  In traditional quantum optics, where the interaction between atoms and light
at optical frequencies is studied, the atoms can be approximated as point-like
when compared to the wavelength of light. So far, this relation has also been
true for artificial atoms made out of superconducting circuits or quantum dots,
interacting with microwave radiation. However, recent and ongoing experiments
using surface acoustic waves show that a single artificial atom can be coupled
to a bosonic field at several points wavelengths apart. Here, we theoretically
study this type of system. We find that the multiple coupling points give rise
to a frequency dependence in the coupling strength between the atom and its
environment, and also in the Lamb shift of the atom. The frequency dependence
is given by the discrete Fourier transform of the coupling point coordinates
and can therefore be designed. We discuss a number of possible applications for
this phenomenon, including tunable coupling, single-atom lasing, and other
effects that can be achieved by designing the relative coupling strengths of
different transitions in a multi-level atom.
