Determining $G$ with Laser Spectroscopy to 38 ppb

A precision measurement is proposed to determine, in a couple hours of integration time, the axion Compton frequency using a modest power (3 mW) tunable external-cavity diode laser at 2458 nm as input to drive a free-space table-top Mach-Zehnder interferometer whose sensing arm passes the expanded beam-waist ($3~{\rm mm}$) light beam through a $1~{\rm T}$ strong, $40~{\rm cm}$ long dipole magnetic field created by a custom-built permanent-magnet assembly with a large but achievable ($6~{\rm mm}$) gap between poles. As the laser frequency is slowly modulated at 1 kHz through a 65 MHz wide window that is well within the 30 GHz fine-tuning range of the laser, a small but readily observable modulation appears in the dark-port optical power of the dark-fringe phase-locked interferometer due to photons converting into axions within the light beam as it passes through the magnetic field. Measuring the axion Compton frequency, $\nu_A\approx{\rm 122~THz}$, where the dark-port power modulation peaks, to within the line-width of the laser, $\Delta \nu_A=1~{\rm MHz}$, then determines $G$ to 38 ppb, a roughly 600-fold improvement, through a relation between $\nu_A$ and $G$, involving $h$, $c$, and nucleon masses.