Laser cooling a centimeter-scale torsion pendulum
Dong-Chel Shin Tina M. Hayward Dylan Fife Rajesh Menon Vivishek Sudhir





dongchel@mit.edu












vivishek@mit.edu





§ ABSTRACT 
Torsion pendula have long been pivotal in the study of classical gravitational forces. Experimental tests of gravity’s fundamental nature call for mechanical systems in the quantum regime while being sensitive to gravity.
We laser cool a centimeter-scale torsion pendulum to a temperature of 10 mK 
(average occupancy of 6000 phonons) starting from room temperature (equivalent to 2·10^8 phonons). 
This is achieved by optical radiation
pressure forces conditioned on a quantum-noise-limited optical measurement of the pendulum's angular displacement
with an imprecision 13 dB below that at the standard quantum limit (SQL).
The measurement sensitivity is the result of a novel `mirrored' optical lever that passively rejects extraneous 
spatial-mode noise by 60 dB.
The high mechanical quality (10^7) and quantum-noise-limited sub-SQL measurement imprecision demonstrate the
necessary ingredients for realizing the quantum ground state of torsional motion — a pre-requisite for 
mechanical tests of gravity's alleged quantum nature.


Laser cooling a centimeter-scale torsion pendulum
    Vivishek Sudhir
    September 9, 2024
=================================================




Introduction. 
Torsion pendula have long been pivotal in the measurement of weak fundamental forces, most notably 
in establishing the electrostatic inverse-square law <cit.>,
precision measurements of classical gravitational forces <cit.>, 
tests of the equivalence principle <cit.>, 
and the first observation of radiation pressure torque <cit.>.
In all these experiments, the torsion pendulum is employed as a sensor for a weak classical force. 

Recent interest in observing gravity's alleged quantum nature calls for
experiments where gravitationally attracting macroscopic mechanical oscillators are simultaneously prepared in
quantum states of their motion <cit.>. Torsion pendula are particularly suited for
such experiments on account of the low thermal Brownian noise of torsional suspensions even 
when mass-loaded <cit.>, and well-understood techniques for isolating gravitational interaction
between them even with masses as small as 100 mg <cit.>.
However, in contrast to the mature array of techniques available for quantum-limited measurement and control of 
linear motion within the field of cavity optomechanics <cit.>, 
experimental realization of similar techniques has remained
elusive for torsional motion.

In this letter we demonstrate laser cooling of a centimeter-scale high-quality torsion pendulum using a 
novel “mirrored optical lever” whose quantum-noise-limited sensitivity of 10^-12 rad/√(Hz) 
is 13 dB below the zero-point motion of the pendulum. Conditioned on this measurement, we apply optical radiation 
pressure torque on the pendulum so as to cool its angular motion to 10 mK from room temperature, corresponding to an average phonon occupation of 5964±39 (starting from ∼ 2×10^8).
In the following, we describe the torsion pendulum, the mirrored optical lever used to measure its angular motion,
its performance in terms of classical noise cancellation, its calibration, and utility in measurement-based 
feedback cooling of torsional motion.



High-Q centimeter-scale torsion pendulum. 
A remarkable advantage of torsional pendula in studies of gravity is that their suspensions can be realized with 
exceptionally high mechanical quality factor (Q) even when mass-loaded <cit.>. 
The reason is two-fold <cit.>: in a doubly-clamped bifilar (or ribbon-shaped) torsional suspension, 
tensile stress leads to dilution of the intrinsic mechanical dissipation, and the bifilar geometry 
is naturally soft-clamped so that loss at the clamps, and in the suspended mass, is suppressed.
Thus the intrinsic quality factor, Q_int, is elevated to <cit.>
Q_0 = Q_intD_Q where the dissipation dilution factor is D_Q ≈ (σ/2E)(w/h)^2;
here E and σ are Young's modulus and tensile stress, and w and h are the width and 
thickness of the ribbon. 
This is in marked contrast to tensile-stressed mass-loaded linear oscillators, where bending curvature due to the
loaded mass <cit.> erases the advantage from dissipation dilution.
This capability positions a macroscopic high-Q torsion pendulum as a unique candidate for both reaching 
its motional ground state and for gravitational experiments.

We fabricated a doubly-clamped 0.9-cm long thin-film (w=0.5mm, h=400nm) 
tensile-stressed (σ = 0.8GPa) torsion pendulum made of
stoichiometric Si_3 N_4. 
The device was fabricated starting from a double-sided Si_3N_4-on-Si 
wafer followed by lithography and reactive ion
etching. A second aligned lithography and etch created the window for optical access from the back-side. 
The device was released using a 24-hour etch in potassium hydroxide. The sample underwent meticulous cleaning with acetone, isopropyl alcohol, deionized water, and oxygen plasma (see SI for more details).
The specific device used in the current study has its fundamental torsional mode resonating at
Ω_0 = 2π·35.95kHz with quality factor Q = 1.4· 10^7 inferred by ringdown measurements 
(in vacuum, at 6e-7mbar) as shown in <ref>(c). 
The measured Q is consistent with the expected dilution factor D_Q ≈ 2300.

Principle of mirrored optical lever. To achieve quantum-limited readout of angular motion, 
we devised a `mirrored' optical lever. Its primary
advantage is passive rejection of classical noises arising from the laser beam's transverse displacement and tilt.
Suppose the output of a laser is predominantly in the fundamental Hermite-Gaussian mode with 
amplitude a̅ = √(P/ħω_ℓ) (P the optical power, ω_ℓ = 
2π c/λ the carrier frequency for a wavelength λ≈1064nm), then the optical field
can be expressed as <cit.> Ê_in(r,t) = (a̅ + a_00(t)) U_00(r) 
+ ∑_n,ma_nm(t) U_nm(r). Here U_nm(r) is the (n,m)-Hermite-Gaussian (HG_nm)
basis function (see SI); the operators {a_nm(t)} represent
fluctuations in the laser field, which in the ideal case, when the field is quantum-noise-limited, 
model the quantum vacuum fluctuations in HG_nm mode.
If this incident field is subjected to a transverse displacement δr = (δ x,0,0) and angular tilt
δθ at its beam waist, the optical field is transformed to 
Ê_out(r,t) ≈Ê_in(r,t) + a̅(δ x/w_0 + 
ikw_0δθ/2)U_10(r),
where w_0 is the waist size. 
That is, transverse or angular motion scatters light from the incident HG_00 mode to the 
HG_10 mode in proportion to the motion.
When the beam waist is placed at the location of the torsion pendulum, its motion θ_sig(t) 
is experienced twice by the reflected optical beam, i.e. 2θ_sig(t); 
the incident beam may also have extraneous classical 
noises in the transverse displacement (x_ext) and tilt (θ_ext). 
When the reflected beam is detected by a split balanced photodetector (SPD) downstream, the resulting
photocurrent fluctuations are described by its (symmetrized single-sided) power spectral density 
(see SI, we neglect correlations between the tilt and transverse displacement <cit.>)

