  We explore the dependence of the performance bounds of heat engines and
refrigerators on the initial quantum state and the subsequent evolution of
their piston, modeled by a quantized harmonic oscillator. Our goal is to
provide a fully quantized treatment of self-contained (autonomous) heat
machines, as opposed to their prevailing semiclassical description that
consists of a quantum system alternately coupled to a hot or a cold heat bath,
and parametrically driven by a classical time-dependent piston or field. Here
by contrast, there is no external time-dependent driving. Instead, the
evolution is caused by the stationary simultaneous interaction of two heat
baths (having distinct spectra and temperatures) with a single two-level system
that is in turn coupled to the quantum piston.
  The fully quantized treatment we put forward allows us to investigate work
extraction and refrigeration by the tools of quantum-optical amplifier and
dissipation theory, particularly, by the analysis of amplified or dissipated
phase-plane quasiprobability distributions. Our main insight is that quantum
states may be thermodynamic resources and can provide a powerful handle, or
control, on the efficiency of the heat machine. In particular, a piston
initialized in a coherent state can cause the engine to produce work at an
efficiency above the Carnot bound in the linear amplification regime. In the
refrigeration regime, the coefficient of performance can transgress the Carnot
bound if the piston is initialized in a Fock state. The piston may be realized
by a vibrational mode, as in nanomechanical setups, or an electromagnetic field
mode, as in cavity-based scenarios.
