Figure 1

(a) Photon loss due to the piezoelectric effect at the interface. A photon traveling in the dielectric waveguide formed by a superconducting microstrip line and ground plane spontaneously decays into an acoustic phonon. The selection rules for photon to phonon conversion ensure the phonon propagates nearly perpendicular to the interface, with emission from the top and bottom interfaces interfering with each other. (b) Calculated $1/Q$ as a function of photon frequency $\mathrm{\Omega }$ for the microstrip line shown in (a). The metal is Al ($0.2\mu \mathrm{m}$ thick) and the dielectric and substrate is sapphire (${\mathrm{Al}}_2{\mathrm{O}}_3$), with $W=20\mu \mathrm{m}$ and $d=2\mu \mathrm{m}$. We used the participation ratios calculated in [25], $f_{\mathrm{MV}}=6.5\times {10}^{-6}$, $f_{\mathrm{DV}}=2.9\times {10}^{-4}$, and $f_{\mathrm{DM}}=2.9\times {10}^{-3}$. The loss is oscillatory as a function of frequency, making evident the presence of phonon interference. These oscillations can be used to distinguish the piezoelectric effect from other sources of loss.