Real-time Kalman filter: Cooling of an optically levitated nanoparticle

We demonstrate that a Kalman filter applied to estimate the position of an optically levitated nanoparticle, and operated in real-time within a field programmable gate array, is sufficient to perform closed-loop parametric feedback cooling of the center-of-mass motion to sub-Kelvin temperatures. The translational center-of-mass motion along the optical axis of the trapped nanoparticle has been cooled by 3 orders of magnitude, from a temperature of 300 K to a temperature of $162\pm 15$ mK.

Figure 1

The three-dimensional position of the particle is detected by interference of the scattered and divergent field by the photodetector. This signal is then passed into an FPGA which is implementing the Kalman filter that estimates the $z$ position and amplifies the estimate. This estimated signal is then sent into a second FPGA which dc shifts it, frequency doubles it, applies a time delay, and multiplies it so as to keep the amplitude approximately constant. This signal is then fed to an AOM to modulate the power of the trapping lasers. The erbium-doped fiber amplifier (EDFA) then amplifies this modulated signal to the power required to trap the nanoparticle; this signal is then used to trap and measure the nanoparticle.

Figure 2

(a) Time trace of unfiltered measured signal from the particle and filtered $z$ signal over 0.5 ms. (b) Time trace of filtered $z$ signal and Kalman-filter-estimated $z$ signal over 0.5 ms. (c) Power spectral density of measured signal from the particle and the Kalman-filtered estimate. The peaks at 100 and 120 kHz are due to the $x$ motion and the $y$ motion, respectively. (d) Same as panel (c) but over a smaller frequency range centered on the $z$ frequency of 38 kHz.

Figure 3

The PSD of the uncooled particle in equilibrium with the environment at 300 K (at a pressure of 3 mbar) and the cooled particle at a pressure of $5.7\times {10}^{-5}$ mbar. The lines represent the Lorentzian functions fitted to the PSD data to calculate the temperature of the cooled state.