Thermodynamics of Minimal Coupling Quantum Heat Engines

Abstract

The minimal-coupling quantum heat engine is a thermal machine consisting of an explicit energy storage system, heat baths, and a working body, which alternatively couples to subsystems through discrete strokes --- energy-conserving two-body quantum operations. Within this paradigm, we present a general framework of quantum thermodynamics, where a work extraction process is fundamentally limited by a flow of non-passive energy (ergotropy), while energy dissipation is expressed through a flow of passive energy. It turns out that small dimensionality of the working body and a restriction only to two-body operations make the engine fundamentally irreversible. Our main result is finding the optimal efficiency and work production per cycle within the whole class of irreversible minimal-coupling engines composed of three strokes and with the two-level working body, where we take into account all possible quantum correlations between the working body and the battery. One of the key new tools is the introduced ``control-marginal state" --- one which acts only on a working body Hilbert space, but encapsulates all features regarding work extraction of the total working body-battery system. In addition, we propose a generalization of the many-stroke engine, and we analyze efficiency vs extracted work trade-offs, as well as work fluctuations after many cycles of the running of the engine.

Work and Fluctuations: Coherent vs. Incoherent Ergotropy Extraction

“…Our first main result is the mentioned before formula for change of the energy-storage variance (during the work extraction process). As it was shown in previous pa-pers [19,25], the first cumulant of the quasi-distribution (i.e., change in the average energy of the weight) is equal to the first cumulant of the work operator but calculated with respect to the control-marginal state. In the formula for the second cumulant (i.e., change in weight's variance), similarly appears an analogous term given by the second cumulant of the work operator; however, an additional term is also present, which is a fully quantum contribution that cannot be captured in non-autonomous frameworks.…”

“…The weight model (defined below) overcame this work definition problem in the quantum regime. Essentially, it was first proven that it could not decrease an entropy of the system [21,23], and later, more precisely, that the optimal work is given by the system's ergotropy [19,25].…”

“…Next, the crucial object for our analysis is the effective density operator σ, which we call the control state, and the marginal operator σS , i.e. the control-marginal state [19,25], in the form:…”

“…[24] where are discussed corrections if translational invariance is violated). Moreover, it has also been shown that the optimal work extracted by the quantum weight is limited by the ergotropy [19,25], which, with the notion of passivity, is another building block of quantum thermodynamics (see, e.g., [26,27]). Hence, taking the weight as a proper model for fluctuation theorems allows us to analyze the quantum coherences, from a thermodynamic perspective, within a fully quantum setup.…”

Second law of thermodynamics for batteries with vacuum state

“…Throughout we adapt a resource-theoretic approach to quantum thermodynamics called thermal operations [27,27,[29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45]. This is a wellestablished framework for studying thermodynamic processes in the quantum regime which gives the experimenter the most freedom in manipulating systems without access to external resources like coherence or asymmetry [25,[46][47][48][49], entanglement [50][51][52] or conserved quantities [53][54][55].…”