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arXiv:2206.14751 · Physical review. E

Strongly coupled quantum Otto cycle with single qubit bath

Sagnik Chakraborty, Arpan Das, Dariusz Chruściński

Published Dec 26, 2022• 12 citations
arXivHTMLPDF

Abstract

We discuss a model of a closed quantum evolution of two qubits where the joint Hamiltonian is so chosen such that one of the qubits acts as a bath and thermalizes the other qubit which is acting as the system. The corresponding exact master equation for the system is derived. Interestingly, for a specific choice of parameters the master equation takes the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) form, with constant coefficients representing pumping and damping of a single qubit system. Based on this model we construct an Otto cycle connected to a single qubit bath and study its thermodynamic properties. Our analysis goes beyond the conventional weak coupling scenario and illustrates the effects of finite baths, including non-Markovianity. We find closed form expressions for efficiency (coefficient of performance), power (cooling power) for the heat engine regime (refrigerator regime), and for different modifications of the joint Hamiltonian.

DOI
10.1103/physreve.106.064133

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We discuss a model of a closed quantum evolution of two qubits where the joint Hamiltonian is so chosen such that one of the qubits acts as a bath and thermalizes the other qubit which is acting as the system. The corresponding exact master equation for the system is derived. Interestingly, for a specific choice of parameters the master equation takes the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) form, with constant coefficients representing pumping and damping of a single qubit system. Based on this model we construct an Otto cycle connected to a single qubit bath and study its thermodynamic properties. Our analysis goes beyond the conventional weak coupling scenario and illustrates the effects of finite baths, including non-Markovianity. We find closed form expressions for efficiency (coefficient of performance), power (cooling power) for the heat engine regime (refrigerator regime), and for different modifications of the joint Hamiltonian.
We study finite-time quantum Otto cycles whose working medium is a harmonic oscillator undergoing a quantum Brownian motion described by the Caldeira-Leggett model when the oscillator is in contact with heat baths in isochoric processes. The time evolution of the Otto cycle is studied by analytically solving the exact Heisenberg-Langevin equations for the system variables and the interaction energy between the system and the bath. This enables us to investigate non-Markovian strong-coupling effects on the quantum Otto cycle. We obtain cyclic steady states and study the thermodynamic properties of the Otto cycle for various values of the parameters describing the heat baths and the coupling between the system and the bath. We compare our results with those obtained in the Markovian limit, where the time evolution is described by the Lindblad equation. We find that the change in the interaction energy during the isochoric process contributes to both work and heat and plays a crucial role in determining the thermodynamic behavior of the cycle. This is in contrast to the behavior of the state observables of the working medium, which show relatively small change compared to the Markovian counterparts. We find that non-Markovian effects only become significant when we consider thermodynamic fluxes such as work and heat with the system-bath interaction fully taken into account. We also compare our results with the power-efficiency trade-off relation recently proposed for the Markovian quantum Otto engine.
Algebraic methods for solving time-dependent Hamiltonians are used to investigate the performance of quantum thermal machines. We investigate the thermodynamic properties of an engine formed by two coupled qubits, performing an Otto cycle. The thermal interaction occurs with two baths at different temperatures, while work is associated with the interaction with an arbitrary time-dependent magnetic field that varies in intensity and direction. For the coupling, we consider the 1D isotropic Heisenberg model, which allows us to describe the system by means of the irreducible representation of the su(2)\mathrm{su}(2)su(2) Lie algebra within the triplet subspace. We also inspect different settings of the temperatures and frequencies of the cycle, and investigate the corresponding operation regimes. We describe how these regimes can change with the speed of the protocol. Finally, we compare the weak and strong coupling regimes, and conclude that for the weak case the efficiency is optimal and can surpass the value of the classical Otto cycle. For low temperatures of the heat baths, we show that losses due to quantum friction are reduced in the strong coupling limit.
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 propose two algorithmic quantum Otto refrigerators (AQOR) in which the standard isochoric cooling stroke is replaced by two distinct heat-bath algorithmic cooling (HBAC) protocols based on the partner-pairing algorithm (PPA). For each protocol, we derive closed-form expressions for the target qubit's asymptotic polarization and explicitly incorporate the work performed by HBAC into the coefficient of performance (COP). We further quantify the thermodynamic cost of the HBAC stroke and Landauer's bound for both protocols. Our analytical COP curves reveal parameter regimes where the AQOR surpasses the conventional Otto refrigerator. Furthermore, we show that, by tuning the auxiliary qubit's transition frequency and adjusting external control parameters, both protocols achieve enhanced COP and extend the operational window across a wider range of bath temperature differences. Therefore, the AQORs present a tradeoff between achieving high performance in specific regimes and maintaining enhanced applicability. These results demonstrate that algorithmic cooling can be seamlessly integrated into Otto cycles to realize quantum thermal machines with improved practicality on near-term hardware.
In this paper we examine some foundational issues of a class of quantum engines where the system consists of a single quantum parametric oscillator, operating in an Otto cycle consisting of four stages of two alternating phases: the isentropic phase is detached from any bath (thus a closed system) where the natural frequency of the oscillator is changed from one value to another, and the isothermal phase where the system (now rendered open) is put in contact with one or two squeezed baths of different temperatures, whose nonequilibrium dynamics follows the Hu-Paz-Zhang (HPZ) master equation for quantum Brownian motion. The HPZ equation is an exact non-Markovian equation which preserves the positivity of the density operator and is valid for (1) all temperatures, (2) arbitrary spectral density of the bath, and (3) arbitrary coupling strength between the system and the bath. Taking advantage of these properties we examine some key foundational issues of theories of quantum open and squeezed systems for these two phases of the quantum Otto engines. This includes (1) the non-Markovian regimes for non-Ohmic, low-temperature baths, (2) what to expect in nonadiabatic frequency modulations, (3) strong system-bath coupling, as well as (4) the proper junction conditions between these two phases. Our aim here is not to present ways for attaining higher efficiency but to build a more solid theoretical foundation for quantum engines of continuous variables covering a broader range of parameter spaces that we hope are of use for exploring such possibilities.
We use the spin-boson model to describe the dynamics of a two-level atom interacting with Fabry-Pérot cavity modes. We solve the Schrödinger equation for the system-bath model without the Born-Markov approximation to derive the non-Markovian reduced dynamics of the qubit. We further construct an exact Lindblad-type master equation for it. Similar to the quantum Otto cycle, we construct a non- Markovian quasi-cyclic process based on the atom-cavity interactions, which we call the quasi-Otto cycle. For judicious choices of input state and parameters, the quasi-cycle can be more efficient as a quantum engine than the Otto cycle. We also showed that if the quasi-cycle is repeated multiple times, the efficiency of the quasi-Otto engine asymptotically approaches that of the Otto engine.