The ultimate quantum limit to the linewidth of lasers

The standard quantum limit to the linewidth of a laser for which the gain medium can be adiabatically eliminated is $\ell_{0}=\kappa/2\bar{n}$. Here $\kappa$ is the intensity damping rate and $\bar{n}$ the mean photon number. This contains equal contributions from the loss and gain processes, so that simple arguments which attribute the linewidth wholly to phase noise from spontaneous gain are wrong. I show that an {\em unstimulated} gain process actually introduces no phase noise, so that the ultimate quantum limit to the laser linewidth comes from the loss alone and is equal to $\ell_{ult}= \kappa/4\bar{n}$. I investigate a number of physical gain mechanisms which attempt to achieve gain without phase noise: a linear atom-field coupling with finite interaction time; a nonlinear atom-field coupling; and adiabatic photon transfer using a counterintuitive pulse sequence. The first at best reaches the standard limit $\ell_{0}$, the second reaches $3/4 \ell_{0}$, while the third reaches the ultimate limit of $\ell_{ult} = 1/2 \ell_{0}$.