1. INTRODUCTION

Torsion pendula have long been pivotal in the measurement of weak fundamental forces, most notably in establishing the electrostatic inverse-square law [1,2], precision measurements of classical gravitational forces [3–7], tests of the equivalence principle [8–11], and the first observation of radiation pressure torque [12]. 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 [13–15]. Torsion pendula are particularly suited for such experiments on account of the low thermal Brownian noise of torsional suspensions even when mass-loaded [16,17] and well-understood techniques for isolating the gravitational interaction between them even with masses as small as 100 mg [18]. 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 [19], the experimental realization of similar techniques has remained elusive for torsional motion. While levitated optomechanics has achieved significant progress [20], including the quantum measurement, control, and ground-state cooling of both linear and librational motions [21–25], these techniques remain constrained to nanoscopic scales.

In this paper, we demonstrate laser cooling of a centimeter-scale high-quality torsional oscillator using a novel “mirrored optical lever” whose quantum-noise-limited sensitivity of ${10^{- 12}} \;{\rm rad/}\sqrt {{\rm Hz}}$ is 9.8 dB below the peak zero-point motion of the torsional mode. Conditioned on this measurement, we apply optical radiation pressure torque on the oscillator so as to cool its angular motion to $10 \;{\rm mK}$ from room temperature, corresponding to an average phonon occupation of $5964 \pm 39$ (starting from ${\sim}2 \times {10^8}$). In the following, we describe the torsional oscillator, the mirrored optical lever used to measure its angular motion, its performance in terms of classical noise cancelation, its calibration, and its utility in measurement-based feedback cooling of torsional motion.

2. HIGH-Q CENTIMETER-SCALE TORSIONAL OSCILLATOR

A remarkable advantage of torsional pendula in studies of gravity is that their suspensions can be realized with an exceptionally high mechanical quality factor ($Q$) even when mass-loaded [16,17]. The reason is twofold [17]: 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, ${Q_{{\rm int}}}$, is elevated to ${Q_0} = {Q_{{\rm int}}}{D_Q}$ [17], where the dissipation dilution factor is ${D_Q} \approx (\sigma /2E)(w/h{)^2}$; here $E$ and $\sigma$ are Young’s modulus and tensile stress, and $w$ and $h$ are the width and thickness of the ribbon, respectively. This is in marked contrast to tensile-stressed mass-loaded linear oscillators, where bending curvature due to the loaded mass undermines the advantage from dissipation dilution [26,27]. This capability positions a macroscopic high-Q torsion pendulum as a unique candidate both for reaching its motional ground state and for gravitational experiments.

We fabricated a doubly clamped 0.9 cm long thin-film ($w = 0.5\;{\rm mm} $, $h = 400\;{\rm nm}$) tensile-stressed ($\sigma = 0.8\;{\rm GPa}$) torsional oscillator made of stoichiometric ${{\rm Si}_{3}}{{\rm N}_{4}}$. The device was fabricated starting from a double-sided ${{\rm Si}_{3}}{{\rm N}_{4}}$-on-Si wafer, followed by lithography and reactive ion etching. A second aligned lithography and etch created the window for optical access from the backside. The device was released using a 24-h etch in potassium hydroxide. The sample underwent meticulous cleaning with acetone, isopropyl alcohol, deionized water, and oxygen plasma (see Supplement 1 for more details). The specific device used in the current study has its fundamental torsional mode resonating at ${\Omega _0} = 2\pi \cdot 35.95\;{\rm kHz}$ with quality factor $Q = 1.4 \times {10^7}$ inferred by ringdown measurements (in vacuum, at $6 \times 10^{-7}\;{\rm mbar}$) as shown in Fig.  1 (c). The measured $Q$ is consistent with the expected dilution factor ${D_Q} \approx 2300$. The moment of inertia of the fundamental torsional mode is measured to be $I = 5.54\;{\rm kg} \cdot \;{{\rm m}^2}$ (see Supplement 1), with an effective mass $m = I{(2/w)^2} = 0.89\; \unicode{x00B5} {\rm g}$ [17]. Importantly, the design principles demonstrated here ensure that when a mass is added, the quality factor will not degrade [28].

