Optimal Feedback Cooling of a Charged Levitated Nanoparticle with Adaptive Control

We use an optimal control protocol to cool one mode of the center-of-mass motion of an optically levitated nanoparticle. The feedback technique relies on exerting a Coulomb force on a charged particle with a pair of electrodes and follows the control law of a linear quadratic regulator, whose gains are optimized by a machine learning algorithm in under 5 s. With a simpler and more robust setup than optical feedback schemes, we achieve a minimum center-of-mass temperature of 5 mK at $3\times {10}^{-7}\mathrm{mbar}$ and transients 10–600 times faster than cold damping. This cooling technique can be easily extended to 3D cooling and is particularly relevant for studies demanding high repetition rates and force sensing experiments with levitated objects.

Figure 1

Experimental setup. (a) Picture of the setup inside the vacuum chamber. The purple glow on the right is a plasma generated to control the particle’s charge. (b) Close-up image of the trapping region with a simulation of the electric field for an applied dc voltage of 1 V. (c) Setup sketch. A microscope objective (OBJ) is used to trap a silica nanoparticle. The scattered light is collected and sent to a balanced photodiode to detect the motion. The $x$-motion signal is bandpass filtered (BPF) and sent to the FPGA, where the LQR signal is calculated. The output is low-pass filtered (LPF) and sent to the electrodes to cool the motion electrically. The LQR parameters are controlled with a ML routine that runs on the board CPU. PFC is used to modulate the laser light with an electrooptic modulator (EOM) and cool the two other axes during all the measurements.

Figure 2

Cold damping: $k_p=0$ and varying $k_d$. (a) PSD of the motion in the $x$ direction recorded from the IL detector signal at various gains for a particle at $p={10}^{-5}\mathrm{mbar}$. For large $k_d$, $S_x(\omega )$ shows noise squashing. (b) PSD of the motion in the $x$ direction simultaneously recorded with the OOL detector. Noise squashing is avoided. (c) Minimum temperature achieved at different pressures for the optimal $k_d$ (OOL detector). The lowest temperature, $T_{\mathrm{eff}}=5.1\pm 0.5\mathrm{mK}$, is reached at $3\times {10}^{-7}\mathrm{mbar}$. (d) Temperature vs $k_d$ gain at $3\times {10}^{-7}\mathrm{mbar}$, displaying an optimal gain minimum at $k_d\simeq 700\mathrm{rad}/\mathrm{s}$, where $T_{\mathrm{eff}}=5.1\pm 0.5\mathrm{mK}$ (OOL detector). The qualitative behavior (see Supplemental Material [33]) is in accordance with data, with optimal $k_d$ a factor of 3.4 larger than measured.

Figure 3

ML algorithm. (a) ML routine adapting to the optimal $k_d$ while the pressure in the chamber is reduced from $p=10$ to $p={10}^{-5}\mathrm{mbar}$. Points in the same segment are measured at a constant pressure. (b) Convergence of the ML routine (started at $t=0$) to the optimal gain in a few steps at $p=3\times {10}^{-6}\mathrm{mbar}$, for five uncooled $x$ initial conditions. All reach similar final energy values (value of $k_d$ in FPGA units is displayed) but slightly differ in the converged gains. Each section represents 3 second iteration steps during which $k_d$ has been left constant (for clarity). Data are smoothed with a moving average filter and separated on the vertical axis by a constant factor for readability. (c) Setup and processing details of the algorithm. The OOL signal is sent to the FPGA (through a different input than the IL feedback signal), where the mode energy is estimated. A stochastic gradient descent algorithm, running on the CPU, optimizes the values of $k_p$ and $k_d$.

Figure 4

LQR. (a) Ratio $t_{\mathrm{LQR}}/t_{\mathrm{CD}}$ of the decay times (energy reduced to $1/e$) of a controller having both $K_p$ and $K_d$ (gain values in arbitrary FPGA units) and of cold damping ($k_p=0$) for a pressure of $p={10}^{-5}\mathrm{mbar}$. The figure displays how the transient times can be reduced by adding a $K_p$ term to the feedback controller. Lines connecting points are added as guides to the eye. (b) Experimental sample traces of the transients after turning on the feedback ($t=0$) for a fixed value of $k_d$. The transients are reduced by increasing $k_p$. (c) Numerical simulation of controllers approaching the $k_p^{\mathrm{LQR}}$ optimal value. From red (smaller $k_p$) to yellow (larger and closer to $k_p^{\mathrm{LQR}}$), the simulations show how the minimum stationary energy can be reached in a fraction of one oscillation period when $k_p$ is properly tuned. In our measurements, we never reach microsecond transients, due to the use of artificial delays to approximate $v(t)$.