    S_I[Ω] = 2/π(2η R     P k w_0 sinζ)^2
        [S_θ^sig[Ω] + 1/4S_θ^ext[Ω]
        + ( ζ/k w_0^2)^2 S_x^ext[Ω] 
        + π/2/2η (a̅k w_0)^2^2ζ],

where R is the responsivity of the detector, η the detection efficiency, 
and ζ is the Gouy phase shift from the torsion pendulum to the SPD. 
The last three terms represent the imprecision in the measurement arising from extraneous tilt noise
(S_θ^ext), extraneous transverse displacement noise (S_x^ext), and the fundamental 
imprecision from quantum vacuum fluctuations (from all odd-order HG modes that the SPD is sensitive to; 
see SI).
Thus, quantum-noise-limited detection of torsional motion is only possible if extraneous 
noise in the spatial mode of the laser beam is suppressed.

Our mirrored optical lever technique (shown in <ref>a,b) cancels classical spatial mode noise 
to achieve quantum-noise-limited detection. 
Specifically, the laser beam is split by a polarizing beam splitter (PBS) into two paths: one arm to the torsion pendulum, and the other to a retroreflector. 
A corner cube retroreflector produces a mirror-image of the laser beam's transverse displacement and tilt, 
i.e. δâ_10→ -δâ_10 after interacting with the retroreflector 
(<ref>(b)(i) shows the effect of this on input noise), whereas the torsion pendulum induces 
pure tilt (<ref>(b)(ii)).
The fields from the two arms are collected by quarter-wave plates at the PBS, and subsequently detected by the 
SPD with a Gouy phase control using a lens.
Careful balancing of the power and Gouy phase in the two arms ensures arbitrarily good cancellation 
of extraneous noises (acquired by the laser beam before the PBS, <ref>(b)(iii)) and
transduction of torsional motion into transverse displacement at the SPD (<ref>(b)(iv)). 
Thus, for the mirrored optical lever

    S_I[Ω] = 2/π(2η R P k w_0 sinζ)^2 [S_θ^sig[Ω] 
        + π/2/η (a̅k w_0)^2^2ζ],

immune to classical noises and thus quantum-noise-limited in its imprecision (see SI for more details).
The trade-off is double the quantum noise from the mirrored beam.

In contrast to interferometric detection of linear motion, our scheme is immune to laser phase noise;
yet, the mirrored optical lever is qualitatively similar to homodyne detection: the quantum-noise-limited imprecision
decreases inversely with optical power (a̅∝ P) and the Gouy phase plays the role of the 
homodyne angle. We place the SPD at Gouy phase ζ = π/2 which maximizes the angular signal.



Performance and calibration of mirrored optical lever. 
In order to investigate the sensitivity of the mirrored optical lever, we first operate it with a flat mirror, 
instead of the torsion pendulum, in the signal arm. 

Independent calibration of the SPD voltage into angular motion is crucial for further investigation.
(Note that estimating the angular displacement from the measured optical lever arm does not 
hold beyond the Rayleigh length.)
We perform direct calibration against a known frequency modulation. To wit, 
an acouto-optic deflector (AOD) is placed at the input plane of the entrance lens of a 4f imaging system
(see <ref>), with the torsion pendulum (and retroreflector) at the output plane of the second lens.
In this configuration, the tilt of the laser beam at the input plane can be modulated by frequency-modulating the AOD drive as 
Δθ_cal = (λ/v_c) Δ f_cal, where Δ f_cal is the frequency-modulation
depth, and v_c ≈5700m/s is the acoustic velocity of the AOD quartz crystal.


This known tilt change manifests as a voltage change (Δ V_cal) in the SPD signal.
By estimating the calibration factor, defined as α_cal=Δθ_cal/2Δ V_cal, the angular displacement spectrum is computed from the SPD spectrum: S_θ[Ω]=α_cal^2 S_V[Ω], where S_V[Ω] is the measured SPD spectrum (see SI for details).
<ref>(a) shows the voltage amplitude detected by the SPD with different frequency modulation depth of the AOD drive. The beam tilt modulation using the AOM ensures the linear calibration of the voltage to the angular spectrum with an R-square value of 0.99996, as revealed by the linear fit.

<ref>(b) demonstrates the calibrated performance of the mirrored optical lever. 
The angle-referred imprecision noise decreases as the reflected optical power increases, and 
reaches 2.6e-12rad/√(Hz) with 5 mW incident power, consistent with the quantum-noise scaling 
in <ref> corresponding to a detection efficiency η≈ 0.75.

The excess noise peaks between 200 Hz and 1 kHz are attributable to seismic noise-induced 
fluctuations (see SI). 
Without the mirrored arm, we observed extraneous low frequency tilt noise above 1 mW 
of optical power that compromised the linearity of the SPD; with the mirrored arm, we characterized input tilt 
noise suppression by ∼ 60 dB over 1 Hz (see SI).
Given that our lab temperature is stabilized to a precision around 20 mK <cit.>, we conjecture
that the low frequency drifts in input laser tilt are due to refractive index fluctuations from air currents.



Imprecision below the standard quantum limit. 
Next, we place the torsion pendulum in the signal arm of the mirrored optical lever. 
<Ref>(a) shows the power spectrum of the measured angular fluctuations of the fundamental torsional
mode as the power in the measurement field increases. 
The observed angle fluctuations 
θ_obs = θ_phys + θ_imp
consists of the physical motion of the pendulum θ_phys and the imprecision noise 
θ_imp of the optical lever. 

