Proposal for a new test of the time independence of the fine structure constant α using orthogonally polarized whispering gallery modes in a single sapphire resonator

A new experiment to test for the time independence of the fine structure constant α is proposed. The experiment utilizes orthogonally polarized transverse electric and transverse magnetic whispering gallery modes in a single sapphire resonator tuned to similar frequencies. When configured as a dual mode sapphire clock, we show that the anisotropy of sapphire makes it is possible to undertake a sensitive measurement from the beat frequency between the two modes. At infrared frequencies this is possible due to the different effect of the lowest phonon frequency on the two orthogonally polarized modes. At microwave frequencies we show that the phonon effect is too small. We show that the electron spin resonance of paramagnetic impurities (such as ${\mathrm{Cr}}^{3+})$ in the lattice affects only one polarization with an ${\alpha }^6$ dependence. This enables an enhancement of the sensitivity to temporal changes in α at microwave frequencies.