We report the use of novel, capacitively terminated coplanar waveguide (CPW)
resonators to measure the quadratic Stark shift of phosphorus donor qubits in
Si. We confirm that valley repopulation leads to an anisotropic spin-orbit
Stark shift depending on electric and magnetic field orientations relative to
the Si crystal. By measuring the linear Stark effect, we estimate the effective
electric field due to strain in our samples. We show that in the presence of
this strain, electric-field sources of decoherence can be non-negligible. Using
our measured values for the Stark shift, we predict magnetic fields for which
the spin-orbit Stark effect cancels the hyperfine Stark effect, suppressing
decoherence from electric-field noise. We discuss the limitations of these
noise-suppression points due to random distributions of strain and propose a
method for overcoming them
