## 3.1 Translational motions

The experimental setup for levitating a nanofabricated nanorod using a metalens is shown in Fig. 3a. A 1550 nm laser, amplified by an erbium-doped fibre amplifier (EDFA), is collimated to a polarizer for obtaining linearly polarized light. A quarter-wave plate (QWP) is utilized to control the light's polarization state. The scattering light is collected by an objective lens and detected by a photodetector to record the nanorod's motion. A polarizing beam splitter (PBS) is utilized before the photodetector for detecting rotation signals. A focused 532 nm laser is used to illuminate the levitated nanorod for capturing the nanorod’s picture (i.e., the inset picture in Fig. 3a). A dichroic mirror (DM) is used to split the 532 nm and 1550 nm lasers before the detection. Nanorods are loaded into the optical trap in ambient conditions via an ultrasonic nebulizer.

Once a nanorod is trapped, we reduce the vacuum chamber's pressure to 4 mbar. Figure 3b shows the power spectral density (PSD) signal of the particle's motions when the trapping laser beam is linearly polarized. The PSD is defined as34

$${S_{qq}}(\omega )=\frac{{{\Gamma _{CM}}{k_B}{T_{CM}}/(\pi M)}}{{{{\left( {{\omega ^2} - \omega _{q}^{2}} \right)}^2}+\Gamma _{{CM}}^{2}{\omega ^2}}}$$

2

where *Γ**CM* is the damping rate of the COM motion, *k**B* is the Boltzmann constant, *T**CM* is the temperature of the COM, *M* is the particle's mass, and *ω**q* is the mechanical oscillation frequency of the trapped nanorod. Using Eq. (2) to fit each peak, we can obtain that the oscillation frequencies in the z, x, and y directions are 43.6 kHz, 78.3 kHz and 122.8 kHz, respectively. The frequency in the z-direction is lower than the other frequencies because the optical field is elongated along the direction of the propagating beam. A second harmonic signal along the z-direction in the PSD (Fig. 3b) arises from the non-perfectly harmonic trapping potential in the axial direction.

As the trapping laser’s focal field depends on the laser beam’s polarization, which can affect the levitated nanorod’s motion. Figure 3c shows that the COM motions of the levitated nanorod can be manipulated by changing the trapping laser beam’s polarization. The waveplate angle in the figure refers to the angle between the optical axis of the QWP and the polarizing axis of the polarizer. The ellipticity of the polarized light increases with increasing angle. We use the frequency difference (Δfx−y) between the x and y directions to show the polarization’s dependency of the translational motions in x and y directions. It can be seen that the Δfx−y gradually reduces when adjusting the input beam's polarization from linear to elliptical polarization, indicating the translational motion frequencies in the x and y directions are becoming closer. The mechanical oscillation frequency in the z direction remains constant as the field distribution in the z direction is not affected by the laser beam’s polarization.

For an optically levitated nanorod, anisotropic damping rates for COM motions in the air are expected5. The ratio of damping rates depends on the aspect ratio of the levitated particle. Thus, we use the ratio of damping rates to identify that the levitated particle is a fabricated nanorod. Figure 3d shows the PSD signals of the detected particle motion with a QWP angle of 30º and pressure of 4 mbar. The red curves are the Lorentzian fittings based on Eq. (2). A maximum damping ratio Γx/Γz of 1.467 is obtained when the QWP angle is 30º. This ratio closely matches the damping ratio calculated for a chain tetramer5, indicating a nanorod is optically levitated. The damping rate in the z direction suggests that the nanorod's long axis is oriented parallel to the optical axis, as shown in the inset of Fig. 3d.

## 3.2 MHz spin rotation and manipulation

In this section, we demonstrate the control over the rotational dynamics of an optically levitated nanorod by a metalens. The pressure inside the vacuum chamber is reduced to 0.11 mbar in order to mitigate the damping effect caused by gas molecules and to observe a clear rotational signal. Figure 4a (4b) show the PSD spectra from 0 to 200KHz (200–1500KHz) for circularly polarized (orange curve) and linearly polarized (blue curve) light beams, respectively. While the central frequencies of COM motions in Fig. 4a are similar to that (Fig. 3b) at 4 mbar pressure, the linewidth of the peaks in Fig. 4a is much narrower than that in Fig. 3b. It means that the air pressure only acts as a damping factor which goes down with lowing the air pressure.

Figure 4(b) clearly shows that two new trapping frequencies (frot and 2frot) appear on the PSD curve for the circularly polarized laser beam, in comparison to the linearly polarized light. As the pressure is reduced further, the frequencies frot and 2frot increase as shown in Fig. 4c. This is a typical feature of the spin rotation of a levitated nanorod. In the PSD spectrum, the amplitude of 2frot is much larger than that of frot, due to the geometrical symmetry of the nanofabricated nanorod.

The rotation of the levitated nanorod can be attributed to the torque exerted by the circularly polarized light. The strength of this optical torque depends on the laser power and the particle’s polarizability, and can be expressed as35

$${\tau _z}=\frac{1}{2}E_{0}^{2}\frac{{{k^3}}}{{6\pi {\varepsilon _0}}}{\left( {\Delta {\alpha _0}} \right)^2}$$

3

where E0 is the amplitude of the optical input field, k is the wavenumber, ε0 is the dielectric constant in vacuum, and Δα0 is given by

$$\Delta {\alpha _0}=\frac{{{\alpha _x}}}{{1+i{k^3}{\alpha _x}/(6\pi {\varepsilon _0})}} - \frac{{{\alpha _y}}}{{1+i{k^3}{\alpha _y}/(6\pi {\varepsilon _0})}}$$

4

with αx and αy being the nanorod’s polarization along the x and y axis, respectively. The maximum steady-state rotation frequency of the particle can be represented as

$${f_{rot}}=\frac{{{\tau _z}}}{{2\pi I\Gamma }}$$

5

where I is the rectangular nanorod’s momentum of inertia, and Γ is the rotational damping rate for diffuse reflection of gas molecules.

Figure 4c shows the calculated rotation frequency (blue curve) of the nanorod at different pressures using Eqs. (2–4). The detailed calculation is shown in the supplementary materials. For complex-shaped nanoparticles, their optical torque can be calculated by combining the finite difference in the time-domain method with the discrete dipole approximation method36. The calculated rotation frequency closely matches the experimentally measured frequency. The difference between measured and calculated frequencies originates from the deviations between the measured and actual sizes of the nanorod.

The dependency of the nanorod’s rotation frequency on the laser beam’s power and polarization is experimentally explored as shown in Figs. 4d and 4e. Figure 4d shows that the measured rotational frequency is proportional to the laser power, which is consistent with Eq. (3). The polarization is tuned by rotating the QWP. As shown in Fig. 4e, when the angle is within the range from 0 to 30º, there is no rotational signal in the PSD spectrum, indicating that the optical torque applied to the nanorod is smaller than the air drag. When the QWP angle is larger than 30º, the nanorod starts to rotate and the rotational PSD signal appears. With furtherly increasing the QWP angle, more spin angular momentum can be transferred to the nanorod, thereby a higher rotation frequency can be obtained. The blue curves in Figs. 4d and 4e show the calculated results, which agree well with the experimental measurements (orange curves).