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arXiv:2004.00323 · Physical Review Applied

Exponential Improvement for Quantum Cooling through Finite-Memory Effects

Philip Taranto, Faraj Bakhshinezhad, Philipp Schüttelkopf, Fabien Clivaz, Marcus Huber

Published Nov 4, 2020• 29 citations
arXivHTMLPDF

Abstract

Practical implementations of quantum technologies require preparation of states with a high degree of purity---or, in thermodynamic terms, very low temperatures. Given finite resources, the third law of thermodynamics prohibits perfect cooling; nonetheless, attainable upper bounds for the asymptotic ground-state population of a system repeatedly interacting with quantum thermal machines have been derived. These bounds apply within a memoryless (Markovian) setting, in which each refrigeration step proceeds independently of those previous. Here, we expand this framework to study the effects of memory on quantum cooling. By introducing a memory mechanism through a generalized collision model that permits a Markovian embedding, we derive achievable bounds that provide an exponential advantage over the memoryless case. For qubits, our bound coincides with that of heat-bath algorithmic cooling, which our framework generalizes to arbitrary dimensions. We lastly describe the adaptive stepwise optimal protocol that outperforms all standard procedures.

DOI
10.1103/physrevapplied.14.054005

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Practical implementations of quantum technologies require preparation of states with a high degree of purity---or, in thermodynamic terms, very low temperatures. Given finite resources, the third law of thermodynamics prohibits perfect cooling; nonetheless, attainable upper bounds for the asymptotic ground-state population of a system repeatedly interacting with quantum thermal machines have been derived. These bounds apply within a memoryless (Markovian) setting, in which each refrigeration step proceeds independently of those previous. Here, we expand this framework to study the effects of memory on quantum cooling. By introducing a memory mechanism through a generalized collision model that permits a Markovian embedding, we derive achievable bounds that provide an exponential advantage over the memoryless case. For qubits, our bound coincides with that of heat-bath algorithmic cooling, which our framework generalizes to arbitrary dimensions. We lastly describe the adaptive stepwise optimal protocol that outperforms all standard procedures.
Landauer's limit on heat dissipation during information erasure is critical as devices shrink, requiring optimal pure-state preparation to minimize errors. However, Nernst's third law states this demands infinite resources in energy, time, or control complexity. We address the challenge of cooling quantum systems with finite resources. Using Markovian collision models, we explore resource trade-offs and present efficient cooling protocols (that are optimal for qubits) for coherent and incoherent control. Leveraging thermodynamic length, we derive bounds on heat dissipation for swap-based strategies and discuss the limitations of preparing pure states efficiently.
Heat-Bath Algorithmic Cooling is a set of techniques for producing highly pure quantum systems by utilizing a surrounding heat-bath and unitary interactions. These techniques originally used the thermal environment only to fully thermalize ancillas at the environment temperature. Here we extend HBAC protocols by optimizing over the thermalization strategy. We find, for any d -dimensional system in an arbitrary initial state, provably optimal cooling protocols with surprisingly simple structure and exponential convergence to the ground state. Compared to the standard ones, these schemes can use fewer or no ancillas and exploit memory effects to enhance cooling. We verify that the optimal protocols are robusts to various deviations from the ideal scenario. For a single target qubit, the optimal protocol can be well approximated with a Jaynes-Cummings interaction between the system and a single thermal bosonic mode for a wide range of environmental temperatures. This admits an experimental implementation close to the setup of a micromaser, with a performance competitive with leading proposals in the literature. The proposed protocol provides an experimental setup that illustrates how non-Markovianity can be harnessed to improve cooling. On the technical side we 1. introduce a new class of states called maximally active states and discuss their thermodynamic significance in terms of optimal unitary control, 2. introduce a new set of thermodynamic processes, called β-permutations , whose access is sufficient to simulate a generic thermalization process, 3. show how to use abstract toolbox developed within the resource theory approach to thermodynamics to perform challenging optimizations, while combining it with open quantum system dynamics tools to approximate optimal solutions within physically realistic setups.
Heat-Bath Algorithmic Cooling is a technique for producing pure quantum systems by utilizing a surrounding heat-bath. Here we connect the study of these cooling techniques to the resource theory of thermal operations, enabling us to derive provably optimal cooling protocols under a variety of experimental restrictions on the available control. For qubit systems, we find that a surprisingly simple, optimal protocol consisting of repeated application of a Pauli XXX unitary and a thermal operation can achieve purity converging exponentially quickly to one. What is more, this thermal operation can be well approximated using a Jaynes Cummings interaction between the system and a single thermal bosonic mode and we consider experimental implementations of this. In addition, we investigate the role of quantum coherence and non-Markovianity in cooling protocols and extend our results to higher dimensional systems. Finally, by considering the role of correlations with auxiliary systems in cooling, we show that purity arbitrary close to one can be achieved in a fixed number of operations. Our results serve to find practical applications for the resource theoretic approach to quantum thermodynamics and suggest relevant experimental implementations.
This paper compares the cooling performance of Markovian and non-Markovian quantum collisional models for nanoscale thermodynamic devices. A minimal setup is considered: a single qubit system repeatedly collides with qubit ancillas acting as a heat bath, while a local cooling “demon” operates on the system after each collision. Two regimes are analyzed: (i) a Markovian protocol where the system interacts with a fresh thermal ancilla at every step and the ancilla is subsequently rethermalized, and (ii) a non-Markovian protocol where the system repeatedly interacts with the same initially thermal environment qubit with no reset, allowing correlations (memory) to build up. A central contribution is an efficiency definition that explicitly includes all external energetic resources, counting both the energy removed by the demon and the otherwise hidden work cost of rethermalizing the ancillas in the Markovian implementation. The demon is modeled as an amplitude damping channel; via an explicit Stinespring dilation, the extracted energy is interpreted as heat dumped into an auxiliary cold reservoir, and the paper discusses how additional demon memory/erasure (Landauer) costs would further reduce operational efficiency. Numerical simulations show that the non-Markovian protocol can cool the system to much lower final energies and achieve substantially higher efficiency than the Markovian protocol across a wide range of collision (coupling) and damping strengths. The enhancement is traced to the finite environment’s initial energy and its capacity to store correlations as usable thermodynamic resources. The results emphasize that bounds and benchmarks derived under Markovian assumptions may significantly underestimate performance in finite, memoryful environments, and that reported efficiencies should be interpreted as upper bounds when demon implementation costs are not fully modeled.
Cooling a quantum system with NNN thermal machines costs more than Landauer's bound. For swap-based collision models the excess is set by thermodynamic length, which is optimal for qubits and was suggested by numerics to be optimal in general. We show that for qudits, permutations of energy eigenstates that use several resonant transitions at once do better for every spectrum, reducing the excess by up to 83% near the ground state. To second order, coherent collisions do no better, and at infinite temperature dMd_MdM​-level machines reduce the excess by at most a factor dM−1d_M-1dM​−1.
We introduce a class of thermal operations based on the collision model, where the system sequentially interacts with uncorrelated bath molecules via energy-preserving unitaries. To ensure finite complexity, each molecule is constrained to be no larger than the system. We identify a necessary condition for cooling below the bath temperature via a single collision: The system must initially lack a well-defined effective temperature, even a negative one. By constructing an iterative protocol, we demonstrate that sub-bath cooling is achievable without a machine under these restricted thermal operations. Moreover, introducing a qubit machine further enhances both the cooling limit and energy efficiency. These findings contribute to the broader study of cooling with finite resources.