Thermalization in quantum systems

A short discussion regarding some ideas about the emergence of thermal equilibrium in isolated quantum systems.

Thermodynamics was originally developed as a phenomenological theory of macroscopic systems: A set of empirical results which, during the years, have been progressively formalised, organised and synthesised into a number of laws which are completely independent on the physical substrate. Because of that, thermodynamics puts severe constraints on the behaviour of macroscopic systems. Most of the technological advances occurred during the industrial revolution have their roots in this simple, and yet powerful, fact.

In the second half of the XIX century Maxwell, Boltzmann, Gibbs and others tried to connect these macroscopic laws with the multi-faceted nature of the microscopic world. The result of this attempt was Statistical Mechanics: a theory which provides mechanical roots to macroscopic concepts as temperature and heat. According to Statistical Mechanics, such macroscopic behaviour is ascribed to the emergence of a single condition: thermal equilibrium. Indeed, the central postulate of the theory is that the state of the system has a specific functional form which we call thermal state. The tools of statistical mechanics are then used to study macroscopic properties of complex systems, from this single assumption. Later, in the first half of the XX century, the rise of quantum mechanics forced us to adapt these tools to include quantum effects. The result was Quantum Statistical Mechanics, a notable improvement of the classical theory which expanded its domain of applicability. Both classical and quantum statistical mechanics enjoy a marvelous success in predicting and explaining the large-scale behaviour of several systems.

Despite an undeniable success, it is well known that there are systems which do not thermalize, thus escaping such description. Two examples are integrable and localised systems. Moreover, the theory is not free from conceptual issues. In particular, a long-standing problem is that the evolution of isolated quantum systems is not compatible with the dynamical emergence of thermal states. We find ourselves in the situation where we have a theory which, despite being based on a seemingly wrong postulate, still works very well in a large number of cases. A possible way out of this problem is to question the practical utility of the concept of isolated systems. This is a legitimate route, based on the successful theory of Open Quantum Systems. However, it is opinion of the author that, from the conceptual point of view, this is not sufficiently satisfactory. Even if our system is interacting with a large environment, in principle we could still include the environment in our description and obtain an isolated system. Here we insist on dealing with isolated quantum systems. This path is undoubtedly more painful as it tackles a broader issue: understanding the interplay between coherent dynamics of complex quantum systems and their equilibrium behaviour. We choose this perspective as, beyond the purpose of the thesis, the outcomes of these investigations are expected to have technological consequences, for example for the task of building a quantum computer.

Moreover, in the last twenty years we witnessed a remarkable progress in the ability to manipulate quantum systems. Thanks to these technological developments, we witnessed a surge of interest for these questions, which now can be experimentally addressed. Modern experiments are able to probe the coherent dynamics of nearly- isolated systems, providing important data about equilibration and thermalization in quantum systems. More recently, the will to understand quantum equilibration and thermalization has met the need to investigate how the rules of thermodynamics are modified at the nanoscale, where quantum, non-equilibrium and collective effects can dominate. These joints efforts gave birth to “Stochastic Thermodynamics” first and then to the field of “Quantum Thermodynamics”, which is addressing the interplay between quantum theory and thermodynamics, both for foundational purposes and with the aim of pushing forward our technology. For all these reasons, to understand the conditions which lead to a thermal behaviour, and how to avoid them, is a topic of both conceptual and technological relevance.

Thermalization of Quantum Observable

With this perspective in mind, part of my Ph.D. was devoted to studying the detailed role played by observables in the emergence of thermal equilibrium, which is often overlooked. Almost all known approaches to the foundations of statistical mechanics focus on the state of a system. While a statement about the form of the state will always reflect on the observables, the converse is not generally true. Thus, observing thermal equilibrium properties for a few observables does not mean that the whole state of the system is thermal. In practical experiments, and sometimes even in our numerical simulations, our conclusions are based on the behaviour of a few observables. In order to trace back such behaviour to a specific form of the whole state of the system we need to have access to all the matrix elements of the density matrix. While this can be done for relatively small systems, it is concretely impossible for systems of modest sizes, which still fall in the category of microscopic systems. For example, a pure state of L spin-1/2 has 2 × 2^L − 1 linearly independent real parameters. This number grows exponentially with the size of the system, while the number of observables we can measure in a laboratory is usually very limited and certainly it does not grow exponentially with the size of the system. We must accept that our experimental observations provide only very little information about the state of the system. Thus, the information we can gather, when we have access to one observable, is given by the probability distribution of the eigenvalues of the observable. For this reason, the emergence of thermal equilibrium for an observable should be ascribed to the behaviour of its eigenvalues probability distribution. This picture presents no contradiction with the unitary dynamics of isolated systems.