Finite-time quantum Otto engine: Surpassing the quasistatic efficiency due to friction

Controlling work output and coherence in finite-time quantum Otto engines through monitoring

We examine the role of diagnostic quantum measurements on the work statistics of a finite-time quantum Otto heat engine operated in the steady state. We consider three pointer-based measurement schemes that differ in the number of system-pointer interactions and pointer measurements. We show that the coherence of the working substance and the work output of the engine can be controlled by tuning the monitoring measurements. Moreover, for a working substance consisting of a two-level system we show that while all three schemes reproduce the predictions of the cycle without any monitoring for the average work in the limit of infinitely weak measurement, only two of the schemes can reproduce the two-point projective measurement results in the limit of strong measurement.

Quantum Otto-type heat engine with fixed frequency

In this work we analyze an Otto-type cycle operating with a working substance composed of a quantum harmonic oscillator (QHO). Unlike other studies in which the work extraction is done by varying the frequency of the QHO and letting it thermalize with a squeezed reservoir, here we submit the QHO to a parametric pumping controlled by the squeezing parameter and let it thermalize with a thermal reservoir. We then investigate the role of the squeezing parameter in our Otto-type engine powered by parametric pumping and show that it is possible to reach the Carnot limit by arbitrarily increasing the squeezing parameter. Notably, for certain squeezing parameters r, e.g., r=0.4, the quasistatic Otto limit can be reached even at nonzero power. We also investigated the role of entropy production in the efficiency behavior during the unitary strokes, showing that positive (negative) changes in entropy production correspond to increases (decreases) in engine efficiency, as expected. Furthermore, we show that under thermal reservoirs a work extraction process that is more efficient than the Carnot engine is impossible, regardless of the quantum resource introduced via the Hamiltonian of the system.

Bounds on nonequilibrium fluctuations for asymmetrically driven quantum Otto engines

For a four-stroke asymmetrically driven quantum Otto engine with working medium modeled by a single qubit, we study the bounds on nonequilibrium fluctuations of work and heat. We find strict relations between the fluctuations of work and individual heat for hot and cold reservoirs in arbitrary operational regimes. Focusing on the engine regime, we show that the ratio of nonequilibrium fluctuations of output work to input heat from the hot reservoir is both upper and lower bounded. As a consequence, we establish a hierarchical relation between the relative fluctuations of work and heat for both cold and hot reservoirs and further make a connection with the thermodynamic uncertainty relations. We discuss the fate of these bounds also in the refrigerator regime. The reported bounds, for such asymmetrically driven engines, emerge once both the time-forward and the corresponding reverse cycles of the engine are considered on an equal footing. We also extend our study and report bounds for a parametrically driven harmonic oscillator Otto engine.

Measurement-based quantum Otto engine with a two-spin system coupled by anisotropic interaction: Enhanced efficiency at finite times

We have studied the performance of a measurement-based quantum Otto engine (QOE) in a working system of two spins coupled by Heisenberg anisotropic interaction. A nonselective quantum measurement fuels the engine. We have calculated thermodynamic quantities of the cycle in terms of the transition probabilities between the instantaneous energy eigenstates, and also between the instantaneous energy eigenstates and the basis states of the measurement, when the unitary stages of the cycle operate for a finite time τ. The efficiency attains a large value in the limit of τ→0 and then gradually reaches the adiabatic value in a long-time limit τ→∞. For finite values of τ and for anisotropic interaction, an oscillatory behavior of the efficiency of the engine is observed. This oscillation can be interpreted in terms of interference between the relevant transition amplitudes in the unitary stages of the engine cycle. Therefore, for a suitable choice of timing of the unitary processes in the short time regime, the engine can have a higher work output and less heat absorption, such that it works more efficiently than a quasistatic engine. In the case of an always-on heat bath, in a very short time, the bath has a negligible effect on its performance.

