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arXiv:1911.11416 · Physical review. B./Physical review. B

Nonadiabatic single-qubit quantum Otto engine

Andrea Solfanelli, Marco Falsetti, Michele Campisi

Published Feb 25, 2020• 83 citations
arXivHTMLPDF

Abstract

According to Clausius formulation of the second law of thermodynamics, for any thermal machine withdrawing heats ${Q}_{1,2}$ from two heat reservoirs at temperatures ${T}_{1,2}$, it holds that ${Q}_{1}/{T}_{1}+{Q}_{2}/{T}_{2}\ensuremath{\le}0$. Combined with the observation that the quantity ${Q}_{1}+{Q}_{2}$ is the work $W$ done by the system, that inequality tells us that only four operation modes are possible for the thermal machine, namely, heat engine, refrigerator, thermal accelerator, and heater. We illustrate their emergence in the finite time operation of a quantum Otto engine realized with a single qubit. We first focus on the ideal case when isochoric and thermally insulated strokes are well separated and give general results as well as results pertaining to the specific finite-time Landau-Zener dynamics. We then present realistic results pertaining to the solid-state experimental implementation proposed by Karimi and Pekola [Phys. Rev. B 94, 184503 (2016)]. That device is nonadiabatic both in the quantum mechanical sense and in the thermodynamical sense. Oscillations in the power extracted from the baths due to coherent LZ tunneling at too low of temperatures are observed that might hinder the robustness of the operation of the device against experimental noise on the control parameters.

