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Caustics occur widely in dynamics and take on shapes classified by catastrophe theory. At finite wavelengths they produce interference patterns containing networks of vortices (phase singularities). Here we investigate caustics in quantized fields, focusing on the collective dynamics of quantum spins. We show that, following a quench, caustics are generated in the Fock space amplitudes specifying the many-body configuration and which are accessible in experiments with cold atoms, ions or photons. The granularity of quantum fields removes all singularities, including phase singularities, converting point vortices into nonlocal vortices that annihilate in pairs as the quantization scale is increased. Furthermore, the continuous scaling laws of wave catastrophes are replaced by discrete versions. Such `quantum catastrophes' are expected to be universal dynamical features of quantized fields., Comment: The original title of this paper was "Morphology of a quantum catastrophe". The title was changed to reflect its relevance to collective spin dynamics. There are 4 figures

Exact analytical results (in terms of Bessel functions) for the bandgaps, reflectance, and transmittance of one-dimensional photonic crystals with a sawtooth refractive index profile on the period are derived for the first time. This extends a group of exactly solvable models of periodic refractive indices. The asymptotic approximations of the above exact results have been also obtained., Comment: 8 pages, 8 figures

Post-exponential decay of the probability density of a quantum particle leaving a trap can be reproduced accurately, except for interference oscillations at the transition to the post-exponential regime, by means of an ensemble of classical particles emitted with constant probability per unit time and the same half-life as the quantum system. The energy distribution of the ensemble is chosen to be identical to the quantum distribution, and the classical point source is located at the scattering length of the corresponding quantum system. A 1D example is provided to illustrate the general argument.

Initial-switching refers to the way in which the decay of an initially confined state begins, as the barrier isolating it from the exterior is relaxed. We study these effects in the context of Longhi's version of the Fano-Anderson model. Most authors assume the sudden approximation where the coupling is turned on instantaneously. We consider a finite rise time T, both numerically and analytically. When the coupling is ramped up linearly over a switching time T, we show that the asymptotic survival amplitude acquires a phase T and is modulated by a factor (sin T)/T. Several other results relating to the solution of the model are obtained. All site amplitudes have the same decay constant during the exponential decay regime. In the asymptotic regime, the amplitude and phase of decay oscillations depend on the initial-switching profile, but the period does not., Comment: 12 pages, 10 figures, accepted by Phys. Rev. A

We study the elusive transition from exponential to post-exponential (algebraic) decay of the probability density of a quantum particle emitted by an exponentially decaying source, in one dimension. The main finding is that the probability density at the transition time, and thus its observability, increases with the distance of the detector from the source, up to a critical distance beyond which exponential decay is no longer observed. Solvable models provide explicit expressions for the dependence of the transition on resonance and observational parameters, facilitating the choice of optimal conditions.

We discuss the time spent by an electron propagating through a finite periodic system such as a semiconductor superlattice. The relation between dwell-time and phase-time is outlined. The envelopes of phase-time at maximum and minimum transmission are derived, and it is shown that the peaks and valleys of phase-time can be well described by parameters fitted at the extrema. For a many-period system this covers most of the allowed band. Comparison is made to direct numerical solutions of the time-dependent Schr\"odinger equation by Veenstra et al. [cond-mat/0411118] who compared systems with and without addition of an anti-reflection coating (ARC). With an ARC, the time delay is consistent with propagation at the Bloch velocity of the periodic system, which significantly reduces the time delay, in addition to increasing the transmissivity., Comment: 8 pages, 10 figures, based on a talk at Theory-Canada 3 in Edmonton AB, June 2007

We study coherent electron dynamics in a biassed undriven ideal semiconductor superlattice coupled to the continuum, near energy level anticrossings. In particular, we examine the dependence of wavepacket dynamical characteristics on electric field detuning, and investigate mixed regimes involving a superposition of energy level anticrossings showing both Rabi oscillations and resonant tunnelling. In earlier work [Phys. Rev. B 75 165421 (2007)], Rabi and Zener resonances were shown to have a common origin, and a criteria for the occurrence of either was proposed. The present results allow a better understanding of the nature of an interminiband resonance, which can be useful in the areas of microwave radiation generation and matter manipulation on the particle level, as well as demonstrate an alternative approach to examining electron level structure of a finite superlattice., Comment: 18 pages, 10 figures, submitted to Phys. Rev. B. submitted to Journal of Physics

Quantal systems are predicted to show a change-over from exponential decay to power law decay at very long times. Although most theoretical studies predict integer power-law exponents, recent measurements by Rothe et al. of decay luminescence of organic molecules in solution {Phys. Rev. Lett. 96 (2006) 163601} found non-integer exponents in most cases. We propose a physical mechanism, within the realm of scattering from potentials with long tails, which produces a continuous range of power law exponents. In the tractable case of the repulsive inverse square potential, we demonstrate a simple relation between the strength of the long range tail and the power law exponent. This system is amenable to experimental scrutiny.

