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 power fluctuations, average power, and thermodynamic efficiency are explicitly obtained for a complete cycle operating with an analytically solvable two-level system. We show that, there is a trade-off between efficiency (or power) and power fluctuations.
We consider the efficiency at maximum power of a quantum Otto engine, which uses a spin or a harmonic system as its working substance and works between two heat reservoirs at constant temperatures $T_h$ and $T_c$ $ (<T_h)$. Although the spin-$1/2$ system behaves quite differently from the harmonic system in that they obey two typical quantum statistics, the efficiencies at maximum power based on these two different kinds of quantum systems are bounded from the upper side by the same expression of the efficiency at maximum power: $η_{mp}\leqη_+\equiv η_C^2/[η_C-(1-η_C)\ln(1-η_C)]$, with $η_C=1-…
Despite its idealizations, thermodynamics has proven its power as a predictive theory for practical applications. In particular, the Curzon-Ahlborn efficiency provides a benchmark for any real engine operating at maximal power. Here we further develop the analysis of endoreversible Otto engines. For a generic class of working mediums, whose internal energy is proportional to some power of the temperature, we find that no engine can achieve the Carnot efficiency at finite power. However, we also find that for the specific example of photonic engines the efficiency at maximal power is larger tha…
We present the spin quantum Otto machine under different optimization criterion when function either as a heat engine or a refrigerator. We examine the optimal performance of the heat engine and refrigerator depending on their efficiency, output power and maximum entropy production. For heat engine case, we obtain the expression for the upper and lower bounds efficiencies at maximum power and maximum ecological function. In addition, the spin quantum Otto refrigerator coefficient of performance is optimized for three different criterion -- cooling power, product of performance and power and ec…
The second law of thermodynamics constrains that the efficiency of heat engines, classical or quantum, cannot be greater than the universal Carnot efficiency. We discover another bound for the efficiency of a quantum Otto heat engine consisting of a harmonic oscillator. Dynamics of the engine is governed by the Lindblad equation for the density matrix, which is mapped to the Fokker-Planck equation for the quasi-probability distribution. Applying stochastic thermodynamics to the Fokker-Planck equation system, we obtain the $\hbar$-dependent quantum mechanical bound for the efficiency. It turns …
Recent experimental breakthroughs produced the first nano heat engines that have the potential to harness quantum resources. An instrumental question is how their performance measures up against the efficiency of classical engines. For single ion engines undergoing quantum Otto cycles it has been found that the efficiency at maximal power is given by the Curzon-Ahlborn efficiency. This is rather remarkable as the Curzon-Alhbron efficiency was originally derived for endoreversible Carnot cycles. Here, we analyze two examples of endoreversible Otto engines within the same conceptual framework as…
We derive the probability distribution of the efficiency of a quantum Otto engine. We explicitly compute the quantum efficiency statistics for an analytically solvable two-level engine. We analyze the occurrence of values of the stochastic efficiency above unity, in particular at infinity, in the nonadiabatic regime and further determine mean and variance in the case of adiabatic driving. We finally investigate the classical-to-quantum transition as the temperature is lowered.
We investigate quantum Otto engine and refrigeration cycles of a time-dependent harmonic oscillator operating under the conditions of maximum $Ω$-function, a trade-off objective function which represents a compromise between energy benefits and losses for a specific job, for both adiabatic and nonadiabatic (sudden) frequency modulations. We derive analytical expressions for the efficiency and coefficient of performance of the Otto cycle. For the case of adiabatic driving, we point out that in the low-temperature regime, the harmonic Otto engine (refrigerator) can be mapped to Feynman's ratchet…
We propose a quantum Otto cycle based on the properties of a two-level system in a realistic out-of-thermal-equilibrium electromagnetic field acting as its sole reservoir. This steady configuration is produced without the need of active control over the state of the environment, which is a non-coherent thermal radiation, sustained only by external heat supplied to macroscopic objects. Remarkably, even for non-ideal finite-time transformations, it largely over-performs the standard ideal Otto cycle, and asymptotically achieves unit efficiency at finite power.
We study internal work optimization over the energy levels of a generic hot quantum Otto engine. We find universal features in the efficiency that resembles the classical external power optimization over the coupling times to the thermal baths. It is shown that in the ultra hot regime the efficiency is determined solely by the optimization constraint, and independent of the engine details. We show that for some constraints the radius of convergence of the perturbative approach used in the classical analysis is zero even for very arbitrarily low efficiencies at small temperature difference.
