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Physica A Statistical Mechanics and its Applications

Monitored quantum unital Otto heat engines: Fluctuations of the released heat and entropy production

Abdelkader El Makouri, Abdallah Slaoui, Rachid Ahl Laamara

Published Apr 26, 2025• 5 citations
Publisher
DOI
10.1016/j.physa.2025.130591

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Recently, we have considered a single-qubit quantum Otto cycle based on arbitrary (non-unitary) unital channels as a heat source. We showed similarly to some works in literature that the ratio of the fluctuations of work (W) and heat absorbed (QM) is lower-and-upper bounded. In this paper, we drive this line of research further and consider the fluctuations of the stochastic heat exchanged (denoted by QC′) with the cold bath. In general, there is another stochastic quantity, which we denote by QC, that gives the same average as QC′− but not higher cumulants. While QC′ is defined through the two-point measurement scheme, i.e., by measuring the energy of the working medium in all four strokes, QC follows from imposing the first of thermodynamics at the stochastic level. Our work has two important results: (i) first, we prove that the measurement of the working medium during three strokes is sufficient to construct the distribution of QC′ (ii) and that the cumulants of the stochastic entropy production denoted by Σ can be inferred from the cumulants of QC.

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Recently, we have considered a single-qubit quantum Otto cycle based on arbitrary (non-unitary) unital channels as a heat source. We showed similarly to some works in literature that the ratio of the fluctuations of work (W) and heat absorbed (QM) is lower-and-upper bounded. In this paper, we drive this line of research further and consider the fluctuations of the stochastic heat exchanged (denoted by QC′) with the cold bath. In general, there is another stochastic quantity, which we denote by QC, that gives the same average as QC′− but not higher cumulants. While QC′ is defined through the two-point measurement scheme, i.e., by measuring the energy of the working medium in all four strokes, QC follows from imposing the first of thermodynamics at the stochastic level. Our work has two important results: (i) first, we prove that the measurement of the working medium during three strokes is sufficient to construct the distribution of QC′ (ii) and that the cumulants of the stochastic entropy production denoted by Σ can be inferred from the cumulants of QC.
Recently, measurement-based quantum thermal machines have drawn more attention in the field of quantum thermodynamics. However, the previous results on quantum Otto heat engines were either limited to special unital and nonunital channels in the bath stages, or a specific driving protocol at the work strokes and assuming the cycle being time-reversal symmetric, i.e., V^{†}=U (or V=U). In this paper, we consider a single spin-1/2 quantum Otto heat engine, by first replacing one of the heat baths by an arbitrary unital channel, and then we give the exact analytical expression of the characteristic function from which all the cumulants of heat and work emerge. We prove that under the effect of monitoring, ν_{2}>ν_{1} is a necessary condition for positive work, either for a symmetric or asymmetric-driven Otto cycle. Furthermore, going beyond the average we show that the ratio of the fluctuations of work and heat is lower and upper-bounded when the system is working as a heat engine. However, differently from the previous results in the literature, we consider the third and fourth cumulants as well. It is shown that the ratio of the third (fourth) cumulants of work and heat is not upper-bounded by unity nor lower-bounded by the third (fourth) power of the efficiency, as is the case for the ratio of fluctuations. Finally, we consider applying a specific unital map that plays the role of a heat bath in a coherently superposed manner, and we show the role of the initial coherence of the control qubit on efficiency, on the average work and its relative fluctuations.
We consider a finite-time quantum Otto heat engine that consists of two isochoric (thermal-contact) process, where the system is alternatively coupled to a hot squeezed and a cold thermal reservoir, and two unitary driven strokes, where the system is isolated from these two baths and its von Neumann entropy keeps constant. Both quantum inner friction and coherence are generated along the driven stroke and coherence cannot be fully erased after the finite-time hot isochore. Using full counting statistics, we present the probability distribution functions of heat injection and total work per cycle, which are dependent on the time duration along each process. With these, we derive the analytical expressions for the thermodynamic quantities of the two-level heat engine, such as total work, thermodynamic efficiency, entropy production, and work fluctuations, in which effects of coherence, squeezing, inner friction and finite-time heat exchange are included. We then numerically determine the thermodynamic quantities and the fluctuations using the parameters employed in the experimental implementation. Our results clarify the role of coherence and squeezing in the performance and fluctuations in the quantum Otto engines.