1. INTRODUCTION

Optomechanics uses light to monitor and control the motion of micro- and macroscale objects [1]. State-of-the-art optical cooling has reached the quantum ground state of translational motion in a number of platforms [2–10], an essential step for fundamental tests of quantum mechanics on massive objects [11–13]. In such a context, larger and more massive systems will enable us to test the limits of current theories [14–17]. Moreover, precise control and transduction of mechanical motion enables metrological applications [18]. Controlling rotational degrees of freedom, however, remains challenging [19], in part because rotation does not couple naturally to an optical cavity.

In this work, we report purely optical feedback cooling [1,20–24] of a 5 mm long torsional resonator with a frequency of $\approx 190\mathrm{kHz}$, reducing the mean-square angular displacement over three orders of magnitude from room temperature. The platform is the fundamental torsional mode of an optical nanofiber (ONF), coupled to the polarization of the guided light [25–27]. We perform in-loop and out-of-loop measurements and observe cooling from the reduction of the angular fluctuations and broadening of the spectral density of the fluctuations. The measured optimal cooling is near the theoretical limit of the technique given by the signal-to-noise ratio (SNR) [28,29], scaling as $\sim 2\sqrt{1/\mathrm{SNR}}\approx 1.2\times {10}^{-3}$. Moreover, the platform presents a torque sensitivity $\approx {10}^{-26}{\mathrm{N}\mathrm{m}\mathrm{Hz}}^{-1/2}$, comparable with the record sensitivities achieved with nanodumbells [30]. Our results demonstrate ONFs to be a fruitful platform for rotational optomechanics, with potential applications in metrology and quantum optomechanics.

2. THEORETICAL MODEL

The ONF is a silica cylinder of diameter $\approx 550\mathrm{nm}$ and length $\approx 5\mathrm{mm}$ created by tapering a length of standard optical fiber, as shown in Fig. 1(b). It has string, compressional, and torsional modes [31], the latter of which couples to the polarization of guided light due to intrinsic birefringence produced during the fabrication process. When linearly polarized light of power $P_{\mathrm{opt}}$ propagates through the fiber, it results in an optically induced torque ${\tau }_{\mathrm{opt}}={\tau }_0{P}_{\mathrm{opt}}\sin (2(\theta -{\theta }_{\mathit{L}}))$, where ${\tau }_0$ has units of torque per unit power, $\theta$ is the angle of the slow axis, and ${\theta }_{\mathrm{L}}$ is the angle of the polarization [27,32].

 

Fig. 1. (a) Apparatus schematic. Probe and drive laser beams counterpropagate with independent polarization control. A beam splitter (BS) separates the probe into out-of-loop and in-loop detection. Each path has a half-wave plate ($\lambda /2$) to set the detection basis, followed by a polarizing beam splitter (PBS) and a balanced photodiode pair. The out-of-loop detection signal goes to a spectrum analyzer, while the in-loop signal splits parts, one to the spectrum analyzer, and the other is amplified, filtered ($\approx 100\mathrm{kHz}$ wide centered on resonance), and then goes to a control unit that slightly rotates the drive polarization closing the feedback loop. (b) ONF schematic with two effective polarization axes: ordinary and extraordinary indices of refraction, aligned with the optical axes but at an angle $\mathrm{\Delta }\theta$ with the input light polarization. The fiber is clamped (not shown) in the unmodified section.

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A fluctuating Langevin torque ${\tau }_{\mathrm{th}}$ with white spectral density and an external torque from applied optical feedback drive the ONF. The equation of motion for the angular coordinate $\theta$ representing the slow axis of the fundamental torsional mode is

When a static optical field is introduced, the torque results in a new equilibrium angle $\overline{\theta }$, found by dropping the time derivatives and fluctuating torque ${\tau }_{\mathrm{th}}$ in Eq. (1), yielding the transcendental equation $\kappa \overline{\theta }={\tau }_0{P}_{\mathrm{opt}}\sin (2(\overline{\theta }-{\theta }_{\mathit{L}}))$. Feedback will be introduced by modulating the polarization angle ${\theta }_{\mathit{L}}$ with a Pockels cell. Taking ${\theta }_{\mathit{L}}={\overline{\theta }}_{\mathit{L}}+\delta {\theta }_{\mathit{L}}$ and $\theta =\overline{\theta }+\delta \theta$ and linearizing Eq. (1) about the steady-state, one obtains

