  The quantum Otto cycle serves as a bridge between the macroscopic world of
heat engines and the quantum regime of thermal devices composed from a single
element. We compile recent studies of the quantum Otto cycle with a harmonic
oscillator as a working medium. This model has the advantage that it is
analytically trackable. In addition, an experimental realization has been
achieved employing a single ion in a harmonic trap. The review is embedded in
the field of quantum thermodynamics and quantum open systems. The basic
principles of the theory are explained by a specific example illuminating the
basic definitions of work and heat. The relation between quantum observables
and the state of the system is emphasized. The dynamical description of the
cycle is based on a completely positive map formulated as a propagator for each
stroke of the engine. Explicit solutions for these propagators are described on
a vector space of quantum thermodynamical observables. These solutions which
employ different assumptions and techniques are compared. The tradeoff between
power and efficiency is the focal point of finite-time-thermodynamics. The
dynamical model enables to study finite time cycles limiting time on the
adiabtic and the thermalization times. Explicit finite time solutions are found
which are frictionless, meaning that no coherence is generated also known as
shortcuts to adiabaticity. The transition from frictionless to sudden adiabats
is characterized by a non-hermitian degeneracy in the propagator. In addition
the influence of noise on the control is illustrated. These results are used to
close the cycles either as engines or as refrigerators.
