Hiroyasu Tajima

Finite-size effect on optimal efficiency of heat engines

The optimal efficiency of quantum (or classical) heat engines whose heat baths are n-particle systems is given by the strong large deviation. We give the optimal work extraction process as a concrete energy-preserving unitary time evolution among the heat baths and the work storage. We show that our optimal work extraction turns the disordered energy of the heat baths to the ordered energy of the work storage, by evaluating the ratio of the entropy difference to the energy difference in the heat baths and the work storage, respectively. By comparing the statistical mechanical optimal efficiency with the macroscopic thermodynamic bound, we evaluate the accuracy of the macroscopic thermodynamics with finite-size heat baths from the statistical mechanical viewpoint. We also evaluate the quantum coherence effect on the optimal efficiency of the cycle processes without restricting their cycle time by comparing the classical and quantum optimal efficiencies.

Measurement-based formulation of quantum heat engines

There exist two formulations for quantum heat engine. One is semi-classical scenario, and the other is full quantum scenario. The former is formulated as a unitary evolution for the internal system (working body and heat baths), and is adopted by the community of statistical mechanics. The latter is formulated as a unitary for the work storage and the internal system. It is adopted by the community of quantum information. However, these formulations do not consider measurement process. In this paper, we formulate the quantum heat engine based on the quantum measurement theory, because the amount of extracted work should be observed in a practical situation. With using our formulation, we derive two trade-off relations that show the semi-classical scenario does not work when the amount of extracted work is observed, i.e., we can hardly know the amount of the extracted work when the time evolution of the internal system is close to unitary. Next, based on our formulation, we derive the optimal efficiency of quantum heat engines with the finite-size heat baths, without assuming the existence of quasi-static processes. Using the strong large deviation theory, we asymptotically expand the optimal efficiency up to the third order. The first term is shown to be Carnot efficiency, and the higher order terms are shown to be the finite-size effects of the heat baths. We can construct the optimal work extraction as an energy-preserving unitary evolution among the internal system and the work storage. During the optimal work extraction, the entropy gain of the work storage is so negligibly small as compared with the energy gain of the work storage.

Efficiency versus speed in quantum heat engines: Rigorous constraint from Lieb-Robinson bound

A long-standing open problem whether a heat engine with finite power achieves the Carnot efficiency is investgated. We rigorously prove a general trade-off inequality on thermodynamic efficiency and time interval of a cyclic process with quantum heat engines. In a first step, employing the Lieb-Robinson bound we establish an inequality on the change in a local observable caused by an operation far from support of the local observable. This inequality provides a rigorous characterization of the following intuitive picture that most of the energy emitted from the engine to the cold bath remains near the engine when the cyclic process is finished. Using this description, we prove an upper bound on efficiency with the aid of quantum information geometry. Our result generally excludes the possibility of a process with finite speed at the Carnot efficiency in quantum heat engines. In particular, the obtained constraint covers engines evolving with non-Markovian dynamics, which almost all previous studies on this topic fail to address.

Second law of information thermodynamics with entanglement transfer.

We present an inequality which holds in the thermodynamical processes with measurement and feedback controls and uses only the Helmholtz free energy and the entanglement of formation: W(ext)≤-ΔF-k(B)TΔE(F). The quantity -ΔE(F), which is positive, expresses the amount of entanglement transfer from system S to probe P through the interaction U(SP) during the measurement. It is easier to achieve the upper bound in this inequality than in the Sagawa-Ueda inequality [Phys. Rev. Lett. 100, 080403 (2008)]. Our inequality has clear physical meaning: in the above thermodynamical processes, the work which we can extract from the thermodynamic system is greater than the upper bound in the conventional thermodynamics by the amount of the entanglement extracted by the measurement.

Quantum Jarzynski equality of measurement-based work extraction

Many studies of quantum-size heat engines assume that the dynamics of an internal system is unitary and that the extracted work is equal to the energy loss of the internal system. Both assumptions, however, should be under scrutiny. In the present paper, we analyze quantum-scale heat engines, employing the measurement-based formulation of the work extraction recently introduced by Hayashi and Tajima [M. Hayashi and H. Tajima, arXiv:1504.06150]. We first demonstrate the inappropriateness of the unitary time evolution of the internal system (namely, the first assumption above) using a simple two-level system; we show that the variance of the energy transferred to an external system diverges when the dynamics of the internal system is approximated to a unitary time evolution. Second, we derive the quantum Jarzynski equality based on the formulation of Hayashi and Tajima as a relation for the work measured by an external macroscopic apparatus. The right-hand side of the equality reduces to unity for natural cyclic processes but fluctuates wildly for noncyclic ones, exceeding unity often. This fluctuation should be detectable in experiments and provide evidence for the present formulation.