Scaling laws for precision in quantum interferometry and the bifurcation landscape of the optimal state

Phase precision in optimal two-channel quantum interferometry is studied in the limit of large photon number N >> 1, for losses occurring in either one or both channels. For losses in one channel an optimal state undergoes an intriguing sequence of local bifurcations as the number of photons (or losses) increase. The optimal state has a continuous form in the Fock state basis for large N. The loss parameter limits any precision improvement over classical light to at most a constant factor independent of N. We determine a crossover value of photon number N-c beyond which supraclassical precision is progressively lost.