We explore the potential of direct spectroscopy of ultra-narrow optical transitions of atoms localized in an optical cavity. In contrast to stabilization against a reference cavity, which is the approach currently used for the most highly stabilized
lasers, stabilization against an atomic transition does not suffer from Brownian thermal noise. Spectroscopy of ultra-narrow optical transitions in a cavity operates in a very highly saturated regime in which non-linear effects such as bistability play an important role. From the universal behavior of the Jaynes-Cummings model with dissipation, we derive the fundamental limits for laser stabilization using direct spectroscopy of ultra-narrow atomic lines. We find that with current lattice clock experiments, laser linewidths of about 1 mHz can be achieved in principle, and the ultimate limitations of this technique are at the 1 $mu$ Hz level.
Efforts to place limits on deviations from canonical formulations of electromagnetism and gravity have probed length scales increasing dramatically over time.Historically, these studies have passed through three stages: (1) Testing the power in the i
nverse-square laws of Newton and Coulomb, (2) Seeking a nonzero value for the rest mass of photon or graviton, (3) Considering more degrees of freedom, allowing mass while preserving explicit gauge or general-coordinate invariance. Since our previous review the lower limit on the photon Compton wavelength has improved by four orders of magnitude, to about one astronomical unit, and rapid current progress in astronomy makes further advance likely. For gravity there have been vigorous debates about even the concept of graviton rest mass. Meanwhile there are striking observations of astronomical motions that do not fit Einstein gravity with visible sources. Cold dark matter (slow, invisible classical particles) fits well at large scales. Modified Newtonian dynamics provides the best phenomenology at galactic scales. Satisfying this phenomenology is a requirement if dark matter, perhaps as invisible classical fields, could be correct here too. Dark energy {it might} be explained by a graviton-mass-like effect, with associated Compton wavelength comparable to the radius of the visible universe. We summarize significant mass limits in a table.