The physical motion, shown as the black line in <ref>(a), is predominantly due to the 
thermal Brownian motion of the mode due to its n_th = k_B T/(ħΩ_0) ≈ 2· 10^8 
average phonons at room temperature, and is described by the fluctuation-dissipation theorem <cit.>, 
S_θ^phys[Ω] = 4ħ(n_th + 12) χ_0[Ω]; here χ_0[Ω] = 
[I(-Ω^2 + Ω_0^2 + i ΩΓ_0[Ω])]^-1 is the susceptibility of the torsional mode
with moment of inertia I and damping rate Γ_0[Ω] = (Ω_0/Q)(Ω_0/Ω).
This model shown as the black line in <ref>(a) is based on independent 
measurement of the frequency, mechanical Q, and 
calibrated mode temperature. The latter is inferred by calibrating each spectra using our frequency
modulation technique, and assuming that the torsional mode is in thermal equilibrium.
We verified (see SI) that the mode temperature is constant across all powers except the highest, which shows a
13% increase, presumably due to optical absorption.

The observed imprecision noise (<ref>(a)) is inversely proportional to the optical power, 
consistent with quantum-noise-limited measurement.
The inset shows the measurement imprecision around resonance in
equivalent phonon units, n_imp = S_θ^imp/2S_θ^zp[Ω_0], where 
S_θ^zp[Ω_0] = 4θ_zp^2/Γ_0 is the peak spectral density of the zero-point motion
θ_zp = √(ħ/(2I Ω_0)) of the torsional mode.

The overall low imprecision of the measurement, n_imp≈ 5· 10^-2, can be best 
contextualized by comparing it against the standard quantum limit (SQL). 
The SQL is the minimum total noise
in the measurement allowed by quantum mechanics without the use of any quantum enhancement: 
it is the sum of the zero-point motion of the torsional mode, and the minimum trade-off between the measurement imprecision and the concomitant quantum back-action of the measurement
due to quantum fluctuations in the radiation torque of the measurement beam.
It can be shown (see SI) that for the measurement scheme we use here, the SQL is given by 
S_θ^SQL = 2S_θ^zp[Ω_0](1+√(π))/2 ≈ 1.38· 2S_θ^zp[Ω_0], which is
the blue line in <ref>(a). 
The imprecision in our most precise measurement is ∼ 13 dB below the SQL.

Laser cooling by radiation torque feedback. 
The ability to measure with an imprecision below the SQL opens the door for laser cooling of torsional motion,
eventually into the ground state, paralleling the development of feedback cooling of linear 
motion <cit.>.

We apply radiation pressure torque (δτ_fb) from a second laser beam, conditioned on the observed motion
δθ_obs, to actuate on the torsional mode.
The observed motion is used to synthesize a signal that drives an amplitude modulator
in the path of an actuation laser. This beam is focused to a 50μ m spot on the edge of the torsion pendulum 
(at an angle with respect to the measurement beam so as to not cause scatter into the measurement beam, see <ref>(a)).
Torque actuation is exclusively driven by radiation pressure and constrained by imprecision noise at the SPD, ensuring that feedback cooling is governed primarily by the observed motion (see SI).
The resulting optical feedback torque, δτ_fb = -χ_fb^-1δθ_obs, can be engineered to
affect damping by adjusting the phase of the feedback filter χ_fb^-1 to be π/2 around resonance. 
The low imprecision of the measurement guarantees that the damped motion is in fact cold.

The torsional mode is cooled by increasing the gain in the feedback loop. <Ref>(b) shows the spectrum S_θ^obs of
the observed motion as the gain is increased.
The effective damping rates (Γ_eff) are estimated by fitting the curves to a model of the apparent motion (see SI for details
of the model); solid lines show these fits. 
Assuming that the torsional mode satisfies the equipartition principle, we estimate the phonon occupation from the model of the 
physical angular motion, using parameters inferred from fits to the observed motion:

    n_eff≈∫S_θ^phys/2θ_zp^2dΩ/2π≈ n_thΓ_0/Γ_eff+n_impΓ_eff/Γ_0.

The inset in <ref>(b) shows the inferred phonon occupation as the effective damping rate is increased by the feedback gain.
Feedback cooling cools the torsional mode from an average phonon occupation of ∼ 2· 10^8 (at room temperature) to about 5964±39, 
ultimately limited by heating by feedback of imprecision noise, consistent with <ref>. 
This sets the limit n_eff≳ 2√(n_thn_imp).

Conclusion. We demonstrated laser cooling of a centimeter-scale torsion pendulum using a measurement 
whose imprecision is 13 dB below the standard quantum limit. 
The performance of cooling is limited by the achievable strength of the measurement given the size of the 
zero-point motion. This limitation can be overcome using lower frequency torsional oscillators laser-cooled
in a strong optical trap <cit.>, for which the requirements are much reduced <cit.>
compared to the current protocol.
Thus, our work opens the door for eventual quantum state control of macroscopic torsional pendula. 
Combined with the pedigree and advantage of torsional pendula in the measurement of weak gravitational 
interactions, this work establishes the necessary step towards mechanical experiments that explore the 
fundamental nature of gravity. 














    Supplementary Information







§ TORSION PENDULUM




 §.§ Fabrication


The torsion pendulum used in this work is fabricated starting
with a double-sided, 400 nm thick Si_3 N_4-on-Si wafer (WaferPro); 
<ref> shows the essential steps in the process.
We coated both sides with 3μ m thick S1813 positive photoresist. Using mask lithography, 
we patterned the top-side resist with the pendulum shape, while the bottom-side resist protected the wafer from 
scratches during handling.
We transferred the pattern to the SiN film using a CF_4-based reactive ion etch. After removing the 
remaining photoresist, we applied a fresh coat to both sides. We then patterned the backside with a square window, 
carefully aligning it with the front pattern. After dry-etching the square window into the wafer, we applied a thick layer of photoresist to both sides for protection. We diced the wafer into individual samples. We removed the photoresist using 
an acetone, iso-propyl alcohol (IPA), and de-ionized (DI) water rinse, followed by an oxygen plasma clean to eliminate any 
lingering residue. We vertically placed each sample in a 74^∘ C potassium hydroxide bath for 24 hours. 
After etching, we rinsed the samples in DI water, followed by IPA, and let them air-dry. 
Finally, we gave the device a final oxygen plasma clean to remove any remaining residue.





 §.§ Ringdown measurements


The mechanical Q of the fundamental torsional mode was estimated from ringdown measurements. The motion is
excited by a radiation pressure torque, and measured using the SPD. The measured signal is demodulated at the
mechanical frequency using a lock-in amplifier (Moku:Pro, Liquid Instruments).
<Ref> shows such measurements at a few different powers of the measurement beam.