Here, we use active feedback such that $\delta {\theta }_{\mathit{L}}[\omega ]=G[\omega ]\delta \theta [\omega ]$, where $G[\omega ]$ describes the collective transfer function of the balanced photodetector, amplifier, proportional integral derivative controller, and Pockels cell shown in Fig. 1. In practice, there will always be measurement noise ${\theta }_n$ to which the feedback will respond as well; taking $\delta \theta [\omega ]\to \delta \theta [\omega ]+{\theta }_n[\omega ]$, we find

The term proportional to $P_{\mathrm{opt}}$ on the left-hand side of Eq. (4) can be chosen, by means of $G[\omega ]$, to maintain the form of a harmonic oscillator while altering its damping rate and/or natural frequency. Neglecting the frequency dependence of ${\tau }_0$, we choose derivative feedback with transfer function $G[\omega ]=i\omega G_D$ in order to add a term (proportional to $i\omega$) corresponding to damping. Equation (4) then takes the form of a harmonic oscillator with optically modified damping rate ${\mathrm{\Gamma }}^{\prime }=\mathrm{\Gamma }-\frac{2\beta }{I}{P}_{\mathrm{opt}}{\tau }_0{G}_D$, frequency ${\omega }_m^{\prime }={({\omega }_m^2-\frac{2\beta }{I}{P}_{\mathrm{opt}}{\tau }_0)}^{1/2}$, and torsional spring constant ${\kappa }^{\prime }=I{\omega }_m^{\prime 2}$ driven by fluctuations with torque spectral density

The spectral density of the angular fluctuations is then given by

Integration over all frequencies yields the mean-square angular fluctuations. Assuming a white spectral density for the measurement noise, the integral can be evaluated analytically:

Defining an effective mode temperature by $k_B{T}_{\mathrm{mode}}=I{\omega }_m^{\prime 2}⟨\delta {\theta }^2⟩$ and a dimensionless feedback gain $g=-\frac{2\beta }{I\mathrm{\Gamma }}{P}_{\mathrm{opt}}{\tau }_0{G}_D$, one finds

The torsional mode cooling will continue as long as the in-loop measurement can resolve the signal of torsional motion from the noise. Hence, the relative temperature reduction depends only on the SNR, the crucial figure of merit for feedback cooling, and it is manifestly independent of $T$, as shown in Eq. (9).

The reduction of the mode temperature can be measured in various ways. For high $Q$ systems, even in the presence of feedback, it is given by $k_B{T}_{\mathrm{mode}}=I{\omega }_{\mathrm{m}}^{\prime 2}⟨\delta {\theta }^2⟩$, where $⟨\delta {\theta }^2⟩$ is determined in terms of the integral of the measured distribution $S_{\delta \theta }$. It is also related to the broadened linewidth ${\mathrm{\Gamma }}^{\prime }$ of $S_{\delta \theta }$ by

3. EXPERIMENTAL SETUP

Figure 1 shows the experimental apparatus. We heat and pull [33] a commercial optical fiber (50 μm cladding, 4.2 μm core) to produce a $\approx 550\mathrm{nm}$ diameter, $\approx 5\mathrm{mm}$ length waist with a $\approx 1\mathrm{mrad}$ taper. For the wavelengths used, it allows the propagation of the fundamental ${\mathrm{HE}}_{11}$ mode [34]. The nanofiber resides in a vacuum chamber at a pressure of ${10}^{-5}\mathrm{Pa}$ (${10}^{-7}\mathrm{mbar}$) to suppress air damping; Ref. [25] shows that the $Q$ saturates at a pressure $P<{10}^{-2}\mathrm{Pa}$ (${10}^{-4}\mathrm{mbar}$). The fundamental torsional resonance of the ONF is at about 190 kHz with a half width at half maximum of 0.75(5) Hz, corresponding to a mechanical quality factor $Q\approx 1.26(8)\times {10}^5$. All the uncertainties reported in the text and Figs. 2 and 3 correspond to one standard deviation as obtained using the ${\chi }^2$ method of the fits.