Numerically "exact" simulations of a quantum Carnot cycle: Analysis using thermodynamic work diagrams

We investigate the efficiency of a quantum Carnot engine based on open quantum dynamics theory. The model includes time-dependent external fields for the subsystems controlling the isothermal and isentropic processes and for the system--bath (SB) interactions controlling the transition between these processes. Numerical simulations are conducted in a nonperturbative and non-Markovian SB coupling regime using the hierarchical equations of motion under these fields at different cycle frequencies. The work applied to the total system and the heat exchanged with the baths are rigorously evaluated. In addition, by regarding quasi-static work as free energy, we compute the quantum thermodynamic variables and analyze the simulation results using thermodynamic work diagrams for the first time. Analysis of these diagrams indicates that, in the strong SB coupling region, the fields for the SB interactions are major sources of work, while in other regions, the field for the subsystem is a source of work. We find that the maximum efficiency is achieved in the quasi-static case and is determined solely by the bath temperatures, regardless of the SB coupling strength, which is a numerical manifestation of Carnot's theorem.

The laws of thermodynamics for quantum dissipative systems: A quasi-equilibrium Helmholtz energy approach

Using the quasi-equilibrium Helmholtz energy (qHE), defined as the thermodynamic work in a quasi-static process, we investigate the thermal properties of both an isothermal process and a transition process between the adiabatic and isothermal states (adiabatic transition). Here, the work is defined by the change in energy from a steady state to another state under a time-dependent perturbation. In particular, the work for a quasi-static change is regarded as thermodynamic work. We employ a system--bath model that involves time-dependent perturbations in both the system and the system--bath interaction. We conduct numerical experiments for a three-stroke heat machine (a Kelvin-Planck cycle). For this purpose, we employ the hierarchical equations of motion (HEOM) approach. These experiments involve an adiabatic transition field that describes the operation of an adiabatic wall between the system and the bath. Thermodynamic--work diagrams for external fields and their conjugate variables, similar to the P-V diagram, are introduced to analyze the work done for the system in the cycle. We find that the thermodynamic efficiency of this machine is zero because the field for the isothermal processes acts as a refrigerator, whereas that for the adiabatic wall acts as a heat engine. This is a numerical manifestation of the Kelvin-Planck statement, which states that it is impossible to derive mechanical effects from a single heat source. These HEOM simulations serve as a rigorous test of thermodynamic formulations because the second law of thermodynamics is only valid when the work involved in the operation of adiabatic wall is treated accurately.

Extrapolating the thermodynamic length with finite-time measurements

The thermodynamic length, though providing a lower bound for the excess work required in a finite-time thermodynamic process, is determined by the properties of the equilibrium states reached by the quasistatic process and is thus beyond the direct experimental measurement. We propose an experimental strategy to measure the thermodynamic length of an open classical or quantum system by extrapolating finite-time measurements. The current proposal enables the measurement of the thermodynamic length for a single control parameter without requiring extra effort to find the optimal control scheme, and is illustrated with examples of the quantum harmonic oscillator with tuning frequency and the classical ideal gas with changing volume. Such a strategy shall shed light on the experimental design of the lacking platforms to measure the thermodynamic length.

Quantum Heat Engines with Complex Working Media, Complete Otto Cycles and Heuristics

Quantum thermal machines make use of non-classical thermodynamic resources, one of which include interactions between elements of the quantum working medium. In this paper, we examine the performance of a quasi-static quantum Otto engine based on two spins of arbitrary magnitudes subject to an external magnetic field and coupled via an isotropic Heisenberg exchange interaction. It has been shown earlier that the said interaction provides an enhancement of cycle efficiency, with an upper bound that is tighter than the Carnot efficiency. However, the necessary conditions governing engine performance and the relevant upper bound for efficiency are unknown for the general case of arbitrary spin magnitudes. By analyzing extreme case scenarios, we formulate heuristics to infer the necessary conditions for an engine with uncoupled as well as coupled spin model. These conditions lead us to a connection between performance of quantum heat engines and the notion of majorization. Furthermore, the study of complete Otto cycles inherent in the average cycle also yields interesting insights into the average performance.

Fluctuations in irreversible quantum Otto engines

We derive the general probability distribution function of stochastic work for quantum Otto engines in which both the isochoric and driving processes are irreversible due to finite time duration. The time-dependent work fluctuations, average work, and thermodynamic efficiency are explicitly obtained for a complete cycle operating with an analytically solvable two-level system. The effects of the irreversibility originating from finite-time cycle operation on the thermodynamic efficiency, work fluctuations, and relative power fluctuations are discussed.

