Adaptive estimation of a time-varying phase with a power-law spectrum via continuous squeezed states

When measuring a time-varying phase, the standard quantum limit and Heisenberg limit as usually defined, for a constant phase, do not apply. If the phase has Gaussian statistics and a power-law spectrum $1/|\omega|^p$ with $p>1$, then the generalized standard quantum limit and Heisenberg limit have recently been found to have scalings of $1/{\cal N}^{(p-1)/p}$ and $1/{\cal N}^{2(p-1)/(p+1)}$, respectively, where ${\cal N}$ is the mean photon flux. We show that this Heisenberg scaling can be achieved via adaptive measurements on squeezed states. We predict the experimental parameters analytically, and test them with numerical simulations. Previous work had considered the special case of $p=2$.