Local PT invariance and supersymmetric parametric oscillators

We introduce the concept of local parity-time symmetric (PT) invariance in optical waveguides (or cavity) structures. Starting from a Lagrangian formalism, we establish the connection between light dynamics in these configurations and the seemingly different physics of supersymmetric parametric oscillators. Using this powerful tool, we present analytical solutions for optical beam propagation in local PT-invariant coupled systems and we show that the intensity tunneling between the two channels critically depends on the initial conditions. For unbalanced inputs, symmetric as well as asymmetric power evolution can be observed depending on the excitation channel. On the other hand, under certain physical conditions, our analysis predicts that for a modal PT-symmetric input, a unidirectional fractional phase exchange can take place. Few cases where analytical solutions cease to exist are also investigated numerically. Finally, by exploiting the supersymmetric nature of the oscillator equations, we show that under certain initial conditions, one can obtain the propagation dynamics of field amplitudes that resides on the supersymmetric eigenfunctions of the system-a phenomenon we call resonant propagation.