Finite-time performance of a single-ion quantum Otto engine

We study how a quantum heat engine based on a single trapped ion performs in finite time. The always-on thermal environment acts like the hot bath, while the motional degree of freedom of the ion plays the role of the effective cold bath. The hot isochoric stroke is implemented via the interaction of the ion with its hot environment, while a projective measurement of the internal state of the ion is performed as an equivalent to the cold isochoric stroke. The expansion and compression strokes are implemented via suitable change in applied magnetic field. We study in detail how the finite duration of each stroke affects the engine performance. We show that partial thermalization can in fact enhance the efficiency of the engine, due to the residual coherence, whereas faster expansion and compression strokes increase the inner friction and therefore reduce the efficiency.

Quantumness and thermodynamic uncertainty relation of the finite-time Otto cycle

To reveal the role of the quantumness in the Otto cycle and to discuss the validity of the thermodynamic uncertainty relation (TUR) in the cycle, we study the quantum Otto cycle and its classical counterpart. In particular, we calculate exactly the mean values and relative error of thermodynamic quantities. In the quasistatic limit, quantumness reduces the productivity and precision of the Otto cycle compared to that in the absence of quantumness, whereas in the finite-time mode, it can increase the cycle's productivity and precision. Interestingly, as the strength (heat conductance) between the system and the bath increases, the precision of the quantum Otto cycle overtakes that of the classical one. Testing the conventional TUR of the Otto cycle, in the region where the entropy production is large enough, we find a tighter bound than that of the conventional TUR. However, in the finite-time mode, both quantum and classical Otto cycles violate the conventional TUR in the region where the entropy production is small. This implies that another modified TUR is required to cover the finite-time Otto cycle. Finally, we discuss the possible origin of this violation in terms of the uncertainty products of the thermodynamic quantities and the relative error near resonance conditions.

Universal two-level quantum Otto machine under a squeezed reservoir

We study an Otto heat machine whose working substance is a single two-level system interacting with a cold thermal reservoir and with a squeezed hot thermal reservoir. By adjusting the squeezing or the adiabaticity parameter (the probability of transition) we show that our two-level system can function as a universal heat machine, either producing net work by consuming heat or consuming work that is used to cool or heat environments. Using our model we study the performance of these machine in the finite-time regime of the isentropic strokes, which is a regime that contributes to make them useful from a practical point of view.

Optimal Control of Hydrogen Atom-Like Systems as Thermodynamic Engines in Finite Time

Nano-size machines are moving from only being topics of basic research to becoming elements in the toolbox of engineers, and thus the issue of optimally controlling their work cycles becomes important. Here, we investigate hydrogen atom-like systems as working fluids in thermodynamic engines and their optimal control in minimizing entropy or excess heat production in finite-time processes. The electronic properties of the hydrogen atom-like system are controlled by a parameter κ reflecting changes in, e.g., the effective dielectric constant of the medium where the system is embedded. Several thermodynamic cycles consisting of combinations of iso-κ, isothermal, and adiabatic branches are studied, and a possible a-thermal cycle is discussed. Solving the optimal control problem, we show that the minimal thermodynamic length criterion of optimality for finite-time processes also applies to these cycles for general statistical mechanical systems that can be controlled by a parameter κ, and we derive an appropriate metric in probability distribution space. We show how the general formulas we have obtained for the thermodynamic length are simplified for the case of the hydrogen atom-like system, and compute the optimal distribution of process times for a two-state approximation of the hydrogen atom-like system.

Thermodynamics of the Coarse-Graining Master Equation

We study the coarse-graining approach to derive a generator for the evolution of an open quantum system over a finite time interval. The approach does not require a secular approximation but nevertheless generally leads to a Lindblad-Gorini-Kossakowski-Sudarshan generator. By combining the formalism with full counting statistics, we can demonstrate a consistent thermodynamic framework, once the switching work required for the coupling and decoupling with the reservoir is included. Particularly, we can write the second law in standard form, with the only difference that heat currents must be defined with respect to the reservoir. We exemplify our findings with simple but pedagogical examples.