Optically levitated rotor at its thermal limit of frequency stability

Optically levitated rotors are prime candidates for torque sensors whose precision is limited by the fluctuations of the rotation frequency. In this work we investigate an optically levitated rotor at its fundamental thermal limit of frequency stability, where rotation-frequency fluctuations arise solely due to coupling to the thermal bath.

Figure 1

(a) Simplified schematic of the experimental setup. Inside a vacuum chamber, an optical trap is formed by focusing a laser beam with an aspheric lens ($\text{NA}=0.77$). The intensity of the laser beam [wavelength $\lambda =1565.0\left(1\right)\mathrm{nm}]$ can be modulated with an electro-optical modulator (EOM). The polarization of the laser beam is set with a quarter-wave plate. The light is collected and split into two paths with a nonpolarizing beamsplitter (BS). One half of the optical power is sent to a center-of-mass (COM) motion detector. The other half is used to measure the rotation in a balanced detection scheme. (b) In the focus, we trap a dumbbell formed by two spherical nanoparticles with diameter $d$. For a circularly polarized trapping beam and at low enough pressure, the particle rotates due to the optical torque exerted by the laser beam [22]. (c) Power spectral densities of the COM motion of a rotor at a pressure $p_{\mathrm{gas}}=7.0\left(7\right)\mathrm{mbar}$. The trapping laser is close to linearly polarized ($x$ axis), which orients the particle along the $x$ axis. The ratio between the damping rates along the $x$ and $y$ axes is $1.20\left(5\right)$, identifying the trapped object as a dumbbell [25].

Figure 2

Effect of COM cooling on rotation frequency $f$. (a) Time trace of $f$ for a rotor at pressure $p_{\mathrm{gas}}=5.0\left(5\right)\times {10}^{-3}\mathrm{mbar}$ without COM cooling. (b) Histogram of time trace shown in (a). (c) Time trace at the same pressure as in (a) but under COM cooling. (d) Histogram of time trace shown in (c). The motion along $x$ and $y$ is cooled below $20\mathrm{K}$, while the motion along $z$ is cooled to $90\mathrm{K}$.

Figure 3

(a) Dependence of rotational fluctuations ${\sigma }_f$ on gas temperature $T$. Measurement of ${\sigma }_f$ as a function of $T$ at $p_{\mathrm{gas}}=9\left(1\right)\times {10}^{-2}\mathrm{mbar}$. The dashed line shows a fit to Eq. (4). (b) Dependence of ${\sigma }_f$ on rotor size $d$. The dashed line shows the theoretical prediction according to Eq. (4) with a scaling factor extracted from (a) and using $T=300\mathrm{K}$.

Figure 4

(a) Plot of the mean rotation frequency $\langle f\rangle$ (green diamonds) and its standard deviation ${\sigma }_f$(blue circles) as a function of gas pressure $p_{\mathrm{gas}}$. While ${\sigma }_f$ remains constant over a pressure range spanning more than two orders of magnitude, $\langle f\rangle$ is proportional to $1/p_{\mathrm{gas}}$ (shown as dashed line). (b) Influence of optical torque ${\tau }_{\mathrm{opt}}$. We measure $\langle f\rangle$ (green diamonds) and ${\sigma }_f$ (blue circles) as a function of quarter-wave plate angle $\varphi$, setting the polarization state of the trapping field, and thus ${\tau }_{\mathrm{opt}}$ [cf. Fig. 1].

Figure 5

Method 1: Extraction of standard deviation ${\sigma }_{f,i}$ from a histogram of a time trace of the rotation frequency $f$. (a) Time trace of the rotation frequency $f$ of a dumbbell at pressure $p_{\mathrm{gas}}=5.0\left(5\right)\times {10}^{-2}\mathrm{mbar}$. (b) Histogram of time trace shown in (a). From a Gaussian fit (red dashed curve) we extract the standard deviation ${\sigma }_{f,i}$ of the time trace.

Figure 6

Method 2: Extraction of standard deviation ${\sigma }_f$ from $S_{ff}\left(\nu \right)$. (a) Time trace of the rotation frequency $f$ of a dumbbell at pressure $p_{\mathrm{gas}}=8.0\left(8\right)\times {10}^{-3}\mathrm{mbar}$. (b) Histogram of time trace shown in (a). Because of slow pressure drifts, the histogram does not have a Gaussian shape. (c) Power spectral density $S_{ff}\left(\nu \right)$ of the time trace shown in (a). The standard deviation is extracted from the area under the Lorentzian fit (red dashed curve).