Nanoscale Torsional Dissipation Dilution for Quantum Experiments and Precision Measurement

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Nanoscale Torsional Dissipation Dilution for Quantum Experiments and Precision Measurement

J. R. Pratt, A. R. Agrawal, C. A. Condos, C. M. Pluchar, S. Schlamminger, and D. J. Wilson

Phys. Rev. X 13, 011018 – Published 15 February 2023

Abstract

The quest for ultrahigh- $Q$ nanomechanical resonators has driven intense study of strain-induced dissipation dilution, an effect whereby vibrations of a tensioned plate are effectively trapped in a lossless potential. Here, we show for the first time that torsion modes of nanostructures can experience dissipation dilution, yielding a new class of ultrahigh- $Q$ nanomechanical resonators with broad applications to quantum experiments and precision measurement. Specifically, we show that torsion modes of strained nanoribbons have $Q$ factors scaling as their width-to-thickness ratio squared (characteristic of “soft clamping”), yielding $Q$ factors as high as $10^{8}$ and $Q$ -frequency products as high as $10^{13} Hz$ for devices made of ${Si}_{3} N_{4}$ . Using an optical lever, we show that the rotation of one such nanoribbon can be resolved with an imprecision 100 times smaller than the zero-point motion of its fundamental torsion mode, without the use of a cavity or interferometric stability. We also show that a strained nanoribbon can be mass loaded without changing its torsional $Q$ . We use this strategy to engineer a chip-scale torsion pendulum with an ultralow damping rate of $7 μ Hz$ and show how it can be used to sense micro- $g$ fluctuations of the local gravitational field. Our findings signal the potential for a new field of imaging-based quantum optomechanics, demonstrate that the utility of strained nanomechanics extends beyond force microscopy to inertial sensing, and hint that the landscape for dissipation dilution remains largely unexplored.

Received 24 May 2022
Accepted 22 December 2022

DOI:https://doi.org/10.1103/PhysRevX.13.011018

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Laboratory studies of gravity Optomechanics Quantum measurements

Micromechanical & nanomechanical oscillators Micromechanical devices Nanomechanical devices

Optical interferometry Precision measurements

Gravitation, Cosmology & AstrophysicsCondensed Matter, Materials & Applied PhysicsQuantum Information, Science & TechnologyAtomic, Molecular & Optical

Authors & Affiliations

J. R. Pratt ^1,*, A. R. Agrawal ^2,*, C. A. Condos ^2,*, C. M. Pluchar ^2,*, S. Schlamminger ¹, and D. J. Wilson^2,†

¹National Institute of Standards and Technology, 100 Bureau Drive, Gaithersburg, Maryland 20899, USA
²Wyant College of Optical Sciences, University of Arizona, Tucson, Arizona 85721, USA

^*These authors contributed equally to this work.
^†Corresponding author. dalziel@arizona.edu

Popular Summary

Tensioning a string increases how many times it will ring when plucked (a measure known as its $Q$ factor). The higher the $Q$ factor, the more sensitive to being plucked. Nanomechanical engineers have harnessed this effect, called dissipation dilution, to an extreme, achieving nanostrings with $Q$ factors such that, were they the size of a guitar string, would ring for days when plucked. Here, we introduce a new twist on dissipation dilution by showing that it also works for the torsion modes of a nanoribbon. The effect is like twisting a stretched rubber band, except our nanofabricated “rubber band” has the aspect ratio of a bridge built across the United States, and under a tensile stress akin to suspending a bowling ball from a human hair.

High- $Q$ nanodevices are sought for their ability to observe quantum behavior or sense ultraweak forces, like gravity or the pressure of light. The nanoribbons we study exhibit room-temperature $Q$ factors as high as $100 \times 10^{6}$ —like guitar strings that ring for more than a month when plucked—paving the way for a new class of quantum experiments and precision measurements.

By reflecting a laser off a ribbon and monitoring its deflection, we show that we can in principle resolve the ribbon’s smallest possible quantum-mechanical motion, a feat never achieved without an interferometer. By mass loading a nanoribbon, we also realize a chip-scale pendulum with a damping rate sufficient to resolve changes in gravity at the level of one part in $10^{6}$ .

Our findings signal the potential for a new field of imaging-based quantum optomechanics, demonstrate that the utility of strained nanomechanics extends beyond force microscopy to inertial sensing, and hint that the landscape for dissipation dilution remains largely unexplored.

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Vol. 13, Iss. 1 — January - March 2023

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Figure 1
Dissipation dilution of a torsion resonator. Left: a torsion pendulum formed by suspending a mass $m$ from a ribbonlike fiber of width $w$ and thickness $h$ . Gravity loads the fiber into tension $m g$ , producing a tensile stress $σ = m g / w h$ . The color gradient represents the transverse displacement $u$ of the fundamental torsion mode. Center: a doubly clamped ribbon under tensile prestress $σ$ . Right: bifilar model of the ribbon’s torsion mode [21]. In both cases, the ribbon length $L$ is preserved to first order, leading to dissipation dilution.
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Figure 2
Ultralow-loss nanomechanical torsion resonators. (a) Simulated torsion modes of a nanoribbon with (green) and without (red) mass loading. (b) Photos of representative devices: (top) a $400 − μ m$ -wide, 7-mm-long, 75-nm-thick ribbon; (middle) a $25 − μ m$ -wide, 7-mm-long, 75-nm-thick ribbon loaded with a $100 − μ m$ -thick Si paddle; and (bottom) a micrograph of the paddle. (c) Ringdowns of the fundamental mode of a $400 − μ m$ -wide ribbon (red) and a mass-loaded, $25 − μ m$ -wide ribbon (green). (d) Compilation of $Q$ versus ribbon width (left) and frequency (right). Red and blue points correspond to thicknesses of 75 and 180 nm, respectively. Dark and light green points correspond to inverted ( $- g$ ) and noninverted ( $+ g$ ) chip orientations, respectively. Solid, dashed, and dotted curves correspond to the lumped mass model, finite element model, and a gas damping model with $Q_{gas} = 1.1 \times 10^{9}$ [21], respectively. Left inset: $Q$ versus thickness for 200- and $50 − μ m$ -wide beams, compared to the lumped model (solid and dashed lines). Right inset: $Q$ versus frequency for mass-loaded ribbons of different orientation, compared to Eq. (4) (solid line).
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Figure 3
Optical lever measurement with an imprecision below that at the standard quantum limit. (a) Schematic of the optical lever technique: A Gaussian beam with waist $w_{0}$ is reflected off a torsion ribbon of width $w$ (the beam waist is at the ribbon [21]). Ribbon deflection $θ$ is monitored using a split photodiode. (b) Displacement of a $w = 400 μ m$ ribbon measured using a $w_{0} \approx 200 μ m$ , $P = 4 mW$ optical lever. Solid curves correspond to data (red), a thermal noise model (blue), and the inferred zero-point spectral density [21] (green). Dotted lines are guides to the eye for the peak thermal (blue), zero-point (green), and imprecision (black) noise. The ideal imprecision [Eq. (2) with $η = 1$ ] is shown as a solid black line.
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Figure 4
Torsion micropendulum gravimeter. (a) Device geometry: a ${Si}_{3} N_{4}$ nanoribbon (blue) loaded with a Si pad (gray). (b) Photographs, (c) displacement spectra, and (d) ringdowns of the device in inverted and noninverted configuration. (e) Free-running (green), driven (red), and temperature-corrected (blue) Allan deviation measurements. Inset: frequency (red) and temperature (blue) of driven device. Dashed lines: thermal noise (green) and drift (red) model.
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