§ NOISE IN AN OPTICAL LEVER


To better understand the quantum noises in optical lever detection, we consider here the decomposition of a 
quantized optical field on an orthonormal Hermite-Gaussian (HG) basis. 
The HG mode U_mn of m-th order in x-axis and n-th order in y-axis is defined as <cit.>

    U_mn(x,y,z,t)    = u_mn(x,y,z)e^iϕ_mn(x,y,z,t),
    
        u_mn(x,y,z)    = √(2π)1w(z)1√(2^m+nm!n!)e^-(x^2+y^2)/w^2(z) 
            H_m(√(2)x/w(z)) H_n√(2)y/w(z)),
    ϕ_mn(x,y,z,t)    = ω_ℓ t-kz-k(x^2+y^2)/2R(z)+(m+n+1)ζ(z);

here, ω_ℓ and k are the angular frequency and wavenumber of the laser beam; 
w(z)=w_0√(1+(z/z_R)^2), R(z)=z(1+(z_R/z)^2), and ζ(z)=artan(z/z_R) are the beam radius, radius of curvature, and Gouy phase, respectively; 
w_0 is the beam width at the waist; z_R=kw_0^2/2 is the Rayleigh length; and
H_n is the n^th Hermite polynomial. 
These bases satisfy the orthonormality relation
∫ x  y  U_mn^* U_m'n'=δ_mm'δ_nn'.

The quantized electric field (in an arbitrary, but static polarization direction) of a laser beam with
power P in the (fundamental) HG_00 mode — such as the emission of an ideal laser — can be modeled by

    Ê(x,y,z,t) = (a̅+ a_00(t)) U_00+∑_mn^δâ_mn(t)U_mn.

Here a̅=√(P/ħω_ℓ) is the mean photon flux amplitude and the operators 
δâ_mn represent fluctuations in U_mn mode, that satisfy the 
canonical commutation relations <cit.>:

    [δâ_mn[Ω], δâ_m'n'[Ω']^†]
        = 2πδ_mm'δ_nn'δ[Ω+Ω'].

In the ideal case, the fluctuations are given by vacuum fluctuations of the HG modes, a_mn^0, derived through the projection of the fluctuations of a plane wave into the HG modes. 
To account for extraneous transverse displacement and tilt noises, we set 

    a_mn = a_mn^0  + δ a_mn^ext,

where δ a_mn^ext represent the extraneous fluctuations in each mode.

Let us consider a case where the laser beam at arbitrary point z=Z is transversely displaced by δ x and tilted by δθ in the xz plane, viz., 

    Ê(x,y,Z,t) →Ê(x-δ x,y,Z,t)e^ikδθ x≈Ê(x,y,Z,t)+a̅(1/w(Z)δx̂+ikw(Z)/2δθ̂)
        e^-iζ(Z)U_10,

where we assume δ x, δθ≪ 1, and have then represented these fluctuations by associating
them with operators.
<ref> shows that the laser beam's transverse displacement and tilt can be interpreted as the scattering of the fundamental mode to the first-order mode. For instance, at the beam waist (z=0), the displacement (tilt) fluctuations are encoded into the amplitude (phase) quadrature of the first-order mode, while in the far field (z≪ z_0 or z≪-z_0) the fluctuations are transposed to the opposite quadratures. Meanwhile, <ref> also explicitly shows how the first-order amplitude and phase quadratures contribute to the displacement and tilt fluctuations along the beam propagation:

    x(Z) = w(Z)/2a̅(δâ_10 e^iζ(Z)+δâ_10^† e^-iζ(Z))
       θ(Z) = 1/ia̅kw(Z)(δâ_10 e^iζ(Z)-δâ_10^† e^-iζ(Z))

This represents the essence of the optical lever technique; the laser tilt produced by a torsion pendulum at the beam waist is converted into the transverse displacement at the far field.



 §.§ Measurement imprecision in split photo-detected optical lever


Thus, a split photodetector (SPD) can used in the far field to detect the transverse displacement of the laser beam,
and thereby infer the tilt that produced it <cit.>. 
Specifically, the SPD takes the difference of the photocurrents from a pair of photodetectors placed next to each
other, which we take to be horizontal in the plane perpendicular to the propagation direction z. 
The photocurrent operator Î(t;z) for the SPD located at a distance z can be computed by the photon number operator, n̂=â^†â, as

    Î(t;z)    = q_e [n̂_R(t)-n̂_L(t)]
       =q_e∫_-∞^∞dx∫_-∞^∞dy  sgn(x)Ê^†(x,y,z,t)Ê(x,y,   z,t)
       =2q_ea̅∑_m,n≠0^1/√(2^m+nm!n!)[δâ_mn(t)e^i(m+n)ζ(z)]∫_-∞^∞dx  sgn(x) ∫_-∞^∞dy 2/π w(z)^2 H_m(x)H_n(y)e^-2(x^2+y^2)/w(z)^2
       =2/πq_ea̅∑_m,n≠0^1/√(2^m+nm!n!)[δâ_mn(t)e^i(m+n)ζ(z)]∫_-∞^∞dx  sgn(x) H_m(x)e^-x^2∫_-∞^∞dy H_n(y)e^-y^2
       =2q_ea̅∑_m≠0^1/√(π2^mm!)[δâ_m,0(t)e^imζ(z)]∫_0^∞dx  (1-(-1)^m) H_m(x)e^-x^2
       =4q_ea̅∑_k=0^∞1/√(π2^2k+1(2k+1)!)[δâ_2k+1,0(t)e^i(2k+1)ζ(z)]∫_0^∞dx H_2k+1(x)e^-x^2
       =4q_ea̅∑_k=0^∞[δâ_2k+1,0(t)e^i(2k+1)ζ(z)](-1)^k/(2k+1)k!√((2k+1)!/π 2^2k+1)
       =2/√(π)q_ea̅∑_k=0^∞[δq̂_2k+1,0(t)cos((2k+1)ζ(z))-δp̂_2k+1,0(t)sin((2k+1)ζ(z))]D_k

where q_e is the electron charge, δq̂_mn(t)=(δâ_mn+δâ_mn^†)/√(2) and δp̂_mn(t)=(δâ_mn-δâ_mn^†)/√(2)i are amplitude and phase quadrature operators, and the coefficient D_k is defined as D_k =(-1)^k/(2k+1)k!√((2k+1)!/2^2k). <ref> shows that the split photodetection leads to the detection of quadratures of the odd-number HG modes δâ_2k+1,0, being amplified by the amplitude a̅ of the fundamental HG mode, and with a rotation determined by the Gouy phase acquired during propagation. 
That is, in analogy with a balanced homodyne detection employed for measurement of longitudinal phase
fluctuations, the amplitude of fundamental mode a̅ and the Gouy phase shift act as the local 
oscillator beam and the homodyne angle, respectively.

