Slow dynamics and thermodynamics of open quantum systems

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.Comment: 10 page

Optimal thermodynamic control in open quantum systems

We apply advanced methods of control theory to open quantum systems and we determine finite-time processes which are optimal with respect to thermodynamic performances. General properties and necessary conditions characterizing optimal drivings are derived, obtaining bang-bang type solutions corresponding to control strategies switching between adiabatic and isothermal transformations. A direct application of these results is the maximization of the work produced by a generic quantum heat engine, where we show that the maximum power is directly linked to a particular conserved quantity naturally emerging from the control problem. Finally we apply our general approach to the specific case of a two level system, which can be put in contact with two different baths at fixed temperatures, identifying the processes which minimize heat dissipation. Moreover, we explicitly solve the optimization problem for a cyclic two-level heat engine driven beyond the linear-response regime, determining the corresponding optimal cycle, the maximum power, and the efficiency at maximum power.Comment: 11 pages, 5 figures; corrected typos, added references, all results unchange

Variational approach to the optimal control of coherently driven, open quantum system dynamics

Quantum coherence inherently affects the dynamics and the performances of a quantum machine. Coherent control can, at least in principle, enhance the work extraction and boost the velocity of evolution in an open quantum system. Using advanced tools from the calculus of variations and reformulating the control problem in the instantaneous Hamiltonian eigenframe, we develop a general technique for minimizing a wide class of cost functionals when the external control has access to full rotations of the system Hamiltonian. The method is then applied both to time and heat loss minimization problems and explicitly solved in the case of a two level system in contact with either bosonic or fermionic thermal environments.Comment: 13 pages, 2 figures, added references, corrected typo

Trasformazioni termodinamiche in sistemi quantistici microscopici

L'interesse per la fisica dei microsistemi è cresciuto in modo notevole negli ultimi decenni, grazie al ruolo sempre più importante che questa disciplina ha assunto in molti campi come l'informatica, la chimica, la biofisica. In quest'ultimo ambito, all'inzio degli anni '90, sono state effettuate le prime osservazioni sperimentali, grazie ad innovazioni tecnologiche come il microscopio a forza atomica e l'optical tweezer, che hanno permesso di controllare accuratamente la dinamica su scale più piccole del micrometro. In questo modo è stata monitorata l'evoluzione dei microsistemi a contatto con un bagno termico, dai processi di dissociazione e deformazione delle macromolecole al moto browniano, e sperimentalmente si è data una descrizione molto accurata del bilancio energetico di queste trasformazioni. Per una comprensione completa di questi sistemi è diventato necessario lo sviluppo di una teoria della microtermodinamica e della termodinamica quantistica, la cui portata comunque si estende ben oltre la chimica e la biofisica. Sin dagli anni '60 è noto il legame tra irreversibilità logica e irreversibilità termodinamica, e quindi il legame tra quest'ultima e l'informatica, ma è solo recentemente che la miniaturizzazione dei computer e lo sviluppo dell' informatica quantistica hanno spinto per una teoria termodinamica che si adattasse alle loro esigenze. In questa tesi cercheremo di quantificare il lavoro che si può estrarre in una trasformazione isoterma di un micro sistema quantistico. In particolare sappiamo che nel caso macroscopico, cioè nel caso in cui si può assumere valido il limite termodinamico, il lavoro estraibile è legato alla differenza di energia libera tra lo stato iniziale e quello finale: L<= F_i - F_f, dove l'uguaglianza è valida solo se la trasformazione è reversibile. Cosa accade invece se non assumiamo il limite termodinamico?Vedremo che la caratteristica più importante degli scambi energetici nei sistemi microscopici è che il lavoro estratto può eccedere la soglia reversibile F_i - F_f, a patto di accettare che l'operazione di estrazione possa fallire con una certa probabilità. Dopo che avremo evidenziato questa proprietà, ci concentreremo sul calcolo delle probabilità di estrazione del lavoro per due classi di processi molto note in letteratura: 1) I processi di Jarzynski. Questa classe di trasformazioni isoterme è di fondamentale importanza dal punto di vista sperimentale. Ad oggi la maggior parte dei sistemi microscopici costruiti in laboratorio evolvono seguendo dei processi di Jarzynski, come è stato ampiamente mostrato dai dati. 2) Le operazioni termiche. Affrontando il problema della microtermodinamica nel modo più generale possibile, si prendano come punti di partenza i postulati della meccanica quantistica e la caratterizzazione di Gibbs dei bagni termici. Studiando le dinamiche previste da queste ipotesi nei sistemi quantistici a contatto con un singolo bagno termico, si ottiene la classe delle operazioni termiche. Nel caso dei processi di Jarzynski non solo riusciremo a calcolare le probabilità di estrazione del lavoro, ma anche a descrivere come sono fatte le trasformazioni che rendono massima questa probabilità. Per farlo introdurremo un modello di termodinamica discreta, in cui decomponiamo l'evoluzione di un sistema a contatto con l'ambiente termico in una successione di trasformazioni elementari

