  We present a fully quantum dynamical treatment of a quantum heat engine and
its baths based on the Hierarchy of Pure States (HOPS), an exact and general
method for open quantum system dynamics. We show how the change of the bath
energy and the interaction energy can be determined within HOPS, for arbitrary
coupling strength and smooth time dependence of the modulation protocol. The
dynamics of all energetic contributions during the operation can be carefully
examined both, in its initial transient phase and also later, in its periodic
steady state. A quantum Otto engine with a qubit as inherently nonlinear work
medium is studied in a regime where the energy associated with the interaction
Hamiltonian plays an important role for the global energy balance and, thus,
must not be neglected when calculating its power and efficiency. We confirm
that the work required to drive the coupling with the baths depends sensitively
on the speed of the modulation protocol. Remarkably, departing from the
conventional scheme of well-separated phases by allowing for temporal overlap,
we discover that one can even gain energy from the modulation of the bath
interactions. We visualize these various work contributions using the analogue
of state change diagrams of thermodynamic cycles. We offer a concise, full
presentation of HOPS with its extension to bath observables, as it serves as a
universal tool for the numerically exact description of general quantum
dynamical (thermodynamic) scenarios far from the weak-coupling limit.