Suppose now that the angular fluctuations are produced by the motion of a torsion pendulum positioned at z=z_0. 
The laser beam reflected off from the torsion pendulum, whose angular motion is denoted as θ_sig, attains a tilt of 2θ_sig. Given that the laser beam is subject to extraneous classical noises in the first-order HG mode, i.e. classical transverse displacement (x_ext) and tilt noise (θ_ext) referred to the position of the torsion pendulum, the optical field can be expressed as

    Ê(r,t) ≈a̅U_00(r,t)+[ a̅/w(z_0)x_ext+ikw(z_0)/2(2θ_sig+θ_ext)]U_10(r,t)+ ∑_n,ma_nm(t) U_nm(r,t).

Then, the symmetrized single-sided spectral density of the photocurrent can be computed from <ref> as

    S_I[Ω] = 2/π(2η R P k w(z_0) sinζ)^2[S_θ^sig[Ω] + 1/4S_θ^ext[Ω] + ( ζ/k w(z_0)^2)^2 S_x^ext[Ω]+ π/2/2η (a̅k w(z_0))^2^2ζ],

where R= q_e/ħ w_l and η are the responsivity and the quantum efficiency of the SPD, and ζ=ζ(z)-ζ(z_0) is the Gouy phase shift from the torsion pendulum to the SPD. In this calculation, the 
single-sided spectral densities of the vacuum fluctuations are S_q^mn,0=1 and S_p^mn,0=1 
(<ref>), and ∑_k=0^∞D_k^2=π/2. 
Note that transduction of angular motion to SPD signal reaches its maximum value when the Gouy phase shift is set to 
π/2. This condition can be achieved either by preparing a sufficient lever length, i.e., z-z_0 ≫ z_R, or by controlling the accumulated Gouy phase shift with a lens.
The last term in <ref> gives the imprecision noise due to quantum fluctuations in the higher-order modes,

    S_θ^imp[Ω] 
        = π/2/2η (a̅k w(z_0))^2^2ζ,

whereas the second and third terms represent the imprecision noise from extraneous fluctuations in the laser.



 §.§ Measurement back action in optical lever

We consider the quantum back-action torque induced by the laser beam in the optical lever. In the main manuscript, it is assumed that the laser beam is incident at the center of the torsion pendulum. Here, we address the scenario where the laser beam is offset by a distance x_0 from the center of rotation of the torsion pendulum. By linearizing the torque fluctuations, the back-action torque can be expressed as

    δτ̂_ba[Ω] = Fδx̂[Ω] + δF̂[Ω] x_0,

where the first term represents the back-action torque due to the transverse displacement fluctuation, and the second term accounts for the force fluctuation. The spectral density of the back-action torque is given by

    S_τ^ba[Ω] = F^2S_x[Ω] + x_0^2 S_F[Ω].

In general, these fluctuations can be decomposed into two contributions: extraneous classical noise and quantum noise. Therefore, we incorporate the classical components of the fluctuations, denoted as S_x^ext[Ω] and S_F^ext[Ω]. The quantum fluctuation of the transverse displacement can be derived from Eq.(<ref>) as S_x[Ω] = w^2(z_0)/2a̅^2, while the quantum radiation pressure force fluctuation is given by S_F^ba[Ω] = 8a̅^2ħ^2k^2, with the mean radiation pressure force F_ba = 2a̅^2ħ^2k^2. Consequently, the total back-action torque is expressed as

    S_τ^ba[Ω] = (√(2)ħa̅kw(z_0))^2[1 + (2x_0/w(z_0))^2] + (2ħa̅^2k)^2S_x^ext + x_0^2 S_F^ext.

The first term arises from quantum fluctuations in the optical beam used for the measurement (with the term ∝ x_0 coming from any
potential mis-centering of the beam on the torsion pendulum), while the second and third terms arise from extraneous fluctuations in it.



 §.§ Total measurement noise and the Standard Quantum Limit


The observed SPD signal, referred to angle, can be understood as the sum of the intrinsic 
(i.e. thermal and zero-point) motion, back-action-driven motion, and measurement imprecision, i.e.

    θ_obs = θ_int + θ_ba + θ_imp.

The intrinsic motion is given by the 
fluctuation-dissipation theorem <cit.>

    S_θ^int[Ω] = 4ħ(n_th[Ω]+12) χ_0 [Ω],

where n_th[Ω] = (e^ħΩ/k_B T-1)^-1 is the average thermal phonon occupation at temperature
T, and χ_0[Ω] = [I (-Ω^2 + Ω_0^2 - i ΩΓ_0[Ω])]^-1 is the mechanical 
susceptibility for the torsion pendulum with moment of inertia I and 
frequency-dependent structural damping rate Γ_0[Ω] = (Ω_0/Q)(Ω_0/Ω).
The back-action-driven motion is θ_ba = χ_0 δτ_ba, whose spectrum is

    S_θ^ba = χ_0^2 S_τ^ba,

with S_τ^ba given by <ref>.

Note from <ref> that as the measurement imprecision due to quantum noise decreases (∝ 1/P), 
quantum back-action increases (∝ P). 
Thus the measured angular noise has a minimum

    S_θ^obs 
        = S_θ^int + S_θ^ba + S_θ^imp≥ S_θ^int + 2√(S_θ^ba S_θ^imp),

whose ideal limit (zero temperature, perfect detection efficiency, beam perfectly centered, no extraneous 
imprecision or back-action) on resonance defines the (resonant) standard quantum limit (SQL) of the measurement: 

    S_θ^SQL[Ω_0] = 2 ħχ_0[Ω_0] (1+√(π))
        = 2 S_θ^zp[Ω_0] 1+√(π)/2.

Here the spectral density of the zero-point motion of the torsional mode is

    S_θ^zp[Ω_0] = S_θ^int[Ω_0]|_T=0 = 4θ_zp^2/Γ_0,

where θ_zp = √(ħ/(2I Ω_0)) is the zero-point motion of the mode.
In contrast to the SQL for the measurement of linear motion, this is larger than twice the resonant 
zero-point motion of the mechanical oscillator by (1+√(π))/2 ≈ 1.386.




