Noise considerations for locking to the center of a Lorentzian line

Many areas of science require locking a probe signal to the center of a Lorentzian line. In this paper we experimentally explore the limits to the resolution for locking a sine-wave modulated probe signal to a Lorentzian line. Surprisingly we find that in some cases relatively high-frequency phase modulation (PM) noise on the probe signal (even 100 times the linewidth of the resonance) can significantly degrade the stability of the lock. The 6.835-GHz hyperfine transition in ${}^{87}\mathrm{Rb}$ was used as a test bed for these studies because it affords very high resolution for investigating this effect. The details presented here should prove useful for testing a complete theory of this effect and give some insight into the requirements of PM noise of local oscillators to be locked to a Lorentzian line.