We study a model of an isothermal steady-state work-to-work converter, where a single quantum two-level system (TLS) driven by time-dependent periodic external fields acts as the working medium and is permanently put in contact with a thermal reservoir at fixed temperature $T$. By combining short-iterative Lanczos (SIL) method and analytic approaches, we study the converter performance in the linear response regime and in a wide range of driving frequencies, from weak to strong dissipation. We show that for our ideal quantum machine several parameter ranges exist where a violation of thermodynamics uncertainty relations (TUR) occurs. We find the violation to depend on the driving frequency and on the dissipation strength, and we trace it back to the degree of coherence of the quantum converter. We eventually discuss the influence of other possible sources of violation, such as non-Markovian effects during the converter dynamics.

Describing the thermodynamic properties of quantum systems far from equilibrium is challenging, in particular when the system is strongly coupled to its environment, or when memory effects cannot be neglected. Here, we address such regimes when the system-bath couplings are periodically modulated in time. We show that the couplings modulation, usually associated with a purely dissipative effect, can be suitably engineered to perform thermodynamic tasks. In particular, asymmetric couplings to two heat baths can be used to extract heat from the cold reservoir and to realize an ideal heat rectifier, where the heat current can be blocked either in the forward or in the reverse configuration by simply tuning the frequency of the couplings modulation. Interestingly, both effects take place in the low-temperature, quantum non-Markovian regime. Our work paves the way for the use of optimal control techniques for heat engines and refrigerators working in regimes beyond standard approaches.

By using the worldline Monte Carlo technique, matrix product state, and a variational approach à la Feynman, we investigate the equilibrium properties and relaxation features of the dissipative quantum Rabi model, where a two level system is coupled to a linear harmonic oscillator embedded in a viscous fluid. We show that, in the Ohmic regime, a Beretzinski-Kosterlitz-Thouless quantum phase transition occurs by varying the coupling strength between the two level system and the oscillator. This is a nonperturbative result, occurring even for extremely low dissipation magnitude. By using state-of-the-art theoretical methods, we unveil the features of the relaxation towards the thermodynamic equilibrium, pointing out the signatures of quantum phase transition both in the time and frequency domains. We prove that, for low and moderate values of the dissipation, the quantum phase transition occurs in the deep strong coupling regime. We propose to realize this model by coupling a flux qubit and a damped LC oscillator.

Starting with the Carnot engine, the ideal efficiency of a heat engine has been associated with quasistatic transformations and vanishingly small output power. Here, we exactly calculate the thermodynamic properties of an isothermal heat engine, in which the working medium is a periodically driven underdamped harmonic oscillator, focusing instead on the opposite, antiadiabatic limit, where the period of a cycle is much shorter than the system's timescales. We show that in that limit it is possible to approach the ideal energy conversion efficiency $\eta =1$, with finite output power and vanishingly small relative power fluctuations. The simultaneous realization of all the three desiderata of a heat engine is possible thanks to the breaking of time-reversal symmetry. We also show that non-Markovian dynamics can further improve the power-efficiency trade-off.

The effectiveness of the variational approach à la Feynman is proved in the spin-boson model, i.e., the simplest realization of the Caldeira-Leggett model able to reveal the quantum phase transition from delocalized to localized states and the quantum dissipation and decoherence effects induced by a heat bath. After exactly eliminating the bath degrees of freedom, we propose a trial, nonlocal in time, interaction between the spin and itself simulating the coupling of the two-level system with the bosonic bath. It stems from a Hamiltonian where the spin is linearly coupled to a finite number of harmonic oscillators whose frequencies and coupling strengths are variationally determined. We show that a very limited number of these fictitious modes is enough to get a remarkable agreement, up to very low temperatures, with the data obtained by using an approximation-free Monte Carlo approach, predicting (1) in the Ohmic regime, a Berezinski-Thouless-Kosterlitz quantum phase transition exhibiting the typical universal jump at the critical value; and (2) in the sub-Ohmic regime ($s\le 0.5$), mean-field quantum phase transitions, with logarithmic corrections for $s=0.5$.

By using worldline and diagrammatic quantum Monte Carlo techniques, matrix product state, and a variational approach à la Feynman, we investigate the equilibrium properties and relaxation features of a quantum system of $N$ spins antiferromagnetically interacting with each other, with strength $J$, and coupled to a common bath of bosonic oscillators, with strength $\alpha$. We show that, in the Ohmic regime, a Beretzinski-Thouless-Kosterlitz quantum phase transition occurs. While for $J=0$ the critical value of $\alpha$ decreases asymptotically with $1/N$ by increasing $N$, for nonvanishing $J$ it turns out to be practically independent on $N$, allowing to identify a finite range of values of $\alpha$ where spin phase coherence is preserved also for large $N$. Then, by using matrix product state simulations, and the Mori formalism and the variational approach à la Feynman jointly, we unveil the features of the relaxation, that, in particular, exhibits a nonmonotonic dependence on the temperature reminiscent of the Kondo effect. For the observed quantum phase transition we also establish a criterion analogous to that of the metal-insulator transition in solids.