Unifying Paradigms of Quantum Refrigeration: A Universal and Attainable Bound on Cooling

Abstract

Cooling quantum systems is arguably one of the most important thermodynamic tasks connected to modern quantum technologies and an interesting question from a foundational perspective. It is thus of no surprise that many different theoretical cooling schemes have been proposed, differing in the assumed control paradigm and complexity, and operating either in a single cycle or in steady state limits. Working out bounds on quantum cooling has since been a highly context dependent task with multiple answers, with no general result that holds independent of assumptions. In this letter we derive a universal bound for cooling quantum systems in the limit of infinite cycles (or steady state regimes) that is valid for any control paradigm and machine size. The bound only depends on a single parameter of the refrigerator and is theoretically attainable in all control paradigms. For qubit targets we prove that this bound is achievable in a single cycle and by autonomous machines.Characterising the ultimate performance limits of quantum thermal machines is directly related to the understanding of energy exchanges at the quantum scale, and hence to the formulation of thermodynamic laws valid in the quantum regime [1]. A central challenge towards this goal is to identify the relevance of different levels of control that can be achieved over quantum mechanical machines and how this determines the thermodynamic limits in the quantum regime. To analytically derive such fundamental limits, it is instructive to observe that the time evolution of closed quantum systems is unitary in quantum mechanics. Discussing nontrivial changes of energy and entropy on a target system thus requires a conceptual separation of a thermodynamic process into a target system, upon which a desired task is performed, and a 'machine', comprised of several other quantum systems, which is used to perform the task upon the target. The total evolution will be a global unitary operation and feature a limited depth/complexity (i.e. machines and the baths constituents they couple to have a finite size) and can either be assumed to be energy changing, if one allows for a coherent control of the process, or alternatively energy preserving in an incoherent control paradigm. Furthermore, machines can operate in a cyclic manner, such that each of these unitary operations can be repeated an arbitrary number of times. To derive self-contained and non-trivial bounds on thermodynamic performance, the state of the machine for each of the repetitions should be thermal with respect to its Hamiltonian. Additionally, to include heat engines, part of the machine could be thermalised with respect to a higher temperature. All these different paradigms are illustrated in Figure 1, and include paradigmatic scenarios, such as autonomous quantum thermal machines [2][3][4], heat bath algorithmic cooling [5][6][7][8][9][10], and quantum otto engines [11,12] for finite size processes. Our framework is a special

Boosting the performance of small autonomous refrigerators via common environmental effects

“…They represent simple models comprised by few qubits (or harmonic oscillators) or a single qutrit weakly interacting with thermal reservoirs at different temperatures. They are self-contained configurations, thus avoiding external sources of coherence and control, which may incur in non-trivial thermodynamic costs [20,37] (for a comparison see [38,39]). Proposals for their implementation in the laboratory include quantum-dot models [40,41], a setup with ions in optical cavities [42] or in circuit QED architectures [43,44].…”

“…The performance of autonomous quantum refrigerators has been extensively studied in most prototypical configurations [24][25][26][30][31][32], in some of which the presence of steady-state entanglement [27] and quantum discord [31] has been found for particular regimes of parameters. Nevertheless, in these refrigerators, the ability to cool is limited by design constraints like the Hilbert space dimension of the machine [28] or the maximum energy gap in the machine Hamiltonian [38,39], and to be surpassed extra resources are needed. In particular, it has been shown that when replacing one of the thermal reservoirs by a squeezed thermal reservoir (or other nonthermal reservoirs) both the cooling power and the efficiency can be greatly improved [32,36] (see also [45]).…”

Work estimation and work fluctuations in the presence of non-ideal measurements

“…To formulate our ideas we adopt a commonplace view in quantum thermodynamics, namely, that work is a central resource that is required to move systems away from freely available thermal equilibrium states [24]-an approach that has staged a diverse range of investigations within the broader field [25][26][27]. In this paradigm, previous research has investigated the work-cost (or gain) of quantum processes [28][29][30][31][32], refrigeration [33,34], or for establishing correlations [35][36][37][38].…”

“…In particular, it was shown in [14] that ideal projective measurements are not exactly realisable in practise because they require the preparation of initially pure pointer states (at least in some nontrivial subspace) to satisfy condition (ii). However, the third law of thermodynamics prevents one from reaching the ground-state of any system with finite resources [15,33,34,[39][40][41][42]. Since any other (pure) state necessarily has higher energy than the ground state, the third law thus excludes ideal measurements.…”

“…Ideal measurements can be approached by increasing the pointer dimension (e.g. measuring a system with an N-qubit pointer and increasing N), or by cooling the pointer to a smaller but non-vanishing temperature (using a desired refrigeration paradigm [33,34,[47][48][49]); these measures all increase the work cost of the measurement [14, A.7]. Achieving the limit C 1 max  , requires infinite time, infinite energy, or infinite control (e.g.…”