§ EXTRANEOUS NOISE CANCELLATION IN MIRRORED OPTICAL LEVER



Here we describe the noise spectral density in the mirrored optical lever. In this configuration (as shown in Fig. 1 in the main text), the input laser beam is prepared to have equal components for polarizations in S and P, so that the P-polarized component passes on to the torsion pendulum to sense the angular motion (signal arm), while the S-polarized component is diverted to a retroreflector (reference arm). A retroreflector is a 90^∘ corner cube that inverts the tilt and transverse displacement of the beam with respect to the normal reflection from a flat mirror, i.e., it alters the sign of the amplitude flux of the first-order HG mode as δâ_10(t) → -δâ_10(t). In each arm, a quarter-wave plate is positioned so that both beams are collected from the output port of the PBS.
The output optical field is written as *Ê_out(r,t) = Ê_sig(r,t)*e_S+Ê_ref(r,t)*e_P, where *e_S and *e_P are the unit vectors for S and P polarizations, respectively. The signal and reference fields are obtained from the Eq.(<ref>) as

    Ê_sig(r,t) = a̅U_00^sig(r,t)+[ a̅/w(z_0)x_ext+ikw(z_0)/2(2θ_sig+θ_ext)]U_10^sig(r,t)+ ∑_n,ma_nm(t) U_nm^sig(r,t),
    Ê_ref(r,t) = a̅U_00^ref(r,t)+[ -a̅/w(z_0)x_ext-ikw(z_0)/2θ_ext]U_10^ref(r,t)+ ∑_n,ma_nm(t) U_nm^ref(r,t).

Here, U^sig(ref)_mn represents the notation for the HG modes coming from the signal (reference) arm.
We assume that the optical power and arm length for each arm are identical.
The combined field is then detected by an SPD, yielding the photocurrent density as

    S_I[Ω] = 2/π(2η R P k w(z_0) sinζ)^2 [S_θ^sig[Ω] 
        + π/2/η (a̅k w_0)^2^2ζ],

This indicates that the reference beam cancels out the classical displacement and tilt noises with the trade-off of the doubled shot noise. The orthogonal polarizations for the optical fields allow us to obviate the requirement for stabilizing laser frequency noise. The sensitivity for the angular motion detection is maximized when the Gouy phase shift ζ̃ is precisely set to π/2 as follows:

    S_I[Ω] = 2/π(2η R P k w(z_0))^2 [S_θ^sig[Ω] 
        + π/2/η (a̅k w(z_0))^2],

This result shows that, in principle, the mirrored optical lever with the use of a retroreflector is immune to the classical noises that the laser beam experiences before the PBS.

In our experiment, the output laser beam is split by a knife-edge right-angle mirror and then collected by a low noise balanced photodetector (HBPR-100M-60K-IN-FS, Femto). The difference in photocurrents between the two photodiodes of the BPD corresponds to the transverse displacement of the laser beam at the knife-edge mirror. The Gouy phase shift in Eq. (<ref>) is defined by the locations of the torsion pendulum and knife-edge mirror.  We set the Gouy phase to be π/2 by manipulating a lens between the torsion pendulum and the knife-edge mirror.

To characterize the classical noise suppression of the mirrored optical lever, we measure the angular fluctuations over time with and without the mirrored arm. In this proof-of-concept experiment, the torsion pendulum is replaced with a flat mirror so that the performance is only limited by the measurement system. <ref>(a) shows the angle fluctuations of the laser beam referred to the location of the flat mirror over 300 s with a sampling rate of 100 Hz. We observe that the tilt fluctuation (blue curve) is suppressed when the mirrored image is simultaneously incident on the SPD, reducing the standard deviation from 620 nrad to 73 nrad.
Furthermore, we characterize the mirrored optical lever system in terms of the frequency response. To this end, we generate the periodic virtual tilt signal swept over 1 Hz – 100 kHz using the acousto-optic deflector. This virtual angular signal is detected by the SPD and subsequently lock-in amplified. As shown in <ref>(b), we confirm that the mirrored optical lever is capable of reducing the angle fluctuations by 50–60 dB in a broad frequency range up to 100 kHz. Thine residual noise may be due to the imperfect polarization division, the finite extinction ratio of the PBS ∼ 3000, and other systematic errors.



In Fig. 2(a) of the main text, excess noise is observed in the low-frequency regime below 1 kHz Fourier frequency. Potential sources of this extraneous noise include seismic noise and residual spatial noise caused by air current-induced refractive index fluctuations. (Laser intensity noise is significantly suppressed by balanced photodetection at the SPD, and the optical lever is insensitive to the phase quadrature of the laser beam.) To elucidate the origins of this excess noise, we examine the coherence spectrum between the angular displacement noise S_θ[Ω] and seismic displacement noise S_x^sei[Ω]. For this purpose, seismic acceleration S_a^sei is measured and low-pass filtered to obtain S_x^sei[Ω]=S_a^sei[Ω]/Ω^4.

<Ref> shows the coherence spectrum, |S_θ x^sei[Ω]|^2/S_θ[Ω]S_x^sei[Ω], between angular displacement and seismic noise. A relatively high coherence is observed in the range of 300 Hz – 1 kHz, resembling the excess noise seen in Fig. 2(a) of the main text. Therefore, the noise peaks in this frequency range are likely attributable to seismic noise, while the long-term noise below 300 Hz may be caused by air current-induced refractive index fluctuations.



§ ANGLE CALIBRATION




 §.§ Calibration of mirrored optical lever



In conventional optical lever techniques, the angular displacement of the torsion pendulum, δθ, is inferred from the geometry of the lever: a laser beam deflected with a tilt of 2δθ propagates over a distance z, thereby amplifying the transverse displacement by the lever arm, i.e., δ x = 2zδθ. 
This suggests that extending the lever arm could infinitely amplify the SPD signal gain. 
However, this naive calibration is only valid within the framework of ray optics; beyond the Rayleigh length of the laser beam, the gain of the SPD signal eventually saturates.

For accurate measurement of the intrinsic thermal motion of a high-Q torsion pendulum, calibration is often performed by fitting the theoretical model of thermal motion to the SPD voltage spectrum. 
However, this method can lead to inaccuracies when extraneous noise, such as photothermal effects or external disturbances, contaminates the angular displacement spectrum.

To overcome this limitation, we employ a direct calibration method that relates the SPD signal to angular displacement using an acousto-optic deflector (AOD). 
As described in the main text, an AOD (M1377-aQ80L-1, ISOMET) is positioned at the input plane of the entrance lens of a 4f imaging system, with the torsion pendulum (and retroreflector) located at the output plane of the second lens. 
The 4f system, composed of two lenses with identical focal lengths, reproduces the angular tilt of the laser beam at the input plane at the output plane.

