Recent years have seen the emergence of a new class of ultrahigh-Q mechanical resonators fashioned from strained thin films [1]. The mechanism behind their performance is dissipation dilution, an effect whereby an elastic body is subjected to a conservative stress field, increasing its stiffness without adding loss [2,3]. Access to extreme dimensions and stresses at the nanoscale has enabled dilution factors (the ratio of final to initial Q) in excess of 10 5 , yielding Q factors in excess of 10 9 for devices made of amorphous glass and Q-frequency products exceeding 10 15 Hz using "soft clamping" [4,5]. Attractive features of these devices include attonewton force sensitivities [6], thermal coherence times of milliseconds [4], and zero-point displacement amplitudes in excess of picometers [5], spurring proposals from room-temperature quantum experiments [7] to ultrafast force microscopy [8].
Despite rapid innovation, a key limitation of dissipation dilution is its restriction to transverse flexural modes, a consequence of its reliance on nonlinear stress-strain coupling. It has been formally shown [3] that breathing modes, such as the longitudinal modes of a cylinder, cannot be diluted by a uniform strain field, ruling out the application of dissipation dilution to a large class of metrologically important mechanical devices, such as the mirrors used in precision optical cavities and gravitational wave interferometers. It is also commonly held that torsion modes of nanostructures are not diluted by strain [3,9,10], despite the prevalence of tensioned microsuspensions in macroscopic torsion pendula [11] and the historic use of torsion microresonators to study mechanical loss [12].
Here, we show that torsion modes of a simple nanobeam can experience massive dissipation dilution due to thin film stress and draw a connection to a century-old theory from the torsion balance community [13] that implies that torsion modes of a beam are naturally soft-clamped [4]. The key insight is that, when tensioned, the torsional stiffness of a beam increases as its width-to-thickness-ratio w=h squared; thus, for ribbonlike nanobeams (not commonly studied in nanomechanics [14]), torsional Q factors should scale as ðw=hÞ 2 , provided that curvature at the clamps is negligible. We confirm this theory by studying high-stress Si 3 N 4 nanoribbons with w=h as high as 10 4 , realizing Q factors as high as 10 8 and Q-frequency products as high as 10 13 Hz at room temperature.
By resolving a common misperception in the nanomechanics community (that torsion modes do not experience dissipation dilution [3]), our findings invite a rethinking of strategies for quantum experiments and precision measurement with nanomechanical resonators. We explore two examples: First, owing to their large angular zero-point motion, strained nanoribbons are promising for imaging-based quantum optomechanics [15][16][17]. We show this by resolving the rotation of a Si 3 N 4 nanoribbon with an imprecision 100 times smaller than its peak zero-point spectral density using an optical lever, setting the stage for observation of radiation pressure shot noise in torque [18]. Second, with precision inertial sensing in mind, we show that strained nanoribbons can be mass loaded without reducing their torsional Q (contrary to what is typically observed for flexural modes [19,20]). We use this strategy to engineer chip-scale torsion pendula with damping rates as low as 7 μHz and describe how such a device can be used to sense micro-g fluctuations of local gravity.
The remainder of the paper provides an essential overview of these findings, followed by the Appendix with key derivations. Comprehensive theoretical and experimental discussion is provided in Supplemental Materials [21].
The history of torsional dissipation dilution can be traced back to the theory of bifilar suspensions by Buckley in 1914 [13]. In brief, loading a torsion strip with a massive plumb bob increases its stiffness without adding loss [11]. This is because, as the strip twists, it lifts the bob and does work against a conservative gravitational field. Vibrating in its fundamental mode, the Q of the loaded strip scales as the ratio of its torsional stiffness due to tensile (k σ ) and internal stress (k E ) [21]:
where w and h are the strip width and thickness, respectively, σ ¼ mg=wh is the tensile stress due to gravity (Fig. 1), and E is the elastic modulus of the strip material. Generalized dissipation dilution theory [3] holds that any form of static tensile stress, such as residual stress in thin films, can give rise to dissipation dilution. Buckley's theory can therefore be generalized to the idea that placing a ribbon under tensile stress increases its stiffness without adding loss. Equation (1) should, thus, hold for a nanoribbon released from a thin film with biaxial stress σ. Remarkably, Eq. ( 1) is also known to be the ideal "soft-clamped" dissipation dilution factor for a thin beam vibrating in its fundamental flexural mode (with width replaced by half length) [5]. In the Appendix, we provide a continuum mechanics model that supports this claim, which we now proceed to investigate experimentally.
