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arXiv:0906.0986

Optimal Performance of Quantum Refrigerators

Tova Feldmann, Ronnie Kosloff

Published Jun 4, 2009• 0 citations
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Abstract

A reciprocating quantum refrigerator is studied with the purpose of determining the limitations of cooling to absolute zero. We find that if the energy spectrum of the working medium possesses an uncontrollable gap, then there is a minimum achievable temperature above zero. Such a gap, combined with a negligible amount of noise, prevents adiabatic following during the demagnetization stage which is the necessary condition for reaching $T_c \to 0$. The refrigerator is based on an Otto cycle where the working medium is an interacting spin system with an energy gap. For this system the external control Hamiltonian does not commute with the internal interaction. As a result during the demagnetization and magnetization segments of the operating cycle the system cannot follow adiabatically the temporal change in the energy levels. We connect the nonadiabatic dynamics to quantum friction. An adiabatic measure is defined characterizing the rate of change of the Hamiltonian. Closed form solutions are found for a constant adiabatic measure for all the cycle segments. We have identified a family of quantized frictionless cycles with increasing cycle times. These cycles minimize the entropy production. Such frictionless cycles are able to cool to $T_c=0$. External noise on the controls eliminates these frictionless cycles. The influence of phase and amplitude noise on the demagnetization and magnetization segments is explicitly derived. An extensive numerical study of optimal cooling cycles was carried out which showed that at sufficiently low temperature the noise always dominates restricting the minimum temperature.

DOI
10.48550/arxiv.0906.0986

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A reciprocating quantum refrigerator is studied with the purpose of determining the limitations of cooling to absolute zero. We find that if the energy spectrum of the working medium possesses an uncontrollable gap, then there is a minimum achievable temperature above zero. Such a gap, combined with a negligible amount of noise, prevents adiabatic following during the demagnetization stage which is the necessary condition for reaching Tc→0T_c \to 0Tc​→0. The refrigerator is based on an Otto cycle where the working medium is an interacting spin system with an energy gap. For this system the external control Hamiltonian does not commute with the internal interaction. As a result during the demagnetization and magnetization segments of the operating cycle the system cannot follow adiabatically the temporal change in the energy levels. We connect the nonadiabatic dynamics to quantum friction. An adiabatic measure is defined characterizing the rate of change of the Hamiltonian. Closed form solutions are found for a constant adiabatic measure for all the cycle segments. We have identified a family of quantized frictionless cycles with increasing cycle times. These cycles minimize the entropy production. Such frictionless cycles are able to cool to Tc=0T_c=0Tc​=0. External noise on the controls eliminates these frictionless cycles. The influence of phase and amplitude noise on the demagnetization and magnetization segments is explicitly derived. An extensive numerical study of optimal cooling cycles was carried out which showed that at sufficiently low temperature the noise always dominates restricting the minimum temperature.
A reciprocating quantum refrigerator is studied with the purpose of determining the limitations of cooling to absolute zero. The cycle is based on demagnetization and magnetization of a working medium. We find that if the energy spectrum of the working medium possesses an uncontrollable gap, and in addition there is noise on the controls, then there is a minimum achievable temperature above zero. The reason is that even a negligible amount of noise prevents adiabatic following during the demagnetization stage. This results with a minimum temperature, Tc(min)>0, which scales with the energy gap. The refrigerator is based on an Otto cycle where the working medium is an interacting spin system with an energy gap. For this system the external control Hamiltonian does not commute with the internal interaction. As a result during the demagnetization and magnetization segments of the operating cycle the system cannot follow adiabatically the temporal change in the energy levels. We connect the nonadiabatic dynamics to quantum friction. An adiabatic measure is defined characterizing the rate of change of the Hamiltonian. Closed-form solutions are found for a constant adiabatic measure for all the cycle segments. We have identified a family of quantized frictionless cycles with increasing cycle times. These cycles minimize the entropy production. Such frictionless cycles are able to cool to Tc=0. External noise on the controls eliminates these frictionless cycles. The influence of phase and amplitude noise on the demagnetization and magnetization segments is explicitly derived. An extensive numerical study of optimal cooling cycles was carried out which showed that at sufficiently low temperature the noise always dominated restricting the minimum temperature.
A reciprocating quantum refrigerator is analyzed with the intention to study the limitations imposed by external noise. In particular we focus on the behavior of the refrigerator when it approaches the absolute zero. The cooling cycle is based on the Otto cycle with a working medium constituted by an ensemble of non interacting harmonic oscillators. The compression and expansion segments are generated by changing an external parameter in the Hamiltonian. In this case the force constant of the harmonic oscillators mω2m ω^2mω2 is modified from an initial to a final value. As a result, the kinetic and potential energy of the system do not commute causing frictional losses. By proper choice of scheduling function ω(t)ω(t)ω(t) frictionless solutions can be obtained in the noiseless case. We examine the performance of a refrigerator subject to noise. By expanding from the adiabatic limit we find that the external noise, gaussian phase and amplitude noises, reduce the amount of heat that can be extracted but nevertheless the zero temperature can be approached.
A reciprocating quantum refrigerator is analyzed with the intention to study the limitations imposed by external noise. In particular we focus on the behavior of the refrigerator when it approaches the absolute zero. The cooling cycle is based on the Otto cycle with a working medium constituted by an ensemble of noninteracting harmonic oscillators. The compression and expansion segments are generated by changing an external parameter in the Hamiltonian. In this case the force constant of the harmonic oscillators mω^{2} is modified from an initial to a final value. As a result, the kinetic and potential energy of the system do not commute causing frictional losses. By proper choice of scheduling function ω(t) frictionless solutions can be obtained in the noiseless case. We examine the performance of a refrigerator subject to noise. By expanding from the adiabatic limit we find that the external noise, Gaussian phase, and amplitude noises reduce the amount of heat that can be extracted but nevertheless the zero temperature can be approached.
The emergence of the laws of thermodynamics from the laws of quantum mechanics is an unresolved issue. The generation of the third law of thermodynamics from quantum dynamics is analysed. The scaling of the optimal cooling power of a reciprocating quantum refrigerator is sought as a function of the cold bath temperature as T c →0. The working medium consists of noninteracting particles in a harmonic potential. Two closed-form solutions of the refrigeration cycle are analyzed, and compared to a numerical optimization scheme, focusing on cooling toward zero temperature. The optimal cycle is characterized by linear relations between the heat extracted from the cold bath, the energy level spacing of the working medium and the temperature. The scaling of the optimal cooling rate is found to be proportional to T c 3/2 giving a dynamical interpretation to the third law of thermodynamics.
In this work we considered the quantum Otto cycle within an optimization framework. The goal was maximizing the power for a heat engine or maximizing the cooling power for a refrigerator. In the field of finite-time quantum thermodynamics it is common to consider frictionless trajectories since these have been shown to maximize the work extraction during the adiabatic processes. Furthermore, for frictionless cycles, the energy of the system decouples from the other degrees of freedom, thereby simplifying the mathematical treatment. Instead, we considered general limit cycles and we used analytical techniques to compute the derivative of the work production over the whole cycle with respect to the time allocated for each of the adiabatic processes. By doing so, we were able to directly show that the frictionless cycle maximizes the work production, implying that the optimal power production must necessarily allow for some friction generation so that the duration of the cycle is reduced.