Supersymmetric Quantum Mechanics and Solitons of the sine-Gordon and Nonlinear Schrödinger Equations

Andrew Koller, Maxim Olshanii

Published 2010-12-13, updated 2011-12-30Version 5

We present a case demonstrating the connection between supersymmetric quantum mechanics (SUSY--QM), reflectionless scattering, and soliton solutions of integrable partial differential equations. We show that the members of a class of reflectionless Hamiltonians, namely, Akulin's Hamiltonians, are connected via supersymmetric chains to a potential-free Hamiltonian, explaining their reflectionless nature. While the reflectionless property in question has been mentioned in the literature for over two decades, the enabling algebraic mechanism was previously unknown. Our results indicate that the multi-solition solutions of the sine-Gordon and nonlinear Schr\"{o}dinger equations can be systematically generated via the supersymmetric chains connecting Akulin's Hamiltonians. Our findings also explain a well-known but little-understood 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 $n$ being an integer and $\tau$ being the pulse duration, it remains in the ground state after the pulse has been applied, for {\it any} choice of the laser detuning.