††thanks: These authors contributed equally.††thanks: These authors
contributed equally.

# High frequency torsional motion transduction using optomechanical coupled
oscillators

Hamidreza Kaviani  Institute for Quantum Science and Technology, University of
Calgary, Calgary, AB, T2N 1N4, Canada  National Research Council of Canada
Nanotechnology Research Centre, Edmonton, AB, T6G 2M9, Canada  Bishnupada
Behera  Institute for Quantum Science and Technology, University of Calgary,
Calgary, AB, T2N 1N4, Canada  National Research Council of Canada
Nanotechnology Research Centre, Edmonton, AB, T6G 2M9, Canada  Ghazal
Hajisalem  Institute for Quantum Science and Technology, University of
Calgary, Calgary, AB, T2N 1N4, Canada  National Research Council of Canada
Nanotechnology Research Centre, Edmonton, AB, T6G 2M9, Canada  Gustavo de
Oliveira Luiz  nanoFAB Centre, University of Alberta, Edmonton, AB, T6G 2V4,
Canada  David P. Lake  Institute for Quantum Science and Technology,
University of Calgary, Calgary, AB, T2N 1N4, Canada  National Research Council
of Canada Nanotechnology Research Centre, Edmonton, AB, T6G 2M9, Canada  Paul
E. Barclay<EMAIL_ADDRESS>Institute for Quantum Science and Technology,
University of Calgary, Calgary, AB, T2N 1N4, Canada  National Research Council
of Canada Nanotechnology Research Centre, Edmonton, AB, T6G 2M9, Canada

###### Abstract

Using light to measure an object’s motion is central to operating mechanical
sensors that probe forces and fields. Cavity optomechanical systems embed
mechanical resonators inside optical resonators. This enhances the sensitivity
of optomechanical measurements, but only if the mechanical resonator does not
spoil the properties of the optical cavity. For example, cavity optomechanical
detection of resonators made from optically absorbing materials, or whose
geometry does not possess suitable spatial symmetry, is challenging. Here we
demonstrate a system that overcomes challenges in measuring high-frequency
twisting motion of a nanodisk by converting them to vibrations of a photonic
crystal cavity. Optomechanical readout of the cavity then enables measurement
of the nanodisk’s torsional resonances with sensitivity $5.1\times
10^{-21}-1.2\times 10^{-19}\,\text{Nm}/\sqrt{\text{Hz}}$ for a mechanical
frequency range of 5–800 MHz. The nanodisk can be outfitted with magnetic
nanostructures or metasurfaces without affecting the optical properties of the
cavity, making the system suitable for torque magnetometry and structured
light sensing.

The enhanced optomechanical interaction between light and mechanical
resonances in nanophotonic cavities allows ultra-precise measurement of force
and displacement Aspelmeyer et al. (2014); Xia et al. (2020); Sun et al.
(2012); Gavartin et al. (2012); Gomis-Bresco et al. (2014); Ho et al. (2015);
Han et al. (2014); Li et al. (2021); Liu et al. (2021), enabling sensors
including accelerometers Krause et al. (2012), ultrasound detectors Basiri-
Esfahani et al. (2019), mass spectrometers Sansa et al. (2020); Li and Zhu
(2012), and atomic force measurement devices Liu et al. (2012). Cavity
optomechanical devices can also sensitively detect torque Wu et al. (2014);
Huang et al. (2017); He et al. (2016); Kim et al. (2013); Kaviani et al.
(2020), leading to advances in torque magnetometry Wu et al. (2017); Kim et
al. (2017), spin detection of electrons Kim et al. (2016), and measurement of
photon spin and orbital angular momentum He et al. (2016); Kaviani et al.
(2020). Ideally, mechanical resonators in these systems are integrated
directly within the optical cavity to maximize transduction strength: the
conversion of mechanical motion to an optical response. However, many of these
applications require metallic or magnetic materials that degrade the cavity’s
properties if they interact with its optical field. For example, torque
magnetometers incorporate permalloy nanostructures Wu et al. (2017), while
mechanical transducers of microwave fields incorporate metal Jiang et al.
(2020). Challenges also arise with resonators whose geometry precludes their
integration directly within nanoscale cavities such as photonic crystals and
whispering gallery mode resonators whose optical confinement relies upon
precisely engineered geometry that can not be perturbed, or whose mechanical
resonance spatial symmetry results in vanishing optomechanical coupling to a
cavity’s optical field.

