FPGA Implementation of a Kalman-Based Motion Estimator for Levitated Nanoparticles

Optically trapped nanoparticles are used in various fields ranging from biophysics to precision sensing. An optically trapped nanoparticle can be regarded as a harmonic oscillator driven by the thermal fluctuations of its environment. Unlocking the potential of optically levitated systems for precision measurements in the classical and the quantum regime requires cooling of the particle motion. In parametric feedback cooling, the center-of-mass motion of a nanoparticle optically levitated in a vacuum is reduced by temporally modulating the optical trapping potential. This technique requires a precise measurement of the particle's motion to derive the feedback signal and is prone to measurement noise and inevitable thermal process noise. In a state-of-the-art implementation, the feedback signal is derived from a simple phase-locked loop (PLL). Kalman filters are regularly deployed in a variety of application scenarios to improve system performance under noisy conditions. In this paper, we investigate theoretically and experimentally the performance of parametric feedback cooling, where the measurement signal is Kalman filtered before entering the PLL. Compared to sole PLL cooling, our numerical full-system simulations show a 20% reduction of the residual motional energy of a trapped nanoparticle in the presence of the Kalman filter. We detail a field-programmable gate array-based implementation of a Kalman filter and evaluate its performance in-field.