Transverse quasi-modes in periodic potentials

We discuss how to find the quasi-modes of the Schrödinger equation when the potential is periodic in time. Our method confirms that the profile of the time-independent (continuous) component of the wavefunction obeys an effective Schrödinger equation where the potential is given by the Kapitza term plus the temporal mean of the original potential, a result originally found by Rahav et al (2003 Phys. Rev. Lett. 91 110404). We then find closed-form expression for the higher order corrections to the quasi-modes, showing how the generic quasi-mode undergoes periodic temporal oscillations and a non-flat phase profile. Validity of our theoretical results is verified against full numerical simulations of the Schrödinger equation. Our findings can be applied both to quantum mechanics and light propagation in the paraxial regime. (paper)