Previously, these challenges have been addressed using mechanical resonators
whose interaction with the cavity optical field is limited to regions not
patterned with lossy material Wu et al. (2017), or with resonators that only
weakly perturb the optical cavity volume Kim et al. (2017), enabling sensitive
transduction of MHz frequency mechanical resonances. Here we demonstrate a new
approach that couples two mechanical resonators–one actuated by an external
signal, and one integrated within an optical cavity–allowing individual
optimization of the system’s actuation and readout properties. We show that
through resonant mechanical coupling, a photonic crystal readout cavity can
sensitively monitor motion of a high mechanical frequency nanodisk positioned
far from the cavity’s optical mode. The nanodisk’s resonances are suited for
actuation by sources of torque commonly encountered in magnetometry Wu et al.
(2017) and structured light Kaviani et al. (2020), and can be designed to
reach frequencies $>700\,\text{MHz}$, nearly two orders of magnitude larger
than previous optomechanical torque sensors Kim et al. (2016); Wu et al.
(2017). The photonic crystal cavity allows sensitive measurement of the
nanodisks’s mechanical resonances via their resonant mechanical coupling to
its mechanical modes. This enables detection of high frequency nanodisk modes,
creating a path towards resonant measurement of nanomagnetic dynamics
Hajisalem et al. (2019), enhancement of the Einstein-de Haas effect Losby et
al. (2018); Mori et al. (2020), studies of spin and orbital angular momentum
of photons and electrons Kim et al. (2016); Kaviani et al. (2020); Delord et
al. (2020), and transduction of radiofrequency signals Bagci et al. (2014);
Pearson et al. (2020); Hajisalem et al. (2019).

Figure 1: (a) Finite element simulation of an in-plane hybridized torsional
resonance of the device. Note that the nanodisk’s displacement $x_{\text{d}}$
is related to radius $r$ and angle of rotation $\theta$ of the nanodisk by
$x_{\text{d}}=r\theta$, using the small angle approximation. (b) SEM image of
the device. Finite difference time domain simulation of electric field profile
of the zipper cavity: (c) fundamental bonded mode with
$\lambda=1559\,\text{nm}$, $Q_{\text{o}}=1.8\times 10^{7}$ and (d) fundamental
anti-bonded mode with $\lambda=1530\,\text{nm}$, $Q_{\text{o}}=9.2\times
10^{7}$. (e) The torque sensing device (represented inside the yellow box) can
be modeled as a coupled oscillator system where the nanodisk and readout
nanobeam are mechanically coupled. The readout nanobeam forms one half of a
photonic crystal zipper cavity. Motion of both the readout nanobeam and the
auxiliary nanobeam of the cavity are optomechanically coupled to the cavity
mode. Dynamics of each mode are described by frequency $\omega$ and damping
rate $\gamma$. Mechanical coupling between the nanodisk and readout nanobeam
resonances is determined by $g_{\text{M}}$. The cavity optomechanical coupling
is described by $g_{\text{OM}}$. Figure 2: (a) Individual simulated mechanical
in-plane flexural resonances of uncoupled readout nanobeam (magenta) and
torque actuated mechanical resonances of uncoupled nanodisk (orange, green,
cyan - for first three resonances respectively, and blue - for rest higher
order resonances) as a function of the length of nanodisk’s support beams
$l_{\text{s}}$. (b) Simulated eigenfrequency of the in-plane hybridized
torsional resonances of the device - coupled nanodisk and readout nanobeam -
as a function of $l_{\text{s}}$. (c) Calculated mechanical coupling strengths
$g_{\text{M}}$ of a typical device for first three harmonics of nanodisk’s
resonances. (d) A zoom in of the region highlighted by a red box in (b)
showing the calculated optomechanical frequency shift per unit applied torque
as indicated by the color scale. (e) The displacement profiles of four
mechanical resonances with different $l_{\text{s}}$, labeled I-IV in (d). The
dashed grey line suggests the displacement profile at the center of the
readout nanobeam, where the optical cavity is localized. Figure 3: (a)
Measured optical transmission profile of the torque sensing device. (b)
Optomechanical spectrum of the device in ambient conditions. Peaks correspond
to thermally driven Brownian motion of the device’s mechanical resonances.
Peaks associated with torsional modes of the disk are identified. Three
specific modes, labeled I-III, are chosen to demonstrate the torque
sensitivity of this device, later in Fig. 4. Figure 4: Calculated displacement
resolution $\sqrt{S_{\text{xx}}}$ and torque sensitivity
$\sqrt{S_{\tau\tau}}$, and their respective fits - red solid line for
displacement resolution and dark blue solid line for torque sensitivity, for
the three specific modes of interest, labeled [I-III] in Fig. 3, of the device
$l_{\text{s}}=5.5\,\mu\text{m}$.

