Observation of Lamb shift in an artificial atom caused by coupling to the phonon vacuum

Recently, it has been shown that surface acoustic waves (SAWs) can be coupled to an artificial atom in the form of a transmon qubit. We have experimentally measured the response of such an artificial atom placed on a piezoelectric lithium niobate substrate. The artificial atom has an interdigitated capacitance which gives multiple coupling points to SAW resulting in a strong frequency dependent coupling to the phononic vacuum. This coupling results in a non-monotonic Lamb shift due to causality, via the Kramers-Kronig relation. We observe a frequency dependent Lamb shift and corresponding coupling variation which agree well with theory.