Globally optimal interferometry with lossy twin Fock probes

Abstract

Parity or quadratic spin (e.g., Jz2) readouts of a Mach–Zehnder (MZ) interferometer probed with a twin Fock (TF) input state allow saturating the optimal sensitivity attainable among all mode-separable states with a fixed total number of particles but only when the interferometer phase θ is near zero. When more general Dicke state probes are used, the parity readout saturates the quantum Fisher information (QFI) at θ = 0, whereas better-than-standard quantum limit performance of the Jz2 readout is restricted to an o(N) occupation imbalance. We show that a method of moments readout of two quadratic spin observables Jz2 and J+2+J−2 is globally optimal for Dicke state probes; i.e., the error saturates the QFI for all θ. In the lossy setting, we derive the time-inhomogeneous Markov process describing the effect of particle loss on TF states, showing that the method of moments readout of four at-most-quadratic spin observables is sufficient for globally optimal estimation of θ when two or more particles are lost. The analysis culminates in a numerical calculation of the QFI matrix for distributed MZ interferometry on the four-mode state |N4,N4,N4,N4〉 and its lossy counterparts, showing that an advantage for the estimation of any linear function of the local MZ phases θ1 and θ2 (compared to independent probing of the MZ phases by two copies of |N4,N4〉) appears when more than one particle is lost.