S-matrix theory of single-channel ballistic transport through coupled quantum dots

We consider single-channel transmission through a double quantum dot system that consists of two single dots coupled by a wire of finite length L. In order to explain the numerically obtained results for a realistic double dot system we explore a simple model. It consists, as the realistic system, of two dots connected by a wire of length L. However, each of the two single dots is characterized by a few energy levels only, and the wire is assumed to have only one level whose energy depends on L. The transmission is described by using S-matrix theory. The model explains in particular the splitting of the resonant transmission peaks and the origin of the transmission zeros. The latter are independent of the length of the wire. When the transmission zeros of the single dots are of first order and both single dots are identical, those of the double dot are of second order. First-order transmission zeros cause phase jumps of the transmission amplitude by $\pi$, while there are no phase jumps related to second-order transmission zeros. In this latter case, a phase jump appears due to a resonance state whose decay width vanishes when crossing the energy of the transmission zero.