To investigate Eq. ( 1), we fabricated a set of high-stress Si 3 N 4 nanoribbons with aspect ratios w=h varying from 10 2 to 10 4 [21] [Fig. 2(a)]. Devices were housed in a roomtemperature high-vacuum chamber (≲10 -7 mbar), and ringdowns [Fig. 2(c)] were performed using optical lever measurements in conjunction with radiation pressure driving [21]. Ringdown-inferred Q factors were then compiled for flexural and torsional modes up to third order, as shown in Fig. 2(d) and Supplemental Material [21].
Considering first the hypothesis that Q ∝ ðw=hÞ 2 , in Fig. 2(d) we compare Q factors of ribbons with widths from 10 to 400 μm to the lumped mass model [Eq. (1)] and a finite element model accounting for bending loss at the clamps [21]. For both models, we assume σ ¼ 0.85 GPa [21], E ¼ 250 GPa, h ¼ 75 nm, and Q 0 ¼ 6000 × h=ð100 nmÞ (an established surface loss model for Si 3 N 4 thin film resonators [22]). We observe quantitative FIG. 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 mg, producing a tensile stress σ ¼ mg=wh. 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. agreement up to a width of 100 μm, beyond which Q begins to drop, consistent with simulated bending loss [21] and gas damping (we estimate Q gas ≈ 10 9 [21]). For widths smaller than 100 μm, we observe a slightly higher Q than predicted by both models. Though a more careful investigation is required, we speculate this may be due to an overestimate of surface loss, since the model in Ref. [22] is inferred from a study of flexural rather than torsional modes.
Further support for Eq. (1) was obtained by inspecting higher-order modes [solid diamonds and triangles in Figs. 2(d) and2(e)]. The continuum model derived in the Appendix predicts that Q n =Q 0 is independent of mode order n for acoustic wavelengths L=n ≫ w, consistent with our observation that fQ n g is bound above by Eq. (1). We also observed that the fundamental torsional mode had consistently high Q. This is contrary to the typical behavior of flexural modes of Si 3 N 4 beams and membranes [22] and suggests that torsional modes may be more resistant to acoustic radiation ("mounting") loss [23].
Finally, we fabricated several ribbons with a different thickness, 180 nm, and observed that the Q of torsion modes scaled inversely with thickness [Fig. 2(d) inset]. By contrast, as shown in Supplemental Material [21], we observed that the Q of flexural modes were roughly independent of width and thickness. These different scalings are telltale signatures of soft clamping (Q=Q 0 ∝ w 2 =h 2 ) and "hard clamping" (Q=Q 0 ∝ w=h) [14], respectively, in the presence of surface loss (Q 0 ∝ h) [22].
