Abstract

We demonstrate feedback cooling of a millimeter-scale, 40 kHz SiN membrane from room temperature to 5 mK (3000 phonons) using a Michelson interferometer, and discuss the challenges to ground-state cooling without an optical cavity. This advance appears within reach of current membrane technology, positioning it as a compelling alternative to levitated systems for quantum sensing and fundamental weak force measurements.

© 2020 Optical Society of America

Strained thin-film resonators (strings and membranes) with millimeter dimensions can support acoustic frequency modes with extremely high quality factors, leveraging the effect of dissipation dilution [14]. It has been speculated that they may enable room-temperature quantum optomechanics [3], ultraprecise force and acceleration sensing [5,6], quantum memories, and detection of fundamental weak signals such as spontaneous waveform collapse [7] and ultralight dark matter [8].

Here we discuss an additional potential of acoustic frequency thin-film resonators, which is to remove the need for a cavity in quantum optomechanics experiments. This possibility is due to the large zero-point fluctuations of an acoustic frequency nanomechanical resonator, as exploited in experiments with levitated nanoparticles [9]. In contrast to levitated nanoparticles, thin-film resonators can be read out with high efficiency by direct reflection or near-field sensing [10]. The main challenge to reaching the quantum regime is technical noise, such as laser relaxation oscillations, which can be far in excess of shot noise at acoustic frequencies. Optical absorption can 2also lead to large bolometric effects in tethered nanostructures, while in principle they can be decoupled from motion of a levitated particle [11].

To illustrate the potential for “cavity-free” quantum optomechanics, we describe an experiment in which the fundamental mode of a 2.5 mm, high-stress silicon nitride (${{\rm Si}_3}{{\rm N}_4}$) trampoline resonator [3,4] is subject to radiation pressure feedback cooling using a Michelson interferometer. The conditions for ground-state cooling are twofold [12]: (1) the oscillator’s thermal decoherence rate ${\Gamma _{{\rm th}}}$ must not exceed its frequency ${\Omega _0}$,

$${\Gamma _{{\rm th}}} = \frac{{{k_{\rm B}}{T_0}}}{{\hbar {Q_0}}} \lt {\Omega _0},$$
and (2) the measurement imprecision $S_\textit{xx}^{{\rm imp}}$ (expressed as a single-sided power spectral density) must be low enough to resolve zero-point motion ${x_{{\rm zp}}}$ in the thermal decoherence time
$$S_\textit{xx}^{{\rm imp,gs}} = \frac{{4x_{{\rm zp}}^2}}{{{\Gamma _{{\rm th}}}}} = \frac{{2{\hbar ^2}}}{{{k_{\rm B}}{T_0}}}\frac{{{Q_0}}}{{m{\Omega _0}}},$$
where ${T_0}$ is the intrinsic device temperature.

In our experiment, operated at room temperature, an optimized trampoline design [3,4] yields a fundamental frequency of ${\Omega _0} = 2\pi \cdot 40\;{\rm kHz} $, a quality factor ${Q_0} = 3 \times {10^7}$, and an effective mass of $m = 12 \;{\rm ng}$, corresponding to a thermal decoherence rate of ${\Gamma _{{\rm th}}} = 5{\Omega _0}$, a zero-point displacement of ${x_{{\rm zp}}} = 4\;{\rm fm}$, and a ground-state cooling requirement of [Eq. (2)] ${(S_\textit{xx}^{{\rm imp,gs}})^{1/2}} \approx {10^{- 17}}\;{\rm m}/\sqrt {{\rm Hz}}$. The latter is 3 orders of magnitude below the sensitivity of our interferometer; nevertheless, using an auxiliary laser field as a radiation pressure actuator, we realize feedback cooling to an effective temperature of ${T_{{\rm eff}}} = 5\;{\rm mK}$, corresponding to a mean phonon number of

$$\langle n\rangle = \frac{{{k_B}{T_{{\rm eff}}}}}{{\hbar {\Omega _0}}} \approx \sqrt {\frac{{S_\textit{xx}^{{\rm imp}}}}{{S_\textit{xx}^{{\rm imp,gs}}}}} \approx 3 \times {10^3}.$$
Below we discuss the design and limitations of this experiment and speculate that $\langle n\rangle \sim 1$ should be possible with simple modifications, including pre-cooling in a helium cryostat and using a common path interferometer topology.

1. MEASUREMENT-BASED FEEDBACK COOLING

In feedback cooling protocols, a continuous position measurement is used to suppress the thermal motion of a mechanical oscillator by derivative feedback (velocity damping). The technique dates back to regulation of electrometers [13] and is commonly used in atomic force microscopes to increase their dynamic range [14]. More recently, feedback cooling has received attention in the cavity optomechanics community as a means to prepare a nanomechanical oscillator in its ground state [15,16]. Cooling to $\langle n\rangle \sim 4$ has been achieved with a free space optically levitated nanoparticle [17], while cavity-enhanced measurements have been used to cool a ${{\rm Si}_3}{{\rm N}_4}$ nanostring to $\langle n\rangle \sim 4$ [12] and, more recently, a ${{\rm Si}_3}{{\rm N}_4}$ membrane to $\langle n\rangle \sim 0.3$ [18].