The coupled resonator system is illustrated in Fig. 1(a) and an example of a
fabricated device is shown in Fig. 1(b). A suspended silicon nanodisk anchored
to an unpatterned silicon-on-insulator (SOI) chip is connected to a photonic
crystal nanobeam that forms one half of an optomechanical zipper cavity Chan
et al. (2009); Eichenfield et al. (2009). The second half of the zipper cavity
is separated by a small gap from the coupled resonators and anchored to the
unpatterned region of the chip; it is labeled as the auxiliary nanobeam in
Fig. 1(b). The nanodisk can be mechanically driven by sources of torque
($\tau$), for example from nanomagnetic structures used in torque magnetometry
Wu et al. (2017); Hajisalem et al. (2019); Losby et al. (2018), or by angular
momentum transfer from structured optical fields Kaviani et al. (2020). Its
design addresses limitations of previous photonic crystal torque sensors that
detect cantilever-like motion of split-beam cavities Wu et al. (2014),
extending both the frequency range and nature of mechanical resonances that
can be routinely measured. This is accomplished by resonantly coupling a wide
spectrum of high frequency mechanical resonances of the nanodisk to mechanical
resonances of the zipper cavity. The zipper cavity combines high optical
quality factor $Q_{\text{o}}$ with large optomechanical coupling. It is formed
following the fabrication process in Ref. Wu et al. (2014). The suspended
photonic crystal nanobeams (width $500\,\text{nm}$, thickness
$220\,\text{nm}$) are separated by a gap $70\,\text{nm}$ and patterned with
holes of radius 110 nm. The hole spacing in each nanobeam tapers quadratically
from $a_{\text{o}}=400\,\text{nm}$ in the cavity’s mirror region to
$a_{\text{c}}=350\,\text{nm}$ at the center of the cavity. The resulting
cavity supports fundamental bonded and anti-bonded optical modes shown in
Figs. 1(c) and 1(d), respectively, with predicted radiation loss limited
$Q_{\text{o}}>10^{7}$ in the 1500 nm wavelength band. Mechanical motion of the
nanobeams can change the gap separating them, shifting the cavity mode
frequencies. The frequency shift per unit displacement is quantified by
optomechanical coupling coefficients $g^{+}_{\text{OM}}/2\pi\approx
85\,\text{GHz/nm}$ and $g^{-}_{\text{OM}}/2\pi\approx 2\,\text{GHz/nm}$
between the fundamental in-plane mechanical flexural resonance of the
nanobeams and the bonded and anti-bonded optical cavity modes, respectively.
The larger optomechanical coupling of the bonded mode arises from its higher
field concentration in the gap, making it more susceptible to vibrations of
the nanobeams. As a result, we focus on the bonded mode in this study.

In the absence of coupling between the nanodisk and the photonic crystal
nanobeam, torque actuated motion of the nanodisk is not transduced by the
cavity. However, transduction becomes possible when mechanical coupling is
present. Intuitively, this transduction will be enhanced if the nanodisk
motion is resonant with a nanobeam mode. This coupled oscillator approach to
transduction of the ‘remote’ and nominally dark nanodisk resonances is
described by the model illustrated in Fig. 1(e). Motion of the readout
nanobeam connected to the nanodisk is converted to an optical signal through
its dispersive cavity optomechanical coupling quantified by $g_{\text{OM}}$.
The dynamics of coupled nanodisk and readout nanobeam are described by
equations of motion for the displacement amplitudes $x_{\text{d}}$ and
$x_{\text{b}}$ of the nanodisk and readout nanobeam resonances, respectively:

$\displaystyle\ddot{x}_{\text{d}}$ $\displaystyle=-\omega_{\text{d}}^{\prime
2}x_{\text{d}}-\gamma_{\text{d}}\dot{x}_{\text{d}}+\sqrt{\frac{m_{\text{b}}}{m_{\text{d}}}}g_{\text{M}}^{2}x_{\text{b}}+\frac{\tau_{\text{t}}}{I_{\text{d}}}f(t)\,,$
(1a) $\displaystyle\ddot{x}_{\text{b}}$
$\displaystyle=-\omega_{\text{b}}^{\prime
2}x_{\text{b}}-\gamma_{\text{b}}\dot{x}_{\text{b}}+\sqrt{\frac{m_{\text{d}}}{m_{\text{b}}}}g_{\text{M}}^{2}x_{\text{d}}\,,$
(1b)

