Imperfect Thermalizations Allow for Optimal Thermodynamic Processes

Elisa Bäumer1, Martí Perarnau-Llobet2, Philipp Kammerlander1, Henrik Wilming1, and Renato Renner1

1Institute for Theoretical Physics, ETH Zurich, Wolfgang-Pauli-Str. 27, 8093 Zürich, Switzerland
2Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany

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Optimal (reversible) processes in thermodynamics can be modelled as step-by-step processes, where the system is successively thermalized with respect to different Hamiltonians by an external thermal bath. However, in practice interactions between system and thermal bath will take finite time, and precise control of their interaction is usually out of reach. Motivated by this observation, we consider finite-time and uncontrolled operations between system and bath, which result in thermalizations that are only partial in each step. We show that optimal processes can still be achieved for any non-trivial partial thermalizations at the price of increasing the number of operations, and characterise the corresponding tradeoff. We focus on work extraction protocols and show our results in two different frameworks: A collision model and a model where the Hamiltonian of the working system is controlled over time and the system can be brought into contact with a heat bath. Our results show that optimal processes are robust to noise and imperfections in small quantum systems, and can be achieved by a large set of interactions between system and bath.

As we know from everyday live, when an object, such as a cup of coffee, is put in contact with a large body at a certain temperature, often called "heat bath", it will assume the temperature of the heat bath.
This is the very basic mechanism used in thermal machines, such as engines or power plants.
Recently, the study of thermal machines in the quantum regime has received great attention.
In this field of study it is customary to model a system, after it has been in contact with the heat bath, by a so-called canonical ensemble.
One says that the system "thermalizes" to the canonical ensemble.
Therefore, many recent results rest on the assumption that the system under consideration thermalizes exactly to the canonical ensemble due the interaction with a heat bath.
We show that a large class of results is still valid when only imperfect thermalization occurs due to the interaction and hence the system cannot be described exactly by the canonical ensemble.
We further show that not even the duration of certain protocols to run engines is modified significantly.
This shows that general arguments about thermodynamics in the quantum regime are not very sensitive to imperfect thermalization processes.

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