We develop a perturbation theory of quantum (and classical) master equations with slowly varying parameters, applicable to systems which are externally controlled on a time scale much longer than their characteristic relaxation time. We apply this technique to the analysis of finite-time isothermal processes in which, differently from quasistatic transformations, the state of the system is not able to continuously relax to the equilibrium ensemble. Our approach allows one to formally evaluate perturbations up to arbitrary order to the work and heat exchange associated with an arbitrary process. Within first order in the perturbation expansion, we identify a general formula for the efficiency at maximum power of a finite-time Carnot engine. We also clarify under which assumptions and in which limit one can recover previous phenomenological results as, for example, the Curzon-Ahlborn efficiency.
We develop a perturbation theory of quantum (and classical) master equations with slowly varying parameters, applicable to systems which are externally controlled on a time scale much longer than their characteristic relaxation time. We apply this technique to the analysis of finite-time isothermal processes in which, differently from quasistatic transformations, the state of the system is not able to continuously relax to the equilibrium ensemble. Our approach allows one to formally evaluate perturbations up to arbitrary order to the work and heat exchange associated with an arbitrary process. Within first order in the perturbation expansion, we identify a general formula for the efficiency at maximum power of a finite-time Carnot engine. We also clarify under which assumptions and in which limit one can recover previous phenomenological results as, for example, the Curzon-Ahlborn efficiency.
We develop a perturbation theory to estimate the finite time corrections around a quasi static trajectory, in which a quantum system is able to equilibrate at each instant with its environment. The results are then applied to non equilibrium thermodynamics, in which context we are able to provide a connection between the irreversible contributions and the microscopic details of the dynamical map generating the evolution. Turning the attention to finite time Carnot engines, we found a universal connection between the spectral density esponent of the hot/cold thermal baths and the efficiency at maximum power, giving also a new interpretation to already known results such as the Curzon-Ahborn and the Schmiedl-Seifert efficiencies.
The finite time performance of quantum heat engines has been examined with emphasis on the classical, high temperature, limit. Two basic engine models were studied, differing by their consistency of working fluid: the harmonic engine, consisting of noninteracting harmonic oscillators, and the spin-j engine, consisting of noninteracting spin-j subsystems. The two models represent two distinct types of engines, with bounded vs unbounded Hamiltonians, and with creation and annihilation operators of the Fermionic vs the Bosonic type. The analysis is based on the time derivatives of the first and second laws of thermodynamics. Explicit relations linking quantum observables to thermodynamic quantities are utilized. The dynamics of the engines was modeled by the semigroup approach. The engines were optimized with respect to various target functions: power, entropy production, and efficiency, while subject to finite cycle duration. The main strategy of optimization was based on the Euler–Lagrange equation, and is similar to that previously applied by Salamon and Nitzan for the investigation of Newtonian engines [J. Chem. Phys. 74, 3546 (1981)]. The optimal cycles obtained at the classical limit are not of the Curzon–Ahlborn type, i.e., the internal temperature along the thermal branches is not constant. This result is in conflict with Newtonian thermodynamics, where the optimal cycles are of the Curzon–Ahlborn type. Nonetheless, some of the main features of Newtonian thermodynamics, such as the Curzon–Ahlborn efficiency at maximum power, are reproduced at the classical limit. This makes it possible to establish a ‘‘thermodynamic correspondence principle.’’ This principle asserts that the phenomenological Newtonian thermodynamic approach provides an asymptote of a theory based upon the more fundamental semigroup quantum approach. It is argued that the asymptotic nature of Newtonian thermodynamics is twofold since its validity is restricted by two demands: that of high temperatures and that of proximity to equilibrium.
We apply advanced methods of control theory to open quantum systems and we determine finite-time processes which are optimal with respect to thermodynamic performances. General properties and necessary conditions characterizing optimal drivings are derived, obtaining bang-bang-type solutions corresponding to control strategies switching between adiabatic and isothermal transformations. A direct application of these results is the maximization of the work produced by a generic quantum heat engine, where we show that the maximum power is directly linked to a particular conserved quantity naturally emerging from the control problem. Finally we apply our general approach to the specific case of a two-level system, which can be put in contact with two different baths at fixed temperatures, identifying the processes that minimize heat dissipation. Moreover, we explicitly solve the optimization problem for a cyclic two-level heat engine driven beyond the linear-response regime, determining the corresponding optimal cycle, the maximum power, and the efficiency at maximum power.
