Torsional Optomechanics of a Levitated Nonspherical Nanoparticle

An optically levitated nanoparticle in vacuum is a paradigm optomechanical system for sensing and studying macroscopic quantum mechanics. While its center-of-mass motion has been investigated intensively, its torsional vibration has only been studied theoretically in limited cases. Here we report the first experimental observation of the torsional vibration of an optically levitated nonspherical nanoparticle in vacuum. We achieve this by utilizing the coupling between the spin angular momentum of photons and the torsional vibration of a nonspherical nanoparticle whose polarizability is a tensor. The torsional vibration frequency can be 1 order of magnitude higher than its center-of-mass motion frequency, which is promising for ground state cooling. We propose a simple yet novel scheme to achieve ground state cooling of its torsional vibration with a linearly polarized Gaussian cavity mode. A levitated nonspherical nanoparticle in vacuum will also be an ultrasensitive nanoscale torsion balance with a torque detection sensitivity on the order of ${10}^{-29}\mathrm{N}\mathrm{m}/\sqrt{\mathrm{Hz}}$ under realistic conditions.

Figure 1

(a) Experimental diagram for detecting torsional (TOR) vibration and center-of-mass (c.m.) motion of a levitated nonspherical nanoparticle (NP). A nanodiamond (represented by an ellipsoid) is levitated by a tightly focused linearly polarized 1550 nm laser beam. The nanoparticle’s motion is monitored by the exiting trapping laser. The exiting beam is split by a beam splitter (BS) to a c.m. detector and a TOR detector. A $\lambda /2$ wave plate balances the power of the beams after the polarizing beam splitter (PBS) for the TOR detector. (b) A SEM image of irregular nanodiamonds. The scale bar is 100 nm. (c) A proposed scheme to cool the torsional vibration of a levitated ellipsoidal nanoparticle with an optical cavity driven by a linearly polarized Gaussian beam. (d) The relation between Cartesian coordinate systems of the nanoparticle ($x_N$, $y_N$, $z_N$), the trapping laser ($x_T$, $y_T$, $z_T$), and the cavity mode ($x_C$, $y_C$, $z_C$). The $x_N$ axis aligns with the longest axis of the nanoparticle. $x_T$ and $x_C$ axes align with the polarization directions of the trapping laser and the cavity mode, respectively. The angle between $x_N$ and $x_T$ is $\theta$, and the angle between $x_T$ and $x_C$ is $\beta$. The $y_C$ axis is the optical axis of the cavity. The $z_T$ axis is the propagation direction of the trapping laser. As we consider only one torsional mode, we assume $z_C$ and $z_N$ are parallel to $z_T$ for simplicity.

Figure 2

(a) Measured power spectrum densities (PSD) of the c.m. motion (labeled with ${\mathrm{\Omega }}_{x,y}/2\pi$) and TOR vibration (labeled with ${\mathrm{\Omega }}_{\theta }/2\pi$) of an optically levitated nanodiamond at 100 Torr. When the power of the trapping laser is about 500 mW, ${\mathrm{\Omega }}_x/2\pi =0.16\mathrm{MHz}$, ${\mathrm{\Omega }}_y/2\pi =0.18\mathrm{MHz}$, and ${\mathrm{\Omega }}_{\theta }/2\pi =1.0\mathrm{MHz}$. The damping coefficient is anisotropic with a ratio ${\mathrm{\Gamma }}_x/{\mathrm{\Gamma }}_y=0.69$ for this nanodiamond. (b) Measured TOR and c.m. frequencies of two different nanodiamonds at 100 Torr as a function of the trapping power. Frequency data are fitted to a function $A\sqrt{P}$ with $P$ being the trapping power and $A$ being a fitting parameter. Solid and open symbols represent different nanodiamonds. (c) ${\mathrm{\Omega }}_{\theta }/⟨{\mathrm{\Omega }}_{\theta }⟩$ of 13 nanodiamonds at different pressures. $⟨{\mathrm{\Omega }}_{\theta }⟩$ is the average over different pressures of a nanodiamond’s torsional frequency. The shaded region is the standard deviation of all ${\mathrm{\Omega }}_{\theta }/⟨{\mathrm{\Omega }}_{\theta }⟩$ data. The inset plot is the histogram of the measured ratios of damping coefficients in $x$ and $y$ directions (${\mathrm{\Gamma }}_x/{\mathrm{\Gamma }}_y$) for these nanodiamonds at many different trapping powers and pressures. ${\mathrm{\Gamma }}_x/{\mathrm{\Gamma }}_y<1$ indicates that the long axes of the nanodiamonds align with the polarization direction ($x$ axis) of the laser beam. (d) The ratio ${\mathrm{\Omega }}_{\theta }/{\mathrm{\Omega }}_{x,y}$ of two levitated nanodiamonds [shown in (b)] at different laser powers.

Figure 3

(a) Trapping potentials in the transverse ($U_y$) and angular ($U_{\theta }$) directions. (b) Frequencies of the TOR and c.m. vibrations as a function of the size of the ellipsoid when its aspect ratio is $r_y/r_x=0.8$. (c) Quality factors of the TOR vibration and c.m. motion of a levitated nanoparticle as a function of the pressure. (d) Enhancement ratio ${\mathrm{\Omega }}_{\theta }/{\mathrm{\Omega }}_y$ as a function of the size of the ellipsoid with different aspect ratios. In the calculations, we assume the waist of the Gaussian optical tweezer to be 600 nm, the laser wavelength to be 1550 nm, and the laser power to be 100 mW. To calculate results shown in (a) and (c), we assume the semiaxes of the ellipsoid to be $r_x=50\mathrm{nm}$ and $r_y=r_z=40\mathrm{nm}$. The corresponding vibration frequencies are ${\mathrm{\Omega }}_{\theta }/2\pi =1.26\mathrm{MHz}$ and ${\mathrm{\Omega }}_y/2\pi =220\mathrm{kHz}$.

Figure 4

(a) The single phonon–single photon coupling strengths of the c.m. motion ($g_y$) and torsional vibration $g_{\theta }$ as a function of cavity length. For the calculation, we assume the size of the diamond ellipsoid to be $r_x=50\mathrm{nm}$ and $r_y=r_z=25\mathrm{nm}$. The trapping laser parameters are the same as in Fig. 3. The calculated vibration frequencies are ${\mathrm{\Omega }}_{\theta }/2\pi =2.6\mathrm{MHz}$ and ${\mathrm{\Omega }}_y/2\pi =248\mathrm{kHz}$. (b) The final phonon numbers of the c.m. mode ($n_y$) and the torsional mode ($n_{\theta }$) as a function of the driving field. The resonant wavelength of the cavity is 1540 nm, which is different from that of the trapping laser. The background pressure is assumed to be ${10}^{-8}\mathrm{Torr}$. The cavity finesse is $\mathcal{F}=10^5$.