  We consider the interaction dynamics of a classical oscillator and a quantum
two-level system for different pure-dephasing Hamiltonians of the type
$\widehat{H}(q,p)=H_C(q,p)\boldsymbol{1}+H_I(q,p)\widehat\sigma_z$. This type
of systems represents a severe challenge for popular hybrid quantum-classical
descriptions. For example, in the case of the common Ehrenfest model, the
classical density evolution is shown to decouple entirely from the
pure-dephasing quantum dynamics. We focus on a recently proposed hybrid wave
equation that is based on Koopman's wavefunction description of classical
mechanics. This model retains quantum-classical correlations whenever a
coupling potential is present. Here, several benchmark problems are considered
and the results are compared with those arising from fully quantum dynamics. A
good agreement is found for a series of study cases involving harmonic
oscillators with linear and quadratic coupling, as well as time-varying
coupling parameters. In all these cases the classical evolution coincides
exactly with the oscillator dynamics resulting from the fully quantum
description. In the special case of time-independent coupling involving a
classical oscillator with varying frequency, the quantum Bloch rotation
exhibits peculiar features that escape from the hybrid description. In
addition, nonlinear corrections to the harmonic Hamiltonian lead to an overall
growth of decoherence at long times, which is absent in the fully quantum
treatment.
