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Vanishing efficiency of a speeded-up ion-in-Paul-trap Otto engine

Ander Tobalina, Ion Lizuain, Juan Gonzalo Muga

Published Aug 28, 2019• 14 citations
Publisher

Abstract

\n\nWe assess the energy cost of shortcuts to adiabatic expansions or compressions of a harmonic oscillator, the power strokes of a quantum Otto engine. Difficulties to identify the cost stem from the interplay between different parts of the total system (the primary system —the particle— and the control system) and definitions of work (exclusive and inclusive). While attention is usually paid to the inclusive work of the microscopic primary system, we identify the energy cost as the exclusive work of the total system, which, for a clear-cut scale disparity between a microscopic primary system and a macroscopic control system, coincides with the exclusive work for the control system alone. We redefine the “engine efficiency” taking into account this cost. Our working horse model is an engine based on an ion in a Paul trap with power strokes designed via shortcuts to adiabaticity. Opposite to the paradigm of slow-cycle reversible engines with vanishing power and maximal efficiency, this fast-cycle engine increases the microscopic power at the price of a vanishing efficiency. The Paul trap fixes the gauge for the primary system, resulting in a counterintuitive evolution of its inclusive power and internal energy. Conditions for which inclusive power of the primary system and exclusive power control system are proportional are found.\n

DOI
10.1209/0295-5075/127/20005/pdf

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\n\nWe assess the energy cost of shortcuts to adiabatic expansions or compressions of a harmonic oscillator, the power strokes of a quantum Otto engine. Difficulties to identify the cost stem from the interplay between different parts of the total system (the primary system —the particle— and the control system) and definitions of work (exclusive and inclusive). While attention is usually paid to the inclusive work of the microscopic primary system, we identify the energy cost as the exclusive work of the total system, which, for a clear-cut scale disparity between a microscopic primary system and a macroscopic control system, coincides with the exclusive work for the control system alone. We redefine the “engine efficiency” taking into account this cost. Our working horse model is an engine based on an ion in a Paul trap with power strokes designed via shortcuts to adiabaticity. Opposite to the paradigm of slow-cycle reversible engines with vanishing power and maximal efficiency, this fast-cycle engine increases the microscopic power at the price of a vanishing efficiency. The Paul trap fixes the gauge for the primary system, resulting in a counterintuitive evolution of its inclusive power and internal energy. Conditions for which inclusive power of the primary system and exclusive power control system are proportional are found.\n
We assess the energy cost of shortcuts to adiabatic expansions or compressions of a harmonic oscillator, the power strokes of a quantum Otto engine.Difficulties to identify the cost stem from the interplay between different parts of the total system (the primary system -the particle-and the control system) and definitions of work (exclusive and inclusive).While attention is usually paid to the inclusive work of the microscopic primary system, we identify the energy cost as the exclusive work of the total system, which, for a clear-cut scale disparity between a microscopic primary system and a macroscopic control system, coincides with the exclusive work for the control system alone.We redefine the "engine efficiency" taking into account this cost.Our working horse model is an engine based on an ion in a Paul trap with power strokes designed via shortcuts to adiabaticity.Opposite to the paradigm of slow-cycle reversible engines with vanishing power and maximal efficiency, this fast-cycle engine increases the microscopic power at the price of a vanishing efficiency.The Paul trap fixes the gauge for the primary system, resulting in a counterintuitive evolution of its inclusive power and internal energy.Conditions for which inclusive power of the primary system and exclusive power control system are proportional are found.focus article
We consider a paradigmatic quantum harmonic Otto engine operating in finite time. We investigate its performance when shortcut-to-adiabaticity techniques are used to speed up its cycle. We compute efficiency and power by taking the energetic cost of the shortcut driving explicitly into account. We analyze in detail three different shortcut methods: counterdiabatic driving, local counterdiabatic driving, and inverse engineering. We demonstrate that all three lead to a simultaneous increase of efficiency and power for fast cycles, thus outperforming traditional heat engines.
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.
Abstract The efficiency of a quantum heat engine is maximum when the unitary strokes of the quantum Otto cycle are adiabatic. On the other hand, this may not be always possible due to small energy gaps in the system, especially at the critical point (CP) where the gap between the ground state and the first excited state vanishes and the system gets excited. With the aim to regain this lost adiabaticity, we modify one of the unitary strokes of the Otto cycle by allowing the system to first evolve with a time dependent Hamiltonian as in the case of a usual Otto cycle, followed by an additional evolution with a different time independent Hamiltonian so that the system reaches a less excited state. This will help in increasing the magnitude of the heat absorbed from the hot bath so that the work output and efficiency of the engine can be increased. We demonstrate this method using an integrable model and a non-integrable model as the working medium and discuss the generality and limitations of this method. In the case of a two spin system, the optimal value for the time till which the system needs to be freely evolved is calculated analytically in the adiabatic limit. The results show that implementing this modified unitary stroke significantly improves the work output and efficiency of the engine, especially when it crosses the CP.
There is recent interest in determining energy costs of shortcuts to adiabaticity (STA), but different definitions of 'cost' have been used.We demonstrate the importance of taking into account the control system (CS) for a fair assessment of energy flows and consumptions.We model the energy consumption and power to transport an ion by a STA protocol in a multisegmented Paul trap.The ion is driven by an externally controlled, moving harmonic oscillator.Even if no net ion-energy is gained at destination, setting the time-dependent control parameters is a macroscopic operation that costs energy and results in energy dissipation for the short time scales implied by the intrinsically fast STA processes.The potential minimum is displaced by modulating the voltages on control (dc) electrodes.A secondary effect of the modulation, usually ignored as it does not affect the ion dynamics, is the time-dependent energy shift of the potential minimum.The non trivial part of the energy consumption is due to the electromotive forces to set the electrode voltages through the low-pass filters required to preserve the electronic noise from decohering the ion's motion.The results for the macroscopic CS (the Paul trap) are compared to the microscopic power and energy of the ion alone.Similarities are found-and may be used quantitatively to minimize costs-only when the CSdependent energy shift of the harmonic oscillator is included in the ion-energy.
Thermodynamic constraints impose a trade-off between power and efficiency in heat engines, preventing the simultaneous achievement of high power and high efficiency. For classical microscopic engines, explicit inequalities have been discovered, demonstrating the inherent inevitability of this power-efficiency trade-off. However, extensions of these results to quantum engines have so far been limited to cases of slow operation. In this study, we derive a power-efficiency trade-off relation for a paradigmatic quantum engine operating within a finite time, specifically the Otto cycle of a quantum harmonic oscillator. By utilizing a phase-space approach based on quasiprobability representations, we establish a universal trade-off relation applicable to arbitrary time-dependent protocols during the adiabatic processes. Our results reveal that the power of the quantum engine vanishes as the efficiency approaches the quantum-mechanical efficiency bound, which is stricter than the Carnot bound. Furthermore, we identify the conditions under which the upper bound is attained, which indicate maximum power is achieved when the generation of quantum coherence is reduced, and the difference in time durations of the isochoric processes increases. These findings are validated through numerical calculations, which confirm their applicability across various types of protocols for heat engine cycles.