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Atti della Accademia Peloritana dei Pericolanti. Classe di Scienze Fisiche, Matematiche e Naturali/Atti della Accademia Peloritana dei Pericolanti, Classe di Scienze Fisiche, Matematiche e Naturali

Quantum finite-time availability

Karl Heinz Hoffmann, Peter Salamon, Kim Schmidt

Published May 1, 2019• 2 citations
Publisher

Abstract

The availability of a thermodynamic system with respect to an environment is the maximum work, which can be gained from bringing it into equilibrium with its environment by a reversible process. If the process has to proceed in finite time, there will be unavoidable losses diminishing the availability; this consequence is captured by the Finite-Time Availability. Here we consider the consequences of an availability extracting process for a paradigmatic quantum system, the parametric harmonic oscillator. Differences and similarities between its Quantum Finite-Time Availability and the classical Finite-Time Availability of an ideal gas in a cylinder with a piston are discussed.

DOI
10.1478/aapp.97s1a10

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The availability of a thermodynamic system with respect to an environment is the maximum work, which can be gained from bringing it into equilibrium with its environment by a reversible process. If the process has to proceed in finite time, there will be unavoidable losses diminishing the availability; this consequence is captured by the Finite-Time Availability. Here we consider the consequences of an availability extracting process for a paradigmatic quantum system, the parametric harmonic oscillator. Differences and similarities between its Quantum Finite-Time Availability and the classical Finite-Time Availability of an ideal gas in a cylinder with a piston are discussed.
Abstract The availability of a thermodynamic system out of equilibrium with its environment describes its ability to perform work in a reversible process which brings it to equilibrium with this environment. Processes in finite time can usually not be performed reversibly thus leading to unavoidable losses. In order to account for these losses the concept of finite time availability was introduced. We here add a new feature through the introduction of quantum finite time availability for an ensemble of parametric oscillators. For such systems there exists a certain critical time, the FEAT time. Quantum finite time availability quantifies the available work from processes which are shorter than the FEAT time of the oscillator ensemble.
Classically, availability refers to the work available in any reversible process that brings about equilibrium between the system and its environment. Here we introduce an additional meaning of availability as the maximum work associated with the change of an external parameter in the Hamiltonian of a quantum system. This availability can be gained in a FEAT process and for times larger than or equal to the FEAT time, there exists an optimal control that achieves the available work. For shorter times, quantum friction effects are unavoidable and the available work is thereby lowered. This finite-time availability is quantified here as a function of the time available.
In this article we use geometric optimal control to completely solve the problem of minimum-time transitions between thermal equilibrium and fixed average energy states of the quantum parametric oscillator, a system which has been extensively used to model quantum heat engines and refrigerators. We subsequently use the obtained results to find the minimum driving time for a quantum refrigerator and the quantum finite-time availability of the parametric oscillator, i.e., the potential work which can be extracted from this system by a very short finite-time process.
This paper considers the problem of obtaining maximum work from a conservative quantum system corresponding to a given change in an external parameter in the Hamiltonian. The example we present is a non-interacting collection of harmonic oscillators with a shared frequency omega which changes from a given initial to a given final value. The example is interesting for its role in experiments at ultra-low temperatures and for probing finite-time versions of the third law of thermodynamics. It is also the simplest system displaying quantum friction, which represents loss mechanisms in any reversible prelude to a thermal process. The example leads to a new type of availability. It is also the first example of a minimum time for transitions between thermal states of a thermodynamic system.
A key concept in quantum thermodynamics is extractable work, which specifies the maximum amount of work that can be extracted from a quantum system. Different quantities are used to measure extractable work, the most prevalent of which are ergotropy and the difference between the non-equilibrium and equilibrium quantum free energies. Using the latter, we investigate the evolution of extractable work when an open quantum system undergoes a general quantum process described by a completely-positive and trace-preserving dynamical map. We derive a fundamental equation of thermodynamics for such processes as a relation between the distinct sorts of energy change in such a way that the first and the second law of thermodynamics are combined. We then identify the contributions from the reversible and irreversible processes in this equation and demonstrate that they are respectively responsible for the evolution of heat and extractable work of the open quantum system. Furthermore, we show how this correspondence between irreversibility and extractable work has the potential to provide a clear explanation of how the quantum properties of a system affect its extractable work evolution. Specifically, we establish this by directly connecting the change in extractable work with the change in standard quantifiers of quantum non-Markovianity and quantum coherence during a general quantum process. We illustrate these results with two examples.