Optimal sensing on an asymmetric exceptional surface

We study the connection between exceptional points (EPs) and optimal parameter estimation by considering a simple system consisting of two counter-propagating traveling wave modes in a micro-resonator. The unknown parameter of interest is the strength of a perturbing cross-coupling between the two modes. Partially retro-reflecting the output of one mode into the other creates a non-Hermitian Hamiltonian which exhibits a family of EPs, creating an exceptional surface (ES). We make a fully quantum treatment of the field inputs and their respective noise sources to quantitatively bound the estimation error, by calculating the quantum Fisher information (QFI) in the output fields. We find that for both a semi-classical coherent state and a highly non-classical NOON state, the QFI is enhanced in the presence of an EP. For each type of input state, we construct experimentally feasible measurements, which achieve the quantum Cramer Rao lower bound for the estimation error.