where $\omega_{\text{d}}$, $\gamma_{\text{d}}$, $m_{\text{d}}$, and
$\omega_{\text{b}}$, $\gamma_{\text{b}}$, $m_{\text{b}}$, are the frequency,
damping rate, and mass, respectively, of each uncoupled mechanical resonator.
Note that
$\omega^{\prime}_{\text{d}}=\omega_{\text{d}}+\sqrt{\frac{m_{\text{b}}}{m_{\text{d}}}}g_{\text{M}}^{2}$
and
$\omega^{\prime}_{\text{b}}=\omega_{\text{b}}+\sqrt{\frac{m_{\text{d}}}{m_{\text{b}}}}g_{\text{M}}^{2}$
are the modified frequencies, of the nanodisk and the readout nanobeam,
respectively, when they are coupled. These equations include an external
torsional drive $f(t)$ with an amplitude $\tau_{\text{t}}$ applied to the
nanodisk, whose moment of inertia $I_{\text{d}}$ is described by an effective
mass $m_{\text{eff}}$ and effective radius $r_{\text{eff}}$ that is determined
from the moment of inertia of the device, which relates the device’s angular
velocity to its energy, and is defined in Hauer et al. (2013). Note that for
generality an arbitrary drive is used here, where $f(t)$ is a real function.
The mechanical susceptibilities
$\chi_{\text{d,b}}(\omega)=(\omega_{\text{d,b}}^{\prime
2}-\omega^{2}-i\gamma_{\text{d,b}}\omega)^{-1}$ and mechanical coupling
$g_{\text{M}}$ determine how effectively this applied torque is converted into
motion of the readout nanobeam. The nanobeam amplitude can be derived by
Fourier transforming Eqs. 1 and solving for $x_{\text{b}}$ to obtain,

$x_{\text{b}}(\omega)=\frac{g_{\text{M}}^{2}}{\sqrt{\frac{m_{\text{b}}}{m_{\text{d}}}}((\chi_{\text{d}}(\omega)\chi_{\text{b}}(\omega))^{-1}-g_{\text{M}}^{4})}\frac{\tau_{\text{t}}}{I_{\text{d}}}f(\omega)\,.$
(2)

Equation 2 shows that the resonantly driven transduced amplitude is enhanced,

$|x_{\text{b}}(\omega_{\text{m}})|=\frac{g_{\text{M}}^{2}}{\sqrt{\frac{m_{\text{b}}}{m_{\text{d}}}}(\gamma_{\text{d}}\gamma_{\text{b}}\omega_{\text{m}}^{2}+g_{\text{M}}^{4})}\frac{\tau_{\text{t}}}{I_{\text{d}}}\,,$
(3)

when the nanodisk and readout nanobeam are resonant, i.e.,
($\omega^{\prime}_{\text{d}}=\omega^{\prime}_{\text{b}}=\omega_{\text{m}}$).
Here, for simplicity we have chosen the drive amplitude to be unity. The
equation also implies that the mechanical mode frequencies become
renormalized, as expected for a coupled system, with a frequency difference
$\Delta\omega^{2}\approx 2g_{\text{M}}^{2}$.

The coupling strength $g_{\text{M}}$ depends on the device geometry and modes
of interest. It can be determined by calculating the mode spectrum of the
coupled resonators for varying device parameters such that the nanodisk and
nanobeam modes are tuned through resonance. An example of this approach is
shown in Fig. 2. We first show in Fig. 2(a) the mechanical frequencies of the
individual components of the device for varying length $l_{\text{s}}$ of the
support beams anchoring the suspended nanodisk. We see that the nanobeam’s in-
plane flexural resonance frequencies remain constant, as expected since the
nanobeam geometry is not affected by the support length, while the resonance
frequencies of the nanodisk’s in-plane twisting modes decrease quadratically
with increasing $l_{\text{s}}$. In-plane twisting modes can be efficiently
actuated by torque and for simplicity are referred to below as torsional
modes.

At certain $l_{\text{s}}$ values, the mechanical resonances of the uncoupled
nanodisk and readout nanobeam are resonant and the resonance frequencies are
renormalized by the mode coupling. This is shown in Fig. 2(b), which plots the
full device’s simulated mode spectrum for varying $l_{\text{s}}$. The anti-
crossing behaviour at points where the nanodisk and readout nanobeam modes are
resonant is a signature of coupling between them. This mechanical coupling is
the key feature that makes transduction of torsional actuation of nanodisk to
in-plane flexural motion of readout nanobeam possible. Figure 2(c) shows
$g_{\text{M}}$ values extracted from the anti-crossing widths as a function of
$l_{\text{s}}$ for the first three harmonics of the nanodisk’s twisting
resonances. The trend of increasing $g_{\text{M}}$ with increasing mechanical
frequency suggests that higher frequency nanodisk resonances have better
overlap with the nanobeam resonances at their clamping point. Similar
increases in coupling between modes with increasing frequency are predicted
when considering phononic clamping loss in nanomechanical resonators Wilson-
Rae (2008).

