Measurement and feedback for cooling heavy levitated particles in low-frequency traps

We consider a possible route to ground-state cooling of a levitated nanoparticle, magnetically trapped by a strong permanent magnet, using a combination of measurement and feedback. The trap frequency of this system is much lower than those involving trapped ions or nanomechanical resonators. Minimization of environmental heating is therefore challenging as it requires control of the system on a timescale comparable to the inverse of the trap frequency. We show that these traps are an excellent platform for performing optimal feedback control via real-time state estimation, for the preparation of motional states with measurable quantum properties.

Figure 1

Sketch of apparatus for measuring the intensity of a standing wave, modulated by a particle's motion along its main trap axis $x$. The trap center is marked a distance $L$ from the mirror, with the probe light incident at an angle $\theta$. The range of motion over which this measurement would be valid is restricted about a node of the standing light field, and has also been marked. A focused beam and a curved mirror that maximizes coupling into the reflected mode could reasonably produce light-collection efficiency $>15\%$.

Figure 2

Simulation of a trapped particle undergoing measurement, using (16, 17, 18, 19, 20). The normalized measurement strength, $\kappa x_0^2/\omega =1$, with $0.2\%$ quantum efficiency, and an initial particle energy corresponding to a temperature of $T=1\mu \mathrm{K}$. The top figure shows a numerically generated example of a position measurement and the middle figure shows the results of continuous-state estimation using the same signal. The estimated mean position plotted beside the true value and the shaded region covers 2 standard deviations in the estimate. The bottom figure shows the improvement in the standard deviation in both position (light line) and momentum (dark line) due to the measurement. The dashed line indicates the width of the motional ground state.

Figure 3

Simulation of particle heating due to measurement over several oscillation cycles, using (16, 17, 18, 19, 20). The normalized measurement strength, $\kappa x_0^2/\omega =1$, with $0.2\%$ quantum efficiency. The particle is initially in its ground state with temperature $T=0\mathrm{K}$. This figure is otherwise organized in the same way as Fig. 2.

Figure 4

Final resolution of the normalized position (light line) and momentum (dark line) variances of a trapped particle, from the steady-state solutions of a Gaussian estimator (32),(33). Variance values of less than 1 are squeezed compared to the harmonic-oscillator ground state. The solid lines correspond to a measurement with perfect efficiency $\eta =1$ and the dashed lines $\eta =0.15$; these values and the measurement strength would vary depending on the nature of the measurement.

Figure 5

Simulation of a damped levitated particle, using (25), (26), and (18, 19, 20). The normalized measurement strength, $\kappa x_0^2/\omega =1$, with $15\%$ quantum efficiency, and initial particle energy corresponding to a temperature of $T=1\mu \mathrm{K}$. The top figure shows a numerically generated example of a position measurement. The bottom figure shows the evolution of the mean position of the true state alongside the estimated position from the measurement record. The estimated position is almost completely damped relative to the fundamental shot noise in the original measurement signal. The standard deviation of the true motion from $t=\pi /2\to 2\pi$ is highlighted and matches the estimated variance (32).

Figure 6

Average steady-state phonon occupancy of a trapped nanoparticle after undergoing active feedback, calculated using the equations for a damped Gaussian state with excess noise (39). The effective damping rate (feedback gain) was chosen to be $\mathrm{\Gamma }/\omega =10$, strong enough to remove almost all stochastic drift due to the measurement disturbance. The quantum efficiencies from the top line down are $\eta =0.05,0.1,0.2,0.5,1.0$. The final occupancies range from $\langle n\rangle <3$, for currently feasible experimental parameters ($\eta =0.2,k=1$), to near zero, with perfect collection efficiency and a weaker measurement.