Quantum chaotic fluctuation-dissipation theorem: effective Brownian motion in closed quantum systems

We analytically describe the decay to equilibrium of generic observables of a non-integrable system after a perturbation in the form of a random matrix. We further obtain an analytic form for the time-averaged fluctuations of an observable in terms of the rate of decay to equilibrium. Our result shows the emergence of a Fluctuation-Dissipation theorem corresponding to a classical Brownian process, specifically, the Ornstein-Uhlenbeck process. Our predictions can be tested in quantum simulation experiments, thus helping to bridge the gap between theoretical and experimental research in quantum thermalization. We test our analytic results by exact numerical experiments in a spin-chain. We argue that our Fluctuation-Dissipation relation can be used to measure the density of states involved in the non-equilibrium dynamics of an isolated quantum system

Ergodicity probes: using time-fluctuations to measure the Hilbert space dimension

Quantum devices, such as quantum simulators, quantum annealers, and quantum computers, may be exploited to solve problems beyond what is tractable with classical computers. This may be achieved as the Hilbert space available to perform such `calculations' is far larger than that which may be classically simulated. In practice, however, quantum devices have imperfections, which may limit the accessibility to the whole Hilbert space. We thus determine that the dimension of the space of quantum states that are available to a quantum device is a meaningful measure of its functionality, though unfortunately this quantity cannot be directly experimentally determined. Here we outline an experimentally realisable approach to obtaining the required Hilbert space dimension of such a device to compute its time evolution, by exploiting the thermalization dynamics of a probe qubit. This is achieved by obtaining a fluctuation-dissipation theorem for high-temperature chaotic quantum systems, which facilitates the extraction of information on the Hilbert space dimension via measurements of the decay rate, and time-fluctuations

Congruence lattices of semilattices

The main result of this paper is that the class of congruence lattices of semilattices satisfies no nontrivial lattice identities. It is also shown that the class of subalgebra lattices of semilattices satisfies no nontrivial lattice identities. As a consequence it is shown that if V is a semigroup variety all of whose congruence lattices satisfy some fixed nontrivial lattice identity, then all the members of V are groups with exponent dividing a fixed finite number

A class of infinite convex geometries

Various characterizations of finite convex geometries are well known. This note provides similar characterizations for possibly infinite convex geometries whose lattice of closed sets is strongly coatomic and lower continuous. Some classes of examples of such convex geometries are given.Comment: 10 page

Ultra-Strong Optomechanics Incorporating the Dynamical Casimir Effect

We propose a superconducting circuit comprising a dc-SQUID with mechanically compliant arm embedded in a coplanar microwave cavity that realizes an optomechanical system with a degenerate or non-degenerate parametric interaction generated via the dynamical Casimir effect. For experimentally feasible parameters, this setup is capable of reaching the single-photon, ultra-strong coupling regime, while simultaneously possessing a parametric coupling strength approaching the renormalized cavity frequency. This opens up the possibility of observing the interplay between these two fundamental nonlinearities at the single-photon level.Comment: 7 pages, 1 figure, 1 tabl

Comparing generalisation in children and adults learning an artificial language

Successful language acquisition involves generalization, but learners must balance this against the acquisition of lexical constraints. Examples occur throughout language. For example, English native speakers know that certain noun-adjective combinations are impermissible (e.g., strong winds, high winds, strong breezes, *high breezes). Another example is the restrictions imposed by verb sub-categorization (e.g., I gave/sent/threw the ball to him; I gave/sent/threw him the ball; I donated/carried/pushed the ball to him; * I donated/carried/pushed him the ball; Baker, 1979). A central debate has been the extent to which learning such patterns depends on semantic cues (Pinker, 1989) and/or distributional statistics (Braine et al., 1990). The current experiments extend previous work which used Artificial Language learning to demonstrate that adults (Wonnacott et al., 2008) and 6 year olds (Wonnacott, 2011) are able to learn lexically based restrictions on generalization using distributional statistics. Here we directly compare the two age groups learning the same artificial language, with a view to exploring maturational differences in language learning. In addition to manipulating frequency (across high and low frequency items) and quantity of exposure (across days), languages were constructed such that a word’s semantic class was helpful for learning the restrictions for some types of lexical items, but potentially misleading for others

Reading comprehension: nature, assessment and teaching.

The goal of reading is understanding. In order to understand print, a child must be able to decode the words on the page and to extract meaning. A large body of research focuses on how children learn to decode text and how best to foster children’s decoding skills. In contrast, we know much less about the process of reading comprehension in children. In this booklet we first consider what is required in order to ‘read for meaning’. We then move on to discuss children who have difficulties with reading comprehension. Our aim is to enable teachers to assess individual differences in reading and to foster the comprehension strategies that characterize fluent reading