Optical detection of nanobeam displacement from torque applied to the nanodisk
depends on the optomechanical coupling strength between the hybridized mode
and the photonic crystal cavity, the mechanical mode compliance, and the
mechanical coupling between the nanodisk and the nanobeam. To predict the
efficacy of a given mode to optomechanically detect torque given all of these
competing factors, we directly calculate the frequency shift of the cavity
mode for a given torque. We achieve this by numerically calculating the
mechanical displacement of the full device per unit applied external torque to
the nanodisk, from which we then determine the resulting torsional
optomechanical frequency shift $\Delta\omega_{\text{o}}\propto
g_{\text{M}}^{2}g_{\text{OM}}$. This shift, which captures both the device’s
cavity optomechanical coupling and the mechanical mode coupling, is shown by
the color of the points in Fig. 2(d) for $l_{\text{s}}$ swept through a
typical anti-crossing. The simulated mechanical displacement profiles at
several points within the anti-crossing are shown in Fig. 2(e). These
simulated mechanical modes illustrate the evolution of the mode profiles
across a mechanical anti-crossing region demonstrating the strong
hybridization of the torsional modes at the center of the anti-crossing. This
observation suggests that for torque sensing these strongly hybridized
torsional modes are modes of interest. However, to effectively readout these
modes, we require a significant non-zero displacement at the center of the
nanobeam where the photonic crystal optical cavity is located. To illustrate
that not all hybridized modes satisfy this criteria, we have added a grey
dashed line to the center of the mode profiles (II) and (III) in Fig. 2(e).
The lower frequency strongly hybridized mode has an anti-node at its center
which results in a stronger interaction with the optical cavity mode, leading
to the optomechanical frequency shift $\Delta\omega_{\text{o}}$ and
sensitivity being maximized at the center of the anti-crossing (II). And as we
move away from this point towards modes (I) and (IV), we observe a decrease in
$\Delta\omega_{\text{o}}$, as clearly suggested by their smaller amplitudes.
On the other hand, the higher frequency strongly hybridized mode has a node at
its center, which results in not only zero displacement at the cavity centre,
but an odd spatial symmetry of the displacement profile interacting with the
cavity mode. This results in a weak interaction with the optical cavity mode,
leading to a small $\Delta\omega_{\text{o}}$ and sensitivity at the center of
the anti-crossing (III). More generally, for our cavity design, nanobeam modes
with an even number of anti-nodes have small $\Delta\omega_{\text{o}}$
compared to modes with an odd number of anti-nodes.

To demonstrate the efficacy of the mode coupling effects discussed above, we
experimentally study a range of devices with varying $l_{\text{s}}$. We
optically measure the optomechanically transduced mechanical modes of our
fabricated devices using a dimpled optical fiber taper waveguide Michael et
al. (2007) to evanescently couple light into and out of the cavity. Figure
3(a) shows the transmission profile of an optical cavity mode from a typical
device. The measured mode has $Q_{\text{o}}\approx 160,000$ and a wavelength
in the 1500 nm wavelength range. In a sideband unresolved system where the
cavity field instantaneously follows the nanobeam’s mechanical motion, the
change in the cavity’s optical intensity response from an applied torque is
$\propto\tau(\omega_{\text{t}})\Delta\omega_{\text{o}}\text{d}H/\text{d}\omega_{\text{L}}$.
Here $\omega_{\text{L}}$ is the frequency of the probe laser input to the
cavity, and $\text{d}H/\text{d}\omega_{\text{L}}$ is the slope of the cavity
mode lineshape, where $H$ is the normalized cavity transmission as a function
of laser frequency. Due to the device’s high sensitivity arising from its
high-$Q_{\text{o}}$, large $\Delta\omega_{\text{o}}$, and low mass and
associated large mechanical susceptibility (see Eq. (2)), thermomechanical
driven motion of the mechanical resonances can be observed. Figure 3(b) shows
a typical mechanical spectrum of a device ($l_{\text{s}}=5.5\,\mu\text{m}$).

Figure 5: Measured mechanical spectrum of the devices for varying support
length. The signal strength is color coded. The red dashed line is guide to
the eye that follows the moving torsional peaks.

A spectrum analyzer is used to convert the temporal fluctuations in fiber-
taper transmission to a power spectral density. Peaks in the spectrum
correspond to thermally driven motion of the nanobeam’s mechanical resonances.
Comparing the frequencies of the peaks in the spectrum with simulated values,
we identified the hybridized torsional modes shown in Figure 3(b) at
frequencies that range from $6.3-784.6\,\text{MHz}$. Following the process in
Wu et al. (2014), one can extract the displacement resolution
$\sqrt{S_{\text{xx}}}$ and torque sensitivity $\sqrt{S_{\tau\tau}}$ from these
measurements as shown in Fig. 4. As the torque sensitivity
$\sqrt{S_{\tau\tau}}$ in our system is limited by thermomechanical noise, it
can be extracted from the thermomechanical displacement resolution
$\sqrt{S_{\text{xx}}}$ by,
$\sqrt{S_{\tau\tau}}=\frac{m_{\text{eff}}r_{\text{eff}}}{|\chi_{\text{m}}(\omega)|}\sqrt{S_{\text{xx}}}$.
This displacement resolution in our system can be calculated by measuring the
signal-to-noise ratio (SNR) of the thermomechanically driven mechanical
resonances using the expression
$\sqrt{S_{\text{xx}}}=\sqrt{\frac{4k_{\text{B}}TQ_{\text{m}}/m_{\text{eff}}\omega_{\text{m}}^{2}}{(SNR)}}$,
where $k_{\text{B}}$, $T$, $Q_{\text{m}}$, are the Boltzmann constant,
temperature and mechanical quality factor, respectively.

