  Experiments near the lock-in region in maximally dissipative non-Hermitian
systems, e.g., conventional laser gyroscopes near the deadband, have run up
against the Petermann limit, where excess noise exactly cancels any
scale-factor enhancement resulting in no overall enhancement in precision. As a
result, one might be tempted to conclude that exceptional points (EPs)
generally cannot be used to increase the precision of laser sensors. Indeed,
using a linear eigenmode analysis we show that the Petermann limit applies not
just to maximally dissipative systems, but for any type of EP, owing to the
fact that EPs are rotationally invariant. It turns out, however, that this
restriction comes from the assumption of linearity. We find that nonlinearity
breaks the rotation symmetry such that the different types of EPs are no longer
equivalent above threshold. In particular, for sufficiently high measurement
frequencies, EPs in conservatively coupled systems having a saturation
imbalance can lead to an increase in the fundamental precision beyond the
Petermann limit. Importantly, we find that only one mode lases under these
conditions. We show that the beat note can be recovered by interference with an
auxiliary mode, but that this has consequences for the quantum and classical
noise that depend on the recovery scheme. Thus, it remains to be seen whether
practical experiments can be designed that can take advantage of this
enhancement.
