Cavity enhanced non-linear spectroscopy of ultra-narrow optical transitions

Optical atomic clocks are continuing to make rapid progress and are penetrating the 1 part in $10^{17}$ decade of fractional stability. A major bottleneck to further improvements of their stability is the linewidth of the clock lasers with which the atomic transitions are interrogated. The linewidth of these clock lasers is typically limited by thermal noise of the reference cavities which is technically challenging to overcome. Here we discuss an alternative laser stabilization scheme that is based on highly non-linear spectroscopy of ultra-narrow optical transitions in a cavity. We discuss the essential non-linear physics underlying this system and we show that laser linewidths in the 10 mHz range could be achieved with such a system using current experimental technology, an improvement of over an order of magnitude over the state-of-the-art. The fundamental limits of this approach are orders of magnitude below 1 mHz.