An important feature of feedback cooling is that, unlike optomechanical sideband cooling [19,20], it does not require a “good” (sideband-resolved) cavity to reach the ground state [21,22]. It is only necessary to achieve sufficient measurement efficiency, which in fact requires a “bad” cavity and in principle requires no cavity at all (enabling use of nonresonant sensors such as single-electron transistors and superconducting quantum interference devices) [17,2326]. To this end, agnostic to the measurement scheme, consider a real-time estimate $y$ of the oscillator displacement $x$ obscured by imprecision noise ${x_{{\rm imp}}}$:

$$y = x + {x_{{\rm imp}}}.$$
Feedback cooling can be understood by including a velocity-proportional feedback force in the Langevin equation [24]
$$m\ddot x + m{\Gamma _{0}}\dot x + m\Omega _{0}^2x = \sqrt {2{k_B}{T_0}m{\Gamma _0}} \xi (t) - gm{\Gamma _0}\dot y,$$
where $\xi (t)$ is a normalized Gaussian white noise process and ${\Gamma _0} = {\Omega _0}/{Q_0}$ is the intrinsic mechanical damping rate. Applying the Wiener–Khinchin theorem, the spectral density of physical $(x)$ and apparent $(y)$ displacement can be expressed as [12,18]
$$\frac{{{S_\textit{xx}}[\Omega]}}{{2S_\textit{xx}^{{\rm zp}}}} = |{\chi _g}[\Omega {]|^2}\left({{n_{{\rm th}}} + {g^2}{n_{{\rm imp}}}} \right),$$
$$\frac{{{S_\textit{yy}}[\Omega]}}{{2S_\textit{xx}^{{\rm zp}}}} = |{\chi _g}[\Omega {]|^2}\left({{n_{{\rm th}}} + {{(1 + g)}^2}|{\chi _0}[\Omega {{]|}^{- 2}}{n_{{\rm imp}}}} \right),$$
where $S_\textit{xx}^{{\rm zp}} = 4x_{{\rm zp}}^2/{\Gamma _0}$ is the zero-point displacement spectral density, ${n_{{\rm th}}} = {k_{\rm B}}{T_0}/m{\Omega _0}$ is the thermal bath occupation, and $\chi _g^{- 1} \approx (1 + g) + 2i(\Omega - {\Omega _0})/{\Gamma _0}$ is the closed-loop mechanical susceptibility. Evidently feedback damping can be “cold” in the sense that ${n_{{\rm imp}}} = S_\textit{xx}^{{\rm imp}}/2S_\textit{xx}^{{\rm zp}} \lt {n_{{\rm th}}}$ when the measurement resolves the thermal motion. Increasing the feedback gain $g$ thus reduces the average displacement of the oscillator $\langle {x^2}\rangle = \int\! {S_\textit{xx}}[\Omega]/2\pi$, resulting in a mean phonon number of
$$\langle n\rangle + \frac{1}{2} = \frac{{\langle {x^2}\rangle}}{{2x_{{\rm zp}}^2}} = \frac{{{n_{{\rm th}}} + {g^2}{n_{{\rm imp}}}}}{{1 + g}} \ge 2\sqrt {{n_{{\rm th}}}{n_{{\rm imp}}}}.$$
Ground-state cooling requires accounting for measurement back-action ${n_{{\rm th}}} \to {n_{{\rm th}}} + {n_{{\rm ba}}} = {n_{{\rm th}}} + \eta /(16{n_{{\rm imp}}})$, where $\eta \in [0,1]$ is the measurement efficiency [12,18]. Equation (7) thus yields Eq. (2) for $\langle n\rangle \lt 1$ and Eq. (3) for $1 \ll \langle n\rangle \ll {n_{{\rm th}}}$ (noting that ${\Gamma _{{\rm th}}} = {\Gamma _0}{n_{{\rm th}}}$). In addition to high efficiency, we emphasize that reaching low occupancy is facilitated by having a high $Q/m{\Omega _0}$ factor, which is equivalent to a high force sensitivity $S_\textit{FF}^{{\rm th}} = 4{k_B}Tm{\Omega _0}/Q$.