Nanomechanical resonators have been probed at the quantum limit using cavity-enhanced interferometry [24,25]. In principle, however, neither a cavity nor interferometry is necessary, provided that the measurement is optimally efficient [26]. Owing to their high torsional Q, strained nanoribbons present a unique opportunity to study the quantum limits of deflectometry, a measurement technique of long-standing inquiry in the field of quantum imaging [27,28] which has received relatively little attention from the quantum optomechanics community [29]. Pursuing an established benchmark [30,31], in our study of Si 3 N 4 nanoribbons, we found that the imprecision of optical lever measurements S imp θ [here, expressed as a power spectral density evaluated at mechanical resonance S θ ðω 0 Þ ≡ S θ ] could be reduced far below the zero-point angular displacement of the ribbon's fundamental torsion mode S ZP θ , satisfying a basic requirement for displacement measurement at the standard quantum limit (at the "SQL,"
, where S ba θ is the displacement produced by backaction torque S BA τ ≥ ℏ 2 =S imp θ ) [16,18,30]. To our knowledge, this represents the first deflectometric displacement measurement with an imprecision below that at the SQL (S imp θ < S ZP θ =2) [30]. Combined with the high Q-f products of our devices (exceeding the threshold for quantum coherence at room temperature, Qf ¼ k B T=h > 6 × 10 12 Hz [32]), access to sub-SQL deflection measurements signals the potential for a new generation of imagingbased quantum optomechanics experiments [15][16][17], in which thermal motion is overwhelmed by radiation pressure shot noise in torque.
To explore the potential for torsional quantum optomechanics, we revisit the optical lever technique with an eye to maximizing the ratio S ZP θ =S imp θ for a torsion ribbon. As shown in Fig. 3(a), the "lever" is formed by reflecting a laser beam off the ribbon and monitoring its deflection on a split photodiode. In the far field, angular displacement of the ribbon θ can be resolved with a shot-noise-limited imprecision of [21,28,33]
where w 0 and P are the waist size and reflected power of the laser beam, respectively, and η ∈ ½0; 1 is an efficiency parameter accounting for optical loss and imperfect transduction. Comparing to the zero-point displacement spectral density of the ribbon's fundamental torsion mode [21]
), we find that maximum "leverage" is achieved by matching the widths of the laser beam and the ribbon (w 0 ≈ w=2), giving access to a favorable scaling
An optical lever measurement with an imprecision below that at the SQL [30,31] is shown in Fig. 3. The measurement was made by reflecting a w 0 ≈ 200-μm-wide laser beam from a 400-μm-wide, 75-nm-thick nanoribbon and detecting the 4 mW reflected field at a distance of 0.4 m. The lateral position of the split photodetector was trimmed to balance out classical intensity noise, and the ribbon position was trimmed to optimize coupling to the fundamental torsion mode [21]. Near the fundamental torsional resonance frequency, ω 1 ≈ 2π × 52.5 kHz, the photocurrent spectrum is dominated by thermal noise with a peak magnitude of S th θ ¼ 2n th S ZP θ , well in excess of zero-point motion due to the large thermal mode occupation, n th ¼ k B T=ℏω 1 ≈ 1.4 × 10 8 (accounting for a small amount of photothermal heating, T ≈ 350 K [21]). Fitting the noise peak to a Lorentzian S θ ½ω ¼ FIG. 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 ¼ 400 μm ribbon measured using a w 0 ≈ 200 μm, P ¼ 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 ¼ 1] is shown as a solid black line.
immunity of deflectometry to various forms of technical noise.
Mass loading a micromechanical resonator is the basis for chip-scale inertial sensing; however, it typically entails increasing mechanical loss, leading to complicated sizesensitivity trade-offs [19,20]. In our study of Si 3 N 4 nanoribbons, we found that mass loading them with a central Si pad (originally a fabrication artifact) had little effect on their torsional Q, enabling us to realize chip-scale torsion pendula with Q-m products as high as kilograms and damping rates as low as microhertz. Owing to their significant gravitational stiffness, these devices can, in fact, possess higher Q than unloaded ribbons, due to gravitational dissipation dilution. They also show promise as chip-scale pendulum gravimeters, achieving spectral resolutions of several micro-g for first-generation devices.
