Single-port homodyne detection achieves optimal multiphase sensing in an $n$-port ring cavity

Abstract

Estimating several phases simultaneously is more than estimating them one at a time: the optimal measurements for different parameters may not commute, and in a multiparameter problem the quantum Cramér-Rao bound is not automatically achievable. Ring cavities are a natural setting for this question, since a single coherent input can be made to interact repeatedly with every internal phase before it is read out. We study the simultaneous local estimation of the $n$ internal phases of a lossless $n$-port ring cavity probed by a fixed-energy multimode coherent state, optimizing over the full coherent-input space and over the operating point of the cavity. We identify a necessary and sufficient condition for the global optimum: the input energy should be distributed equally among the ports, and the internal phases and input phases satisfy a simple phase-matching condition for every port. The optimum is unique up to a common optical phase and minimizes both the symmetric logarithmic derivative (SLD) cost and the sloppiness of the model. At this point the quantum Fisher information matrix is full rank and its spectrum is obtained in closed form, the Uhlmann curvature vanishes, and no incompatibility penalty has to be paid. The precision retains the coherent-state scaling and the dependence on the number of phases, but the cavity feedback reduces the prefactor relative to $n$ independent single-pass experiments. Finally, we show that this bound may be achieved in practice: the Fisher information of a balanced homodyne measurement at each output port, with local-oscillator phases locked to the prepared input phases is equal to the optimal quantum Fisher information. Our results establish the ring cavity as a platform for optimal multiphase sensing and provide the ultimate benchmark against which nonclassical probes may be evaluated for further precision enhancement.