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]

We consider the Kitaev chain model with finite and infinite range in the hopping and pairing parameters, looking in particular at the appearance of Majorana zero energy modes and massive edge modes. We study the system both in the presence and in the absence of time reversal symmetry, by means of topological invariants and exact diagonalization, disclosing very rich phase diagrams. In particular, for extended hopping and pairing terms, we can get as many Majorana modes at each end of the chain as the neighbors involved in the couplings. Finally we generalize the transfer matrix approach useful to calculate the zero-energy Majorana modes at the edges for a generic number of coupled neighbors.Comment: 14 pages, 16 figure

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....