In Fig. 2, we present a study of three mass-loaded nanoribbons (green) alongside unloaded ribbons (red). Each ribbon is 75 nm thick with a width of w ¼ 25, 50, or 100 μm. To load the ribbon, as shown in Fig. 2(b), a central defect is intentionally underetched [21], producing a 100-μm-thick, 600 × 600-μm 2 -wide Si pad with mass m pad ≈ 0.1 mg. As shown in Fig. 2(d The reduced frequency and increased Q of the "torsion micropendula" in Fig. 2 suggest that their dynamics are strongly influenced by the local acceleration of gravity g. To confirm this, as shown in Fig. 4, the pendula were inverted and their frequency and Q were compared to the lumped mass model:
where k g ¼ m p gh p =2 and I are the gravitational stiffness and moment of inertia of the pad (mass m p and thickness h p ), respectively, and -ðþÞ indicates the inverted (noninverted) orientation. As shown in Figs. (ω 35), in good agreement with Eq. ( 4) for k g ≈ k σ ≈ 200k E . Similar agreement was observed for wider, stiffer ribbons, as highlighted by the inset in Fig. 2(e).
Owing to their large Q-k g product, it is intriguing to consider using mass-loaded Si 3 N 4 nanoribbons as pendulum "clock" gravimeters. The potential of our first-generation devices can be seen by noting that their spectral resolution to gravity (the gravity change producing a frequency shift equal to the pendulum damping rate) is at the micro-g level, viz.
where g 0 is standard gravity. This resolution can be improved by driving the pendulum and averaging its frequency, yielding an ideal thermal-noise-limited Allan deviation [34,35] of
where τ is the averaging time,
þ are the frequency Allan deviation and energy damping time of the pendulum, respectively, and ϵ ¼ hθ 2 i=hθ 2 th i is the ratio of the driven to thermal energy. To explore its potential as a gravimeter, the 34 Hz pendulum shown in Fig. 4 was mounted on a 1 Hz vibration isolation stage (atop a floated optical table), and its resonance frequency was tracked using a digital algorithm [21]. Allan deviations were then recorded with the device free running (green) and in free decay (red, with an energy ringdown time of τ þ ¼ 1.1 × 10 4 s) after an impulsive radiation pressure drive. As shown in Fig. 4(e), the freerunning Allan deviation was within a factor of 2 of the thermal noise limit
] (dashed green), for averaging times τ ≲ 1000 s, saturating at σ ω þ =2π ≈ 70 μHz and a gravity resolution of σ g ¼ Δg ffiffiffiffiffiffiffiffiffi ffi τ þ =τ p ≈ 20μg 0 . The driven Allan deviation was 10 times lower, reaching σ ω þ =2π ≈ 18 μHz (σ g ≈ 4 μg) at 300 s; however, both cases are limited by a frequency drift of dω þ =dτ ≈ 2π × 0.3 mHz=h at late times (dashed red). As shown in the inset, we find this drift to be well correlated with the 0.3 K=h temperature drift of the device holder, implying a frequency sensitivity of dω þ =dT ≈ 2π × 1 mHz=K. The blue curve in Fig. 4(e) is obtained by subtracting a scaled temperature measurement [21], yielding σ ω þ =2π ≈ 12 μHz (σ g ≈ 2.5 μg) at τ ≈ 800 s.
In summary, we studied high-stress Si 3 N 4 nanoribbons with width-to-thickness (w=h) ratios as large as 10 4 and found that their torsion modes experienced dissipation dilution scaling as ðw=hÞ 2 , yielding Q factors as high as 10 8 and Q-f products as high as 10 13 Hz. Owing to their large zero-point angular displacement S ZP θ ∼ ðnrad= ffiffiffiffiffiffi Hz p Þ 2 , these devices are promising for quantum-limited deflectometry studies. We demonstrated this by showing that an optical lever measurement could resolve the rotation of a nanoribbon with an imprecision 20 dB smaller than its zero-point motion, paving the way for the investigation of radiation pressure shot noise in torque. We also found that strained nanoribbons could be mass loaded without reducing their torsional Q, enabling us to fabricate chip-scale torsion pendula with damping rates as low as 7 μHz and Q-m factors as high as 1 kg. We explored how these devices might be used as "clock" gravimeters, achieving a resolution of several micro-g in 10 min, limited by thermal drift.