 

Fig. 2. Mode temperature, calculated from the integral of $S_{\delta \theta }(\omega )$ from out-of-loop measurements, as a function of the drive laser power. The continuous line is a fit to Eq. (8). The inset shows the evolution of $S_{\delta \theta }(\omega )$ as the drive power is raised.

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Fig. 3. Mode temperature from the measured statistical distribution of the mean-square angular fluctuation $⟨\delta {\theta }^2⟩$ as a function of drive laser power. The continuous line is a fit to Eq. (8). The inset shows representative measured distributions $p({\theta }^2)$ and their fits to Eq. (11). The fits would appear linear on semilogarithmic axes, but making the horizontal axis logarithmic as well facilitates the visualization of the reduction of $⟨\delta {\theta }^2⟩$.

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In order to measure the angular fluctuations of the ONF, 250 μW of linearly polarized probe laser light at 852 nm is coupled in, and the transmitted polarization is redundantly analyzed by two pairs of balanced photodetectors (BPDs). Rotation of the ONF causes a linear rotation of the output polarization due to the ONF birefringence, and the balanced detection scheme provides a signal proportional to $\delta \theta$ while removing common-mode laser intensity fluctuations. One BPD is used as an “in-loop” detector for feedback; the other is used for “out-of-loop” detection. The signal from either BPD can be sent to a spectrum analyzer, used either to observe the spectral density of the signal, or in zero-span mode as a fixed programmable bandpass filter to eliminate technical noise in a measurement of the squared angular fluctuations $\delta {\theta }^2(t)$ as a function of time. Calibration is accomplished by assuming that the observed fluctuations in the absence of feedback are of thermal origin at room temperature. Feedback is applied by means of a linearly polarized “drive” laser [Fig. 1(a)] whose polarization angle ${\theta }_L$ is controlled by a Pockels cell, generating torque on the ONF. The output of one of the “in-loop” BPD is amplified and filtered to produce a signal corresponding approximately to derivative feedback and then applied to the Pockels cell. The loop gain can be controlled by means of the mean polarization ${\overline{\theta }}_{\mathit{L}}$, drive laser power, or electronic gain. We observe similar results in each case but vary the drive laser power in this work.

The whole experiment is placed atop a mechanically isolating optical table to improve stability. External vibrations mainly couple to the string modes of the ONF. However, we have observed mechanical coupling of the torsional modes to the environment from the turbomolecular pump. For this reason, we turn off the pump for the duration of the measurements, which is typically about 10 min. Small temperature instabilities in the laboratory can cause light polarization to drift in the hour time-scale since the ONF is not polarization-maintaining. For this reason, we monitor all outputs of the BPD, searching for polarization drifts and correcting them when needed.

4. RESULTS

Figure 2 shows the ratio of the mode temperature $T_{\mathrm{mode}}$ to the ambient temperature, $T\approx 300\mathrm{K}$, as a function of drive power (feedback). The mode temperature is inferred from the integral of the out-of-loop angular spectral density $S_{\delta \theta }(\omega )$, and the solid line shows a fit to Eq. (8). The inset shows the evolution of $S_{\delta \theta }(\omega )$ as the drive power increases. The amplitude drops toward the noise floor, the width broadens, and the mean-square angular fluctuation $⟨\delta {\theta }^2⟩$ given by the integral of $S_{\delta \theta }$ diminishes. Drive powers higher than 1.5 mW do not result in better cooling. Indeed, “squashing” [1,28] (not shown) appears in the in-loop signal as the $g^2$ term in Eq. (8) takes over, rendering the cooling less effective. The lowest mode temperature that we infer from these data is $T_{\mathrm{mode}}/T=4.0(1)\times {10}^{-3}$, corresponding to $T_{\mathrm{mode}}\approx 1.2\mathrm{K}$.