 

Fig. 1. (a) Experimental setup: 1064 nm light in the fundamental spatial mode is used to measure and actuate a cm-scale torsional oscillator [panel (c)]. The probe beam is split into two paths: one arm to the torsional oscillator (signal arm) and the other to a retroreflector (reference arm). The fields from the two arms are collected and subsequently detected by the split photodetector (SPD). Here $\lambda /4$, quarter-wave plate; L, lens; PBS, polarizing beam splitter. (b) Principle of spatial mode noise suppression in the mirrored optical lever. The retroreflector (i) produces a mirror image of the laser beam’s spatial fluctuations (blue arrow), while the torsional oscillator (ii) induces a pure tilt motion (red arrow) in the signal arm. At the SPD (iv), the classical spatial mode noise of the laser beam is canceled. (c) A cm-scale torsional oscillator fabricated in tensile-stressed SiN (inset) features a mechanical quality factor of ${10^7}$ due to torsional dissipation dilution. (d) Ambient angle fluctuation over time, measured with and without the mirrored arm. (e) Magnitude response of the mirrored optical lever: a sinusoidal tilt modulation is generated by the acousto-optic deflector and detected by lock-in amplification of the split photodetector signal. The magnitude of this quantity, with and without the mirrored arm, characterizes the suppression of input classical tilt noise, here at the level of 60 dB.

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3. QUANTUM-LIMITED OPTICAL LEVER DETECTION

To achieve quantum-limited readout of angular motion, we devised a “mirrored” optical lever. Its primary advantage is the 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 $\bar a = \sqrt {P/\hbar {\omega _\ell}}$ ($P$ is the optical power, ${\omega _\ell} = 2\pi c/\lambda$ represents the carrier frequency for a wavelength of $\lambda \approx 1064\;{\rm nm}$); then the optical field can be expressed as [29,30] ${\hat E_{{\rm in}}}({\textbf r},t) = \def\LDeqbreak{}(\bar a + \delta {\hat a_{00}}(t)){U_{00}}({\textbf r}) + \sum\nolimits_{n,m} \delta {\hat a_{\textit{nm}}}(t){U_{\textit{nm}}}({\textbf r})$. Here ${U_{\textit{nm}}}({\textbf r})$ is the $(n,m) -$ Hermite–Gaussian (${{\rm HG}_{\textit{nm}}}$) basis function (see Supplement 1); the operators $\{\delta {\hat a_{\textit{nm}}}(t)\}$ represent fluctuations in the laser field, which in the ideal case, when the field is quantum-noise-limited, model the quantum vacuum fluctuations in ${{\rm HG}_{\textit{nm}}}$ mode. If this incident field is subjected to a transverse displacement $\delta {\textbf r} = (\delta x,0,0)$ and angular tilt $\delta \theta$ at its beam waist, the optical field is transformed to ${\hat E_{{\rm out}}}({\textbf r},t) \approx\def\LDeqbreak{} {\hat E_{{\rm in}}}({\textbf r},t) + \bar a(\delta x/{w_0} + ik{w_0}\delta \theta /2){U_{10}}({\textbf r})$, where ${w_0}$ is the waist size. That is, transverse or angular motion scatters light from the incident ${{\rm HG}_{00}}$ mode to the ${{\rm HG}_{10}}$ mode in proportion to the motion. When the beam waist is placed at the location of the torsional oscillator, its physical motion $\delta {\hat \theta _{{\rm phys}}}(t)$ is experienced twice by the reflected optical beam, i.e., $\delta {\hat \theta _{{\rm phys}}}(t)$; the incident beam may also have extraneous classical noises in the transverse displacement ($\delta {\hat x_{{\rm ext}}}$) and tilt ($\delta {\hat \theta _{{\rm ext}}}$). 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 Supplement 1; we neglect correlations between the tilt and transverse displacement [30]) as follows:

Our mirrored optical lever technique [shown in Figs.  1 (a) and 1 (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 torsional oscillator (signal arm) and the other to a retroreflector (reference arm). A corner cube retroreflector produces a mirror image of the laser beam’s transverse displacement and tilt, i.e., $\delta {\hat a_{10}} \to - \delta {\hat a_{10}}$, after interacting with the retroreflector [Fig.  1 (b-i)] showing the effect of this on input noise, whereas the torsional oscillator induces pure tilt [Fig.  1 (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 cancelation of extraneous noises [acquired by the laser beam before the PBS, Fig.  1 (b-iii)] and transduction of torsional motion into transverse displacement at the SPD [Fig.  1 (b-iv)]. Thus, for the mirrored optical lever, the photocurrent density,

We evaluate the suppression of classical spatial mode noise with the mirrored optical lever in both the time and frequency domains. Figure  1 (d) illustrates the tilt angle fluctuations of the laser beam, referenced to the flat mirror’s position, recorded over 300 s at a sampling rate of 100 Hz. Using the mirrored image produced by the reference arm, the tilt fluctuations (blue curve) are significantly suppressed, reducing the standard deviation from 620 to 73 nrad. Further analysis of this suppression was performed in the frequency domain. Figure  1 (e) shows the frequency response of the mirrored optical lever, measured using lock-in amplification with beam tilt modulation introduced via an acousto-optic deflector within a $4f$ lens system [this system, which is part of our calibration method, is detailed in the next section; see Fig.  2 (a)]. The measurements reveal a 60 dB reduction in angular fluctuations across a wide frequency range, up to 100 kHz, when the mirrored arm is employed. Residual noise is attributed to factors such as imperfect polarization division, the finite extinction ratio of the PBS (${\sim}3000$), and other systematic errors.