Looking forward, we highlight several routes to improvement. Starting with the devices themselves, we note that the Q ∼ ðw=hÞ 2 soft-clamping behavior of nanoribbons [Fig. 2(d)] is highly sensitive to clamp geometry. Our use of diagonal fillets was inspired by optimized Si 3 N 4 nanotrampolines [6,37]; as discussed in Supplemental Material [21], this choice turns out to be serendipitous, as square fillets [38] and unfilleted ribbons would have experienced increased bending loss and even buckling [39] beyond w > 100 μm, according to simulations (Fig. S3). Confirming this behavior and exploring routes to further clamp optimization-a task which seems ripe for an emerging set of optimization algorithms [40,41]-will be important to realizing Q > 10 8 devices. The robustness of torsion modes of nanoribbons to anchor loss [42], mass loading [20], and surface loss [22] is also an open question, with potential for substantial improvement over flexural modes.
With regards to deflectometry-based quantum optomechanics, a key next step will be to search for motion due to radiation pressure shot noise in torque, S BA θ ≃ ðS ZP θ Þ 2 =S imp θ [18]. Starting with the measurement in Fig. 3, for which S BA θ =S th θ ∼ 10 -4 , a hypothetical route to S BA θ =S th τ ∼ 1 would be to precool the nanoribbon to 4 K, decrease its mass by a factor of 10 (e.g., by thinning and shortening the ribbon), and increase the reflected optical power by 10 dB, leveraging reduced absorption at telecommunications wavelengths [43], a photonic crystal mirror [44], and/or cavity enhancement [45], to mitigate photothermal heating. An alternative approach would be to look for signatures of S BA θ in optomechanical quantum correlations [46], which in principle could be discerned with S BA θ ≪ S th θ . Interestingly, for an optical lever measurement, the correlations are produced between the angular and lateral displacement of the deflected beam (or, equivalently, the HG 00 and HG 01 components of the deflected beam) [16] rather than phase and amplitude as in the case of an interferometric measurement [47]. While such spatial mode squeezing has been realized in multimode quantum optics experiments [15], it has only recently been explored in the optomechanical domain [48] and offers fertile analogies to ponderomotive squeezing [47].
Finally, to improve the performance of mass-loaded Si 3 N 4 nanoribbons as gravimeters, it will be important to study their frequency stability. A natural target is σ g < 10 -6 g 0 , which for the device in Fig. 4 could, in principle, be achieved with tenfold larger drive amplitude ffiffi ffi ϵ p or tenfold smaller drift. The corresponding fractional Allan deviation σ ω þ =ω þ ≲ 10 -7 has been widely achieved with megahertz flexural modes of Si 3 N 4 nanobeams [34,36,49,50]; however, it is an open question whether the lower frequency of our mass-loaded devices will pose new challenges. One constraint, on ϵ, is the intrinsic spring-softening nonlinearity of the pendulum ðk g ∝ cos θÞ, which, in principle, might be compensated by the Duffing nonlinearity of the nanoribbon [21]. With regards to drift, we note that the temperature sensitivity dω þ =dT of our devices agrees well with the predicted strain sensitivity dσ=dT of the Si 3 N 4 nanoribbon due to thermal expansion mismatch with the Si substrate [21,49]. Thermally invariant strain engineering has been used to reduce this source of drift by 2 orders of magnitude [49] and could be directly applied here. A compelling alternative would be to fabricate the nanoribbon from single-crystal strained Si [51], which could also enable higher Q 0 , especially at cryogenic temperatures. Ultimately, a combination of these strategies may be necessary to compete with state-of-the-art σ g ∼ 10 ng MEMS gravimeters [52][53][54]; nevertheless, the large dynamic range (AE2g 0 , enabled by the lossless tensile stiffness k σ > k g ), submilligram mass, and exceptional simplicity of the micropendulum approach invites further investigation.
Equation ( 1) is given by setting k 0 ¼ k E and k 0 ¼ k σ .
