Noise-aware derivative feedback of a harmonic oscillator under quantum-limited position measurement

Abstract

We study measurement-based feedback control of a damped harmonic oscillator, motivated by optomechanical and interferometric position measurements, in which the feedback force is synthesized directly from a displacement measurement. For a band-limited derivative (“dirty-derivative”) controller, the feedback loop simultaneously shapes the mechanical susceptibility and reinjects measurement noise back onto the oscillator, creating a nontrivial tradeoff between stabilization and noise amplification. We first present a compact classical formulation that isolates this tradeoff in the closed-loop displacement spectrum and identifies the controller parameters that determine resonance suppression versus noise-driven actuation. We then extend the same topology to a quantum-limited position detector, enforcing the imprecision–backaction constraint and introducing a finite measurement bandwidth that both regularizes high-frequency noise reinjection and adds phase lag in the feedback path. In the high-Q regime, we obtain a compact analytic design rule for the optimal feedback gain at fixed controller and measurement bandwidths and compare it directly with full-spectrum calculations. The comparison shows that a high mechanical quality factor alone does not guarantee quantitative agreement: the accuracy of the near-resonant approximation also depends strongly on the controller and measurement bandwidths. Because finite measurement bandwidth changes both the magnitude and phase of the feedback loop, the analysis separates the near-resonant damping contribution from imprecision-noise reinjection and shows that positive near-resonant feedback damping requires the product of the controller cutoff and measurement bandwidth to exceed the square of the mechanical resonance frequency. Finally, we present a dimensionless map of the gain-optimized high-Q occupation and compare its predictions with full-spectrum optimization. All analytical gains evaluated in this map satisfy the full closed-loop stability criterion for the stated parameters.