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Journal of Applied Physics

Quantum refrigerators in quest of the absolute zero

Ronnie Kosloff, Eitan Geva, Jeffrey M. Gordon

Published Jun 1, 2000• 87 citations
Publisher

Abstract

The second and third laws of thermodynamics can be used to establish a fundamental bound for the maximum possible cooling rate in approaching the absolute zero of temperature. In modeling the behavior of the molecular refrigerators geared toward attaining ultralow temperatures, only quantum mechanical, as opposed to classical physics, models can be admissible. As a simple model, we analyze a three-level quantum refrigerator, and in particular its irreversible thermodynamic performance as absolute zero is approached.

DOI
10.1063/1.373503

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The second and third laws of thermodynamics can be used to establish a fundamental bound for the maximum possible cooling rate in approaching the absolute zero of temperature. In modeling the behavior of the molecular refrigerators geared toward attaining ultralow temperatures, only quantum mechanical, as opposed to classical physics, models can be admissible. As a simple model, we analyze a three-level quantum refrigerator, and in particular its irreversible thermodynamic performance as absolute zero is approached.
One of the formulations of the third laws of thermodynamics is that a processes become more isentropic as one approaches the absolute zero temperatures. We examine this prediction by studying an operating model of a quantum refrigerator pumping heat from a cold to a hot reservoir. The working medium consists of a gas of noninteracting harmonic oscillators. The model can be solved in closed form in the quasi-static limit or numerically for general conditions. It is found that the isentropic limit for Tc → 0 is approached only on the expansion segment of the refrigeration cycle. The scaling of the cooling rate with temperature is shown to be consistent with the second law of thermodynamics. This scaling is also consistent with the unattainability principle which is an alternative formulation of the third law of thermodynamics.
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.
The quantum-mechanical and thermodynamic properties of a three-level molecular cooling cycle are derived. An inadequacy of earlier models is rectified in accounting for the spontaneous emission and absorption associated with the coupling to the coherent driving field via an environmental reservoir. This additional coupling need not be dissipative, and can provide a thermal driving force-the quantum analog of classical absorption chillers. The dependence of the maximum attainable cooling rate on temperature, at ultralow temperatures, is determined and shown to respect the recently established fundamental bound based on the second and third laws of thermodynamics.
The rate of temperature decrease of a cooled quantum bath is studied as its temperature is reduced to absolute zero. The third law of thermodynamics is then quantified dynamically by evaluating the characteristic exponent ζ of the cooling process dT(t)/dt∼-T^{ζ} when approaching absolute zero, T→0. A continuous model of a quantum refrigerator is employed consisting of a working medium composed either by two coupled harmonic oscillators or two coupled two-level systems. The refrigerator is a nonlinear device merging three currents from three heat baths: a cold bath to be cooled, a hot bath as an entropy sink, and a driving bath which is the source of cooling power. A heat-driven refrigerator (absorption refrigerator) is compared to a power-driven refrigerator. When optimized, both cases lead to the same exponent ζ, showing a lack of dependence on the form of the working medium and the characteristics of the drivers. The characteristic exponent is therefore determined by the properties of the cold reservoir and its interaction with the system. Two generic heat bath models are considered: a bath composed of harmonic oscillators and a bath composed of ideal Bose/Fermi gas. The restrictions on the interaction Hamiltonian imposed by the third law are discussed. In the Appendices, the theory of periodically driven open systems and its implication for thermodynamics are outlined.
The derivation of general performance benchmarks is important in the design of highly optimized heat engines and refrigerators. To obtain them, one may model phenomenologically the leading sources of irreversibility ending up with results that are model independent, but limited in scope. Alternatively, one can take a simple physical system realizing a thermodynamic cycle and assess its optimal operation from a complete microscopic description. We follow this approach in order to derive the coefficient of performance at maximum cooling rate for any endoreversible quantum refrigerator. At striking variance with the universality of the optimal efficiency of heat engines, we find that the cooling performance at maximum power is crucially determined by the details of the specific system-bath interaction mechanism. A closed analytical benchmark is found for endoreversible refrigerators weakly coupled to unstructured bosonic heat baths: an ubiquitous case study in quantum thermodynamics.
Thermodynamics is a branch of science blessed by an unparalleled combination of generality of scope and formal simplicity. Based on few natural assumptions together with the four laws, it sets the boundaries between possible and impossible in macroscopic aggregates of matter. This triggered groundbreaking achievements in physics, chemistry and engineering over the last two centuries. Close analogues of those fundamental laws are now being established at the level of individual quantum systems, thus placing limits on the operation of quantum-mechanical devices. Here we study quantum absorption refrigerators, which are driven by heat rather than external work. We establish thermodynamic performance bounds for these machines and investigate their quantum origin. We also show how those bounds may be pushed beyond what is classically achievable, by suitably tailoring the environmental fluctuations via quantum reservoir engineering techniques. Such superefficient quantum-enhanced cooling realises a promising step towards the technological exploitation of autonomous quantum refrigerators.