A continuum mechanics model for dissipation dilution of an elastic body is provided in Ref. [3] and recounted in Supplemental Material. In this model, the Q factor of a vibrational mode is related to the time-varying strain field it produces, viz.,
where (using Cartesian coordinates and index notation)
is the time-varying strain tensor and
is the time-varying displacement field of the mode with amplitude A, eigenfrequency ω 0 , and mode shape ϕ i . To derive Eq. ( 1), we assume a mode shape of the form
where θ denotes the angle of rotation about the ribbon axis z, k 1 ¼ π=L, and λL ≡ h ffiffiffiffiffiffiffiffiffiffiffiffiffi E=12σ p characterizes the rapidly varying mode curvature at the clamps [ϕ θ ðzÞ is obtained by solving the Euler-Bernoulli equation with boundary conditions ϕ θ ðzÞ ¼ ½ϕ θ ðzÞ 0 z ¼ 0 [3]]. Following the torsion theory of Saint-Venant [55] with ϕ θ ≪ 1 and "warping function" Wðx; yÞ ≈ -xy [appropriate for a thin beam (h ≪ w) with a constant twist rate ðu θ Þ 0 z ¼ 0], the mode shape can be expressed in Cartesian coordinates as [56][57][58] ϕ x ðx; y; zÞ ¼ -yϕ θ ðzÞ;
Applying the approximate form of Eq. (A8) to Eqs. (A4) and (A5) yields, after some cumbersome algebra (see Supplemental Material [21])
which reduces to Eq. ( 1) for a sufficiently long ribbon. In Supplemental Material [21], we show Eq. ( 1) agrees with numerical simulations for a rectangular ribbon and that the right-hand term can be significantly reduced by filleting the ribbon at clamps. We also extent Eq. (A9) to higher-order torsional modes.
To derive Eq. ( 2), we consider the optical lever setup shown in Fig. 3, in which a laser beam is reflected off the ribbon and directed toward a split photodiode. The nominal incidence angle is taken to be zero (for illustrative purposes, Fig. 3 is sketched with a nonzero incidence angle), and the laser beam waist (with 1=e 2 radius w 0 ) is assumed to coincide with the ribbon. Assuming the laser is in a TEM 00 Gaussian mode, the optical power difference recorded by the split photodiode is given by the familiar "knife edge" formula
where x is the lateral displacement of the laser beam from the photodiode midline (in the plane of the photodiode),
is the beam width at the photodiode, z is the distance of the photodiode from the ribbon, and z 0 ¼ πw 2 0 =λ is the Rayleigh length of the laser beam. Equation ( 2) is obtained by referring laser intensity shot noise
to an apparent fluctuation of the angular ribbon displacement
using the small angle approximation x ¼ 2θz and the small displacement approximation ðdΔP=dxÞj x≪wðzÞ ¼ ffiffiffiffiffiffiffi ffi 8=π p ½P=wðzÞ. Equation (2) assumes that the photodetector is placed in the far field, z ≫ z 0 .
The zero-point angular displacement spectral density of a torsion oscillator (green curve in Fig. 3) can be expressed as
where χðωÞ ≈ ½1 þ ω 2 1 ðωω 1 Þ 2 =Q 2 1 is the relative susceptibility of the oscillator (well approximated by a Lorentzian near resonance ω ≈ ω 1 ), hθ 2 zp i ¼ ℏ=2I 1 ω 1 is its zero-point displacement variance, and I 1 is its moment of inertia.
To obtain Eq. ( 3), we define I 1 ¼ m 1 r 2 ⊥ , where m 1 is the effective mass of the fundamental torsion mode of the ribbon defined for a point at the position of maximum transverse displacement, ϕ max y , and r ⊥ ¼ w=2 is the distance from the torsion axis to the point of maximum displacement. Ignoring mode curvature at the clamps, Eqs. (A7) and (A8) yield
where m phys ¼ ρhwL is the physical mass of the ribbon.