Supersymmetric Origins of the Properties of sech-Pulses and sine-Gordon Solitons

In this thesis, we show that the members of a class of reflectionless Hamiltonians, namely, Akulin\u27s Hamiltonians, are connected via a supersymmetric (SUSY) chain. While the reflectionless property in question (vanishing reflection coefficients at all values of the spectral parameter, e.g. energy) has been mentioned in the literature for over two decades, the enabling algebraic mechanism was previously unknown. We show that the supersymmetric connection of the Akulin\u27s Hamiltonians to a potential-free Hamiltonian is the origin of this property. As the first application for our findings, we show that the SUSY decomposition of Akulin\u27s Hamiltonians explains a well-known effect in laser physics: when a two-level atom, initially in the ground state, is subjected to a laser pulse of the form V(t) = (n&hbar/&tau)/cosh(t/&tau), with n an integer and &tau the pulse duration, it remains in the ground state after the pulse has been applied, for any choice of the laser detuning. The second application concerns the sine-Gordon equation: we demonstrate that the first member of the Akulin\u27s chain is related to the L-operator of the Lax pair for the one-soliton solution of the sine-Gordon equation: its reflectionless nature is now explained by supersymmetry

The Online Revolution: Education at Scale

PDF of presentation used by Andrew Ng of Stanford University and Coursera for his 23 July 2012 Invited Talk "Teaching Machine Learning to 100,000 Students" at the 2012 conference of the American Association for the Advancement of Artificial Intelligence - http://www.webcitation.org/69jee6a8D . Published by kind permission of Andrew Ng. Note that a June 2012 TED talk by Daphne Koller - http://goo.gl/IRJqr - covers similar but different ground

Realizing Exactly Solvable SU(N) Magnets with Thermal Atoms

We show that $n$ thermal fermionic alkaline-earth atoms in a flat-bottom trap allow one to robustly implement a spin model displaying two symmetries: the $S_n$ symmetry that permutes atoms occupying different vibrational levels of the trap and the SU($N$) symmetry associated with $N$ nuclear spin states. The high symmetry makes the model exactly solvable, which, in turn, enables the analytic study of dynamical processes such as spin diffusion in this SU($N$) system. We also show how to use this system to generate entangled states that allow for Heisenberg-limited metrology. This highly symmetric spin model should be experimentally realizable even when the vibrational levels are occupied according to a high-temperature thermal or an arbitrary non-thermal distribution.Comment: 12 pages, 5 figures (including supplemental materials