We consider a finite-time Otto engine operating on a quantum harmonic oscillator and driven by shortcut-to-adiabaticity (STA) techniques to speed up its cycle. We study its efficiency and power when internal friction, time-averaged work, and work fluctuations are used as quantitative figures of merit, showing that time-averaged efficiency and power are useful cost functions for the characterization of the performance of the engine. We then use the minimum allowed time for validity of STA protocol relation to establish a physically relevant bound to the efficiency at maximum power of the STA-dr…
We investigate a heat engine with a finite length Kitaev chain in an Otto cycle. Finite size effects are taken into account using method of Hill's nano-thermodynamics as well as using the method of temperature dependent energy levels. We distinguish the bulk and boundary contributions to the efficiency and the work output of Kitaev chain engine and identify them as non-Otto heat engine and refrigerator cycles, respectively. Possibility of separately running Otto engine cycles associated with the bulk and the whole system, and an Otto refrigerator at the boundary is pointed out. It is found tha…
We study two-mode bosonic engines undergoing an Otto cycle. The energy exchange between the two bosonic systems is provided by a tunable unitary bilinear interaction in the mode operators modeling frequency conversion, whereas the cyclic operation is guaranteed by relaxation to two baths at different temperature after each interacting stage. By means of a two-point-measurement approach we provide the joint probability of the stochastic work and heat. We derive exact expressions for work and heat fluctuations, identities showing the interdependence among average extracted work, fluctuations and…
We consider the optimization of the work output and fluctuations of a finite-time quantum Otto heat engine cycle consisting of compression and expansion work strokes of unequal duration. The asymmetry of the cycle is characterized by a parameter $r_u$ giving the ratio of the times for the compression and expansion work strokes. For such an asymmetric quantum Otto engine cycle, with working substance chosen as a harmonic oscillator or a two-level system, we find that the optimal values of $r_u$ maximising the work output and the reliability (defined as the ratio of average work output to its st…
In finite-time quantum heat engines, some work is consumed to drive a working fluid accompanying coherence, which is called `friction'. To understand the role of friction in quantum thermodynamics, we present a couple of finite-time quantum Otto cycles with two different baths: Agarwal versus Lindbladian. We solve them exactly and compare the performance of the Agarwal engine with that of the Lindbladian engine. In particular, we find remarkable and counterintuitive results that the performance of the Agarwal engine due to friction can be much higher than that in the quasistatic limit with the…
The quantum heat engines have drawn much attention due to miniaturization of devices recently. We study the dynamics of the quantum Otto heat engine using the driven-dissipative Schrödinger equation. Starting from different initial states, we simulate the time evolutions of the internal energy, power and heat-work conversion efficiency. The initial state impacts on these thermodynamic quantities before the Otto cycle reaches stable. In the transition period, the efficiency and power may be higher or lower than the corresponding values in the cyclostationary state. Remarkably, the efficiency co…
Graphene is perhaps the most prominent "Dirac material," a class of systems whose electronic structure gives rise to charge carriers that behave as relativistic fermions. In multilayer graphene several crystal sheets are stacked such that the honeycomb lattice of each layer is displaced along one of the lattice edges. When subject to an external magnetic field, the scaling of the multilayer energy spectrum with the magnetic field, and thus the system's thermodynamic behavior, depends strongly on the number of layers. With this in mind, we examine the performance of a finite-time endoreversible…
In this paper, we study the role and relevance of the cost for an invariant-based shortcut to adiabaticity enabled qubit heat engine operates in a quantum Otto cycle. We consider a qubit heat engine with Landau-Zener Hamiltonian and improve its performance using the Lewis-Riesenfeld invariant-based shortcut to adiabaticity method. Addressing the importance of cost for better performance, the paper explores its relationship with the work and efficiency of the heat engine. We analyze the cost variation with the time duration of non-adiabatic unitary processes involved in the heat engine cycle. T…
