States for phase estimation in quantum interferometry

Abstract

Ramsey interferometry allows the estimation of the phase φ of rotation of the pseudospin vector of an ensemble of two-state quantum systems. For φ small, the noise-to-signal ratio scales as the spin-squeezing parameter ξ, with ξ < 1 possible for an entangled ensemble. However states with minimum ξ are not optimal for single-shot measurements of an arbitrary phase. We define a phase-squeezing parameter, ζ, which is an appropriate figure of merit for this case. We show that (unlike the states that minimize ξ) the states that minimize can be created by evolving an unentangled state (coherent spin state) by the well known two-axis counter-twisting Hamiltonian. We analyse these and other states (for example the maximally entangled state, analogous to the optical 'NOON' state |ψ) = (|N, 0) + |0, N))/√2) using several different properties, including ξ, <, the coefficients in the pseudo-angular momentum basis (in the three primary directions) and the angular Wigner function W(θ,Φ). Finally, we discuss the experimental options for creating phase-squeezed states and doing single-shot phase estimation.