Supersymmetric Mechanism for Inversionless Property of n/cosh(t) Laser Pulse

It is known that a two-level atom subjected to a laser pulse of the form $V(t) = V_0/\cosh(\alpha t)$ has the rare property that, for a discrete series of pulse heights, it shows no transfer of population between the levels, for any detuning of the pulse from resonance.\footnote{V. M. Akulin, ``Coherent Dynamics of Complex Quantum Systems'' (Springer, Heidelberg, 2006).} In particular, pulses of the form $V(t) = \hbar \alpha n/\cosh(\alpha t)$ (with $n$ an integer) have this inversionless property. We show that the problem is analogous to reflectionless scattering in a stationary wave problem, and is linked to a potential-free Hamiltonian via a quantum-mechanical supersymmetric (QM SUSY) chain. We also explore the connection between this supersymmetric chain and the multi-soliton solutions of the sine-Gordon equation.