Criticality in two-mode interferometers

Abstract

To achieve the optimal phase sensitivity in two-mode interferometers, one key problem is how to distribute photons between the two input ports, given a fixed total average photon number. In both the $\text{SU}(2)$ and $\text{SU}(1,1)$ interferometers with a product of coherent and Fock states as inputs, we find there exists a critical value of the total photon number, below which the optimal phase sensitivity can be achieved with the second port being the vacuum state. The critical points can be analytically fixed. We further consider a more general input state as a product of coherent and squeezed Fock states. Then we analytically identify three regions in the parameter space, similar to a phase diagram, and find that critical phenomena occur in two regions. If the total average input photon number is less than the critical value, the optimal distribution tends to minimize the average photon number in the second port. Our finding indicates that, in order to get interferometers with high sensitivity, one must carefully choose the photon number distribution according to the value of the input total average photon number. This central result will be useful in theoretical studies in the field of quantum parameter estimation.