 

Fig. 2. (a) Schematic of the acousto-optic deflector (AOD)-based calibration method. The small tilt angle produced by the AOD emulates the angular displacement of the torsional oscillator within the $4f$ lens system. (b) Calibration curve for converting the observed SPD signal, measured via the mirrored optical lever (without the reference arm), into tilt angles corresponding to each modulation depth applied to drive the AOD (see text for details). (c) Measurement sensitivity of the mirrored optical lever as the optical power is changed. Above 1 kHz, the observed noise is consistent with quantum vacuum fluctuations in the higher-order spatial modes.

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4. CALIBRATION AND PERFORMANCE OF MIRRORED OPTICAL LEVER

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

Independent calibration of the SPD voltage into angular motion is crucial for further investigation [31]. (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 using an acousto-optic deflector (AOD) within a $4f$ lens system. To wit, an AOD is placed at the input plane of the entrance lens of a $4f$ imaging system, with the torsional oscillator (and retroreflector) at the output plane of the second lens [Fig.  2 (a)]. In this configuration, the tilt of the laser beam at the input plane can be modulated by frequency-modulating the AOD drive as $\Delta {\theta _{{\rm cal}}} = (\lambda /{v_c})\Delta {f_{{\rm cal}}}$, where $\Delta {f_{{\rm cal}}}$ is the frequency-modulation depth, and ${v_c} \approx 5700\;{\rm m/s}$ is the acoustic velocity of the AOD quartz crystal. This known tilt change manifests as a voltage change ($\Delta {V_{{\rm cal}}}$) in the SPD signal. By estimating the calibration factor, defined as ${\alpha _{{\rm cal}}} = \Delta {\theta _{{\rm cal}}}/2\Delta {V_{{\rm cal}}}$, the angular displacement spectrum is computed from the SPD spectrum: ${S_\theta}[\Omega] = \alpha _{{\rm cal}}^2{S_V}[\Omega]$, where ${S_V}[\Omega]$ is the measured SPD spectrum (see Supplement 1 for details). Figure  2 (b) shows the voltage amplitude detected by the SPD with different frequency-modulation depths of the AOD drive. The beam tilt modulation using the AOD enables a highly linear calibration of the voltage to the angular spectrum, achieving an R-square value of 0.99996 and a calibration factor error of 1.5%, as determined by the linear fit.

Figure  2 (c) demonstrates the calibrated performance of the mirrored optical lever. The angle-referred imprecision noise decreases as the reflected optical power increases and reaches $2.6 \times {10^{- 12}}\;{\rm rad}/\sqrt {{\rm Hz}}$ with 5 mW incident power, consistent with the quantum-noise scaling in Eq. ( 2 ) corresponding to a detection efficiency of $\eta \approx 0.75$.

The excess noise peaks between 200 Hz and 1 kHz are attributable to seismic noise-induced fluctuations (see Supplement 1). Without the mirrored arm, we observed an extraneous low-frequency tilt noise above 1 mW of optical power that compromised the linearity of the SPD; with the mirrored arm, this low-frequency noise was effectively suppressed, as shown in Fig.  1 (e). Given that our lab temperature is stabilized to a precision around 20 mK [32], we conjecture that the low-frequency drifts in input laser tilt are due to refractive index fluctuations from air currents.

5. LASER COOLING OF TORSIONAL OSCILLATOR

A. Imprecision below the Zero-Point Motion

Next, we place the torsional oscillator in the signal arm of the mirrored optical lever. Figure  3 (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 $\delta {\hat \theta _{{\rm obs}}} = \delta {\hat \theta _{{\rm phys}}} + \delta {\hat \theta _{{\rm imp}}}$ consist of the physical motion of the oscillator $\delta {\hat \theta _{{\rm phys}}}$ and the imprecision noise $\delta {\hat \theta _{{\rm imp}}}$ of the optical lever.

 

Fig. 3. (a) Thermal Brownian motion of the fundamental torsional mode measured with increasing optical power. The dashed blue line shows the peak zero-point motion of the fundamental torsional mode. The inset shows the phonon-equivalent imprecision, whose scaling is consistent with quantum-noise-limited detection, and reaches a minimum of ${n_{{\rm imp}}} \approx 5 \times {10^{- 2}}$. (b) At the lowest imprecision, the measurement record is used in a feedback loop [see Fig.  1 (a)] to cool the torsional mode via radiation torque to a final phonon occupation of ${n_{{\rm eff}}} \approx 6 \times {10^3}$.

