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We investigate the optimal charging processes for several models of quantum batteries, finding how to maximize the energy stored in a given battery with a finite-time modulation of a set of external fields. We approach the problem using advanced tools of optimal control theory, highlighting the universality of some features of the optimal solutions, for instance the emergence of the well-known Bang-Bang behavior of the time-dependent external fields. The technique presented here is general, and we apply it to specific cases in which the energy is both pumped into the battery by external forces (direct charging) or transferred into it from an external charger (mediated charging). In this article we focus on particular systems that consist of coupled qubits and harmonic oscillators, for which the optimal charging problem can be explicitly solved using a combined analytical-numerical approach based on our optimal control techniques. However, our approach can be applied to more complex setups, thus fostering the study of many-body effects in the charging process., 17 pages, 5 figures

We study the optimization of the performance of arbitrary periodically driven thermal machines. Within the assumption of fast modulation of the driving parameters, we derive the optimal cycle that universally maximizes the extracted power of heat engines, the cooling power of refrigerators, and in general any linear combination of the heat currents. We denote this optimal solution as ``generalized Otto cycle'' since it shares the basic structure with the standard Otto cycle, but it is characterized by a greater number of fast strokes. We bound this number in terms of the dimension of the Hilbert space of the system used as working fluid. The generality of these results allows for a widespread range of applications, such as reducing the computational complexity for numerical approaches, or obtaining the explicit form of the optimal protocols when the system-baths interactions are characterized by a single thermalization scale. In this case, we compare the thermodynamic performance of a collection of optimally driven non-interacting and interacting qubits. Remarkably, for refrigerators the non-interacting qubits perform almost as well as the interacting ones, while in the heat engine case there is a many-body advantage both in the maximum power, and in the efficiency at maximum power. Additionally, we illustrate our general results studying the paradigmatic model of a qutrit-based heat engine. Our results strictly hold in the semiclassical case in which no coherence is generated by the driving, and finally we discuss the non-commuting case., Comment: 15+13 pages, 8 figures

We perform direct thermovoltage measurements in a single-electron transistor, using on-chip local thermometers, both in the linear and non-linear regimes. Using a model which accounts for co-tunneling, we find excellent agreement with the experimental data with no free parameters even when the temperature difference is larger than the average temperature (far-from-linear regime). This allows us to confirm the sensitivity of the thermovoltage on co-tunneling and to find that in the non-linear regime the temperature of the metallic island is a crucial parameter. Surprisingly, the metallic island tends to overheat even at zero net charge current, resulting in a reduction of the thermovoltage., 10 pages, 4 figures

We present a comprehensive and systematic study of thermal rectification in a prototypical low-dimensional quantum system -- a non-linear resonator: we identify necessary conditions to observe thermal rectification and we discuss strategies to maximize it. We focus, in particular, on the case where anharmonicity is very strong and the system reduces to a qubit. In the latter case, we derive general upper bounds on rectification which hold in the weak system-bath coupling regime, and we show how the Lamb shift can be exploited to enhance rectification. We then go beyond the weak-coupling regime by employing different methods: i) including co-tunneling processes, ii) using the non-equilibrium Green's function formalism and iii) using the Feynman-Vernon path integral approach. We find that the strong coupling regime allows us to violate the bounds derived in the weak-coupling regime, providing us with clear signatures of high order coherent processes visible in the thermal rectification. In the general case, where many levels participate to the system dynamics, we compare the heat rectification calculated with the equation of motion method and with a mean-field approximation. We find that the former method predicts, for a small or intermediate anharmonicity, a larger rectification coefficient., Comment: 31 pages, 16 figures

We propose a method to perform accurate and fast charge pumping in superconducting nanocircuits. Combining topological properties and quantum control techniques based on shortcuts to adiabaticity, we show that it is theoretically possible to achieve perfectly quantised charge pumping at any finite-speed driving. Model-specific errors may still arise due the difficulty of implementing the exact control. We thus assess this and other practical issues in a specific system comprised of three Josephson junctions. Using realistic system parameters, we show that our scheme can improve the pumping accuracy of this device by various orders of magnitude. Possible metrological perspectives are discussed., 13 pages, 6 figures

We experimentally realize protocols that allow to extract work beyond the free energy difference from a single electron transistor at the single thermodynamic trajectory level. With two carefully designed out-of-equilibrium driving cycles featuring kicks of the control parameter, we demonstrate work extraction up to large fractions of $k_BT$ or with probabilities substantially greater than 1/2, despite zero free energy difference over the cycle. Our results are explained in the framework of nonequilibrium fluctuation relations. We thus show that irreversibility can be used as a resource for optimal work extraction even in the absence of feedback from an external operator., Comment: Submitted, SI included

We demonstrate high-temperature thermoelectric conversion in InAs/InP nanowire quantum dots by taking advantage of their strong electronic confinement. The electrical conductance G and the thermopower S are obtained from charge transport measurements and accurately reproduced with a theoretical model accounting for the multi-level structure of the quantum dot. Notably, our analysis does not rely on the estimate of co-tunnelling contributions since electronic thermal transport is dominated by multi-level heat transport. By taking into account two spin-degenerate energy levels we are able to evaluate the electronic thermal conductance K and investigate the evolution of the electronic figure of merit ZT as a function of the quantum dot configuration and demonstrate ZT ~ 35 at 30 K, corresponding to an electronic effciency at maximum power close to the Curzon- Ahlborn limit., 7 pages, 4 figures

