In this paper we examine some foundational issues of a class of quantum engines where the system consists of a single quantum parametric oscillator, operating in an Otto cycle consisting of 4 stages of two alternating phases: the isentropic phase is detached from any bath (thus a closed system) where the natural frequency of the oscillator is changed from one value to another, and the isothermal phase where the system (now rendered open) is put in contact with one or two squeezed baths of different temperatures, whose nonequilibrium dynamics follows the Hu-Paz-Zhang (HPZ) master equation for quantum Brownian motion. The HPZ equation is an exact nonMarkovian equation which preserves the positivity of the density operator and is valid for a) all temperatures, b) arbitrary spectral density of the bath, and c) arbitrary coupling strength between the system and the bath. Taking advantage of these properties we examine some key foundational issues of theories of quantum open and squeezed systems for these two phases of the quantum Otto engines. This include, i) the nonMarkovian regimes for non-Ohmic, low temperature baths, ii) what to expect in nonadiabatic frequency modulations, iii) strong system-bath coupling, as well as iv) the proper junction conditions between these two phases. Our aim here is not to present ways for attaining higher efficiency but to build a more solid theoretical foundation for quantum engines of continuous variables covering a broader range of parameter spaces hopefully of use for exploring such possibilities.
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