Coquelin et al. studied biperiodic semiconductor superlattices, which consist of alternating cell types, one with wide wells and the other narrow wells, separated by equal strength barriers. If the wells were identical, it would be a simply periodic system of $N = 2n$ half-cells. When asymmetry is introduced, an allowed band splits at the Bragg point into two disjoint allowed bands. The Bragg resonance turns into a transparent state located close to the band edge of the lower(upper) band when the first(second) well is the wider. Analysis of this system gives insight into how band splitting occurs. Further we consider semi-periodic systems having $N= 2n+1$ half-cells. Surprisingly these have very different transmission properties, with an envelope of maximum transmission probability that crosses the envelope of minima at the transparent point., Comment: 12 pages, 10 figures Version 2: improved figures using colour, and some small improvements in the text, in response to referee comments Version 3: incorporates changes which arose in proofs stage

Carrier dynamics at energy level anticrossings in biased semiconductor superlattices, was studied in the time domain by solving the time-dependent Schroedinger equation. The resonant nature of interminiband Rabi oscillations has been explicitly demonstrated to arise from interference of intrawell and Bloch oscillations. We also report a simulation of direct Rabi oscillations across three minibands, in the high field regime, due to interaction between three strongly coupled minibands., Comment: 13 pages, 16 figures

We study surface states of 1D photonic crystals using a semiclassical coupled wave theory. Both TE and TM modes are treated. We derive analytic approximations that clarify the systematics of the dispersion relations, and the roles of the various parameters defining the crystal., Comment: 7 pages, 8 figures

The confinement mechanism of electrons in gated circular quantum dots is studied in a sequence of models, from self-consistent 3D Hartree calculations to the semiclassical model of Shikin et al. Separation of the vertical from transverse confinement allows accurate 2D Hartree calculations. For moderate size dots containing $\sim 50$ electrons, 2D Thomas-Fermi can also be accurate. Finally, a Shikin type parameterization of the electron density allows quantitative study of the importance of the many contributions to the confinement potential., Comment: original: 16 pages, 14 figures revision: reduced size to 14 pages, 13 figures

Accurate analytic approximations are developed for the band gap boundaries and surface waves of a 1D photonic crystal, making use of the semiclassical theory recently developed by the authors: [Phys. Rev. E {69} (2004) 016612 and {70} (2004) 016606]. These analytic results provide useful insight on systematics of surface states., Comment: 14 pages, 11 figures

Phonon-assisted tunneling rates are evaluated for a well isolated double dot system defined in a GaAs semiconductor heterostructure of finite thickness. A separable model for the confining potential allows accurate determinations of doublet electron wavefunctions and energies. It is found that at small doublet energies the piezoelectric rates due to flexural modes give the dominant contribution. For small slab thicknesses the predicted rates are up to two orders of magnitude higher than for very thick slabs., Comment: 7 pages, 12 figures, shortened text and corrected typos. 8 pages, 14 figures

Time delay in electron propagation through a finite periodic system such as a semiconductor superlattice is studied by direct numerical solution of the time-dependent Schr\"odinger equation. It is found that addition of an anti-reflection coating significantly reduces the time delay, in addition to increasing the transmissivity., Comment: Version 3: revised sec III b., added appendix and refs. 10 pages, 11 figs. Version 2: major revision, especially pages 5-9; additional references, 9 pages, 11 figures. Version 1 was 7 pages, 11 figures

We examine shape invariant potentials (excluding those that are obtained by scaling) in supersymmetric quantum mechanics from the stand-point of periodic orbit theory. An exact trace formula for the quantum spectra of such potentials is derived. Based on this result, and Barclay's functional relationship for such potentials, we present a new derivation of the result that the lowest order SWKB quantisation rule is exact., Comment: 4 pages, 1 figure, corrected typos, minor revisions, added references

We discuss the relation between continuum bound states (CBS) localized on a defect, and surface states of a finite periodic system. We use the transfer matrix method to model an experiment of Capasso, and find all continuum bound and anti-bound states. We compute the rate for intra-subband transitions from the ground state to the CBS and derive a sum rule. Finally we show how to improve the confinement of a CBS while keeping the energy fixed., Comment: 6 tables, 13 figures, 19 pages in all

Recent measurements of electron escape from a non-equilibrium charged quantum dot are interpreted within a 2D separable model. The confining potential is derived from 3D self-consistent Poisson-Thomas-Fermi calculations. It is found that the sequence of decay lifetimes provides a sensitive test of the confining potential and its dependence on electron occupation., Comment: 9 pages, 10 figures

Recent improvements in the determination of the running of the fine-structure constant also allow an update of the hadronic vacuum-polarization contribution to the Lamb shift. We find a shift of -3.40(7) kHz to the 1S level of hydrogen. We also comment on the contribution of this effect to the determination by elastic electron scattering of the r.m.s. radii of nuclei., Comment: 7 pages, latex, 1 figure -- Submitted to Phys. Rev. A -- epsfig.sty required

Darwin-Foldy nuclear-size corrections in electronic atoms and nuclear radii are discussed from the nuclear-physics perspective. Interpretation of precise isotope-shift measurements is formalism dependent, and care must be exercised in interpreting these results and those obtained from relativistic electron scattering from nuclei. We strongly advocate that the entire nuclear-charge operator be used in calculating nuclear-size corrections in atoms, rather than relegating portions of it to the non-radiative recoil corrections. A preliminary examination of the intrinsic deuteron radius obtained from isotope-shift measurements suggests the presence of small meson-exchange currents (exotic binding contributions of relativistic order) in the nuclear charge operator, which contribute approximately 1/2%., Comment: 17 pages, latex, 1 figure -- Submitted to Phys. Rev. A -- epsfig.sty required

Ballistic transport of electrons through a quantum wire with a constriction is studied in terms of Bohm's interpretation of quantum mechanics, in which the concept of a particle orbit is permitted. The classical bouncing ball trajectories, which justify the name ``ballistic transport'', are established in the large wave number limit. The formation and the vital role of quantum vortices is investigated., Comment: 14 pages, revtex, 4 postscript figures

We present computational approaches as alternatives to the recent microwave cavity experiment by S. Sridhar and A. Kudrolli (Phys. Rev. Lett. {\bf 72}, 2175 (1994)) on iso-spectral cavities built from triangles. A straightforward proof of iso-spectrality is given based on the mode matching method. Our results show that the experiment is accurate to 0.3% for the first 25 states. The level statistics resemble those of GOE when the integrable part of the spectrum is removed., Comment: 15 pages, revtex, 5 postscript figures