2. TRAMPOLINE RESONATOR

Our mechanical resonator is a modified version of the ${{\rm Si}_3}{{\rm N}_4}$ trampoline introduced by Reinhardt [4] and Norte et al. [3]. Trampoline resonators, like strings [2729], exhibit quality factors scaling as $Q \propto {Q_{\text{mat}}}(h)\sqrt \sigma L/h$, where $Q_{{\rm mat}}^{- 1}$ is the material loss tangent, $L$ is the tether length, $h$ is the film thickness, and $\sigma$ is the tensile stress in the film. Since ${\Omega _0} \propto \sqrt \sigma /L$ and $m \propto hL$, the implication is that $Q/m{\Omega _0} \propto {Q_{{\rm mat}}}(h)L/{h^2}$. Counterintuitively, larger devices can have larger zero-point fluctuations.

 

Fig. 1. ${{\rm Si}_3}{{\rm N}_4}$ trampoline resonator. (Top left) Camera image of a typical device. (Top right) Microscope image of the trampoline used in the experiment. (Bottom right) Finite element simulation of the fundamental 40 kHz vibrational mode. (Bottom left) Energy ringdown of the fundamental mode before (red) and after (blue) deposition of a dust particle onto a tether.

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The trampoline used in our experiment is shown in Fig. 1. The device is suspended from a $h \approx 90\,{\rm nm}$ thick ${{\rm Si}_3}{{\rm N}_4}$ film on a $200\,\unicode{x00B5}{\rm m}$ thick Si wafer (WaferPro) using a standard two-sided photolithography and wet etching technique [3,4,29]. A $200\,\unicode{x00B5}{\rm m}$ pad and $L \approx 1.7$ mm long, $w = 4.2\,\,\unicode{x00B5}{\rm m}$ wide tethers were chosen, as well as an “optimal” [30] $50\,\,\unicode{x00B5}{\rm m}$ radius fillet for this window size (2.5 mm) and tether width. Mechanical ringdown measurements performed in high vacuum [${ \lt\! 10^{- 7}}\; {\rm mbar}$, Fig. 1(c)] reveal a fundamental frequency of ${\Omega _0} = 2\pi \times 39.9\;{\rm kHz} $ and a quality factor as high as $Q = 4.4 \times {10^7}$, which agrees with finite element simulations (COMSOL) assuming a film stress of $\sigma = 0.9 \; {\rm GPa}$ and internal quality factor of ${Q_{{\rm mat}}} = 6 \times {10^3}$. (The latter is consistent with the ${Q_{\text{mat}}} \propto h$ surface loss model of Villanueva et al. [28], suggesting that our device is not limited by clamping loss.) For the experiments described below, dust deposited on the tether resulted in a reduced quality factor of $Q = 2.6 \times {10^7}$. Together with a simulated effective mass of $m = 12 \; {\rm ng}$, this implies a force sensitivity of $S_\textit{FF}^{{\rm th}} = (43\;{\rm aN}/\sqrt {{\rm Hz}} {)^2}$, a zero-point displacement of $S_\textit{xx}^{{\rm zp}} = (86 \; {\rm fm}/\sqrt {{\rm Hz}} {)^2}$, and a ground-state cooling requirement $S_\textit{xx}^{{\rm imp,gs}} = S_\textit{xx}^{{\rm zp}}/{n_{{\rm th}}} \approx {(0.68 \times {10^{- 17}} \;{\rm m}/\sqrt {{\rm Hz}})^2}$.

 

Fig. 2. Setup for probing the trampoline, consisting of a confocal microscope embedded in a balanced Michelson interferometer. Electronics for stabilizing the interferometer path length (PI = proportional integral controller, Newport LB1005) and for radiation pressure feedback cooling (see main text for details) are indicated in black. An image of the focused optical beam on the trampoline pad is shown at bottom left.

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3. INTERFEROMETRIC READOUT

Displacement of the trampoline was read out using a confocal microscope integrated into a Michelson interferometer [10]. Details are shown in Fig. 2. To minimize gas damping, the device chip is mounted in a high vacuum chamber operating at ${ \lt\! 10^{- 7}}\; {\rm mbar}$. This is enabled by a long-working-distance microscope objective (Mitutoyo M Plan APO 10X) with a spot diameter of ${\lt}5\,\,\unicode{x00B5}{\rm m}$. The light source used for the experiment was an 850 nm external cavity diode laser (Newport TLB-6716). To mitigate laser frequency and intensity noise, both the arm length and power of the interferometer were carefully balanced.

 

Fig. 3. Characterization of interferometer sensitivity. Upper plot: imprecision in noise quanta units versus power, compared to Eq. (8) (blue line). Lower plot: apparent displacement spectrum of the trampoline versus frequency for different optical powers. The dashed line is a model for ${S_\textit{xx}}$. The blue line is obtained by blocking the signal arm of the interferometer at highest power. (Inset: broadband spectrum for highest power).

