Wannier–Stark resonances in optical and semiconductor superlattices

In this work, we discuss the resonance states of a quantum particle in a periodic potential plus a static force. Originally, this problem was formulated for a crystal electron subject to a static electric field and it is nowadays known as the Wannier–Stark problem. We describe a novel approach to the Wannier–Stark problem developed in recent years. This approach allows to compute the complex energy spectrum of a Wannier–Stark system as the poles of a rigorously constructed scattering matrix and solves the Wannier–Stark problem without any approximation. The suggested method is very efficient from the numerical point of view and has proven to be a powerful analytic tool for Wannier–Stark resonances appearing in different physical systems such as optical lattices or semiconductor superlattices.