To analyze the torque sensitivity, we study three specific modes of interest,
labeled [I-III] in the mechanical spectrum of the
$l_{\text{s}}=5.5\,\mu\text{m}$ device shown in Fig. 3(b), and plot them in
Fig. 4. Based on the fits, we find a torque sensitivity range of $5.1\times
10^{-21}-1.2\times 10^{-19}\,\text{Nm}/\sqrt{\text{Hz}}$ depending on the
mechanical mode. In particular, the weakly hybridized torsional mode with
frequency $6.3\,\text{MHz}$, which is labeled [I] in Fig. 3(b) has the highest
sensitivity value given above, while the hybridized torsional mode with
frequency $784.6\,\text{MHz}$, which is labeled [III] in Fig. 3(b), is at the
lower end of this sensitivity range. The strongly hybridized torsional mode
with frequency $31.7\,\text{MHz}$, which is labeled [II] in Fig. 3(b) also has
a high sensitivity value of $5.3\times 10^{-21}\,\text{Nm}/\sqrt{\text{Hz}}$.
Though mode [II] is more strongly hybridized than mode [I], because of
relatively higher frequency and lower SNR, it has a slightly lesser torque
sensitivity than that of mode [I]. Note that these measurements were made at
room temperature and in ambient pressure. Operating in vacuum we expect the
mechanical quality factors to increase, which will enhance the torque
sensitivity Wu et al. (2014). Spectra were typically measured over an 83s
acquisition time. Further studies are required to determine how drift or other
timescale dependent noise processes will impact the quoted sensitivities.

To further study the hybridization between the nanodisk and nanobeam
resonances, the optomechanical spectrum was measured for devices with varying
$l_{\text{s}}$. Figure 5 shows this measurement for $l_{\text{s}}$ swept from
$1\,\mu\text{m}$ to $10\,\mu\text{m}$ in increments of $1\,\mu\text{m}$. As
shown by the red trend line, the torsional peaks shift with the expected anti-
crossing pattern predicted by the theory discussed in Figs. 2(a) and 2(b).
Note that each measurement involves a unique device, and the contribution to
the optical transduction related to the optical cavity mode and its coupling
to the fiber taper will vary between devices.

Figure 6: Measured mechanical frequencies of the devices with varying support
length (blue). The small red dots are simulated eigenfrequencies of the
devices.

To compare the measured results with theory more quantitatively, Fig. 6 plots
the frequencies of the hybridized torsional resonances measured in Fig. 5 as a
function of support length, together with the simulated values. The
experimentally observed frequency dependence on support length is in good
agreement with the simulated values, with variations most likely attributed to
fabrication imperfections. Most crucially, the width of the observed
anticrossings closely match the predicted values. It is worth noting that
peaks corresponding to nanobeam flexural resonances with an even numbers of
anti-nodes do not appear in the spectrum. This can be understood from the fact
that optomechanical couplings for these modes is near zero due to their odd
symmetry around the center of the optical cavity.

The device demonstrated here shows how the high sensitivity of nanoscale
optomechanical cavities can be harnessed to detect motion of nanomechanical
resonators not integrated directly within the optical cavity. By coupling
remote and optically dark mechanical resonances to resonators integrated
within a photonic crystal nanocavity, mechanical resonances with a wide range
of frequencies, mode profiles and symmetries can be sensitively transduced.
Since the nanomechanical resonator does not interact directly with the cavity
field, it can be decorated with optically lossy metals or magnetic materials.
This is of particular interest for advancing the performance and frequency
range of optomechanical torque magnetometers used to probe nanomagnetic
properties of permalloy Wu et al. (2017). To this end, the device has
demonstrated torque sensitivity of $5.1\times 10^{-21}-1.2\times
10^{-19}\,\text{Nm}/\sqrt{\text{Hz}}$ for a mechanical frequency range of
$5-800\,\text{MHz}$. Previous nanoscale cavity optomechanical torque sensing
devices Wu et al. (2014); Kim et al. (2013, 2016) achieve similar or better
sensitivities, but are limited to frequencies below 20 MHz. To the best of our
knowledge, the current state-of-the-art in torque sensitivity in an on-chip
platform is $\sim 10^{-24}\,\text{Nm}/\sqrt{\text{Hz}}$ at cryogenic
temperatures in the $<20$ MHz mechanical frequency range Kim et al. (2016),
while the current state-of-the-art in a levitated nanoparticle in vacuum
platform is $\sim 10^{-27}\,\text{Nm}/\sqrt{\text{Hz}}$ at room temperature
with a rotation frequency $>5$ GHz Ahn et al. (2020). The sensitivity of our
devices can be further improved by orders of magnitude by performing the
experiment in ultra high vacuum and cryogenic temperatures where mechanical
quality factors in silicon devices can reach $Q_{\text{m}}>10^{8}$ MacCabe et
al. (2020). Operation at higher ($>\text{GHz}$) frequencies may be possible
using this approach by combining torque actuated mechanical resonators with
optomechanical crystal cavities and waveguides Fang et al. (2017).