DOI
10.1103/physrevb.101.054513

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According to Clausius formulation of the second law of thermodynamics, for any thermal machine withdrawing heats Q1,2{Q}_{1,2}Q1,2​ from two heat reservoirs at temperatures T1,2{T}_{1,2}T1,2​, it holds that Q1/T1+Q2/T2\ensuremath≤0{Q}_{1}/{T}_{1}+{Q}_{2}/{T}_{2}\ensuremath{\le}0Q1​/T1​+Q2​/T2​\ensuremath≤0. Combined with the observation that the quantity Q1+Q2{Q}_{1}+{Q}_{2}Q1​+Q2​ is the work WWW done by the system, that inequality tells us that only four operation modes are possible for the thermal machine, namely, heat engine, refrigerator, thermal accelerator, and heater. We illustrate their emergence in the finite time operation of a quantum Otto engine realized with a single qubit. We first focus on the ideal case when isochoric and thermally insulated strokes are well separated and give general results as well as results pertaining to the specific finite-time Landau-Zener dynamics. We then present realistic results pertaining to the solid-state experimental implementation proposed by Karimi and Pekola [Phys. Rev. B 94, 184503 (2016)]. That device is nonadiabatic both in the quantum mechanical sense and in the thermodynamical sense. Oscillations in the power extracted from the baths due to coherent LZ tunneling at too low of temperatures are observed that might hinder the robustness of the operation of the device against experimental noise on the control parameters.
Experimental demonstration of a truly quantum mechanical solid-state heat engine or refrigerator remains elusive up to now. Here, a four-stroke quantum refrigerator that can be realized using a superconducting qubit coupled alternately to two resonators with embedded reservoirs is analyzed theoretically. Besides being a practically feasible thermal machine, this concept presents a number of fundamentally interesting features. At high frequencies, the system exhibits nearly coherent dynamics, leading to quantum oscillations in cooling power versus operation frequency. At intermediate frequencies, an ideal Otto cycle can be realized. In the nearly adiabatic regime, the cooling power is quadratic in frequency and can be separated to classical and quantum contributions. Interestingly, it is shown here that quantum coherence leads always to performance that is inferior as compared to that arising from purely classical dynamics. Furthermore, in the nearly adiabatic regime the efficiency of the refrigerator exceeds clearly that of the standard Otto cycle.
The quantum Otto cycle serves as a bridge between the macroscopic world of heat engines and the quantum regime of thermal devices composed from a single element. We compile recent studies of the quantum Otto cycle with a harmonic oscillator as a working medium. This model has the advantage that it is analytically trackable. In addition, an experimental realization has been achieved, employing a single ion in a harmonic trap. The review is embedded in the field of quantum thermodynamics and quantum open systems. The basic principles of the theory are explained by a specific example illuminating the basic definitions of work and heat. The relation between quantum observables and the state of the system is emphasized. The dynamical description of the cycle is based on a completely positive map formulated as a propagator for each stroke of the engine. Explicit solutions for these propagators are described on a vector space of quantum thermodynamical observables. These solutions which employ different assumptions and techniques are compared. The tradeoff between power and efficiency is the focal point of finite-time-thermodynamics. The dynamical model enables the study of finite time cycles limiting time on the adiabatic and the thermalization times. Explicit finite time solutions are found which are frictionless (meaning that no coherence is generated), and are also known as shortcuts to adiabaticity.The transition from frictionless to sudden adiabats is characterized by a non-hermitian degeneracy in the propagator. In addition, the influence of noise on the control is illustrated. These results are used to close the cycles either as engines or as refrigerators. The properties of the limit cycle are described. Methods to optimize the power by controlling the thermalization time are also introduced. At high temperatures, the Novikov–Curzon–Ahlborn efficiency at maximum power is obtained. The sudden limit of the engine which allows finite power at zero cycle time is shown. The refrigerator cycle is described within the frictionless limit, with emphasis on the cooling rate when the cold bath temperature approaches zero.
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.
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.
Abstract This paper proposes a theoretical model for a heat engine operating on the Otto cycle, employing a two-qubit Heisenberg chain as its working substance. This system is subjected to an external, time-dependent homogeneous magnetic field and a symmetric cross-spin interaction. The interaction is configured to occur along two distinct orientations, contingent upon the direction of Zeeman energy and symmetric interaction: either exclusively along the X -direction or exclusively along the Z -direction. Furthermore, the thermal reservoirs integral to the Otto engine are considered to engage with the working substance via two discrete modes: a localized interaction or a collective interaction. The two qubits exhibit weak coupling to a non-equilibrium thermal environment during the isochoric heating and cooling strokes, thereby validating the application of a Markovian model as a suitable approximation for describing the time evolution of the working substance. This study investigates the influence of various system parameters on the thermodynamic quantities and the resultant power output. A principal finding is that refrigeration functionality is the predominant operational characteristic when the system engages in collective interaction with the reservoirs. Conversely, localized interactions facilitate dual operational modes, with the sign of the Zeeman energy determining the function as either a refrigerator or a heat engine. Significantly, positive power output is achieved in the engine regime, and we find that the choice of interaction axis acts as a powerful amplification mechanism for the thermodynamic output, while the system-environment coupling scheme proves to be the most decisive factor in determining the machine’s fundamental purpose.
Measurement-based quantum heat engines have attracted significant interest as alternatives to conventional thermal engines, as they replace the hot thermal reservoir with quantum measurements, thereby offering greater controllability and simpler implementation. Motivated by these advantages, we investigate a measurement-driven quantum Otto engine with a qubit working substance and study the optimal work extractable from such engines, including whether their performance can surpass that of conventional quantum Otto cycles. We analyze the engine in both the infinite-time (adiabatic) and finite-time (non-adiabatic) regimes, considering two distinct implementations obtained through optimization over all projection-valued measurements (PVMs) and over all two-outcome positive operator-valued measurements (POVMs). We show that measurement-based engines can outperform conventional quantum Otto engines within specific parameter regimes and that POVM-based engines can yield higher optimal work extraction than PVM-based ones. Furthermore, by incorporating the thermodynamic cost associated with resetting the auxiliary system required for POVM implementation, we demonstrate that the resulting net work output can still exceed that of PVM-based engines under suitable conditions on the spectral gaps and cold bath temperature. We also identify regimes in which non-adiabatic implementations can yield higher work output and efficiency than their adiabatic counterparts. Our study provides operational guidelines for designing improved measurement-driven quantum Otto engines.