  Dynamical phases are obtained for a quantum thermal engine, whose working
medium is a single harmonic oscillator. The dynamics of this engine is obtained
by using four steps where in two steps the time dependent frequency is
changing. In the other two steps, the thermal engine is coupled alternatively
to hot and cold heat baths. Similar dynamical phases are obtained in a quantum
thermal engine whose working medium is spin 1/2 system. The role of times
durations of such steps in the quantum engines for getting maximal efficiency
is analyzed. The dynamic of charge pumping in a quantum dot coupled to two
reservoirs is studied. The effects of many modulation parameters including
their fluctuations for getting geometric phases are analyzed. Since the
separate steps in thermal engines describe non-cyclic circuits, we propose to
use a special method for measuring geometric phases in thermal engines for
non-cyclic circuits which is gauge invariant.
