We analyze the dynamics of spin-mixing interactions generated by coupling spin-1 atoms to the mode of a high-finesse optical cavity. We show that the dynamics can be understood in terms of generators of the noncompact Lie group SU(1, 1) and introduce a set of SU(1, 1) coherent states which are preserved under Hamiltonian evolution. In terms of these coherent states the resulting dynamics may be interpreted as classical motion on the unit disk. We explicitly compute the trajectories of this classical motion and show that the motion is equivalent to spin-nematic squeezing in the atomic ensemble. Non-uniform coupling between the atomic ensemble and the cavity mode leads to departures from this simple behavior; we introduce a toy model that captures this non-uniformity and solve it exactly.
Proc. SPIE. 10118, Advances in Photonics of Quantum Computing, Memory, and Communication X
KEYWORDS: Signal to noise ratio, Metrology, Chemical species, Spectroscopy, Particles, Interference (communication), Optical resonators, Signal detection, Tolerancing, Entangled states, Quantum information
Detection noise poses a challenge for achieving Heisenberg-limited phase estimation. We discuss a "twisting echo" protocol<sup>1</sup> that addresses this problem by using interactions to amplify a spectroscopic signal. The echo protocol enables phase sensitivity near the Heisenberg limit while permitting detection noise on the order of the quantum noise of an unentangled state. For comparison with conventional schemes requiring direct detection of entangled states, we calculate the dependence of metrological gain on detection noise in Ramsey spectroscopy with squeezed, twin Fock, and GHZ states. The twisting echo outperforms all of these alternatives if the detection uncertainty is at or above the single-atom level.