplay
between the mass of the unstable particle and the quench parameter, separating
a non-interacting regime described by free fermions from an interacting regime
where the unstable particle is present. This separation of scales leads to a
double-plateau structure of many functions, where the relative height of the
plateaux is related to the ratio of central charges of the UV fixed points
associated with the two regimes, in full agreement with conformal field theory
predictions. The properties of several other functions of the entropy and its
growth rate are also studied in detail, both for fixed quench parameter and
varying unstable particle mass and viceversa.
