A probability distribution encodes all the statistics of its corresponding random variable, hence it encodes all one can learn about the system via the random variable. In the theoretical part of this thesis, we discuss several equivalent representations of probability distributions in physics which establishes their roles as complete information measures in their own contexts. In particular, we show how Rényi entropies (as well as relative Rényi entropies) completely characterise the uncertainty of a system, hence implies the insufficiency of Shannon entropy (as well as relative entropy) in capturing the information in the presence of strong fluctuation. We also generalise Jarzynski equality in terms of relative Rényi entropies and show the equivalence between the generalised equality and the work distribution. As a consequence, an equivalence between a monotonic property of relative Rényi entropies and a kind of second law relations is revealed. In the experimental part, we study the non-equilibrium dynamics of an optically levitated nanosphere system in Knudsen regime and discover that, even though the system is not in equilibrium, the dynamics can still be treated as a Brownian motion with an effective temperature and an effective coefficient. Due to the Gaussian statistics of the levitated sphere, we show how some relevant non-equilibrium properties of the system, including several local temperatures, can be analysed from its power spectrum.
A probability distribution encodes all the statistics of its corresponding random variable, hence it encodes all one can learn about the system via the random variable. In the theoretical part of this thesis, we discuss several equivalent representations of probability distributions in physics which establishes their roles as complete information measures in their own contexts. In particular, we show how Rényi entropies (as well as relative Rényi entropies) completely characterise the uncertainty of a system, hence implies the insufficiency of Shannon entropy (as well as relative entropy) in capturing the information in the presence of strong fluctuation. We also generalise Jarzynski equality in terms of relative Rényi entropies and show the equivalence between the generalised equality and the work distribution. As a consequence, an equivalence between a monotonic property of relative Rényi entropies and a kind of second law relations is revealed. In the experimental part, we study the non-equilibrium dynamics of an optically levitated nanosphere system in Knudsen regime and discover that, even though the system is not in equilibrium, the dynamics can still be treated as a Brownian motion with an effective temperature and an effective coefficient. Due to the Gaussian statistics of the levitated sphere, we show how some relevant non-equilibrium properties of the system, including several local temperatures, can be analysed from its power spectrum.
To understand in detail the relation between unitary quantum theory that describes our world at the microscopic scale and thermodynamics, which was long believed only to apply to macroscopic objects, is one of the most interesting and long-standing problems in physics. Recently, this problem has received renewed attention, in particular from the community of quantum information theory, but also from the field of statistical mechanics, inspired from stochastic thermodynamics. These results suggest that thermodynamics is also relevant for individual quantum systems, provided that they can be brought into contact with thermal baths. In this thesis, I use recently developed tools to provide new results both on fundamental questions, but also on questions which are of practical relevance for potential miniaturized thermal machines. In terms of fundamental questions, the results in this thesis contribute to understanding how the basic laws of thermodynamics and statistical mechanics can be understood directly from unitary quantum mechanics. In particular, I discuss and answer the following questions: i) How can we quantify the third law of thermodynamics using information theoretic methods? ii) How can we quantify the "thermodynamic value" of a state-transition in quantum systems? iii) How can we axiomatically characterize the non-equilibrium free energy and relative entropy? iv) How can we justify statistical ensembles from an operational perspective, without having to introduce either some probability measures or an information theoretic entropy measure? v) How can we understand the equilibration of closed quantum systems, how long does it take and how difficult is it to avoid? In the second part of the thesis I discuss in detail how experimental restrictions, which become important at the quantum scale, influence the ultimate thermodynamic bounds for thermal machines. In particular, the results in this thesis provide thermodynamic bounds on work-extraction and efficiencies of thermal machines in situations where 1.) an experimenter only has limited field strengths available, 2.) an experimenter cannot control the interactions between particles, but external fields arbitrarily well, and 3.) situations in which a small quantum system can only be strongly coupled to heat baths. These bounds are tight and I provide explicit examples illustrating the different behaviours. Finally, I come back to a classic problem in statistical physics: The emergence of spontaneous symmetry breaking. Here, I provide general and rigorous new results that show how symmetry-breaking stationary states emerge from fluctuations in order parameters in dissipative lattice models.
