Irreversible Work and Inner Friction in Quantum Thermodynamic Processes

Abstract

We discuss the thermodynamics of closed quantum systems driven out of equilibrium by a change in a control parameter and undergoing a unitary process. We compare the work actually done on the system with the one that would be performed along ideal adiabatic and isothermal transformations. The comparison with the latter leads to the introduction of irreversible work, while that with the former leads to the introduction of inner friction. We show that these two quantities can be treated on equal footing, as both can be linked with the heat exchanged in thermalization processes and both can be expressed as relative entropies. Furthermore, we show that a specific fluctuation relation for the entropy production associated with the inner friction exists, which allows the inner friction to be written in terms of its cumulants.PACS numbers: 05.70. Ln, With the increasing ability to manufacture and control microscopic systems, we are approaching the limit where quantum fluctuations, as well as thermal ones, become important when trying to put nanomachines and quantum engines to useful purposes [1, 2]. To discuss engines performances, e.g. for heat-to-work conversion, one typically starts by considering reversible transformations that drive the system from an equilibrium configuration to another one. However, if the system is pushed faster than the thermalization time, such transformations are irreversible, and can lead outside the manifold of equilibrium states [3][4][5]. Nonetheless, these processes are of interest as the reversible protocols, despite enjoying very good efficiencies, give rise to very small output powers [6]. The irreversibility of a process is hence related both to better performances and to lack of control, leading to entropy production [7].To analyze irreversibility and entropy production in the quantum realm, we consider a system initially kept in equilibrium and subject to a finite time adiabatic transformation. While its initial state is prepared by keeping it in contact with a thermal bath, the system is then thermally isolated and subject to a parametric change of its Hamiltonian from an initial H i = H[λ i ] to a final H f = H[λ f ] in a finite time τ . The process is defined by the time variation of the work parameter λ(t), changing from λ(t = 0) = λ i to λ(τ ) = λ f .The work w performed on the system during such a process is a stochastic variable with an associated probability density p(w) [4,8,9], which can be reconstructed experimentally [10,11]

Irreversibility in a unitary finite-rate protocol: the concept of internal friction

“…Then, let the system be detached from the heat bath and undergo a parametric unitary change of its Hamiltonian from an initial H i to a final value H f in a time interval, τ . If we differ the average work done on the system in a finite-time, w τ , and the one performed in an infinite time, w τ →∞ , the difference is non-negative and introduces the non-adiabatic work performed on the system by the driving agent, i.e., w f ric = w τ − w τ →∞ ≥ 0 [1]. This indeed defines the internal friction in the system.…”

“…Therefore, w f ric can be considered as a quantitative measure from the deviation of adiabaticity. If the final density matrix for the reversible adiabatic process (τ → ∞) has a well-defined temperature, for example β −1 , then the irreversible work is directly related to the quantum relative entropy between the relevant states through the relation, < w f ric >= β −1 S(ρ τ ||ρ τ →∞ ) [1].…”

“…If we are to use these quantum systems for useful purposes, such as heat to work conversion (i.e, as a quantum heat engine), we have to deal with the limitations imposed by quantum mechanics, such as quantum friction, on the thermodynamical transformations and cycles [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]. Quantum friction is the hallmark of finite-time thermodynamical transformations.…”

“…When the transformation lasts in a finite time, the system state is unable to follow the temporary changes in the Hamiltonian and therefore develops coherences in the energy frame. This leads to store additional parasitic internal energy in the system, corresponding to the "waste energy", which should be released to a heat bath if we want to thermalize the system at the temperature β −1 by a subsequent process [1].…”

One- and two-dimensional quantum models: Quenches and the scaling of irreversible entropy

“…Treating a sudden quench (i.e., a change) of a parameter of the Hamiltonian as a thermodynamic transformation and decoupling the system from the bath so that the evolution of the system following a quench is perfectly unitary, a generalized second law (or Clausius inequality) for an isolated quantum system has been posited with the entropy generated in the process being defined in terms of the irreversible work done in the process of quenching 3,4,8 . These studies have been generalized to the context of Bures metric 9 and Uhlmann fidelity 10 , single qubit interferrometry 11 , open quantum systems 12 and inner friction of quantum thermodynamic processes 13 .…”