We investigate a quantum heat engine with a working substance of two particles, one with a spin $1/2$ and the other with an arbitrary spin (spin $s$), coupled by Heisenberg exchange interaction, and subject to an external magnetic field. The engine operates in a quantum Otto cycle. Work harvested in the cycle and its efficiency are calculated using quantum thermodynamical definitions. It is found that the engine has higher efficiencies at higher spins and can harvest work at higher exchange interaction strengths. The role of exchange coupling and spin $s$ on the work output and the thermal eff…
In this paper, we analyze the total work extracted and the efficiency of the magnetic Otto cycle in its classic and quantum versions. As a general result, we found that the work and efficiency of the classical engine is always greater than or equal to that of its quantum counterpart independent of the working substance. In the classical case, this is due to the fact that the working substance is always in thermodynamic equilibrium at each point of the cycle, maximizing the energy extracted in the adiabatic paths. We apply this analysis to the case of a two-level system, finding that the work a…
We investigate a heat engine with a finite-length Kitaev chain in an ideal Otto cycle. It is found that the critical point of the topological phase transition coincides with the maxima of the efficiency and work output of the total Otto engine. Finite-size effects are taken into account using the method of Hill's nanothermodynamics, as well as using the method of temperature-dependent energy levels. We identify the bulk and boundary thermal cycles of the Kitaev chain engine and find that they are non-ideal Otto cycles. The physics of deviation from ideal Otto cycle is identified as a finite si…
We use the general formulation of irreversible thermodynamics and study the minimally nonlinear irreversible model of heat engines operating between a time-varying hot heat source of finite size and a cold heat reservoir of infinite size. We find the criterion in which the optimized efficiency obtained by this minimally nonlinear irreversible heat engine can reach the reversible efficiency under the tight coupling condition: a condition of no heat leakage between the system and the reservoirs. We assume the rate of heat transfer from hot to cold heat reservoir obeys Fourier law and discuss phy…
We propose a quantum Otto cycle in a two spin-$1/2$ anisotropic XY model in a transverse external magnetic field. We first characterize the parameter regime that the working medium operates as an engine in the adiabatic regime. Then, we consider finite-time behavior of the engine with and without utilizing a shortcut to adiabaticity (STA) technique. STA schemes guarantee that the dynamics of a system follows the adiabatic path, at the expense of introducing an external control. We compare the performance of the non-adiabatic and STA engines for a fixed adiabatic efficiency but different parame…
For finite-time quantum Otto heat engine with working fluid consisting of either a (i) qubit or (ii) a harmonic oscillator, we show that the relative fluctuation of output work is always greater than the corresponding relative fluctuation of input heat absorbed from the hot bath. As a result, the ratio between the work fluctuation and the input heat fluctuation receives a lower bound in terms of the square value of the average efficiency of the engine. The saturation of the lower bound is received in the quasi-static limit of the engine and can be shown for a class of working fluids that follo…
Quantum heat engine with ideal gas has been well studied, yet the role of interaction was seldom explored. We construct a quantum Otto heat engine with N repulsive Bosonic particles in a 1D hard wall box. With the advantage of exact solution using Bethe Ansatz, we obtain not only the exact numerical result of efficiency in all interacting strength c, but also analytical results for strong interaction. We find the efficiency ηrecovers to the one of non-interacting case $η_{\mathrm{non}}=1-(L_{1}/L_{2})^{2}$ for strong interaction with asympotic behavior $η\simη_{\mathrm{non}}-4(N-1)L_{1}\left(L…
We study the minimally nonlinear irreversible heat engines in which the time-reversal symmetry for the systems may b e broken. The expressions for the power and the efficiency are derived, in which the effects of the nonlinear terms due to dissipations are included. We show that, as within the linear responses, the minimally nonlinear irreversible heat engines enable attainment of Carnot efficiency at positive power. We also find that the Curzon-Ahlborn limit imposed on the efficiency at maximum power can be overcomed if the time-reversal symmetry is broken.
We derive the general relations between the maximum power, maximum efficiency and minimum dissipation for the irreversible heat engine in nonlinear response regime. In this context, we use the minimally nonlinear irreversible model and obtain the lower and upper bounds of the above relations for the asymmetric dissipation limits. These relations can be simplified further when the system possesses the time-reversal symmetry or anti-symmetry. We find that our results are the generalization of various such relations obtained earlier for different heat engines.
