Date of Award
Open Access Thesis
Master of Science (MS)
In this thesis, we show that the members of a class of reflectionless Hamiltonians, namely, Akulin's 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's 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's 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's 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.
Koller, Andrew Phillip, "Supersymmetric Origins of the Properties of sech-Pulses and sine-Gordon Solitons" (2011). Graduate Masters Theses. 53.