Recently, the possibility of investigating single molecule, or single spin observables, as well as the necessity of a better understanding of the mechanisms underlying quantum dynamics in order to obtain nanoscale devices and nanostructered materials suitable for quantum computing tasks, have revived the interest in foundational aspects of quantum statistical mechanics. This thesis aims to give a contribution to this field by re-considering the statistical characterization of a quantum system at the light of some paradigmatic changes in our understanding of quantum theory which have taken place in the last two decades. In particular the impressive development of quantum information theory has changed the perceptions of quantum entanglement: for a long time it has been considered a somewhat paradoxical property of the matter at the atomic scale, but now it is regarded as an essential and ubiquitous phenomenon whose consequences are affecting the very macroscopic world that we experience. Still the decoherence program has brought out the importance of considering a quantum system together with its environment in order to clarify some key aspects of quantum dynamics. Thus, we start from the idea that quantum correlations are ubiquitous and somewhat uncontrollable in systems with many degrees of freedom which are typically considered in statistical mechanics. As a consequence we assume the standpoint that quantum statistical mechanics has not to be based on the underlying idea of a collection of many, independent quantum systems but rather it has to emerge at the level of a global wavefunction (pure state) which describes the system as well as its environment as a whole. In order to investigate the consequences of these assumptions we study the equilibrium distribution of an isolated quantum system. This is defined, in analogy with the ergodic foundations of classical statistical mechanics, on the basis of the time evolution of the quantum state. Then, we study the emergence of thermodynamic properties in a quantum system by studying the probability distribution of some function of interests, as the entropy and the equilibrium state of a subsystem, on Ensembles of Pure States. Such a probability distribution is derived from the geometry of the Hilbert space, and the theoretical tools suitable for its characterization are developed. On the one hand we perform a numerical sampling of the ensemble distributions by employing Monte Carlo techniques, on the other hand simpler analytical approximation of the geometrical distributions are derived by means of a maximum entropy principle. Model systems composed of an ensemble of spins are chosen to illustrate the salient features which emerge from the developed theoretical framework: the main point is that the Ensemble Distributions of “thermodynamic observables” (entropy or equilibrium state of a subsystem) are sharply peaked around a typical value. From the analysis it emerges that each of the overwhelming majority of the wavefunctions which has appreciable weight in the considered ensemble, is characterized by the same value of the “macroscopic” functions. This is a striking evidence of the “typicality” of these properties. In the essence, our impossibility to know the state of the system in detail does not matter, just for the remarkable fact that almost all quantum states behave essentially in the same way. By virtue of this typicality the study of the behaviour of the typical values of the thermodynamic function become meaningful. Notably, under certain conditions, one recovers the results of standard statistical mechanics, that is, the equilibrium average of the state of a subsystem can be cast in the Boltzmann canonical form at the temperature given by the usual thermodynamical relation . In the second part of the thesis we consider the dynamical aspects of the equilibrium state of a subsystem interacting with its environment. The fluctuations around the equilibrium average critically depends on the entanglement between the system and the environment and on the form of the interaction Hamiltonian. The connection between the dynamics of the fluctuations of an observable at the equilibrium and the relaxation toward the equilibrium from a “non typical” initial value is also investigated with the aid of simple model systems. The study presented in this thesis was partly motivated by a critical analysis of the statistical methods available for the theoretical modelling of magnetic resonance experiments. One of these, the Stochastic Liouville Equation, has been employed in a work completed during the first year of my Ph.D. program in order to interpret some feature of a two dimensional electron spin resonance experiment, [Fresch B., Frezzato D., Moro G. J., Kothe G., Freed J. H.; J. Phys. Chem. B., 110, 24238, (2006)].
We combine the formalisms of diagonal entropy and Jarzynski Equality to study the thermodynamic properties of closed quantum systems. Applying this approach to a quantum harmonic oscillator, the diagonal entropy offers a notion of temperature for closed systems away from equilibrium, and allows computing free-energy profiles. We also apply this approach to a hard-core boson lattice model, and discuss measures how to estimate temperature, entropy and measure work distribution functions. This technique offers a path to investigate the non-equilibrium quantum dynamics by means of performing work in a series of quenches.