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The physical motion, shown as the black line in Fig.  3 (a), is the sum of the intrinsic motion due to thermal and zero-point fluctuations and a subdominant contribution due to the quantum back-action of the measurement. The intrinsic motion is predominantly the thermal motion due to its ${n_{{\rm th}}} = {k_B}T/(\hbar {\Omega _0}) \approx 2 \times {10^8}$ average phonons at room temperature and is described by the fluctuation–dissipation theorem [33], $S_\theta ^{{\rm int}}[\Omega] = \def\LDeqbreak{}4\hbar ({n_{{\rm th}}} + {\textstyle{1 \over 2}})\text{Im} {\chi _0}[\Omega]$; here ${\chi _0}[\Omega] = [I(- {\Omega ^2} + \Omega _0^2 + i\Omega {\Gamma _0}[\Omega {])]^{- 1}}$ is the susceptibility of the torsional mode with moment of inertia $I$ and damping rate ${\Gamma _0}[\Omega] =({\Omega _0}/Q)({\Omega _0}/\Omega)$. The quantum back-action of the measurement is primarily due to (see Supplement 1) quantum fluctuations in the transverse displacement of the measurement beam, which cause quantum radiation torque fluctuations on the oscillator, $S_\tau ^{{\rm ba}}[\Omega] \approx\def\LDeqbreak{} 2(\hbar \bar ak{w_0}{)^2}$. The black line in Fig.  3 (a) is a model of the physical motion $S_\theta ^{{\rm phys}}$ based on independent measurement of the frequency, mechanical $Q$, and calibrated mode temperature. The latter is inferred by calibrating each spectrum using our frequency-modulation technique and assuming that the torsional mode is in thermal equilibrium. We verified (see Supplement 1) 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 [Fig.  3 (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, ${n_{{\rm imp}}} = S_\theta ^{{\rm imp}}/\def\LDeqbreak{}2S_\theta ^{{\rm zp}}[{\Omega _0}]$, where $S_\theta ^{{\rm zp}}[{\Omega _0}] = 4\theta _{{\rm zp}}^2/{\Gamma _0}$ is the peak spectral density of the zero-point motion ${\theta _{{\rm zp}}} = \sqrt {\hbar /(2I{\Omega _0})}$ of the torsional mode.

The overall low imprecision of the measurement, ${n_{{\rm imp}}} \approx 5 \times {10^{- 2}}$, is 9.8 dB below the peak zero-point motion of the fundamental torsional mode. This performance can also be compared against the ideal performance achievable in weak continuous measurement of angular displacements. The minimum total noise achievable in such a measurement, without the use of any quantum correlations, is (see Supplement 1)

B. Laser Cooling by Radiation Torque Feedback

The ability to measure with an imprecision far below the peak zero-point motion directly informs the efficacy of laser cooling of torsional motion, eventually into the ground state, paralleling the development of feedback cooling of linear motion [35–43]. The primary requirement is that the measurement imprecision, relative to the peak zero-point motion, be comparable to the inverse of the thermal occupation.

We apply radiation pressure torque ($\delta {\tau _{{\rm fb}}}$) from a second laser beam, conditioned on the observed motion $\delta {\theta _{{\rm obs}}}$, 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 $50\;\unicode{x00B5} {\rm m}$ spot on the edge of the torsional oscillator [at an angle with respect to the measurement beam so as to not cause scatter into the measurement beam; see Fig.  1 (a)]. Torque actuation is exclusively driven by the radiation pressure and constrained by the imprecision noise at the SPD, ensuring that feedback cooling is governed primarily by the observed motion (see Supplement 1). The resulting optical feedback torque, $\delta {\tau _{{\rm fb}}} = - \chi _{{\rm fb}}^{- 1}\delta {\theta _{{\rm obs}}}$, can be engineered to affect the damping by adjusting the phase of the feedback filter $\chi _{{\rm fb}}^{- 1}$ to be $\pi /2$ 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. Figure  3 (b) shows the spectrum $S_\theta ^{{\rm obs}}$ of the observed motion as the gain is increased. The effective damping rates (${\Gamma _{{\rm eff}}}$) are estimated by fitting the curves to a model of the apparent motion (see Supplement 1 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:

6. CONCLUSION

We demonstrated laser cooling of a centimeter-scale torsion oscillator using a measurement whose imprecision is $9.8\;{\rm dB} $ below the peak zero-point motion. 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 in two stages: first, through coherent angular signal amplification [44] using a degenerate optical cavity [45] where both ${{\rm HG}_{00}}$ and ${{\rm HG}_{10}}$ modes resonate simultaneously, and second, by implementing laser cooling in a strong optical trap [41], where the requirements are significantly reduced [46] 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 toward mechanical experiments that explore the fundamental nature of gravity.