Foliated Lie systems: Theory and applications

A $\mathcal{F}$- foliated Lie system is a first-order system of ordinary differential equations whose particular solutions are contained in the leaves of the foliation $\mathcal{F}$ and all particular solutions within any leaf can be written as a certain function, a so-called foliated superposition rule, of a family of particular solutions of the system within the same leaf and several parameters. We analyse the properties of such systems and we illustrate our results by studying Lax pairs and a class of $t$-dependent Hamiltonian systems. We finally study foliated Lie systems through Poisson structures and $r$-matrices.Comment: 24 page

Use of cohesive elements in fatigue analysis

Cohesive laws describe the resistance to incipient separation of material surfaces. A cohesive finite element is formulated on the basis of a particular cohesive law. Cohesive elements are placed at the boundary between adjacent standard volume finite elements to model fatigue damage that leads to fracture at the separation of the element boundaries per the cohesive law. In this work, a cohesive model for fatigue crack initiation is taken to be the irreversible loadingunloading hysteresis that represents fatigue damage occuring due to cyclic loads leading to the initiation of small cracks. Various cohesive laws are reviewed and one is selected that incorporates a hysteretic cyclic loading that accounts for energetic dissipative mechanisms. A mathematical representation is developed based on an exponential effective load-separation cohesive relationship. A three-dimensional cohesive element is defined using this compliance relationship integrated at four points on the mid-surface of the area element. Implementation into finite element software is discussed and particular attention is applied to numerical convergence issues as the inflection point between loading and 'unloading in the cohesive law is encountered. A simple example of a displacementcontrolled fatigue test is presented in a finite element simulation. Comments are made on applications of the method to prediction of fatigue life for engineering structures such as pressure vessels and piping

Injection locking of two frequency-doubled lasers with 3.2 GHz offset for driving Raman transitions with low photon scattering in 43^{43}Ca+^+

We describe the injection locking of two infrared (794 nm) laser diodes which are each part of a frequency-doubled laser system. An acousto-optic modulator (AOM) in the injection path gives an offset of 1.6 GHz between the lasers for driving Raman transitions between states in the hyperfine split (by 3.2 GHz) ground level of $^{43}$Ca$^+$. The offset can be disabled for use in $^{40}$Ca$^+$. We measure the relative linewidth of the frequency-doubled beams to be 42 mHz in an optical heterodyne measurement. The use of both injection locking and frequency doubling combines spectral purity with high optical power. Our scheme is applicable for providing Raman beams across other ion species and neutral atoms where coherent optical manipulation is required.Comment: 3 pages, 3 figure

Exact Relations for a Strongly-interacting Fermi Gas from the Operator Product Expansion

The momentum distribution in a Fermi gas with two spin states and a large scattering length has a tail that falls off like 1/k^4 at large momentum k, as pointed out by Shina Tan. He used novel methods to derive exact relations between the coefficient of the tail in the momentum distribution and various other properties of the system. We present simple derivations of these relations using the operator product expansion for quantum fields. We identify the coefficient as the integral over space of the expectation value of a local operator that measures the density of pairs.Comment: 4 pages, 2 figure

Exact Relations for a Strongly-interacting Fermi Gas near a Feshbach Resonance

A set of universal relations between various properties of any few-body or many-body system consisting of fermions with two spin states and a large but finite scattering length have been derived by Shina Tan. We derive generalizations of the Tan relations for a two-channel model for fermions near a Feshbach resonance that includes a molecular state whose detuning energy controls the scattering length. We use quantum field theory methods, including renormalization and the operator product expansion, to derive these relations. They reduce to the Tan relations as the scattering length is made increasingly large.Comment: 25 pages, 8 figure

Experimental recovery of a qubit from partial collapse

We describe and implement a method to restore the state of a single qubit, in principle perfectly, after it has partially collapsed. The method resembles the classical Hahn spin-echo, but works on a wider class of relaxation processes, in which the quantum state partially leaves the computational Hilbert space. It is not guaranteed to work every time, but successful outcomes are heralded. We demonstrate using a single trapped ion better performance from this recovery method than can be obtained employing projection and post-selection alone. The demonstration features a novel qubit implementation that permits both partial collapse and coherent manipulations with high fidelity.Comment: 5 pages, 3 figure