We develop a perturbation theory of quantum (and classical) master equations with slowly varying parameters, applicable to systems which are externally controlled on a time scale much longer than their characteristic relaxation time. We apply this technique to the analysis of finite-time isothermal processes in which, differently from quasi-static transformations, the state of the system is not able to continuously relax to the equilibrium ensemble. Our approach allows to formally evaluate perturbations up to arbitrary order to the work and heat exchange associated to an arbitrary process. Within first order in the perturbation expansion, we identify a general formula for the efficiency at maximum power of a finite-time Carnot engine. We also clarify under which assumptions and in which limit one can recover previous phenomenological results as, for example, the Curzon-Ahlborn efficiency.A central result in the study of open quantum systems [1, 2] is the Markovian Master Equation (MME) approach which, under realistic assumptions, describes the temporal evolution of a system of interest A induced by a weak coupling with a large external environment E. This consists in a first order linear differential equatioṅ ρ(t) = L[ρ(t)] where ρ(t) is the density matrix of A and where the generator of the dynamics is provided by a quantum Liouvillian superoperator L that can be casted in the so called Gorini-Kossakowski-Sudarshan-Lindblad form [3][4][5]. For autonomous systems the latter does not exhibit an explicit time dependence and the dynamics of A exponentially relaxes to a (typically unique) equilibrium steady state ρ 0 identified by the null eigenvector equation L[ρ 0 ] = 0. MMEs can also be employed to describe the temporal evolution of A when it is tampered by the presence of slow varying, external driving forces. Indeed, as long as these operate on a time scale which is much larger than the characteristic bath correlations times and the inverse frequencies of the system of interest, the effective coupling between A and E adapts instantaneously to the driving control, resulting on a MME governed by a time-dependent Liouvillian generator, i.e.An explicit integration of this equation is in general difficult to obtain. Yet, if the control forces are so slow that their associated time scale is also larger with respect to the relaxation time of the system induced by the interaction with E, one expects ρ(t) to approximately follow the instantaneous equilibrium state ρ 0 (t) that nullifies L t . Our aim is to estimate quantitative deviations from this ultra-slow driving regime. For this purpose we develop a perturbation theory valid in the limit of slowly varying Liouvillians L t and derive a formal solution of Eq. (1) which allows one to evaluate non-equilibrium corrections up to arbitrary order. The main motivation of our analysis is to model thermodynamic processes and cycles beyond the usual reversible limit which is strictly valid only for infinitely long quasi-static transformations. Finite-time thermodynamics [7,8]...

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We study how to achieve the ultimate power in the simplest, yet non-trivial, model of a thermal machine, namely a two-level quantum system coupled to two thermal baths. Without making any prior assumption on the protocol, via optimal control we show that, regardless of the microscopic details and of the operating mode of the thermal machine, the maximum power is universally achieved by a fast Otto-cycle like structure in which the controls are rapidly switched between two extremal values. A closed formula for the maximum power is derived, and finite-speed effects are discussed. We also analyze the associated efficiency at maximum power showing that, contrary to universal results derived in the slow-driving regime, it can approach Carnot's efficiency, no other universal bounds being allowed.'fast-driving' regime, i.e. when the driving frequency is faster than the typical dissipation rate induced by the baths, which has received little attention in literature [50][51][52].By applying our optimal protocol to heat engines and refrigerators, we find new theoretical bounds on the efficiency at maximum power (EMP). Many upper limits to the EMP, strictly smaller than Carnot's efficiency, have been derived in literature, such as the Curzon-Ahlborn and Schmiedl-Seifert efficiencies. The Curzon-Ahlborn efficiency emerges in various specific models [53][54][55], and it has been derived by general arguments from linear irreversible thermodynamics [56]. The Schmiedl-Seifert efficiency has been proven to be universal in cyclic Brownian heat engines [57] and for any driven system operating in the slow-driving regime [24]. By studying the efficiency of our system at the ultimate power, i.e. in the fast-driving regime, we prove that there is no fundamental upper bound to the EMP. Indeed, we show that the Carnot efficiency is reachable at maximum power through a suitable engineering of the bath couplings. This is our second main results, illustrated in figures 2(b), (c) and figure 3. In view of experimental implementations, we assess the impact of finite-time effects on our optimal protocol, finding that the maximum power does not decrease much if the external driving is not much slower than the typical dissipation rate induced by the baths [58,59]. Furthermore, we apply our optimal protocol to two experimentally accessible models, namely photonic baths coupled to a qubit [22, 60-63] and electronic leads coupled to a quantum dot [21,23,58,59,64,65].C , and Γ in (c) denotes *  G( ) H . 4 In principle, one can consider a broader family of controls including the possibility of rotating the Hamiltonian eigenvectors; however there is evidence that such an additional freedom does not help in two-level systems [45, 46]. ≔ [ ] ( ) J J t p t J J t p t

Single-qubit thermometry presents the simplest tool to measure the temperature of thermal baths with reduced invasivity. At thermal equilibrium, the temperature uncertainty is linked to the heat capacity of the qubit, however the best precision is achieved outside equilibrium condition. Here, we discuss a way to generalize this relation in a non-equilibrium regime, taking into account purely quantum effects such as coherence. We support our findings with an experimental photonic simulation.Introduction:-Identifying strategies for improving the measurement precision by means of quantum resources is the purpose of Quantum Metrology [1][2][3]. In particular, through the Quantum Cramér-Rao Bound (QCRB), it sets ultimate limits on the best accuracy attainable in the estimation of unknown parameters even when the latter are not associated with observable quantities. These considerations have attracted an increasing attention in the field of quantum thermodynamics, where an accurate control of the temperature is highly demanding [4][5][6][7][8]. Besides the emergence of primary and secondary thermometers based on precisely machined microwave resonators [9,10], recent efforts have been made aiming at measuring temperature at even smaller scales, where nanosize thermal baths are higly sensitives to disturbances induced by the probe [11][12][13][14][15][16][17]. Some paradigmatic examples of nanoscale thermometry involve nanomechanical resonators [19], quantum harmonic oscillators [20] or atomic condensates [21][22][23] (also in conjunction with estimation of chemical potential [24]). In this context the analysis of quantum properties needs to be taken into account in order to establish, and eventually enhance, metrological precision [18,[25][26][27][28][29].In a conventional approach to thermometry, an external bath B at thermal equilibrium is typically indirectly probed via an ancillary system, the thermometer S, that is placed into weak-interaction with the former. Assuming hence that the thermometer reaches the thermal equilibrium configuration without perturbing B too much, the Einstein Theory of Fluctuations (ETF) can be used to characterize the sensitivity of the procedure in terms of the heat capacity of S which represents its thermal susceptibility to the perturbation imposed by the bath [30][31][32]. Since this last is an equilibrium property, one should not expect it to hold in non-equilibrium regimes. However thermometry schemes that do not need a full thermalization of the probe have been recognized to offer higher sensitivities in temperature estimation [33].Thus, if on the one hand the QCRB can still be used as the proper tool to gauge the measurement uncertainty on the bath temperature, on the other hand establishing a direct link between this approach and the thermodynamic properties of the probe is still an open question. Furthermore, the advantages pointed out in [33] are conditional on precisely addressing the probe during its evolution, a task which might be demanding in real experiments [28]. Here S i...