Quantum Otto cycle with inner friction: finite-time and disorder effects

Abstract

The concept of inner friction, by which a quantum heat engine is unable to follow adiabatically its strokes and thus dissipates useful energy, is illustrated in an exact physical model where the working substance consists of an ensemble of misaligned spins interacting with a magnetic field and performing the Otto cycle. The effect of this static disorder under a finite-time cycle gives a new perspective of the concept of inner friction under realistic settings. We investigate the efficiency and power of this engine and relate its performance to the amount of friction from misalignment and to the temperature difference between heat baths. Finally we propose an alternative experimental implementation of the cycle where the spin is encoded in the degree of polarization of photons.Inner friction is a fully quantum phenomenon, whose consequences are similar to those of the mechanical friction occurring when displacing a piston in compressing/expanding a gas in a classical thermodynamic setting. Its origin, however, is completely different: when the external control Hamiltonian does not commute with the internal one, the states of the working fluid cannot follow the instantaneous energy levels, leading to additional energy stored in the working medium. Inner friction is thus associated to diabatic transitions, i.e., changes of populations which occur during the time-dependent adiabatic (here referring to a closed system) strokes if they are performed at finite speed.So far inner friction occurring in specific cycles and transformations has been analyzed by adopting phenomenodlogical and physically motivated assumptions about the explicit time dependence of the population changes (e.g., in [15], a friction coefficient is introduced, giving rise to a constant dissipated power). Our treatment, instead, does not rely on any ad hoc assumption, but rather on the exact dynamics of the working substance. This is important because it has been shown [12] that inner friction is not only an indicator of irreversibility of a quantum process, but also a quantitative measure of its amount. It is therefore crucial to identify and highlight its role in the efficiency reduction of finite-time cycles by analyzing the full quantum dynamics that produces it.In particular, we will explore the quantum friction arising from disorder within the sample playing the role of a working substance. We will consider an ensemble of qubits in a setting in which their Hamiltonian parameters are not homogeneous and connect the presence of these static errors to the appearance of friction and losses during the implementation of the Otto cycle. Explicitly, we provide a quantitative analysis of the amount of losses due to the inner friction as a function of the degree of disorder.Indeed, the performance of the heat machine is negatively affected by inner friction, and the cycle's outputs, such as extracted work, power, and efficiency, are gradually suppressed as disorder and friction increase.The remainder of the paper is organized as follows. In sections 2....

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