Ultrahigh Quality Factor of a Levitated Nanomechanical Oscillator

A levitated nanomechanical oscillator under ultrahigh vacuum is highly isolated from its environment. It has been predicted that this isolation leads to very low mechanical dissipation rates. However, a gap persists between predictions and experimental data. Here, we levitate a silica nanoparticle in a linear Paul trap at room temperature, at pressures as low as $7\times {10}^{-11}\mathrm{mbar}$. We measure a dissipation rate of $2\pi \times 69(22)\mathrm{nHz}$, corresponding to a quality factor exceeding ${10}^{10}$, more than 2 orders of magnitude higher than previously shown. A study of the pressure dependence of the particle’s damping and heating rates provides insight into the relevant dissipation mechanisms.

Figure 1

Particle trapping and detection. (a) Schematic of the ultrahigh-vacuum (UHV) setup and detection schemes. A nanoparticle is detected in two ways: it is imaged with a camera, and its position is measured with an avalanche photodiode (APD) using a confocal technique. The camera inset shows particle images before and after excitation, as well as position histograms. The APD inset shows a time trace of the particle’s position. (b) Camera snapshots of the particle with decaying oscillation amplitude at $P=5.4\times {10}^{-8}\mathrm{mbar}$. Scale bar: $20\mathrm{\mu }\mathrm{m}$.

Figure 2

Measurements of the damping rate $\gamma$. (a) Ring-up measurement at $P_1=1.2\times {10}^{-4}\mathrm{mbar}$. Gray: 400 individual data traces. Three examples are highlighted in dark gray. Black: ensemble average. Red: fit of the ensemble-averaged data with Eq. (2), where $T_{\mathrm{fb}}$ is fixed to 1 K. The dashed line $t=t_{\mathrm{fb}}=0.5\mathrm{s}$ indicates the time at which feedback cooling is switched off. (b) Ring-down measurement at $P_2=5.4\times {10}^{-8}\mathrm{mbar}$, (c) $P_3=5\times {10}^{-9}\mathrm{mbar}$, and (d) $P_4=7\times {10}^{-11}\mathrm{mbar}$, with a logarithmic scale used for the ordinate axis. Error bars on the data are smaller than the diamond symbols; they correspond to the uncertainty in determining amplitudes from camera images. Solid lines represent fits with Eq. (1) in logarithmic scale, with $⟨z(0{)}^2⟩$ and $\gamma$ as fit parameters. Light shaded regions indicate $1\sigma$ error bars on fit parameters. Fits for higher pressures are reproduced in the lower-pressure plots for comparison.

Figure 3

Quality factor of the levitated oscillator. (a) Damping rate $\gamma$ as a function of pressure. Error bars for $\gamma$ represent the uncertainty of the fit parameter, while error bars for pressure represent the imprecision of the pressure gauge. Error bars are smaller than the diamond symbols. Colors indicate the corresponding fits in Fig. 2. The solid line shows a linear fit $\gamma /(2\pi )=aP$ to the data in logarithmic scale, where the parameter $a$ depends on the particle and the background gas properties [13, 19]. Since non-negligible uncertainties are present in both $\gamma$ and $P$, we use a total least squares regression to fit the data [51]. The fit yields $a=0.9(2)\mathrm{kHz}{\mathrm{mbar}}^{-1}$. See Ref. [43] for a comparison with theory. (b) Quality factor at four pressures. Error bars are propagated from the uncertainties of $\gamma$ shown in (a). The solid line represents the function $Q={\mathrm{\Omega }}_z/(2\pi aP)$, where $a$ is obtained from the fit in (a). Dashed lines indicate values for the product $Q{f}_z$, where $f_z={\mathrm{\Omega }}_z/(2\pi )$ is the oscillation frequency.

Figure 4

(a) Allan deviation of the particle oscillation frequency at pressure $P_4$. Lines connect neighboring data points. Error bars represent the standard error of the frequency mean. Inset: frequency drift as a function of time. (b) Heating-rate measurements at $P_4$ for continuous (blue) and stroboscopic (red) illumination of the particle with the detection laser. Error bars on the stroboscopic measurements represent the standard deviation of individual temperature measurements. The dashed line $t=t_{\mathrm{fb}}=5\mathrm{s}$ indicates the time at which feedback cooling is switched off. The dark-blue line is a fit to the data for continuous illumination, the slope of which yields the heating rate ${\mathrm{\Gamma }}_{\mathrm{tot}}=3.3(2)\times {10}^4\text{phonon}/\mathrm{s}$. The fit for stroboscopic illumination is not shown since it coincides with the fit for the continuous case.