We study a driven harmonic oscillator operating an Otto cycle between two thermal baths of finite size. By making extensive use of the tools of Gaussian quantum mechanics, we directly simulate the dynamics of the engine as a whole, without the need to make any approximations. This allows us to understand the non-equilibrium thermodynamics of the engine not only from the perspective of the working medium, but also as it is seen from the thermal baths' standpoint. For sufficiently large baths, our engine is capable of running a number of ideal cycles, delivering finite power while operating very…
Quantum thermal machines make use of non-classical thermodynamic resources, one of which is 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 earlier shown that the said interaction provides an enhancement of cycle efficiency for two spin-1/2 particles, with an upper bound which is tighter than the Carnot efficiency. However, the necessary conditions g…
To optimize the performance of a heat engine in finite-time cycle, it is important to understand the finite-time effect of thermodynamic processes. Previously, we have shown that extra work is needed to complete a quantum adiabatic process in finite time, and proved that the extra work follows a \mathcal{C}/τ^{2} scaling for long control time τ. There the oscillating part of the extra work is neglected due to the complex energy-level structure of the particular quantum system. However, such oscillation of the extra work can not be neglected in some quantum systems with simple energy-level stru…
The Carnot theorem, one expression of the second law of thermodynamics, places a fundamental upper bound on the efficiency of heat engines operating between two heat baths. The Carnot theorem can be stated in a more generalized form for heat engines operating with multiple heat baths, where the maximum efficiency is achieved for reversible heat engines operating quasistatically between two heat baths. In this study, we determine the irreversible efficiency of heat engines operating with multiple heat baths in a linear response regime, i.e., under small temperature differences and a slow variat…
It is investigated whether non-Markovianity, i.e., the memory effects resulting from the coupling of the system to its environment, can be beneficial for the performance of quantum heat engines. Specifically, two physical models are considered. The first one is a well known single-qubit Otto engine; the non-Markovian behaviour is there implemented by replacing standard thermalization strokes with so-called extremal thermal operations which cannot be realized without the memory effects. The second one is a three-stroke engine in which the cycle consists of two extremal thermal operations and a …
The efficiency at maximum power (EMP) of irreversible Carnot-like heat engines is investigated based on the weak endoreversible assumption and the phenomenologically irreversible thermodynamics. It is found that the weak endoreversible assumption can reduce to the conventional one for the heat engines working at maximum power. Carnot-like heat engines are classified into three types (linear, superlinear, and sublinear) according to different characteristics of constitutive relations between the heat transfer rate and the thermodynamic force. The EMPs of Carnot-like heat engines are proved to b…
We formulate the work output and efficiency for linear irreversible heat engines working between a finite-sized hot heat source and an infinite-sized cold heat reservoir until the total system reaches the final thermal equilibrium state with a uniform temperature. We prove that when the heat engines operate at the maximum power under the tight-coupling condition without heat leakage the work output is just half of the exergy, which is known as the maximum available work extracted from a heat source. As a consequence, the corresponding efficiency is also half of its quasistatic counterpart.
We consider a quasi-static quantum Otto cycle using two effectively two-level systems with degeneracy in the excited state. The systems are coupled through isotropic exchange interaction of strength $J>0$, in the presence of an external magnetic field $B$ which is varied during the cycle. We prove the positive work condition, and show that level-degeneracy can act as a thermodynamic resource, so that a larger amount of work can be extracted than from the non-degenerate case, both with and without coupling. We also derive an upper bound for the efficiency of the cycle. This bound is the same as…
A quantum Otto heat engine is studied with multilevel identical particles trapped in one-dimensional box potential as working substance. The symmetrical wave function for Bosons and the anti-symmetrical wave function for Fermions are considered. In two-particle case, we focus on the ratios of $W^i$ ($i=B,F$) to $W_s$, where $W^B$ and $W^F$ are the work done by two Bosons and Fermions respectively, and $W_s$ is the work output of a single particle under the same conditions. Due to the symmetric of the wave functions, the ratios are not equal to $2$. Three different regimes, low temperature regi…