In the near future, the device’s ability to operate at frequencies $>$ 100 MHz
will enable high frequency torque magnetometry studies of nanomagnetic
systems, opening the door to observation of the Einstein-de Haas effect Losby
et al. (2018); Mori et al. (2020), as well as resonant coupling between
nanomagnetic spin dynamics and the optomechanical system. Signatures of the
Einstein-de Hass effect in driven magnetic spin-mechanic systems become more
pronounced at high frequencies Mori et al. (2020). Studies of magnetic
resonances typically occur at frequencies above 100 MHz, and one needs to be
at a similar or higher mechanical detection bandwidth to study their
magnetization dynamics. In cases where the detection bandwidth is much
narrower than the rate of angular momentum transfer between the magnetic and
mechanical system, the ability to detect a torque will give insight into spin-
lattice relaxation processes, which are of fundamental importance to condensed
matter physics research. Furthermore, with the integration of the torsional
device with magnetic materials it will also become possible to generate
effective couplings between light and magnetic spins, with applications in
development of hybrid quantum systems Wallquist et al. (2009).

###### Acknowledgements.

The authors acknowledge helpful discussion with M. Mitchell, P. Parsa, D.
Sukachev, M. Freeman and J. Losby. This work was supported by the Natural
Sciences and Engineering Research Council Discovery Grant and Strategic
Partnership Grant programs, the Alberta Innovates Strategic Research Project
program, the Canadian Foundation for Innovation, and the National Research
Council Nanotechnology Initiative.

