text: Quantum versions of most of the common thermodynamic cycles have been studied, for example the Carnot cycle, Stirling cycle and Otto cycle.
  The Otto cycle  can serve as a template for other reciprocating cycles. 
  It is composed of the following four segments:
  *Segment     isomagnetic or isochoric process, partial equilibration with the cold bath under constant Hamiltonian. The dynamics of the working medium is characterized by the  propagator .
  *Segment   magnetization  or  adiabatic compression, the external field  changes expanding the gap between energy levels of the Hamiltonian. The dynamics is characterized by the propagator .
  *Segment  isomagnetic, or isochoric process partial equilibration with the hot bath described by the propagator .
  *Segment  demagnetization or  adiabatic expansion reducing the energy gaps in the Hamiltonian, characterized by the  propagator .
  The propagator of the four stroke cycle 
  becomes ,  which is the ordered product of the segment propagators:
  The propagators are linear operators defined  on a vector space which completely determines the state of the working medium.
  Common to all thermodynamic cycles the consecutive segment propagators do not commute .
  Commuting propagators will lead to zero power.
  In a reciprocating quantum heat engine the working medium is a quantum system such as spin systems or an harmonic oscillator. For maximum power the cycle time should be optimized. 
  There are two basic timescales in the reciprocating refrigerator the cycle time  and the internal
  timescale . In general when  the 
  engine operates in quasi-adiabatic conditions. The only quantum effect can be found at low temperatures
  where the unit of energy of the device becomes  instead of .
  The efficiency at this limit is , always smaller than the Carnot efficiency . At high temperature and for the harmonic working medium the efficiency at maximum power becomes  which is the endoreversible thermodynamics result.
  For shorter cycle times the working medium cannot follow adiabatically the change in the external parameter.
  This leads to friction-like phenomena. Extra power is required to drive the system faster.
  The signature of such dynamics is the development of coherence causing extra dissipation.
  Surprisingly the dynamics leading to friction is quantized meaning that frictionless solutions to the adiabatic expansion/compression
  can be found in finite time.
   As a result, optimization has to be carried out only with respect to the time allocated
  to heat transport. In this regime the quantum feature of coherence degrades the performance.
  Optimal frictionless performance is obtained when the coherence can be cancelled.
  The shortest cycle times , sometimes termed sudden cycles, have universal features. In this case coherence contributes to the cycles power.
  A two-stroke engine quantum cycle equivalent to the Otto cycle based on two qubits has been proposed.
  The first qubit has frequency  and the second . The cycle is composed
  of a first stroke of partial equilibration of the two qubits with the hot and cold bath in parallel.
  The second power stroke is composed of a partial or full swap between the qubits.
  The swap operation is generated by a unitary transformation which preserves the entropy 
  as a result it is a pure power stroke.
  The quantum Otto cycle refrigerators shares the same cycle with magnetic refrigeration.