  We demonstrate a new method of light phase shift measurement using a
high-finesse optical ring cavity which exhibits reduced phase noise due to
cavity length fluctuations. Two laser beams with a frequency difference of one
cavity free spectral range are simultaneously resonant with the cavity,
demonstrating noise correlations in the error signals due to the common-mode
cavity length fluctuations. The differential error signal shows a 30 dB
reduction in cavity noise down to the noise floor in a frequency range up to
half the cavity linewidth ($\delta\nu/2 \simeq 30$ kHz). Various noise sources
are analyzed and their contributions to the noise floor are evaluated.
Additionally, we apply this noise-reduced phase shift measurement scheme in a
simulated spin-squeezing experiment where we have achieved a factor of 40
improvement in phase sensitivity with a phase resolution of 0.7 mrad, which may
remove one important barrier against attaining highly spin-squeezed states. The
demonstrated method is the first reported measurement using an optical ring
cavity and two independent beams, a flexible situation. This method can find
direct application to non-destructive measurements in quantum systems, such as
for the generation of spin-squeezed states in atom interferometers and atomic
clocks.