Figure 3 shows a complementary measurement of the mode temperature, made by using the spectrum analyzer as a combination of square-law detector and bandpass filter to infer the squared fluctuating amplitude $\delta {\theta }^2(t)$. Again, the measurements use out-of-loop data, and the solid line is a fit to Eq. (8). Representative statistics of $\delta {\theta }^2$ for time-series measurements of 10 s are shown in the inset, along with fits to the Boltzmann distribution given in Eq. (11). The limiting temperature that we observe here is $T_{\mathrm{mode}}/T=1.11(1)\times {10}^{-3}$, or $T_{\mathrm{mode}}\approx 330\mathrm{mK}$.

While we have followed the performance of the first torsional mode of the nanofiber carefully, the higher-order ones also show cooling. We believe the bandwidth of the feedback is sufficiently broad to have a direct effect, rather than other possible coupling mechanisms that we have left unexplored. Further, all high-order torsional modes show optomechanical coupling and can be cooled individually with the presented method. We also reproduce our results for different ONF radii.

5. DISCUSSION AND OUTLOOK

It is of interest to compare the degree of cooling we achieve to the expected limit given by Eq. (9) from the SNR. The noise floor is dominated by classical laser intensity noise and electronic (dark) noise, both of similar amplitudes for a 180 μW probe. We measure the contribution of both noise sources to the system by measuring the electronic voltage noise without the ONF, both with and without probe light striking the detectors. We refer the voltage noise back to effective angular noise by dividing it by the same calibration constant used to interpret the data with the ONF. We currently use commercial detector units operating near their maximum responsivity, corresponding to quantum efficiencies $>90\%$. Upon coupling the probe into the fiber, the overall detection efficiency of the system should be greater than 80%. The overall SNR could be further improved by using lower-noise electronics and a shot-noise-limited probe laser.

The amplitude of the signal for the data set shown in Fig. 2 measured on resonance is $S_s=4.50(9)\times {10}^{-9}{\mathrm{rad}}^2/\mathrm{Hz}$, while for the data set shown in Fig. 3 is $S_s=2⟨\delta {\theta }^2⟩/\mathrm{\Gamma }=3.90(3)\times {10}^{-8}{\mathrm{rad}}^2/\mathrm{Hz}$. The difference is due to systematic experimental variations typically observed. The nanofiber system is made from a nonpolarization-maintaining optical fiber, which causes the light polarization at the ONF to drift and imposes a technical challenge to set truly linearly polarized light at the ONF waist [35]. The differences in the signals from the data sets shown here highlight the role of the SNR in the cooling performance. The corresponding limiting mode temperature, from Eq. (9), is $T_{\mathrm{mode}}/T=4.90(4)\times {10}^{-3}$ and $T_{\mathrm{mode}}/T=1.664(4)\times {10}^{-3}$, respectively, in close agreement with the observed values. To increase the SNR, it would be desirable to enhance the mechanical transduction. Since the birefringence supplying the transduction in the ONF was an unintended artifact of the fabrication process, increasing it by a modification of the process seems plausible.

An independent study, performed in parallel to ours, shows similar results using electrodes for feedback-cooling an ONF torsional mode [36].

Once the detection quantum shot-noise limit is reached, one can apply standard techniques from the detection of squeezing. Further, it is possible to become less sensitive to the overall losses and efficiencies in propagation by applying conditional measurements [37]. Under those conditions, the field of quantum feedback opens new avenues on the platform [38,39]. In such a regime, one must address the effects of the back-action from the probe and the detection. On the one hand, the probe exerts torque on the ONF while sensing it, a common feature in many optomechanical systems [40]. Fluctuations in this torque will manifest themselves by inducing angular noise. We have varied the probe power by a factor of 5 and found no significant change in the SNR nor in the lowest achievable temperature, as the noise is dominated by classical intensity noise. This indicates that back-action from the probe is not a problem in the out-of-loop detection.

