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International Journal of Thermodynamics

Quantum thermal engine with spin 1/2 system and geometric phases and interference obtained by unitary transformations of mixed states

Y. Ben-Aryeh

Published Jun 1, 2025• 0 citations
Publisher

Abstract

Quantum Carnot engine whose working medium is a two-dimensional spin 1/2 system, with a time-dependent magnetic field in the symmetric z direction is described. The dynamic of this engine is obtained by using four steps, where in two steps the system is coupled alternatively to hot and cold heat baths, and in the other two steps the time development is adiabatic and isentropic (with constant entropy). The conditions for getting a reversible Carnot cycle and the role of time duration for its irreversibility are discussed. Since the calculations are made for the expectation values of the Hamiltonian, only dynamical phases are obtained which cannot be used for interference effects. An alternative method is developed for getting geometric phases, which can be used in interferometry. Parallel transport equations for pure states are generalized to mixed states by which dynamical phases are eliminated. The geometric phases are derived by unitary SU(2) transformations, including time-dependent parameters which are a function of the magnetic fields interactions. A special form of the unitary transformation for the mixed thermal states is developed, by which geometric phases are obtained, which are different from those obtained in NMR and neutron interferometry.

DOI
10.5541/ijot.1395683

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Quantum Carnot engine whose working medium is a two-dimensional spin 1/2 system, with a time-dependent magnetic field in the symmetric z direction is described. The dynamic of this engine is obtained by using four steps, where in two steps the system is coupled alternatively to hot and cold heat baths, and in the other two steps the time development is adiabatic and isentropic (with constant entropy). The conditions for getting a reversible Carnot cycle and the role of time duration for its irreversibility are discussed. Since the calculations are made for the expectation values of the Hamiltonian, only dynamical phases are obtained which cannot be used for interference effects. An alternative method is developed for getting geometric phases, which can be used in interferometry. Parallel transport equations for pure states are generalized to mixed states by which dynamical phases are eliminated. The geometric phases are derived by unitary SU(2) transformations, including time-dependent parameters which are a function of the magnetic fields interactions. A special form of the unitary transformation for the mixed thermal states is developed, by which geometric phases are obtained, which are different from those obtained in NMR and neutron interferometry.
Quantum mechanical methods for getting geometric phases for mixed states are analyzed. Parallel transport equations for pure states are generalized to mixed states by which dynamical phases are eliminated. The geometric phases are derived by SU(2) transformations of mixed states wgeometric phase,hich are different from those used in NMR and neutron interference experiments. The mixed states are obtained as two-dimensional thermal states. The SU(2) transformations include time dependent parameters which are function of the magnetic fields interactions in times which are short relative to relaxation times. These conditions lead to a special form of the unitary transformation of the mixed thermal states by which results for geometric phase and for interference intensity are derived
Quantum mechanical methods for getting geometric phases for mixed states are analyzed. Parallel transport equations for pure states are generalized to mixed states by which dynamical phases are eliminated. The geometric phases are derived by SU(2) transformations of mixed states wgeometric phase,hich are different from those used in NMR and neutron interference experiments. The mixed states are obtained as two-dimensional thermal states. The SU(2) transformations include time dependent parameters which are function of the magnetic fields interactions in times which are short relative to relaxation times. These conditions lead to a special form of the unitary transformation of the mixed thermal states by which results for geometric phase and for interference intensity are derived
Quantum mechanical methods for getting geometric phases for mixed states are analyzed. Parallel transport equations for pure states are generalized to mixed states by which dynamical phases are eliminated. The geometric phases of mixed states are obtained as Pancharatnam phases which are valid also for open cycles. The geometric phases are derived here by SU(2) transformations of mixed thermal states which are different from those used in NMR and neutron interference experiments. While the zeroth order Hamiltonian is given by the interaction of a magnetic moment and constant magnetic field in the z direction, the high order perturbations assumed in the present article are composed of two oscillating magnetic fields in the same z direction. These assumptions lead to a special form of the SU(2) unitary transformation of the mixed thermal states by which results for geometric phase and for interference intensity are derived.
Dynamical phases are obtained for a quantum thermal engine, whose working medium is a single harmonic oscillator. The dynamics of this engine is obtained by using four steps where in two steps the time dependent frequency is changing. In the other two steps, the thermal engine is coupled alternatively to hot and cold heat baths. Similar dynamical phases are obtained in a quantum thermal engine whose working medium is spin 1/2 system. The role of times durations of such steps in the quantum engines for getting maximal efficiency is analyzed. The dynamic of charge pumping in a quantum dot coupled to two reservoirs is studied. The effects of many modulation parameters including their fluctuations for getting geometric phases are analyzed. Since the separate steps in thermal engines describe non-cyclic circuits, we propose to use a special method for measuring geometric phases in thermal engines for non-cyclic circuits which is gauge invariant.
Heat engines constitute the major building blocks of modern technologies. However, conventional heat engines with higher power yield lesser efficiency and vice versa and respect various power-efficiency trade-off relations. This is also assumed to be true for the engines operating in the quantum regime. Here we show that these relations are not fundamental. We introduce quantum heat engines that deliver maximum power with Carnot efficiency in the one-shot finite-size regime. These engines are composed of working systems with a finite number of quantum particles and are restricted to one-shot measurements. The engines operate in a one-step cycle by letting the working system simultaneously interact with hot and cold baths via semilocal thermal operations. By allowing quantum entanglement between its constituents and, thereby, a coherent transfer of heat from hot to cold baths, the engine implements the fastest possible reversible state transformation in each cycle, resulting in maximum power and Carnot efficiency. Finally, we propose a physically realizable engine using quantum optical systems.
Quantum Otto and Carnot engines have recently been receiving attention due to their ability to achieve high efficiencies and powers based on the laws of quantum mechanics. This paper discusses the theory, progress, and possible applications of quantum Otto and Carnot engines, such as energy production, cooling, and nanoscale technologies. In particular, we investigate a two-spin Heisenberg system that works as a substance in quantum Otto and Carnot cycles while exposed to an external magnetic field with both Dzyaloshinsky-Moriya and dipole-dipole interactions. The four stages of engine cycles are subject to analysis with respect to the heat exchanges that occur between the hot and cold reservoirs, alongside the work done during each stage. The operating conditions of the heat engine, refrigerator, thermal accelerator, and heater are all achieved. Moreover, our results demonstrate that the laws of thermodynamics are strictly upheld and the Carnot cycle produces more useful work than that of the Otto cycle.