19 research outputs found
New equilibrium ensembles for isolated quantum systems
The unitary dynamics of isolated quantum systems does not allow a pure state
to thermalize. Because of that, if an isolated quantum system equilibrates, it
will do so to the predictions of the so-called "diagonal ensemble" .
Building on the intuition provided by Jaynes' maximum entropy principle, in
this paper we present a novel technique to generate progressively better
approximations to . As an example, we write down a hierarchical set
of ensembles which can be used to describe the equilibrium physics of small
isolated quantum systems, going beyond the "thermal ansatz" of Gibbs ensembles.Comment: Paper written for the Special Issue "Thermalization in Isolated
Quantum Systems" of the Journal Entrop
Information-theoretic equilibrium and observable thermalization
To understand under which conditions thermodynamics emerges from the
microscopic dynamics is the ultimate goal of statistical mechanics. Despite the
fact that the theory is more than 100 years old, we are still discussing its
foundations and its regime of applicability. A point of crucial importance is
the definition of the notion of thermal equilibrium, which is given as the
state that maximises the von Neumann entropy. Here we argue that it is
necessary to propose a new way of describing thermal equilibrium, focused on
observables rather than on the full state of the quantum system. We
characterise the notion of thermal equilibrium, for a given observable, via the
maximisation of its Shannon entropy and highlight the thermal properties that
such a principle heralds. The relation with Gibbs ensembles is brought to
light. Furthermore, we apply such a notion of equilibrium to a closed quantum
systems and prove that there is always a class of observables which exhibits
thermal equilibrium properties and we give a recipe to explicitly construct
them. Eventually, we bring to light an intimate connection of such a principle
with the Eigenstate Thermalisation Hypothesis.Comment: Accepted by Scientific Repor
Pure states statistical mechanics: On its foundations and applications to quantum gravity
The project concerns the interplay among quantum mechanics, statistical
mechanics and thermodynamics, in isolated quantum systems. The underlying goal
is to improve our understanding of the concept of thermal equilibrium in
quantum systems. First, I investigated the role played by observables and
measurements in the emergence of thermal behaviour. This led to a new notion of
thermal equilibrium which is specific for a given observable, rather than for
the whole state of the system. The equilibrium picture that emerges is a
generalization of statistical mechanics in which we are not interested in the
state of the system but only in the outcome of the measurement process. I
investigated how this picture relates to one of the most promising approaches
for the emergence of thermal behaviour in isolated quantum systems: the
Eigenstate Thermalization Hypothesis. Then, I applied the results to study some
equilibrium properties of many-body localised systems. Despite the localization
phenomenon, which prevents thermalization of subsystems, I was able to show
that we can still use the predictions of statistical mechanics to describe the
equilibrium of some observables. Moreover, the intuition developed in the
process led me to propose an experimentally accessible way to unravel the
interacting nature of many-body localised systems. Second, I exploited the
"Concentration of Measure" phenomenon to study the macroscopic properties of
the basis states of Loop Quantum Gravity. These techniques were previously used
to explain why the thermal behaviour in quantum systems is such an ubiquitous
phenomenon, at the macroscopic scale. I focused on the local properties, their
thermodynamic behaviour and interplay with the semiclassical limit. This was
motivated by the necessity to understand, from a quantum gravity perspective,
how and why a classical horizon exhibits thermal properties.Comment: PhD Thesis - University of Oxford - Comments and questions are
welcome and encouraged
Geometric Quantum State Estimation
Density matrices capture all of a quantum system's statistics accessible
through projective and positive operator-valued measurements. They do not
completely determine its state, however, as they neglect the physical
realization of ensembles. Fortunately, the concept of geometric quantum state
does properly describe physical ensembles. Here, given knowledge of a density
matrix, possibly arising from a tomography protocol, we show how to estimate
the geometric quantum state using a maximum entropy principle based on a
geometrically-appropriate entropy.Comment: 5 pages, 2 figures; Supplementary Material: 2 pages;
http://csc.ucdavis.edu/~cmg/compmech/pubs/qgse.ht
Geometric Quantum Thermodynamics
Building on parallels between geometric quantum mechanics and classical
mechanics, we explore an alternative basis for quantum thermodynamics that
exploits the differential geometry of the underlying state space. We develop
both microcanonical and canonical ensembles, introducing continuous mixed
states as distributions on the manifold of quantum states. We call out the
experimental consequences for a gas of qudits. We define quantum heat and work
in an intrinsic way, including single-trajectory work, and reformulate
thermodynamic entropy in a way that accords with classical, quantum, and
information-theoretic entropies. We give both the First and Second Laws of
Thermodynamics and Jarzynki's Fluctuation Theorem. The result is a more
transparent physics, than conventionally available, in which the mathematical
structure and physical intuitions underlying classical and quantum dynamics are
seen to be closely aligned.Comment: 10 pages, 1 figure; Supplementary Material: 7 pages;
http://csc.ucdavis.edu/~cmg/compmech/pubs/gqt.ht
Beyond Density Matrices: Geometric Quantum States
A quantum system's state is identified with a density matrix. Though their
probabilistic interpretation is rooted in ensemble theory, density matrices
embody a known shortcoming. They do not completely express an ensemble's
physical realization. Conveniently, when working only with the statistical
outcomes of projective and positive operator-valued measurements this is not a
hindrance. To track ensemble realizations and so remove the shortcoming, we
explore geometric quantum states and explain their physical significance. We
emphasize two main consequences: one in quantum state manipulation and one in
quantum thermodynamics.Comment: 7 pages, 1 figure; Supplementary Material: 2 pages;
http://csc.ucdavis.edu/~cmg/compmech/pubs/rgqs.ht
Loop gravity, twisted geometries and torsion
It has been shown that discretization of General Relativity admit a geometric interpretation via the so-called twisted geometries, that differ from the Regge calculus because they lack of the shape- matching conditions. However it is still not clear whether or not they are full dynamical objects or just a kinematical arena and the dynamics will deal with a Regge-like geometry.
The main goal of the thesis is to investigate the dynamical arising of these conditions as secondary constraint in a simple twisted-geometries’ discretisation of a flat space-time, triangulated with the graph of a single 4- simplex. The stability procedure pointed out the presence of secondary constraints which are equivalent to the well-known shape-matching conditions
A quantum particle in a box with moving walls
We analyze the non-relativistic problem of a quantum particle that bounces
back and forth between two moving walls. We recast this problem into the
equivalent one of a quantum particle in a fixed box whose dynamics is governed
by an appropriate time-dependent Schroedinger operator.Comment: 12 pages, 0 figure