We use the first-order diffraction beam from the AOD, with its tilt angle precisely controlled by the relation Δθ_cal = (λ/v_c) Δ f_cal, where Δ f_cal is the drive frequency change and v_c ≈5700m/s is the acoustic velocity of the AOD quartz crystal.
Using this relation, we determine the calibration factor for converting the SPD voltage spectrum into an angular displacement spectrum.
Since the laser beam acquires an angular displacement that amounts to twice the torsional motion of the pendulum upon reflection, the calibration factor is defined as α_cal=Δθ_cal/2Δ V_cal, where Δ V_cal represents the voltage change in the SPD signal.
To precisely attain the calibration factor, we frequency modulate the AOD drive with a known depth (Δ f_cal), and observe the corresponding SPD voltage modulation amplitude (Δ V_cal) in the time domain. 
In this process, we trigger and average the calibration tone 100 times to mitigate extraneous noises in the SPD signal. 
Then, the calibration factor is used to convert the SPD spectrum to the angular displacement spectrum: S_θ[Ω]=α_cal^2 S_V[Ω], where S_V[Ω] is the measured SPD spectrum. 
Our typical choice for the AOD calibration is a 10 kHz frequency modulation tone with a 100 kHz frequency depth (the corresponding tilt amplitude of 18.7 rad), centered around 80 MHz (AOD operation frequency).
Note that, although this calibration tone can be detected in the power spectral density (PSD), the measured voltage amplitude may be inaccurate or underestimated due to the broadening of the linewidth in the PSD analysis.

<ref> illustrates the application of AOD-based calibration in our experiments. 
We vary the frequency modulation depth of the AOD drive and monitor the SPD signal in the time domain. The triggering and averaging of the modulation signal is performed by an oscilloscope (Moku:Pro, Liquid Instruments).
The voltage amplitude is then estimated from these averaged data. 
<ref>(b) displays the voltage amplitudes as a function of modulation depths, corresponding to beam tilt amplitudes, which provides the calibration factor for converting the SPD spectrum into the angular displacement spectrum. 
The linear relationship is confirmed with a coefficient of determination R^2 = 0.99996. 
Additionally, <ref>(c) demonstrates the linear dependence of voltage amplitude on incident optical power.




Figure 2(b) in the main text is calibrated from the SPD voltage spectrum to the angular displacement spectrum using the calibration method described above. 
To verify that the imprecision noise in the spectrum is quantum-noise-limited, we measure the noise floor of the photocurrent density, referred back to the optical power, at a frequency of Ω = 2π·80kHz under varying optical power levels.
As shown in <ref>(a), at power levels below 20 µW, the dark noise is dominant, while shot noise starts to prevail beyond this threshold. 
The curve fitting reveals that the quantum efficiency of the balanced photodetector used in this study is 0.81. 
<ref>(b) shows the imprecision noise in the angular displacement obtained by implementing the calibration method.
The beam width at the position of the flat mirror is 587 m measured by a beam profiler (BC207VIS, Thorlabs). 
The angular sensitivity improves as the optical power increases, and, at the maximum power of 5 mW, the imprecision noise reaches 2.56 prad/√(Hz). 
This value is 16% higher than the shot noise level (2.21 prad/√(Hz)) calculated under the assumption of ideal detection (<ref>). Therefore, the detection efficiency is 0.75.



 §.§ Calibration of thermal noise spectra


When measuring the torsion pendulum, we reduced the beam size from 587 µm to 180 µm to minimize clipping loss of the measurement beam at the finite surface width of the pendulum (500 m). 
This reduction was achieved by focusing the beam onto the AOD at the input plane of the 4f system. 
However, it was observed that the AOD calibration method became inaccurate as the beam was focused onto the AOD crystal: despite applying a pure tone frequency modulation, the SPD signal was significantly distorted with harmonics. 
This appears to be due to the incoherent addition of diffraction angles as the beam is focused on the AOD crystal. 
To address this, we adopted a two-step calibration process: (1) calibrating the thermal motion of the torsion pendulum using a reliable, larger beam size (w_0 ≈587 m), and (2) using this thermal motion as a reference to calibrate the SPD spectrum obtained with the smaller beam size (w_0 ≈180 m).
The second step was performed with 1 mW optical power, which kept the pendulum at room 
temperature (290 K) (see <ref>). 
The calibration factor was determined by fitting the spectra to the thermal noise at room temperature, resulting in a calibration factor of α_cal = 1.65×10^-9 rad/V with a 2% error.

Through two-step calibration, the imprecision noise at the SPD was found to be 
S_θ^imp=1.06×10^-22 at 10 mW optical power, corresponding to a detection efficiency of 0.244.
The lower sensitivity is largely due to the reduced beam size (<ref>).

The ideal imprecision derived in (<ref>), which assumes an intact spatial mode profile upon reflection, does not hold for the torsion pendulum measurement: that derivation assumes that tilt is induced by an infinite
plane, whereas the finite extent of the torsion pendulum clips the wings of the Gaussian beam, leading to 
distortion of the reflected beam beyond the simple model. 
Nonetheless, the imprecision noise shown in Fig. 3(a) of the main text is quantum-noise-limited, as confirmed by its good agreement with the expected shot noise scaling (<ref>).



§ MEASUREMENT-BASED FEEDBACK COOLING




 §.§ Theory


The physical angular displacement of the torsion pendulum reacts to external torques, consisting of thermal, back-action, and feedback torques: 

    χ_0^-1[Ω]θ_phys[Ω]=τ_th[Ω]+τ_ba[Ω]
        +τ̂_fb[Ω],

where χ_0[Ω] is the susceptibility of the torsion pendulum. 
In measurement-based feedback control <cit.>, the observed motion [<ref>]

    θ_obs[Ω]=θ_phys[Ω] + θ_imp[Ω]

is used to synthesize a feedback force, in this case a torque,

    τ_fb[Ω]=-χ_fb^-1[Ω]θ_obs[Ω]

so that the physical motion

    θ_phys[Ω] = χ_eff[Ω](τ_tot[Ω] 
        -χ_fb^–1[Ω]θ_imp[Ω])

is modified via the effective susceptibility
χ_eff=(χ_0^-1+χ_fb^-1)^–1; 
here τ_tot=τ_th+τ_ba+τ_fb.
In the presence of feedback, the observed motion is also modified:

