Green's Functions and the Adiabatic Hyperspherical Method

We address the few-body problem using the adiabatic hyperspherical representation. A general form for the hyperangular Green's function in $d$-dimensions is derived. The resulting Lippmann-Schwinger equation is solved for the case of three-particles with s-wave zero-range interactions. Identical particle symmetry is incorporated in a general and intuitive way. Complete semi-analytic expressions for the nonadiabatic channel couplings are derived. Finally, a model to describe the atom-loss due to three-body recombination for a three-component fermi-gas of $^{6}$Li atoms is presented.Comment: 14 pages, 8 figures, 2 table

Three-Body Recombination in One Dimension

We study the three-body problem in one dimension for both zero and finite range interactions using the adiabatic hyperspherical approach. Particular emphasis is placed on the threshold laws for recombination, which are derived for all combinations of the parity and exchange symmetries. For bosons, we provide a numerical demonstration of several universal features that appear in the three-body system, and discuss how certain universal features in three dimensions are different in one dimension. We show that the probability for inelastic processes vanishes as the range of the pair-wise interaction is taken to zero and demonstrate numerically that the recombination threshold law manifests itself for large scattering length.Comment: 15 pages 7 figures Submitted to Physical Review

Ultrafast optical control using the Kerr nonlinearity in hydrogenated amorphous silicon microcylindrical resonators

Microresonators are ideal systems for probing nonlinear phenomena at low thresholds due to their small mode volumes and high quality (Q) factors. As such, they have found use both for fundamental studies of light-matter interactions as well as for applications in areas ranging from telecommunications to medicine. In particular, semiconductor-based resonators with large Kerr nonlinearities have great potential for high speed, low power all-optical processing. Here we present experiments to characterize the size of the Kerr induced resonance wavelength shifting in a hydrogenated amorphous silicon resonator and demonstrate its potential for ultrafast all-optical modulation and switching. Large wavelength shifts are observed for low pump powers due to the high nonlinearity of the amorphous silicon material and the strong mode confinement in the microcylindrical resonator. The threshold energy for switching is less than a picojoule, representing a significant step towards advantageous low power silicon-based photonic technologies

Scattering of two particles in a one-dimensional lattice

This study concerns the two-body scattering of particles in a one-dimensional periodic potential. A convenient ansatz allows for the separation of center-of-mass and relative motion, leading to a discrete Schrodinger equation in the relative motion that resembles a tight-binding model. A lattice Green's function is used to develop the Lippmann-Schwinger equation, and ultimately derive a multiband scattering K matrix which is described in detail in the two-band approximation. Two distinct scattering lengths are defined according to the limits of zero relative quasimomentum at the top and bottom edges of the two-body collision band. Scattering resonances occur in the collision band when the energy is coincident with a bound state attached to another higher or lower band. Notably, repulsive on-site interactions in an energetically closed lower band lead to collision resonances in an excited band

Effect of core size on nonlinear transmission in silicon optical fibers

The nonlinear transmission properties of two hydrogenated amorphous silicon fibers with core diameters of 5.7µm and 1.7µm are characterized. The measured Kerr nonlinearity, two-photon absorption and free-carrier parameters will be discussed in relation to device performance

Large Deviations of Extreme Eigenvalues of Random Matrices

We calculate analytically the probability of large deviations from its mean of the largest (smallest) eigenvalue of random matrices belonging to the Gaussian orthogonal, unitary and symplectic ensembles. In particular, we show that the probability that all the eigenvalues of an (N\times N) random matrix are positive (negative) decreases for large N as \exp[-\beta \theta(0) N^2] where the parameter \beta characterizes the ensemble and the exponent \theta(0)=(\ln 3)/4=0.274653... is universal. We also calculate exactly the average density of states in matrices whose eigenvalues are restricted to be larger than a fixed number \zeta, thus generalizing the celebrated Wigner semi-circle law. The density of states generically exhibits an inverse square-root singularity at \zeta.Comment: 4 pages Revtex, 4 .eps figures included, typos corrected, published versio

On Nori's Fundamental Group Scheme

We determine the quotient category which is the representation category of the kernel of the homomorphism from Nori's fundamental group scheme to its \'etale and local parts. Pierre Deligne pointed out an error in the first version of this article. We profoundly thank him, in particular for sending us his enlightning example reproduced in Remark 2.4 2).Comment: 29 page

Towards in-fiber silicon photonics

The state of the art of silicon optical fibers fabricated via the high pressure chemical deposition technique will be reviewed. The optical transmission properties of step index silicon optical fibers will be presented, including investigations of the nonlinearities that can be used for all-optical signal processing. In addition, alternative complex fiber geometries that permit sophisticated control of the propagating light will be introduced