Adaptive optimal measurements under time uncertainty for optomechanical gravimetry

Investigate and develop adaptive measurement protocols—such as dynamically adjusting the local oscillator phase in homodyne detection based on prior outcomes—that can implement the Symmetric Logarithmic Derivative–optimal measurement and thereby attain or closely approach the quantum Cramér–Rao bound for estimating gravitational acceleration g in the cavity–optomechanical gravimeter defined by the Hamiltonian H_g (Eq. (HamGrav)), when time is treated as a nuisance parameter at generic evolution times.

Background

The paper derives the two-parameter quantum Fisher information (QFI) matrix treating time as a nuisance parameter and applies it to a cavity–optomechanical gravimeter. It shows that time uncertainty couples to the signal generator and degrades sensitivity except at special interrogation times, with a derived decoupling condition for continuous measurements.

In the homodyne detection analysis, the authors find that homodyne is optimal (classical Fisher information equals QFI) at stroboscopic times where time–signal coupling vanishes, but at generic times homodyne is not optimal. Although the SLD formalism guarantees that some measurement saturates the QFI, the practically realizable optimal measurement may be nontrivial and may require adaptive schemes. The authors explicitly state that exploring such adaptive strategies under time uncertainty remains open.