    θ_obs[Ω] = 
        χ_eff[Ω](τ_tot[Ω]+χ_0^–1[Ω]
        θ_imp[Ω])



Cooling by this kind of feedback can be affected by the choice -χ_fb = i I ΩΓ_fb around
mechanical resonance Ω_0.
In this case, the spectrum of the observed motion assumes the form

    S_θ^obs[Ω] = 
        S_τ^tot[Ω]/I^2[(Ω_0^2-Ω^2)^2+(ΩΓ_eff)^2]+(Ω_0^2-Ω^2)^2+(ΩΓ_0[Ω])^2/(Ω_0^2-Ω^2)^2+(ΩΓ_eff)^2S_θ^imp[Ω];

this model is used to fit the data in Fig. 3 of the main text.
The spectrum of the physical motion

    S_θ^phys[Ω] = S_τ^tot[Ω]/I^2[(Ω_0^2-Ω^2)^2+(ΩΓ_eff)^2]+(ΩΓ_fb)^2/(Ω_0^2-Ω^2)^2+(ΩΓ_eff)^2S_θ^imp[Ω],

allows inference of physical properties such as the average phonon occupation achieved by feedback cooling.

If we take the torsional motion to be a harmonic oscillator with creation/annihilation operators 
b̂,b̂^†, then the angular motion is θ̂ = θ_zp(b̂+b̂^†), 
where θ_zp = √(ħ/(2I Ω_0)). Then the variance of the motion is 
*θ̂^2 = 2θ_zp^2 (*b̂^†b̂+12); thus the phonon occupation 
n_eff = *b̂^†b̂
can be inferred from the variance of the (physical) angular motion via

    n_eff + 1/2 = *θ̂^2/2θ_zp^2 
        = ∫S_θ^phys[Ω]/2θ_zp^2Ω/2π.

Using the model in <ref> gives

    n_eff + 1/2 = ( n_th+n_ba + 1/2)Γ_0/Γ_eff 
            + n_impΓ_eff/Γ_0,

where n_ba = S_τ^ba[Ω_0]/(4ħχ_0^-1[Ω_0]) is the effective back-action phonon occupation,
and n_imp = S_θ^imp/(2S_θ^zp[Ω_0]) is the phonon-equivalent measurement imprecision. 




 §.§ Implementation of feedback




The motion of the pendulum is actuated by radiation torque from a secondary laser beam (“actuation beam”)
focused on the edge of the pendulum’s backside.
A fiber-coupled electro-optic Mach-Zehnder amplitude modulator (LNX1020A, Thorlabs) is employed to 
modulate the radiation torque around the mechanical frequency.
The modulator’s response to the external voltage is sinusoidal due to the nature of the Mach-Zehnder 
interferometer in it; we bias the modulator around its linear operating point, which corresponds to the
bias voltage at which half the maximum optical power is transmitted.
At this operating point, the actuation beam exhibits a sensitivity of 15.7 mW/V in response to the external voltage
applied on the modulator.

<Ref>(a) is the measured frequency response of the feedback filter χ_fb^-1, optimized for feedback gain to achieve a phonon number of approximately 6000. This configuration is implemented using an FPGA-based controller (Moku:Pro, Liquid Instruments). The feedback filter comprises a bandpass filter (34–40 kHz) centered around the resonance frequency of the fundamental mode, and a phase delay that aligns the total loop delay to be π/2. As depicted in <ref>(a), the delay at the resonance frequency Ω_0/2π=35.95kHz is precisely set to π/2.



 §.§ Extraneous noise in feedback


Extraneous noise in the feedback loop can limit the performance of feedback cooling if they are larger than the
imprecision noise in the measurement that drives the actuator.
We investigate this possibility by measuring and budgeting the noise in the actuator beam. 
<Ref>(b) shows this budget referred to angular displacement at the SPD.
Two contributions can be distinguished: (a) voltage noise from the feedback controller, 
referred to angle (blue); and, (b) intensity noise in the actuation beam referred to angle (red). 
Grey shows the motional signal at the SPD, with black dashed showing the quantum-noise-limited imprecision.
Clearly, the extraneous noise in the feedback loop lies more than 30 dB below the imprecision noise.
Thus, feedback cooling is governed primarily by the observed motion.



Next, we investigate whether the actuation beam induces mechanical torques 
in excess of that due to radiation pressure. For example, thermoelastic torques <cit.> 
arising from concentrated photothermal heating at the edge of the torsion pendulum. 
The nature of the thermoelastic effect can be understood through the absorption and diffusion of heat within 
the pendulum.
This suggests that the thermoelastic effect can be distinguished from pure radiation pressure 
torque <cit.>: the angular displacement response to thermoelastic torque follows a 
single-pole low-pass filter characteristic at a specific cutoff frequency, while the radiation pressure 
torque remains frequency-independent.

<Ref> is the frequency-response of the torsion pendulum as the intensity in the actuator beam 
is modulated. 
The response remains frequency-independent from 1 kHz up to the resonance for various beam positions, 
both at the edge and near the center of the pendulum. 
This suggests that no thermoelastic effect is observed during feedback actuation, and that the actuation is
dominated by radiation torque.



 §.§ Estimation of mode temperature


To investigate the role of photothermal heating, we measured the mode temperature as a function of optical 
power (from now on, the optical power is referred to as that reflected from the torsion pendulum). For this, we fit the torsion pendulum’s intrinsic motion to a Lorentzian model: 
S_θ^obs[Ω]=S_θ^imp+S_θ^th[Ω_0]/(1+4Q_0^2(Ω-Ω_0)^2/Ω_0^2). 
Since the resolution bandwidth of the spectrum (0.25 Hz) is larger than the linewidth of the torsional 
mode (Γ_0/2π = 2.6mHz), we excluded the vicinity of the peak from the fitting range.

Using these fittings, we estimated the mode temperatures shown in <ref>. 
For optical powers below 5 mW, no signs of photothermal heating were observed.
The fits allow us to infer the effective moment of inertia I = 5.54e-17kg· m^2 with the independently 
measured values of Ω_0=2π· 35.95 kHz, Q=1.365·10^7, and T = 290K. 
This value is 30% smaller than that obtained by finite element simulation 
(COMSOL Multiphysics), (7.87e-17kg· m^2) and  larger by 12% compared to the analytical model
<cit.> I=ρ L h w^3/24 ≈4.91e-17kg· m^2 of an ideal rectangular beam (which 
our device is not). This provides an independent consistency check of our angle calibration procedure and
mode temperature.

