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

Visual processing of words in a patient with visual form agnosia: A behavioural and fMRI study

Patient D.F. has a profound and enduring visual form agnosia due to a carbon monoxide poisoning episode suffered in 1988. Her inability to distinguish simple geometric shapes or single alphanumeric characters can be attributed to a bilateral loss of cortical area LO, a loss that has been well established through structural and functional fMRI. Yet despite this severe perceptual deficit, D.F. is able to “guess” remarkably well the identity of whole words. This paradoxical finding, which we were able to replicate more than 20 years following her initial testing, raises the question as to whether D.F. has retained specialized brain circuitry for word recognition that is able to function to some degree without the benefit of inputs from area LO. We used fMRI to investigate this, and found regions in the left fusiform gyrus, left inferior frontal gyrus, and left middle temporal cortex that responded selectively to words. A group of healthy control subjects showed similar activations. The left fusiform activations appear to coincide with the area commonly named the visual word form area (VWFA) in studies of healthy individuals, and appear to be quite separate from the fusiform face area. We hypothesize that there is a route to this area that lies outside area LO, and which remains relatively unscathed in D.F

Bellman functions and their method in harmonic analysis

This work uses the method of the Bellman function to show a new proof of Hardy's inequality for Carleson measures. Bellman functions come from the theory of stochastic optimal control and there is a method to prove theorems about inequalities over dyadic trees (which have applications in harmonic analysis) that takes inspiration from concepts from the theory of the Bellman functions. The work will display the important concepts of the theory of Bellman functions in stochastic analysis, will show how to use the method of the Bellman function to prove the estimate over dyadic trees for Carleson measures for Hardy spaces (while also showing the connections between the stochastic theory and the harmonic analysis) and will give a new proof of Hardy's inequality for dyadic trees (which is related to the characterization of Carleson measures in Besov spaces) using the Bellman function method

Bellman function for Hardy's inequality over dyadic trees

In this article we use the method of the Bellman function to characterize the measures for which the weighted dual Hardy's inequality holds on dyadic trees. We also give an explicit interpretation of the corresponding Bellman function in terms of the theory of stochastic optimal control.Comment: 21 page

The Federico Zeri Foundation: An International Research Centre for Art History

Unlike previous seminars where fellows of the Italian Academy presented research they were engaged on, I am privileged today to report on a reality that has kept the University of Bologna and a small band of scholars hard at work these last few years. Guided by a series of images, I would like to document Federico Zeri’s 1998 bequest to Bologna University, relate some cultural and biographical episodes featuring this great art historian (1921- 1998), and illustrate the library, the photo archive, and some of the Federico Zeri Foundation's achievements and projects

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