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Dynamical phases are obtained for a quantum thermal engine, whose workingmedium is a single harmonic oscillator. The dynamics of this engine is obtainedby using four steps where in two steps the time dependent frequency ischanging. In the other two steps, the thermal engine is coupled alternativelyto hot and cold heat baths. Similar dynamical phases are obtained in a quantumthermal engine whose working medium is spin 1/2 system. The role of timesdurations of such steps in the quantum engines for getting maximal efficiencyis analyzed. The dynamic of charge pumping in a quantum dot coupled to tworeservoirs is studied. The effects of many modulation parameters includingtheir fluctuations for getting geometric phases are analyzed. Since theseparate steps in thermal engines describe non-cyclic circuits, we propose touse a special method for measuring geometric phases in thermal engines fornon-cyclic circuits which is gauge invariant.

Quantum dynamic and geometric phases in harmonic oscillator, spin 1/2 and two-level thermal engines systems