Regular Tunnelling Sequences in Mixed Systems

We show that the pattern of tunnelling rates can display a vivid and regular pattern when the classical dynamics is of mixed chaotic/regular type. We consider the situation in which the dominant tunnelling route connects to a stable periodic orbit and this orbit is surrounded by a regular island which supports a number of quantum states. We derive an explicit semiclassical expression for the positions and tunnelling rates of these states by use of a complexified trace formula.Comment: submitted to Physica E as a contribution to the workshop proceedings of "Dynamics of Complex Systems" held at the Max Planck Institute for the Physics of Complex Systems in Dresden from March 30 to June 15, 199

A Matrix Element for Chaotic Tunnelling Rates and Scarring Intensities

It is shown that tunnelling splittings in ergodic double wells and resonant widths in ergodic metastable wells can be approximated as easily-calculated matrix elements involving the wavefunction in the neighbourhood of a certain real orbit. This orbit is a continuation of the complex orbit which crosses the barrier with minimum imaginary action. The matrix element is computed by integrating across the orbit in a surface of section representation, and uses only the wavefunction in the allowed region and the stability properties of the orbit. When the real orbit is periodic, the matrix element is a natural measure of the degree of scarring of the wavefunction. This scarring measure is canonically invariant and independent of the choice of surface of section, within semiclassical error. The result can alternatively be interpretated as the autocorrelation function of the state with respect to a transfer operator which quantises a certain complex surface of section mapping. The formula provides an efficient numerical method to compute tunnelling rates while avoiding the need for the exceedingly precise diagonalisation endemic to numerical tunnelling calculations.Comment: Submitted to Annals of Physics. This work has been submitted to Academic Press for possible publicatio

Scarring and the statistics of tunnelling

We show that the statistics of tunnelling can be dramatically affected by scarring and derive distributions quantifying this effect. Strong deviations from the prediction of random matrix theory can be explained quantitatively by modifying the Gaussian distribution which describes wavefunction statistics. The modified distribution depends on classical parameters which are determined completely by linearised dynamics around a periodic orbit. This distribution generalises the scarring theory of Kaplan [Phys. Rev. Lett. {\bf 80}, 2582 (1998)] to describe the statistics of the components of the wavefunction in a complete basis, rather than overlaps with single Gaussian wavepackets. In particular it is shown that correlations in the components of the wavefunction are present, which can strongly influence tunnelling-rate statistics. The resulting distribution for tunnelling rates is tested successfully on a two-dimensional double-well potential.Comment: 20 pages, 4 figures, submitted to Ann. Phy

Nanotechnology Solutions for Global Water Challenges

The lack of clean and safe drinking water is responsible for more deaths than war, terrorism and weapons of mass destruction combined. This suggests contaminated water poses a significant threat to human health and welfare. In addition, standard water disinfection approaches such as sedimentation, filtration, and chemical or biological degradation are not fully capable of destroying emerging contaminants (e.g. pesticides, pharmaceutical waste products) or certain types of bacteria (e.g. Cryptosporidium parvum). Nanomaterials and nanotechnology based devices can potentially be employed to solve the challenges posed by various contaminants and microorganisms. Nanomaterials of different shapes, namely nanoparticles, nanotubes, nanowires and fibers have the ability to function as adsorbents and catalysts. These possess an expansive array of physicochemical characteristics deeming them highly attractive for the production of reactive media for water membrane filtration, a vital step in the production of potable water. As a result of their exceptional adsorptive capacity for water contaminants, graphene based nanomaterials have emerged as an area of significant importance in the area of membrane filtration and water treatment. In addition, Advanced Oxidation Processes (AOPs) together with or without sources of light irradiation or ultrasound, have been found to be promising alternatives for water treatment at near ambient temperature and pressure. Furthermore, the uses of visible light active titanium dioxide photocatalysts and photo-Fenton processes have shown significant potential for water purification. A wide variety of nanomaterial based sensors, for the monitoring of water quality, have also been reviewed in detail. In conclusion, the rapid and continued growth in the area of nanomaterial based devices offers significant hope for addressing future water quality challenges

Edge Diffraction, Trace Formulae and the Cardioid Billiard

We study the effect of edge diffraction on the semiclassical analysis of two dimensional quantum systems by deriving a trace formula which incorporates paths hitting any number of vertices embedded in an arbitrary potential. This formula is used to study the cardioid billiard, which has a single vertex. The formula works well for most of the short orbits we analyzed but fails for a few diffractive orbits due to a breakdown in the formalism for certain geometries. We extend the symbolic dynamics to account for diffractive orbits and use it to show that in the presence of parity symmetry the trace formula decomposes in an elegant manner such that for the cardioid billiard the diffractive orbits have no effect on the odd spectrum. Including diffractive orbits helps resolve peaks in the density of even states but does not appear to affect their positions. An analysis of the level statistics shows no significant difference between spectra with and without diffraction.Comment: 25 pages, 12 Postscript figures. Published versio

Semiclassical Trace Formulas for Noninteracting Identical Particles

We extend the Gutzwiller trace formula to systems of noninteracting identical particles. The standard relation for isolated orbits does not apply since the energy of each particle is separately conserved causing the periodic orbits to occur in continuous families. The identical nature of the particles also introduces discrete permutational symmetries. We exploit the formalism of Creagh and Littlejohn [Phys. Rev. A 44, 836 (1991)], who have studied semiclassical dynamics in the presence of continuous symmetries, to derive many-body trace formulas for the full and symmetry-reduced densities of states. Numerical studies of the three-particle cardioid billiard are used to explicitly illustrate and test the results of the theory.Comment: 29 pages, 11 figures, submitted to PR

Complex Periodic Orbits and Tunnelling in Chaotic Potentials

We derive a trace formula for the splitting-weighted density of states suitable for chaotic potentials with isolated symmetric wells. This formula is based on complex orbits which tunnel through classically forbidden barriers. The theory is applicable whenever the tunnelling is dominated by isolated orbits, a situation which applies to chaotic systems but also to certain near-integrable ones. It is used to analyse a specific two-dimensional potential with chaotic dynamics. Mean behaviour of the splittings is predicted by an orbit with imaginary action. Oscillations around this mean are obtained from a collection of related orbits whose actions have nonzero real part

Myth or Memory? Recollections of Penal Times in Irish Folklore

Stories of priests being hunted down and murdered at Mass Rocks by priest catchers and soldiers during the Penal era in Ireland persist to the present day. Using Ó Ciosáin’s (2004) tripartite taxonomy of memory this paper explores the reasons why these images continue to dominate and reflect persecuted nature of Catholicism