We report on the first measurement of the Seebeck coefficient in a tunnel-contacted and gate-tunable individual single-quantum dot junction in the Kondo regime, fabricated using the electromigration technique. This fundamental thermoelectric parameter is obtained by directly monitoring the magnitude of the voltage induced in response to a temperature difference across the junction, while keeping a zero net tunneling current through the device. In contrast to bulk materials and single molecules probed in a scanning tunneling microscopy (STM) configuration, investigating the thermopower in nanoscale electronic transistors benefits from the electric tunability to showcase prominent quantum effects. Here, striking sign changes of the Seebeck coefficient are induced by varying the temperature, depending on the spin configuration in the quantum dot. The comparison with numerical renormalization group (NRG) calculations demonstrates that the tunneling density of states is generically asymmetric around the Fermi level in the leads, both in the cotunneling and Kondo regimes.

We analyze a simple implementation of an absorption refrigerator, a system that requires heat and not work to achieve refrigeration, based on two Coulomb coupled single-electron systems. We analytically determine the general condition to achieve cooling-by-heating, and we determine the system parameters that simultaneously maximize the cooling power and cooling coefficient of performance (COP) finding that the system displays a particularly simple COP that can reach Carnot's upper limit. We also find that the cooling power can be indirectly determined by measuring a charge current. Analyzing the system as an autonomous Maxwell demon, we find that the highest efficiencies for information creation and consumption can be achieved, and we relate the COP to these efficiencies. Finally, we propose two possible experimental setups based on quantum dots or metallic islands that implement the non-trivial cooling condition. Using realistic parameters, we show that these systems, which resemble existing experimental setups, can develop an observable cooling power., 11 pages, 6 figures

We study the electronic thermal drag in two different Coulomb-coupled systems, the first one composed of two Coulomb blockaded metallic islands and the second one consisting of two parallel quantum wires. The two conductors of each system are electrically isolated and placed in the two circuits (the drive and the drag) of a four-electrode setup. The systems are biased, either by a temperature $\Delta T$ or a voltage $V$ difference, on the drive circuit, while no biases are present on the drag circuit. In the case of a pair of metallic islands we use a master equation approach to determine the general properties of the dragged heat current $I^{\rm (h)}_{\rm drag}$, accounting also for co-tunneling contributions and the presence of large biases. Analytic results are obtained in the sequential tunneling regime for small biases, finding, in particular, that $I^{\rm (h)}_{\rm drag}$ is quadratic in $\Delta T$ or $V$ and non-monotonous as a function of the inter-island coupling. Finally, by replacing one of the electrodes in the drag circuit with a superconductor, we find that heat can be extracted from the other normal electrode. In the case of the two interacting quantum wires, using the Luttinger liquid theory and the bosonization technique, we derive an analytic expression for the thermal trans-resistivity $\rho^{\rm (h)}_{12}$, in the weak-coupling limit and at low temperatures. $\rho^{\rm (h)}_{12}$ turns out to be proportional to the electric trans-resistivity $\rho^{\rm (c)}_{12}$, in such a way that their ratio (a kind of Wiedemann-Franz law) is proportional to $T^3$. We find that the thermal trans-resistivity is proportional to $T$ for low temperatures and decreases like $1/T$ for intermediate temperatures or like $1/T^3$ for high temperatures. We complete our analyses by performing numerical simulations that confirm the above results and allow to access the strong coupling regime., Comment: 21 pages, 17 figures

We study the thermoelectric properties and heat-to-work conversion performance of an interacting, multi-level quantum dot (QD) weakly coupled to electronic reservoirs. We focus on the sequential tunneling regime. The dynamics of the charge in the QD is studied by means of master equations for the probabilities of occupation. From here we compute the charge and heat currents in the linear response regime. Assuming a generic multi-terminal setup, and for low temperatures (quantum limit), we obtain analytical expressions for the transport coefficients which account for the interplay between interactions (charging energy) and level quantization. In the case of systems with two and three terminals we derive formulas for the power factor Q and the figure of merit ZT for a QD-based heat engine, identifying optimal working conditions which maximize output power and efficiency of heat-to-work conversion. Beyond the linear response we concentrate on the two-terminal setup. We first study the thermoelectric non-linear coefficients assessing the consequences of large temperature and voltage biases, focusing on the breakdown of the Onsager reciprocal relation between thermopower and Peltier coefficient. We then investigate the conditions which optimize the performance of a heat engine, finding that in the quantum limit output power and efficiency at maximum power can almost be simultaneously maximized by choosing appropriate values of electrochemical potential and bias voltage. At last we study how energy level degeneracy can increase the output power., Comment: 30 pages, 15 figures. Published version

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 analyse the associated efficiency at maximum power (EMP) showing that, contrary to universal results derived in the slow-driving regime, it can approach Carnot's efficiency, no other universal bounds being allowed., Comment: 25 pages, 4 figures