  In quantum physics, two prototypical model systems stand out due to their
wide range of applications. These are the two-level system (TLS) and the
harmonic oscillator. The former is often an ideal model for confined charge or
spin systems and the latter for lattice vibrations, i.e., phonons. Here, we
couple these two systems, which leads to numerous fascinating physical
phenomena. Practically, we consider different optical excitations and decay
scenarios of a TLS, focusing on the generated dynamics of a single phonon mode
that couples to the TLS. Special emphasis is placed on the entropy of the
different parts of the system, predominantly the phonons. While, without any
decay, the entire system is always in a pure state, resulting in a vanishing
entropy, the complex interplay between the single parts results in
non-vanishing respective entanglement entropies and non-trivial dynamics of
them. Taking a decay of the TLS into account leads to a non-vanishing entropy
of the full system and additional aspects in its dynamics. We demonstrate that
all aspects of the entropy's behavior can be traced back to the purity of the
states and are illustrated by phonon Wigner functions in phase space.
