Optical levitation of 10-ng spheres with nano-$g$ acceleration sensitivity

We demonstrate optical levitation of ${\mathrm{SiO}}_2$ spheres with masses ranging from 0.1 to 30 ng. In high vacuum, we observe that the measured acceleration sensitivity improves for larger masses and obtain a sensitivity of $0.4\times {10}^{-6}g/\sqrt{\mathrm{Hz}}$ for a 12-ng sphere, more than an order of magnitude better than previously reported for optically levitated masses. In addition, these techniques permit long integration times and a mean acceleration of $(-0.7\pm 2.4\left[\text{stat}\right]\pm 0.2\left[\text{syst}\right])\times {10}^{-9}g$ is measured in $1.4\times {10}^4$ s. Spheres larger than 10 ng are found to lose mass in high vacuum where heating due to absorption of the trapping laser dominates radiative cooling. This absorption constrains the maximum size of spheres that can be levitated and allows a measurement of the absorption of the trapping light for the commercially available spheres tested here. Spheres consisting of material with lower absorption may allow larger objects to be optically levitated in high vacuum.

Figure 1

Simplified schematic of the experimental setup. The microsphere is levitated by a 1064-nm laser (dark red) and imaged by two 532-nm beams (light green). Signals coming from the imaging sensors are fed into an FPGA that provides active feedback using the AOM and piezo deflection mirror. The insets show the front and side view of the levitated sphere with respect to the electrodes used to generate the electric field for calibration. The vertical direction is defined as Z, while X denotes the direction perpendicular to the electrode's surface. The electrode diameters are $25.4\mathrm{mm}$ and the electrode separation used in these measurements is $\sim 2\mathrm{mm}$.

Figure 2

Microscope images of microspheres with several different diameters levitated in the optical trap. From left to right the sphere diameters are measured to be $23.0\pm 1.1$, $14.4\pm 0.8$, $11.1\pm 0.7$, and $5.0\pm 0.5\mu \mathrm{m}$. All images have the same scale, indicated on the bottom right, and were obtained with the spheres levitated at the same equilibrium position above the focus.

Figure 3

ASD measured for a sphere with diameter $14.0\pm 0.8\mu \mathrm{m}$ at high and low pressure. The blue line (top) shows the ASD measured at $1\mathrm{mbar}$ with no feedback in the X direction. The orange line (top) shows the damping of the motion near the resonance peak when the feedback in the X direction is turned on, which is necessary to keep the sphere stably trapped at pressures $\lesssim 0.1$ mbar. The black line (bottom) shows the measured ASD at $\approx {10}^{-6}\mathrm{mbar}$ where the corresponding acceleration sensitivity is $\lesssim 2\mu \mathit{g}/\sqrt{\mathrm{Hz}}$ in the frequency range between 10 and $200\mathrm{Hz}$. The light green curve (bottom) shows the measured spectral density of the pointing fluctuations of the 1064-nm laser before it enters the vacuum chamber. These pointing-induced fluctuations are converted to an expected acceleration using the measured transfer function of the sphere (red bottom).

Figure 4

Top: Required laser power entering the vacuum chamber to levitate ${\mathrm{SiO}}_2$ spheres of varying masses at $1\mathrm{mbar}$. The black curve indicates the predicted power in the ray-optics limit ($d\gg \lambda$) using the measured beam parameters. Bottom: Measured ASD for different sphere sizes at a frequency of 50 Hz.

Figure 5

Distribution of the acceleration measured for a 12-ng sphere for data acquired in individual 50-s periods over a total integration time of $1.4\times {10}^4\mathrm{s}$. The black line shows a Gaussian fit to the data.

Figure 6

Two spheres with initial diameters of 28 and $33\mu \mathrm{m},$ respectively (left column from top to bottom) are observed to decrease in size as the pressure is reduced from $10\mathrm{mbar}$ (left column) to ${10}^{-5}\mathrm{mbar}$ (right column). The final diameters are 22 and $24\mu \mathrm{m},$ respectively (right column from top to bottom). The yellow circles in the low-pressure images represent the corresponding size of spheres at $10\mathrm{mbar}\mathrm{s}$.