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

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.