The finite-time operation of a quantum-mechanical heat engine with a working fluid consisting of many noninteracting spin-1/2 systems is considered. The engine is driven by an external, time-dependent and nonrotating magnetic field. The cycle of operation consists of two adiabats and two isotherms. The analysis is based on the time derivatives of the first and second laws of thermodynamics. Explicit relations linking quantum observables to thermodynamic quantities are developed. The irreversible operation of this engine is studied in three cases: (1) The sudden limit, where the performance is found to be the same as that of the spin analog of the Otto cycle. This case provides the lower bound of efficiency. (2) The step-cycle operation scheme. Here, the optimization of power is carried out in the high-temperature limit (the ‘‘classical’’ limit). The results obtained are similar to those of Andresen et al. [Phys. Rev. A 15, 2086 (1977)]. (3) The Curzon–Ahlborn operation scheme. The semigroup approach is used to model the dynamics. Then the power production is optimized. All the results obtained for Newtonian engines operating by the same scheme, such as the Curzon–Ahlborn efficiency, apply in the high-temperature limit. These results are obtained without the additional assumption of proximity to thermal equilibrium, implicitly implied by the use of Newtonian heat conduction in the original derivation. It seems that the results of the Curzon–Ahlborn analysis are always obtained in the high-temperature limit, irrespective of the details of the model. The performance beyond the classical limit is optimized numerically. The classical approximation is found to be valid for most of the spin-polarization range. The deviations from the classical limit depend heavily upon the specific nature of both the working fluid and the heat baths and exhibit great diversity and complexity.
Abstract We study the thermodynamic performance of a finite-time non-regenerative quantum Stirling-like cycle used as a heat engine. We consider specifically the case in which the working substance (WS) is a two-level system (TLS). The Stirling cycle is made of two isochoric transformations separated by a compression and an expansion stroke during which the WS is in contact with a thermal reservoir. To describe these two strokes we derive a non-Markovian master equation which allows to study the real-time dynamics of a driven open quantum system with arbitrary fast driving. Following the real-time dynamics of the WS using this master equation, the endpoints of the isotherms can deviate from the equilibrium thermal states. The role of this deviation in the performance of the heat engine is addressed. We found that the finite-time dynamics and thermodynamics of the cycle depend non-trivially on the different time scales at play. In particular, driving the WS at a time scale comparable to the resonance time of the bath enhances the performance of the cycle and allows for an efficiency higher than the efficiency of the quasistatic cycle, but still below the Carnot bound. However, by adding thermalization of the WS with the baths at the end of compression/expansion processes one recovers the conventional scenario in which efficiency decreases by speeding up the processes. In addition, the performance of the cycle is dependent on the compression/expansion speeds asymmetrically, which suggests new freedom in optimizing quantum heat engines. The maximum output power and the maximum efficiency are obtained almost simultaneously when the real-time endpoints of the compression/expansion processes are considered instead of the equilibrium thermal endpoint states. However, the net extractable work always declines by speeding up the drive.
We investigate the efficiency at maximum power (EMP) of irreversible quantum Carnot engines that perform finite-time cycles between two temperature tunable baths. The temperature form we adopt can be experimentally realized in squeezed baths in the high temperature limit, which makes our proposal of practical relevance. Focusing on low dissipation engines, we first generalize the pervious upper as well as lower bounds for the EMP to temperature tunable cases in which they are solely determined by a generalized Carnot limit. As an illustrative example, we then consider a minimal heat engine model with a two-level spin as the working medium. It mimics a low dissipation engine as confirmed by finite time thermodynamic optimization results. The so-obtained EMP, being constrained by the generalized bounds, is well described by a generalized Curzon- Ahlborn efficiency as consequences of a left/right symmetry for a rate constant and low dissipations. Intriguing features of this minimal heat engine under optimal power output are also demonstrated.