Two-mode bosonic quantum metrology with number fluctuations

We search for the optimal quantum pure states of identical bosonic particles for applications in quantum metrology, in particular, in the estimation of a single parameter for the generic two-mode interferometric setup. We consider the general case in which the total number of particles is fluctuating around an average N with variance Delta N-2. By recasting the problem in the framework of classical probability, we clarify themaximal accuracy attainable and show that it is always larger than the one reachable with a fixed number of particles (i.e., Delta N = 0). In particular, for larger fluctuations, the error in the estimation diminishes proportionally to 1/Delta N, below the Heisenberg-like scaling 1/N. We also clarify the best input state, which is a quasi-NOON state for a generic setup and, for some special cases, a two-mode Schrodinger-cat state with a vacuum component. In addition, we search for the best state within the class of pure Gaussian states with a given average N, which is revealed to be a product state (with no entanglement) with a squeezed vacuum in one mode and the vacuum in the other.