Quantum feedback cooling of a trapped nanoparticle by using a low-pass filter

TL;DR

This paper addresses ground-state cooling of a trapped nanoparticle under continuous measurement by introducing a low-pass-filter (LPF) feedback scheme within a quantum-trajectory framework. By shifting the potential minimum according to an LPF-filtered measurement signal, the authors derive a fundamental energy bound $E \ge \frac{\hbar \omega}{2} \sqrt{\frac{1}{\eta} + \frac{\tilde{\gamma}^2}{2}}$ and an associated phonon occupation $n_{LPF} = \frac{1}{2}\left(\sqrt{\frac{1}{\eta} + \frac{\tilde{\gamma}^2}{2}} - 1\right)$. They show that LPF cooling outperforms cold damping with a band-pass filter, delayed feedback, and LQG control, with the gains growing as the detection efficiency $\eta$ increases; for example, at $\eta \approx 1$ and $\tilde{\gamma} = 0.1$, LPF yields near-zero phonon occupation while others remain above ~0.025–0.05. The results connect theory to experiment, providing practical mappings from experimental parameters to $\tilde{\gamma}$ and highlighting LPF as a viable route to nearly absolute ground-state cooling in levitated systems.

Abstract

We propose a low-pass-filter (LPF) feedback control for cooling a trapped particle with a low-pass filter, which utilizes a shift of the potential caused by the feedback operation. By incorporating this shift in the energy cost function, we show that the LPF control can achieve the minimum phonon occupation number that is lower than cold damping with a band-pass filter, that with delayed feedback, and linear--quadratic--Gaussian (LQG) control, the last two of which are the standard methods of ground-state cooling of a levitated nanoparticle. For the detection efficiency of $90\%$, the achievable phonon occupation number with the LPF control is about one third, two fifths and one half of that of cold damping with a band-pass filter, that with delayed feedback, and LQG control, respectively. Thus our method has a decisive advantage to reach the absolute ground state.

Figures (7)

• Figure 1: Schematic diagram of the system under feedback control. An optically trapped particle scatters an incident light with strength $\gamma$. The scattered light is detected with detection efficiency $\eta$ and the detected light produces a measurement outcome $\xi_1$. The undetection of scattered light is modeled as virtual detection with detection efficiency $1-\eta$ and a measurement outcome $\xi_2$ which is not fed back to the feedback control system.
• Figure 2: Mechanism of cooling by the LPF feedback, where the center of the potential is made to follow the position of the particle. (a) In the absence of the LPF feedback, the change of the kinetic energy (K.E.) is exactly compensated for by that of the potential energy (P.E.). In this case, the total energy of the system remains unchanged ($\Delta E = 0$). (b) Under the LPF feedback, the center of the potential is shifted on the basis of the measurement outcome so that the potential energy of the particle is removed. Hence the total energy decreases in time and the particle is cooled ($\Delta E < 0$).
• Figure 3: Schematic diagram of the LPF feedback. The signal, which carries information about the position of the particle, is processed by a low-pass filter characterized by the transfer function $G(\sigma)$. Then the potential is shifted on the basis of the filtered signal. The low-pass filter is comprised of a resistor and a capacitor (top right). We note that in a real experiment additional processes are needed such as the processing of the raw signal before the low-pass filtration and an appropriate amplification of the output signal from the low-pass filter.
• Figure 4: Phonon occupation number $n=\frac{E}{\hbar\omega} - \frac{1}{2}$ of achievable energies under LPF feedback (Eq. \ref{['mainresult_LPF']}, solid curves), cold damping with a band-pass filter (Eq. \ref{['eq:CDapp']}, dashed curves), LQG control (Eq. \ref{['eq:LQG']}, dotted curves) and cold damping with delayed feedback (numerical simulation (see Appendix \ref{['sec:CD-DaleyedFB']}), triangles) for $\eta=0.3\text{ (orange)},0.4\text{ (blue)}$ and $0.5\text{ (dark green)}$. Every method can cool the system down to the fundamental limit $\frac{1}{2}(\frac{1}{\sqrt{\eta}} -1)$doherty2012quantumbowen2015quantum in the limit of $\tilde{\gamma} \to 0$. However, for small but nonzero $\tilde{\gamma}$, the achievable energy of the LPF feedback is the lowest of the four, followed by LQG control, cold damping by delayed feedback, and then by cold damping with a band-pass filter.
• Figure 5: Phonon occupation number $n=\frac{E}{\hbar\omega} - \frac{1}{2}$ of achievable energies under LPF feedback (Eq. \ref{['mainresult_LPF']}, solid curves), cold damping with a band-pass filter (Eq. \ref{['eq:CDapp']}, dashed curves), LQG control (Eq. \ref{['eq:LQG']}, dotted curves) and cold damping with delayed feedback (numerical simulation (see Appendix \ref{['sec:CD-DaleyedFB']}), triangles) for $\eta=0.9\text{ (light green)},0.95\text{ (magenta)}$ and $1.0\text{ (black)}$. The dashed curves are calculated from Eq. \ref{['eq:CDapp']}. For high measurement efficiency ($\eta \approx 1$), the achievable phonon occupation numbers of two versions of cold damping and LQG control increase dramatically with increasing the dimensionless measurement strength $\tilde{\gamma}$ in sharp contrast with the LPF feedback where the achievable phonon occupation number stays almost constant with respect to $\tilde{\gamma}$.