Investigation on foundational aspects of quantum statistical mechanics recently entered a renaissance period due to novel intuitions from quantum information theory and to increasing attention on the dynamical aspects of single quantum systems. In the present contribution a simple but effective theoretical framework is introduced to clarify the connections between a purely mechanical description and the thermodynamic characterization of the equilibrium state of an isolated quantum system. A salient feature of our approach is the very transparent distinction between the statistical aspects and the dynamical aspects in the description of isolated quantum systems. Like in the classical statistical mechanics, the equilibrium distribution of any property is identified on the basis of the time evolution of the considered system. As a consequence equilibrium properties of quantum system appear to depend on the details of the initial state due to the abundance of constants of the motion in the Schrodinger dynamics. On the other hand the study of the probability distributions of some functions, such as the entropy or the equilibrium state of a subsystem, in statistical ensembles of pure states reveals the crucial role of typicality as the bridge between macroscopic thermodynamics and microscopic quantum dynamics. We shall consider two particular ensembles: the random pure state ensemble and the fixed expectation energy ensemble. The relation between the introduced ensembles, the properties of a given isolated system, and the standard quantum statistical description are discussed throughout the presentation. Finally we point out the conditions which should be satisfied by an ensemble in order to get meaningful thermodynamical characterization of an isolated quantum system.
Thermodynamics typically concerns the physical behaviour of macroscopic systems comprised of many particles. However, recent theoretical progress has extended the theory to both the classical-stochastic and quantum regimes, where systems are comprised of just a small number of particles. In this thesis I investigate a range of situations in which new thermodynamic phenomena emerge due to the reduced size of the systems involved. One central assumption in macroscopic thermodynamics is the weak coupling approximation, which posits that the equilibrium properties of a system are not influenced by the interactions with its surrounding environment. However, for microscopic systems this assumption can break down, and I derive new fluctuation relations that provide a refined form of the second law of thermodynamics in this strong-coupling regime, taking into account corrections stemming from these interactions. In this work I provide a characterisation of stochastic heat and entropy production for small scale classical systems that are defined regardless of the strength of interaction. I then show that these definitions lead to a consistent thermodynamic framework valid beyond the usual weak-coupling regime. The thesis also concerns the effect of interactions on the equilibrium properties of strongly-coupled quantum systems, and I investigate how these interactions can influence the resulting temperature fluctuations in this regime. Using tools from quantum estimation theory, I derive an uncertainty relation between energy and temperature valid at all coupling strengths and system sizes. The relation reveals how quantum energy coherences contribute to statistical fluctuations in the estimated temperature of small- scale systems. Finally, I investigate how quantum fluctuations influence the statistics of work extracted from slowly-driven quantum systems. I prove that, unlike in classical systems, the work dissipated by a general quantum system is no longer proportional to its statistical fluctuations. This result reveals new subtleties involved in designing optimal quantum thermodynamic processes.
Quantum thermodynamics has emerged as a central field for understanding how energy conversion processes occur in microscopic systems. In these systems, effects such as coherence, entanglement, and non-Markovianity play key roles. In this thesis, we explore different ways to describe quantum thermodynamics, using two main approaches: one based on entropy and the other on ergotropy. First, we introduce a generalized approach to quantify non-Markovianity through the breakdown of monotonicity in thermodynamic functions. In this context, the entropy-based heat flow serves as a practical tool to witness and measure quantum memory in unital maps that do not reverse the sign of the internal energy. Next, we analyze the dynamics of ergotropy in open qubits under both Markovian and non-Markovian evolutions. We identify phenomena such as freezing and sudden death of ergotropy, and we establish an analytical relation between the change in ergotropy and the environment-induced work . This provides a clear physical interpretation for the additional term in the first law in the entropy-based formulation. Finally, we propose an ergotropy-based thermodynamic formulation, in which heat is reinterpreted in terms of the change of the passive state associated with the density operator governing the quantum dynamics. This approach allows one to measure non-Markovianity of unital maps more generally and accurately than the entropy-based heat flow. This advantage comes from the direct link between heat and von Neumann entropy, a property ensured by invariance under passive transformations. Moreover, the out-of-equilibrium temperature naturally remains non-negative, similarly to equilibrium thermodynamics.