We propose the minimally nonlinear irreversible heat engine as a new general theoretical model to study the efficiency at the maximum power $η^*$ of heat engines operating between the hot heat reservoir at the temperature $T_h$ and the cold one at $T_c$ ($T_c \le T_h $). Our model is based on the extended Onsager relations with a new nonlinear term meaning the power dissipation. In this model, we show that $η^*$ is bounded from the upper side by a function of the Carnot efficiency $η_C\equiv 1-T_c/T_h$ as $η^*\le η_C/(2-η_C)$. We demonstrate the validity of our theory by showing that the low-d…
We study a limit cycle of a quantum Otto engine whose each cycle consists of two finite-time quantum isochoric (heating or cooling) processes and two quantum adiabatic work-extracting processes. Considering a two-level system as a working substance that weakly interacts with two reservoirs comprising an infinite number of bosons, we investigate the non-Markovian effect (short-time behavior of the reduced dynamics in the quantum isochoric processes (QIPs)) on work extraction after infinite repetition of the cycles. We focus on the parameter region where energy transferred to the reservoir can c…
In the recent progress in nonequilibrium thermodynamics, information has been recognized as a kind of thermodynamic resource that can drive thermodynamic current without any direct energy injection. In this paper, we establish the framework of linear irreversible thermodynamics for a broad class of autonomous information processing. In particular, we prove that the Onsager reciprocity holds true with information: The linear response matrix is well-defined, and is shown symmetric with both of the information affinity and the conventional thermodynamic affinity. As an application, we derive a un…
Power-efficiency trade-off for finite-time quantum harmonic Otto heat engine via phase-space approach
School of Physics, Korea Institute for Advanced Study（韩国高等科学研究院物理学院）；Asia Pacific Center for Theoretical Physics（亚太理论物理中心）；Department of Physics, POSTECH（韩国科学技术院物理系）
We study the 1-d isotropic Heisenberg model of two spin-1/2 systems as a quantum heat engine. The engine undergoes a four-step Otto cycle where the two adiabatic branches involve changing the external magnetic field at a fixed value of the coupling constant. We find conditions for the engine efficiency to be higher than the uncoupled model; in particular, we find an upper bound which is tighter than the Carnot bound. A new domain of parameter values is pointed out which was not feasible in the interaction-free model. Locally, each spin seems to effect the flow of heat in a direction opposite t…
In existing linear response theories for adiabatically driven cyclic heat engines, Onsager symmetry is identified only phenomenologically, and a relation between global and local Onsager coefficients, defined over one cycle and at any instant of a cycle, respectively, is not derived. To address this limitation, we develop a linear response theory for the speed of adiabatically changing parameters and temperature differences in generic Gaussian heat engines obeying Fokker--Planck dynamics. We establish a hierarchical relationship between the global linear response relations, defined over one cy…
Unlike classical systems, a measurement performed on a quantum system always alters its state. In this work, the impacts of two diagnostic schemes to determine the performance of quantum Otto heat engines are compared: In one scheme, the energy of the engine's working substance is measured after each stroke (repeated measurements), and in the other one, the energies after each stroke are recorded in one or two pointer states and measured only after the completion of a prescribed number of cycles (repeated contacts). A single pointer state suffices if one is only interested in either work or he…
Heat engines usually operate by exchanging heat with thermal baths at different (positive) temperatures. Nonthermal baths may, however, lead to a significant performance boost. We here experimentally analyze the power output of a single-atom quantum Otto engine realized in the quasi-spin states of individual Cesium atoms interacting with an atomic Rubidium bath. From measured time-resolved populations of the quasi-spin state, we determine the dynamics during the cycle of both the effective spin temperature and of the quantum fluctuations of the engine, which we quantify with the help of the Sh…
We use fast periodic control to realize finite-time Otto cycles exhibiting quantum advantage. Such periodic modulation of the working medium - bath interaction Hamiltonian during the thermalization strokes can give rise to non-Markovian anti-Zeno dynamics, and corresponding reduction in the thermalization times. Faster thermalization can in turn significantly enhance the power output in engines, or equivalently, the rate of refrigeration in refrigerators. This improvement in performance of dynamically controlled Otto thermal machines arises due to the time-energy uncertainty relation of quantu…
The stochastic efficiency [G. Verley et al., Nat. Commun. 5, 4721 (2014)] was introduced to evaluate the performance of energy-conversion machines in micro-scale. However, such an efficiency generally diverges when no heat is absorbed while work is produced in a thermodynamic cycle. As a result, any statistical moments of the efficiency do not exist. In this study, we come up with a different version of the definition for the stochastic efficiency which is always finite. Its mean value is equal to the conventional efficiency, and higher moments characterize the fluctuations of the cycle. In ad…