Maximum power and corresponding efficiency for two-level heat engines and refrigerators: optimality of fast cycles

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 figure

Entropy production and asymptotic factorization via thermalization: a collisional model approach

The Markovian evolution of an open quantum system is characterized by a positive entropy production, while the global entropy gets redistributed between the system and the environment degrees of freedom. Starting from these premises, we analyze the entropy variation of an open quantum system in terms of two distinct relations: the Clausius inequality, that provides an intrinsic bound for the entropy variation in terms of the heat absorbed by the system, and an extrinsic inequality, which instead relates the former to the corresponding entropy increment of the environment. By modeling the thermalization process with a Markovian collisional model, we compare and discuss the two bounds, showing that the latter is asymptotically saturated in the limit of large interaction time. In this regime not only the reduced density matrix of the system reaches an equilibrium configuration, but it also factorizes from the environment degrees of freedom. This last result is proven analytically when the system-bath coupling is sufficiently strong and through numerical analysis in the weak-coupling regime.Comment: 10 pages, 2 figure

Work statistics across a quantum phase transition

We investigate the statistics of the work performed during a quench across a quantum phase transition using the adiabatic perturbation theory. It is shown that all the cumulants of work exhibit universal scaling behavior analogous to the Kibble-Zurek scaling for the average density of defects. Two kinds of transformations are considered: quenches between two gapped phases in which a critical point is traversed, and quenches that end near the critical point. In contrast to the scaling behavior of the density of defects, the scaling behavior of the work cumulants are shown to be qualitatively different for these two kinds of quenches. However, in both cases the corresponding exponents are fully determined by the dimension of the system and the critical exponents of the transition, as in the traditional Kibble-Zurek mechanism (KZM). Thus, our study deepens our understanding about the nonequilibrium dynamics of a quantum phase transition by revealing the imprint of the KZM on the work statistics

Thermodynamic consistency of quantum master equations

Starting from a microscopic system-baths description, we derive the general conditions for a time-local quantum master equation (QME) to satisfy the first and second law of thermodynamics at the fluctuating level. Using counting statistics, we show that the fluctuating second law can be rephrased as a Generalized Quantum Detailed Balance condition (GQDB), i.e., a symmetry of the time-local generators which ensures the validity of the fluctuation theorem. When requiring in addition a strict system-bath energy conservation, the GQDB reduces to the usual notion of detailed balance which ensures QMEs with Gibbsian steady states. However, if energy conservation is only required on average, QMEs with non Gibbsian steady states can still maintain a certain level of thermodynamic consistency. Applying our theory to commonly used QMEs, we show that the Redfield equation breaks the GQDB, and that some recently derived approximation schemes based on the Redfield equation (which hold beyond the secular approximation and allow to derive a QME of Lindblad form) satisfy the GQDB and the average first law. We find that performing the secular approximation is the only way to ensure the first and second law at the fluctuating level

Quantum bath statistics tagging

The possibility of discriminating the statistics of a thermal bath using indirect measurements performed on quantum probes is presented. The scheme relies on the fact that, when weakly coupled with the environment of interest, the transient evolution of the probe toward its final thermal configuration, is strongly affected by the fermionic or bosonic nature of the bath excitations. Using figures of merit taken from quantum metrology such as the Holevo-Helstrom probability of error and the Quantum Chernoff bound, we discuss how to achieve the greatest precision in this statistics tagging procedure, analyzing different models of probes and different initial preparations and by optimizing over the time of exposure of the probe

A time-dependent regularization of the Redfield equation

We introduce a new regularization of the Redfield equation based on a replacement of the Kossakowski matrix with its closest positive semidefinite neighbor. Unlike most of the existing approaches, this procedure is capable of retaining the time dependence of the Kossakowski matrix, leading to a completely positive (CP) divisible quantum process. Using the dynamics of an exactly-solvable three-level open system as a reference, we show that our approach performs better during the transient evolution, if compared to other approaches like the partial secular master equation or the universal Lindblad equation. To make the comparison between different regularization schemes independent from the initial states, we introduce a new quantitative approach based on the Choi-Jamiolkoski isomorphism