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arXiv:1107.1044 · Physical Review E

Thermodynamic limits of dynamic cooling

ARMEN E. ALLAHVERDYAN, Karen V. Hovhannisyan, Dominik Janzing, G. Mahler

Published Oct 10, 2011• 75 citations
arXivHTMLPDF

Abstract

We study dynamic cooling, where an externally driven two-level system is cooled via reservoir, a quantum system with initial canonical equilibrium state. We obtain explicitly the minimal possible temperature T(min)>0 reachable for the two-level system. The minimization goes over all unitary dynamic processes operating on the system and reservoir and over the reservoir energy spectrum. The minimal work needed to reach T(min) grows as 1/T(min). This work cost can be significantly reduced, though, if one is satisfied by temperatures slightly above T(min). Our results on T(min)>0 prove unattainability of the absolute zero temperature without ambiguities that surround its derivation from the entropic version of the third law. We also study cooling via a reservoir consisting of N≫1 identical spins. Here we show that T(min)∝1/N and find the maximal cooling compatible with the minimal work determined by the free energy. Finally we discuss cooling by reservoir with an initially microcanonic state and show that although a purely microcanonic state can yield the zero temperature, the unattainability is recovered when taking into account imperfections in preparing the microcanonic state.

DOI
10.1103/physreve.84.041109

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We study dynamic cooling, where an externally driven two-level system is cooled via reservoir, a quantum system with initial canonical equilibrium state. We obtain explicitly the minimal possible temperature T(min)>0 reachable for the two-level system. The minimization goes over all unitary dynamic processes operating on the system and reservoir and over the reservoir energy spectrum. The minimal work needed to reach T(min) grows as 1/T(min). This work cost can be significantly reduced, though, if one is satisfied by temperatures slightly above T(min). Our results on T(min)>0 prove unattainability of the absolute zero temperature without ambiguities that surround its derivation from the entropic version of the third law. We also study cooling via a reservoir consisting of N≫1 identical spins. Here we show that T(min)∝1/N and find the maximal cooling compatible with the minimal work determined by the free energy. Finally we discuss cooling by reservoir with an initially microcanonic state and show that although a purely microcanonic state can yield the zero temperature, the unattainability is recovered when taking into account imperfections in preparing the microcanonic state.
We study the asymptotic dynamics of arbitrary linear quantum open systems that are periodically driven while coupled with generic bosonic reservoirs. We obtain exact results for the heat flowing from each reservoir, and these results are valid beyond the weak-coupling or Markovian approximations. We prove the validity of the dynamical third law of thermodynamics (Nernst unattainability principle), showing that the ultimate limit for cooling is imposed by a fundamental heating mechanism that dominates at low temperatures, namely the nonresonant creation of excitation pairs in the reservoirs induced by the driving field. This quantum effect, which is missed in the weak-coupling approximation, restores the unattainability principle, the validity of which was recently challenged.
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.
This work presents a constructive thermodynamic protocol that reaches exactly absolute zero using finite resources, within the rigorous framework of the resource theory of thermal operations. The core result is a General Zero‑Work Cooling Theorem: for any heat bath with strictly convex microcanonical entropy that contains a local two‑level subsystem matching the target qubit gap, a cascade of energy‑conserving SWAP operations cools the qubit from a finite initial temperature to any lower temperature in the bath’s accessible range with identically zero work and a finite number of steps. When the microcanonical temperature can be made arbitrarily small, as in an explicit quadratic‑entropy construction; the cascade alone reaches arbitrarily low temperatures work‑free. A single quantum non‑demolition measurement and a conditional π‑pulse then deterministically map the qubit to its exact ground state at a small, constant work cost, achieving exactly absolute zero with finite total work, a finite number of operations, and a finite predetermined time. The result demonstrates that the unattainability of absolute zero is not a universal law of nature but a contingent restriction to concave‑entropy baths and measurement‑free unitaries. It elevates the escape clause of Masanes-Oppenheim to a generalized unattainability theorem for convex‑entropy systems, with direct connections to black hole thermodynamics, massive gravity, and quantum gravity models. The paper includes a fully explicit quadratic‑entropy cascade, rigorous bounds on all finite‑size corrections, a thermodynamic embedding of the measurement step via Landauer erasure, and a systematic refutation of all known objections.
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 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.
We solve the closed-system problem of cooling MMM qubits, selected from NNN identical thermal qubits, to the lowest common local temperature allowed by unitarity. The optimal protocol consists of two conceptually distinct steps. First, a passive rearrangement assigns the largest eigenvalues of the initial state to target sectors of lowest Hamming weight, thereby minimizing the total target energy. Second, a target-only complex-Hadamard transformation within each fixed-Hamming-weight subspace equalizes the one-qubit target marginals without changing any target-sector probability or the total energy. Consequently, imposing a common local temperature costs neither cooling depth nor additional work: the constrained optimum coincides with the unconstrained passive minimum for every N>MN>MN>M and every initial temperature. The complex-Hadamard correction may nevertheless be costly at the circuit level. We therefore derive an exact arithmetic criterion for when the same optimum can be attained by a temperature-independent computational-basis permutation alone, and exhaustively classify the resulting finite-size islands of feasibility for M+2≤N≤128M+2\leq N\leq 128M+2≤N≤128. At isolated temperatures, further optimal permutations can arise through numerical cancellations between different thermal eigenvalue shells. These alternative realizations may reduce implementation complexity, but they cannot improve the cooling curve already attained by the universal protocol. We also derive the exact cooling curve, prove that at least two ancillary qubits are necessary and sufficient for nontrivial cooling, and show that joint many-target cooling can strictly outperform parallel single-target strategies.