## References

  * Aspelmeyer et al. (2014) M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014), URL http://link.aps.org/doi/10.1103/RevModPhys.86.1391.
  * Xia et al. (2020) J. Xia, Q. Qiao, G. Zhou, F. S. Chau, and G. Zhou, Applied Sciences 10, 7080 (2020).
  * Sun et al. (2012) X. Sun, J. Zhang, M. Poot, C. Wong, and H. Tang, Nano Lett. 12, 2299 (2012).
  * Gavartin et al. (2012) E. Gavartin, P. Verlot, and T. Kippenberg, Nat. Nano. 7, 509 (2012).
  * Gomis-Bresco et al. (2014) J. Gomis-Bresco, D. Navarro-Urrios, M. Oudich, S. El-Jallal, A. Griol, D. Puerto, E. Chavez, Y. Pennec, B. Djafari-Rouhani, F. Alzina, et al., Nature communications 5, 1 (2014).
  * Ho et al. (2015) C. P. Ho, P. Pitchappa, B. W. Soon, and C. Lee, Optics express 23, 10598 (2015).
  * Han et al. (2014) K. Han, K. Zhu, and G. Bahl, Applied Physics Letters 105, 014103 (2014).
  * Li et al. (2021) B.-B. Li, L. Ou, Y. Lei, and Y.-C. Liu, Nanophotonics 10, 2799 (2021).
  * Liu et al. (2021) X. Liu, W. Liu, Z. Ren, Y. Ma, B. Dong, G. Zhou, and C. Lee, International Journal of Optomechatronics 15, 120 (2021).
  * Krause et al. (2012) A. G. Krause, M. Winger, T. D. Blasius, W. Lin, and O. Painter, Nat. Photon. 6, 768 (2012).
  * Basiri-Esfahani et al. (2019) S. Basiri-Esfahani, A. Armin, S. Forstner, and W. P. Bowen, Nature communications 10, 1 (2019).
  * Sansa et al. (2020) M. Sansa, M. Defoort, A. Brenac, M. Hermouet, L. Banniard, A. Fafin, M. Gely, C. Masselon, I. Favero, G. Jourdan, et al., Nature Communications 11, 1 (2020).
  * Li and Zhu (2012) J.-J. Li and K.-D. Zhu, Applied Physics Letters 101, 141905 (2012).
  * Liu et al. (2012) Y. Liu, H. Miao, V. Aksyuk, and K. Srinivasan, Opt. Express 20, 18268 (2012).
  * Wu et al. (2014) M. Wu, A. C. Hryciw, C. Healey, D. P. Lake, H. Jayakumar, M. R. Freeman, J. P. Davis, and P. E. Barclay, Phys. Rev. X 4, 021052 (2014), URL http://link.aps.org/doi/10.1103/PhysRevX.4.021052.
  * Huang et al. (2017) J. G. Huang, H. Cai, Y. Gu, L. Chin, J. Wu, T. Chen, Z. Yang, Y. Hao, and A. Liu, Applied Physics Letters 111, 111102 (2017).
  * He et al. (2016) L. He, H. Li, and M. Li, Science advances 2, e1600485 (2016).
  * Kim et al. (2013) P. Kim, C. Doolin, B. Hauer, A. MacDonald, M. Freeman, P. Barclay, and J. Davis, Appl. Phys. Lett. 102, 053102 (2013).
  * Kaviani et al. (2020) H. Kaviani, R. Ghobadi, B. Behera, M. Wu, A. Hryciw, S. Vo, D. Fattal, and P. Barclay, Optics Express 28, 15482 (2020).
  * Wu et al. (2017) M. Wu, N. L.-Y. Wu, T. Firdous, F. F. Sani, J. E. Losby, M. R. Freeman, and P. E. Barclay, Nature nanotechnology 12, 127 (2017).
  * Kim et al. (2017) P. Kim, B. Hauer, T. Clark, F. F. Sani, M. Freeman, and J. Davis, Nature Communications 8, 1355 (2017).
  * Kim et al. (2016) P. Kim, B. Hauer, C. Doolin, F. Souris, and J. Davis, Nature communications 7, 1 (2016).
  * Jiang et al. (2020) W. Jiang, C. J. Sarabalis, Y. D. Dahmani, R. N. Patel, F. M. Mayor, T. P. McKenna, R. Van Laer, and A. H. Safavi-Naeini, Nature Communications 11, 1 (2020).
  * Hajisalem et al. (2019) G. Hajisalem, J. E. Losby, G. de Oliveira Luiz, V. T. Sauer, P. E. Barclay, and M. R. Freeman, New Journal of Physics 21, 095005 (2019).
  * Losby et al. (2018) J. E. Losby, V. T. Sauer, and M. R. Freeman, Journal of Physics D: Applied Physics 51, 483001 (2018).
  * Mori et al. (2020) K. Mori, M. Dunsmore, J. Losby, D. Jenson, M. Belov, and M. Freeman, Physical Review B 102, 054415 (2020).
  * Delord et al. (2020) T. Delord, P. Huillery, L. Nicolas, and G. Hétet, Nature 580, 56 (2020).
  * Bagci et al. (2014) T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, et al., Nature 507, 81 (2014).
  * Pearson et al. (2020) A. Pearson, K. Khosla, M. Mergenthaler, G. A. D. Briggs, E. Laird, and N. Ares, Scientific reports 10, 1 (2020).
  * Chan et al. (2009) J. Chan, M. Eichenfield, R. Camacho, and O. Painter, Opt. Express 17, 3802 (2009).
  * Eichenfield et al. (2009) M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, Nature 459, 550 (2009).
  * Hauer et al. (2013) B. Hauer, C. Doolin, K. Beach, and J. Davis, Annals of Physics 339, 181 (2013).
  * Wilson-Rae (2008) I. Wilson-Rae, Phys. Rev. B 77, 245418 (2008), URL https://link.aps.org/doi/10.1103/PhysRevB.77.245418.
  * Michael et al. (2007) C. P. Michael, M. Borselli, T. J. Johnson, C. Chrystala, and O. Painter, Opt. Express 15, 4745 (2007).
  * Ahn et al. (2020) J. Ahn, Z. Xu, J. Bang, P. Ju, X. Gao, and T. Li, Nature Nanotechnology 15, 89 (2020).
  * MacCabe et al. (2020) G. S. MacCabe, H. Ren, J. Luo, J. D. Cohen, H. Zhou, A. Sipahigil, M. Mirhosseini, and O. Painter, Science 370, 840 (2020), eprint https://www.science.org/doi/pdf/10.1126/science.abc7312, URL https://www.science.org/doi/abs/10.1126/science.abc7312.
  * Fang et al. (2017) K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, Nature Physics 13, 465 (2017).
  * Wallquist et al. (2009) M. Wallquist, K. Hammerer, P. Rabl, M. Lukin, and P. Zoller, Phys. Scr. T 137, 014001 (2009).