(b) Write out the equations for the time dependence of ρ11, ρ22, ρ12 and ρ21 assuming that a light field interaction V = -μ12Ee iωt + c.c. couples only levels |1> and |2>, and that the excited levels exhibit spontaneous decay. (8 marks) (c) Under steady-state conditions, find the ratio of populations in states |2> and |3>. (3 marks) (d) Find the slowly varying amplitude ̃ ρ 12 of the polarization ρ12 = ̃ ρ 12e iωt . (6 marks) (e) In the limiting case that no decay is possible from intermediate level |3>, what is the ground state population ρ11(∞)? (2 marks) 2. (15 marks total) In a 2-level atom system subjected to a strong field, dressed states are created in the form |D1(n)> = sin θ |1,n> + cos θ |2,n-1> |D2(n)> = cos θ |1,n> sin θ |2,n-1>

The Quantum Theory of Light

Random-matrix theories in quantum physics : common concepts

We review the development of random-matrix theory (RMT) during the last decade. We emphasize both the theoretical aspects, and the application of the theory to a number of fields. These comprise chaotic and disordered systems, the localization problem, many-body quantum systems, the Calogero-Sutherland model, chiral symmetry breaking in QCD, and quantum gravity in two dimensions. The review is preceded by a brief historical survey of the developments of RMT and of localization theory since their inception. We emphasize the concepts common to the above-mentioned fields as well as the great diversity of RMT. In view of the universality of RMT, we suggest that the current development signals the emergence of a new "statistical mechanics": Stochasticity and general symmetry requirements lead to universal laws not based on dynamical principles.

Random Matrix Theories in Quantum Physics: Common Concepts

We review the current status of single-photon-source and single-photon-detector technologies operating at wavelengths from the ultraviolet to the infrared. We discuss applications of these technologies to quantum communication, a field currently driving much of the development of single-photon sources and detectors.

/pdf/invited-review-article-single-photon-sources-and-detectors-5bsn8auo37.pdf

Invited review article: Single-photon sources and detectors

Quantum theory: Concepts and methods

This paper explains how a popular commercially available software package for solving partial-differential-equations (PDEs), as based on the finite-element method, can be configured to efficiently calculate the frequencies and fields of the whispering-gallery (WG) modes of axisymmetric dielectric resonators. The approach is traceable; it exploits the PDE-solver's ability to accept the definition of solutions to Maxwell's equations in so-called weak form. Associated expressions and methods for estimating a WG mode's volume, filling factor(s), and in the case of closed (open) resonators, its wall (radiation) loss, are provided. As no transverse approximation is imposed, the approach remains accurate even for quasi-transverse-magnetic/electric modes of low finite azimuthal mode order. The approach's generality and utility are demonstrated by modeling several nontrivial structures, i.e., 1) two different optical microcavities (one toroidal made of silica, the other an AlGaAs microdisk), 2) a third-order sapphire:air Bragg cavity, and 3) two different cryogenic sapphire WG-mode resonators; both 2) and 3) operate in the microwave X-band. By fitting one of 3) to a set of measured resonance frequencies, the dielectric constants of sapphire at liquid-helium temperature have been estimated.

/pdf/traceable-2-d-finite-element-simulation-of-the-whispering-1hetvc5gca.pdf

Traceable 2-D Finite-Element Simulation of the Whispering-Gallery Modes of Axisymmetric Electromagnetic Resonators

This paper explains how a popular, commercially-available software package for solving partial-differential-equations (PDEs), as based on the finite-element method (FEM), can be configured to calculate, efficiently, the frequencies and fields of the whispering-gallery (WG) modes of axisymmetric dielectric resonators. The approach is traceable; it exploits the PDE-solver's ability to accept the definition of solutions to Maxwell's equations in so-called `weak form'. Associated expressions and methods for estimating a WG mode's volume, filling factor(s) and, in the case of closed(open) resonators, its wall(radiation) loss, are provided. As no transverse approximation is imposed, the approach remains accurate even for quasi-transverse magnetic/electric modes of low, finite azimuthal mode order. The approach's generality and utility are demonstrated by modeling several non-trivial structures: (i) two different optical microcavities [one toroidal made of silica, the other an AlGaAs microdisk]; (ii) a 3rd-order sapphire:air Bragg cavity; (iii) two different cryogenic sapphire WG-mode resonators; both (ii) and (iii) operate in the microwave X-band. By fitting one of (iii) to a set of measured resonance frequencies, the dielectric constants of sapphire at liquid-helium temperature have been estimated.

Traceable 2D finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators

Random tilings that comprise squares and equilateral triangles can model quasicrystals with twelvefold symmetry. A (phason) elastic theory for such tilings is constructed, whose order parameter is the phason field, and whose entropy density includes terms up to third order in the phason strain. Due to an unusual constraint, the phason field of any square-triangle tiling is irrotational and, as a result, the form of the entropy density is simpler than the general form that is required by twelvefold symmetry alone. Using an update move, which rearranges a closed, nonlocal, one-dimensional chain of squares and triangles, the unknown parameters of the elastic theory are estimated via Monte Carlo simulations: (i) One of the two second-order elastic constants and the third-order elastic constant are found by measuring phason fluctuations; athermal systems (maximally random ensembles) with the same background phason strain but different sizes of unit cell are simulated to distinguish the effects of a finite background phason strain from the effects of finite unit-cell size. (ii) The entropy per unit area at zero phason strain and the other second-order elastic constant are found from the entropies that thermal systems (canonical ensembles) gain between zero and infinite temperature, which are estimated using Ferrenberg and Swendsen's histogram method. A way to set up transfer-matrix calculations for random square-triangle tilings is also presented.

Random square-triangle tilings : a model for twelvefold-symmetric quasicrystals

A pair of optical cavities are designed and set up so as to be insensitive to both temperature fluctuations and mechanical vibrations. With the influence of these perturbations removed, a fundamental limit to the frequency stability of the optical cavity is revealed. The stability of a laser locked to the cavity reaches a floor $l2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}15}$ for averaging times in the range $0.5\ensuremath{-}100$ s. This limit is attributed to Brownian motion of the mirror substrates and coatings.

Thermal-noise-limited optical cavity

An optical cavity is designed and implemented that is insensitive to vibration in all directions. The cavity is mounted with its optical axis in the horizontal plane. A minimum response of 0.1 $(3.7)\phantom{\rule{0.3em}{0ex}}\mathrm{kHz}∕{\mathrm{ms}}^{\ensuremath{-}2}$ is achieved for low-frequency vertical (horizontal) vibrations.

Mark Oxborrow

Papers

Traceable 2-D Finite-Element Simulation of the Whispering-Gallery Modes of Axisymmetric Electromagnetic Resonators

Traceable 2D finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators

Random square-triangle tilings : a model for twelvefold-symmetric quasicrystals

Thermal-noise-limited optical cavity

Vibration insensitive optical cavity