Analytically exploiting noise correlations inside the feedback loop to improve locked-oscillator performance

We introduce concepts from optimal estimation to the stabilization of precision frequency standards limited by noisy local oscillators. We develop a theoretical framework casting various measures for frequency standard variance in terms of frequency-domain transfer functions, capturing the effects of feedback stabilization via a time series of Ramsey measurements. Using this framework, we introduce an optimized hybrid predictive feedforward measurement protocol that employs results from multiple past measurements and transfer-function-based calculations of measurement covariance to improve the accuracy of corrections within the feedback loop. In the presence of common non-Markovian noise processes these measurements will be correlated in a calculable manner, providing a means to capture the stochastic evolution of the local oscillator frequency during the measurement cycle. We present analytic calculations and numerical simulations of oscillator performance under competing feedback schemes and demonstrate benefits in both correction accuracy and long-term oscillator stability using hybrid feedforward. Simulations verify that in the presence of uncompensated dead time and noise with significant spectral weight near the inverse cycle time predictive feedforward outperforms traditional feedback, providing a path towards developing a class of stabilization software routines for frequency standards limited by noisy local oscillators.