High-$Q$ Milligram-Scale Monolithic Pendulum for Quantum-Limited Gravity Measurements

We present the development of a high-$Q$ monolithic silica pendulum weighing 7 milligram. The measured $Q$ value for the pendulum mode at 2.2 Hz was $2.0\times {10}^6$. To the best of our knowledge this is the lowest dissipative milligram-scale mechanical oscillator to date. By employing this suspension system, the optomechanical displacement sensor for gravity measurements we recently reported in Matsumoto et al. [Phys. Rev. Lett. 122, 071101 (2019)] can be improved to realize quantum-noise-limited sensing at several hundred hertz. In combination with the optical spring effect, the amount of intrinsic dissipation measured in the pendulum mode is enough to satisfy requirements for measurement-based quantum control of a massive pendulum confined in an optical potential. This paves the way for not only testing dark matter via quantum-limited force sensors, but also Newtonian interaction in quantum regimes, namely, between two milligram-scale oscillators in quantum states, as well as improving the sensitivity of gravitational-wave detectors.

Figure 1

Picture of the welding point (left) at the test mass and the pendulum (right). The image of the welding point was taken with an optical microscope (SELMIC, SE-1300 microscope; SEL-80 objective lens).

Figure 2

Logarithmic plot of ring-down measurements (a) showing a comparison between the required quality factor (blue dashed), the previously obtained value in [24] (black), a pendulum with the disk and fiber welded but clamped at the top (green), and the completely monolithic pendulum (red). The requirement is calculated assuming an optical spring effective frequency of 280 Hz. (b) Averaged ring-down measurements of the yaw mode of a different 5 cm long clamped pendulum (yellow) after adding the mass flow control, and fabricated with the same pulling rig, vs the ring-down measurement of the previous experiment (black). (c) The mechanical mode’s dissipation ${\gamma }_m/2\pi$ vs mass for published experiments with macroscopic oscillators. This experiment is depicted with the star symbol. Data adapted from [8, 24, 26, 33, 34, 35, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53].

Figure 3

Noise budget for a cavity-optomechanics experiment with this pendulum as suspension for the movable mirror. Suspension thermal noise is calculated using the measured dissipation of the pendulum mode. Only the pitching mode and the first two violin modes are shown. Mirror thermal noise was calculated with a substrate loss angle of $1\times {10}^{-6}$, coating loss angle of $3\times {10}^{-5}$, and a beam radius of $184\mu \mathrm{m}$.