States for phase estimation in quantum interferometry

Abstract

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