On the other hand, back-action effects from detection are already manifest in the squashing observed on the in-loop detection at the limit when the system achieves its lowest temperature. Previous studies in quantum optics have proposed using this to enhance signal-to-noise ratios in in-loop measurements [41]. Therefore, one could explore applying similar techniques to an optomechanical system limited by detection shot-noise, improving sensitivity to torque measurements in the nanofiber platform.

Beyond the optomechanical cooling capabilities of the platform, its high sensitivity to rotations makes ONFs a viable candidate for a torque sensor once systematic effects are controlled. The tensile strength of an ONF could allow its use as a torsional pendulum for precision force measurement [17]. Torsional modes also couple to external electric fields [25,36] presenting a potential field sensor. The sensitivity of the platform is ultimately defined by the noise floor, $S_{{\theta }_n}$, corresponding to a rotational sensitivity of $\approx 1.6\times {10}^{-7}\mathrm{rad}/\sqrt{\mathrm{Hz}}$. The conversion from rotation to torque depends on the modulus of the angular displacement susceptibility, which on resonance is $\chi ({\omega }_0)=1/{\omega }_0\gamma$. We thus obtain a torque sensitivity of $\approx 2.9\times {10}^{-26}{\mathrm{N}\mathrm{m}\mathrm{Hz}}^{-1/2}$, which is competitive with state-of-the-art rotational sensors [30]. The large scale of the system could allow for a larger interaction region of the sensor, improving the overall sensitivity and enabling measurements of quantum vacuum friction of polarizable objects near surfaces [30].

Since ONFs are known to be compatible with mK dilution refrigerators while guiding a few mW of optical power [42], ground-state cooling appears feasible. Indeed, at an ambient temperature of $T\approx 8\mathrm{mK}$, the mode temperature in Eq. (9) achievable with our current SNR yields a phonon occupation number $⟨n⟩={(e^{-h\nu /k_B{T}_{\mathrm{mode}}}-1)}^{-1}\approx 1$. Improvements to the SNR would allow starting from a higher ambient temperature, only limited by the necessary condition $Q{\nu }_m>k_{B}T/h$, which requires $T<1.1\mathrm{K}$. In this regime, techniques more appropriate to assessing $⟨n⟩$ than those employed here are available. In particular, the polarization rotation at ${\omega }_m$ induced by the nanofiber imparts phase modulation sidebands on the transmitted probe light at frequencies $\pm {\omega }_m$, in addition to modulating the differential intensity measured by the polarization analyzers shown in Fig. 1. A simple modification of our experimental apparatus as done earlier [26], beating the transmitted light with a local oscillator, would allow the sidebands to be readily measured in the rf domain. The ratio of the power in the sidebands is proportional to $⟨n⟩/(1+⟨n⟩)$ [43], which differs from unity as the quantum regime is approached.

Ground-state cooling of massive objects has implications in quantum technologies [44] and quantum foundations and tests of gravitational effects in quantum theory [45,46]. For example, by preparing the mechanical state in a quantum superposition, one could test gravity-induced quantum collapse models [47], which benefit from using massive objects over large spatial extensions. In that regard, ONFs offer promise as candidates to study quantum torsional optomechanics of relatively massive ($\approx 1\mathrm{ng}$) and large-scale ($\approx 1\mathrm{cm}$ size) objects.

6. CONCLUSION

In summary, we demonstrate optical feedback cooling of the fundamental torsional mode of a 5 mm long optical nanofiber, reducing the effective mode temperature by three orders of magnitude, reaching a mode temperature of $\approx 320\mathrm{mK}$ using optomechanical transduction in a cavityless system. The polarization of the guided light couples to the fiber via its intrinsic birefringence, enabling a sensitive probe of its rotation and a mechanism to optically apply torque for control purposes. We characterize the cooling in the frequency and time domains and find results that are near the limit imposed by the SNR. Finally, we discuss the possibilities of utilizing the platform as a torque sensor and suggest that reaching the quantum regime should be possible with a combination of cryogenic and optical feedback cooling.