  We show that the methods for quantification of system-environment
entanglement that were recently developed for interactions that lead to pure
decoherence of the system can be straightforwardly generalized to
time-dependent Hamiltonians of the same type. This includes the if-and-only-if
criteria of separability, as well as the entanglement measure applicable to
qubit systems, and methods of detection of entanglement by operations and
measurements performed solely on the system without accessing the environment.
We use these methods to study the nature of the decoherence of a
qubit-oscillator system. Qubit-oscillator entanglement is essential for
developing bosonic quantum technology with quantum non-Gaussian states and its
applications in quantum sensing and computing. The dominating bosonic
platforms, trapped ions, electromechanics, and superconducting circuits, are
based on the time-dependent gates that use such entanglement to achieve new
quantum sensors and quantum error correction. The step-like time-dependence of
the Hamiltonian that is taken into account allows us to capture complex
interplay between the build-up of classical and quantum correlations, which
could not be replicated in time-independent scenarios.
