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The Mpemba effect is a counterintuitive relaxation phenomenon whereby a system with a higher initial temperature may cool down to the thermal state faster than an identical system that was initially prepared at a lower temperature. Here, we investigate heat and work in a Markovian state transition system with cyclic switching hot-cold temperatures, which operates as an Otto heat engine working in long but finite time, either with or without the Mpemba effect. Under the condition of the periodic steady state having been reached, the time durations of the heating and cooling relaxation processes are determined by exploring a distance-from-equilibrium equivalent to the Kullback-Leibler divergence. We then numerically evaluate and compare the averages and variances of both the work and the power output of two scenarios with and without the Mpemba effect. The results show that the Markovian Mpemba effect can enhance the machine performance by significantly increasing the power output for a given efficiency without sacrificing the stability.

A quantum Otto engine using two-interacting spins as its working medium is analyzed within framework of stochastic thermodynamics. The time-dependent power fluctuations and average power are explicitly derived for a complete cycle of engine operation. We find that the efficiency and power fluctuations are affected significantly by interparticle interactions, but both of them become interaction-independent under maximal power via optimizing the external control parameter. The behavior of the efficiency at maximum power is further explained by analyzing the optimal protocol of the engine.

We investigate the finite-power performance of quantum heat engines working in the linear response regime where the temperature gradient is small. The engine cycles with working substances of ideal harmonic systems consist of two heat transfer and two adiabatic processes, such as the Carnot cycle, Otto cycle, and Brayton cycle. By analyzing the optimal protocol under maximum power we derive the explicitly analytic expression for the irreversible entropy production, which becomes the low dissipation form in the long duration limit. Assuming the engine to be endoreversible, we derive the universal expression for the efficiency at maximum power, which agrees well with that obtained from the phenomenological heat transfer laws holding in the classical thermodynamics. Through appropriate identification of the thermodynamic fluxes and forces that a linear relation connects, we find that the quantum engines under consideration are tightly coupled, and the universality of efficiency at maximum power is confirmed at the linear order in the temperature gradient.

We consider the finite-time performance of a quantum Otto engine working between a hot squeezed and a cold thermal bath at inverse temperatures β_{h} and β_{c}(>β_{h}) with (k_{B}≡1)β=1/T. We derive the analytical expressions for work, efficiency, power, and power fluctuations, in which the squeezing parameter is involved. By optimizing the power output with respect to two frequencies, we derive the efficiency at maximum power as η_{mp}=(η_{C}^{gen})^{2}/[η_{C}^{gen}-(1-η_{C}^{gen})ln(1-η_{C}^{gen})], where the generalized Carnot efficiency η_{C}^{gen} in the high-temperature or small squeezing limit simplifies to an analytic function of squeezing parameter γ: η_{C}^{gen}=1-β_{h}/[β_{c}cosh(2γ)]. Within the context of irreversible thermodynamics, we demonstrate that the expression of efficiency at maximum power satisfies a general form derived from nonlinear steady state heat engines. We show that, the power fluctuations are considerably increased, although the engine efficiency is enhanced by squeezing.

We analyze general models of quantum heat engines operating a cycle of two adiabatic and two isothermal processes. We use the quantum master equation for a system to describe heat transfer current during a thermodynamic process in contact with a heat reservoir, with no use of phenomenological thermal conduction. We apply the endoreversibility description to such engine models working in the linear response regime and derive expressions of the efficiency and the power. By analyzing the entropy production rate along a single cycle, we identify the thermodynamic flux and force that a linear relation connects. From maximizing the power output, we find that such heat engines satisfy the tight-coupling condition and the efficiency at maximum power agrees with the Curzon-Ahlborn efficiency known as the upper bound in the linear response regime.