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Ideally, the interferometer sensitivity is limited by shot noise

$$S_\textit{xx}^{{\rm imp,shot}} = \frac{{\hbar c\lambda}}{{16\pi\! \eta}}\frac{{{R_{\rm m}}}}{P},$$
where $P$ is the power incident on the membrane, $\lambda$ is the laser wavelength, ${R_{\rm m}}$ is the membrane reflectance, and $\eta \in [0,1]$ is the detection efficiency. We investigated this limit by recording apparent displacement spectra ${S_\textit{yy}}$ at different optical powers, where $y$ is proportional to the voltage signal produced by the balanced photoreceiver (Newport 1807). Results are shown in Fig. 3. To calibrate each measurement, a piezo underneath the device chip was used to drive the trampoline near its fundamental resonance with a coherent amplitude of ${x_{{\rm cal}}} = 0.8$ pm (inferred by bootstrapping to the area beneath the thermal noise peak, $\langle {x^2}\rangle \approx 2x_{{\rm zp}}^2{n_{{\rm th}}}$, after an averaging time ${\gt}{Q_0}/{\Omega _0}$). At low powers ($P \lt 100\; \unicode{x00B5}{\rm W}$), $S_\textit{xx}^{{\rm imp}}$ scales inversely with power, as expected for shot noise, with an apparent efficiency of $\eta \approx 10\%$ (using ${R_{\rm m}} = 0.3$ [31]). This value is consistent with the return loss of our microscope objective (which employs a free-space-to-fiber coupler), and in principle allows cooling to $\langle n\rangle \approx 1.1$.

In practice, the interferometer is limited by extraneous noise at sufficiently high power. This is seen in Fig. 3 for powers above 1 mW, where the noise floor saturates to $S_\textit{xx}^{{\rm imp,ext}} \approx 10 \;{\rm fm}/\sqrt {{\rm Hz}}$. Broadband measurements (Fig. 3, inset) suggest that $S_\textit{xx}^{{\rm imp,ext}}$ is related to differential polarization or path-length fluctuations, possibly exacerbated by peaking of the homodyne phase lock. (We note that extraneous laser frequency noise was ruled out by introducing a path length imbalance of several millimeters, to no apparent effect.) Although an impressive 2 orders of magnitude below the intrinsic zero-point motion $({n_{{\rm imp,ext}}} \approx 0.01)$, this extraneous noise practically limits feedback cooling of the fundamental trampoline mode to $\langle n\rangle \approx 2.5 \times {10^3}$ starting at room temperature (${n_{{\rm th}}} = 1.6 \times {10^8}$), according to Eq. (7).

Heating from optical absorption is another important consideration at high powers. To investigate this effect, we consider the off-resonant thermal noise in Fig. 3, which in the presence of heating should increase linearly with power (${S_\textit{xx}}[\Omega - {\Omega _0} \gg \;g{\Gamma _0}] \propto {n_{{\rm th}}}{\Gamma _0}/(\Omega - {\Omega _0}{)^2}$). The variation observed is within the ${\sim}10\%$ statistical error of the spectral density estimate, suggesting that heating is less than $10 \; {\rm K}/{\rm mW}$. This value is consistent with a simple heat conduction model $dT/dP \approx \alpha L/4wh\kappa$, assuming a thermal conductivity of $\kappa = 3 \; {\rm W}/{\rm m} {\rm K}$ and a conservative optical absorption coefficient of $\alpha = 10\;{\rm ppm} $ [31].

4. RADIATION PRESSURE FEEDBACK COOLING

Feedback cooling was carried out using radiation pressure actuation. The main advantage of this approach is its high bandwidth; however, we note that other methods such as piezoelectric [32] and dielectric [33] actuation are in principle equally viable and may be simpler to interface with feedback electronics.

To implement radiation pressure feedback, we introduce a second laser beam into the microscope, which is intensity modulated by an amplified copy of the photosignal. Specifically, we use a 670 nm laser diode (Hitachi HL6712) modulated by dithering its drive current about the threshold value. To approximate derivate feedback while suppressing feedback to higher-order modes, the photosignal is passed through a 10–50 kHz bandpass filter and a delay line, resulting in an approximately $\phi = 90^ \circ$ phase shift for frequencies near mechanical resonance. The feedback force can in this case be approximated as

$$\delta\! {F_{{\rm fb}}} \approx - g\!m{\Gamma _0}(\dot y + {\Omega _0}\cot (\phi)y),$$
corresponding to a normalized susceptibility ${\chi _g}{[\Omega]^{- 1}} \approx (1 + g) + 2i(\Omega - {\Omega _0})/{\Gamma _0} + ig\cot (\phi)$, where $ig\cot (\phi)$ is a residual feedback stiffening term that contributes negligibly to cooling.

The results of feedback cooling with a 3 mW read out beam and a 60 µW feedback beam are shown in Fig. 4. The feedback gain is tuned electronically using a voltage preamplifier (Stanford Research Systems SR560). To estimate $\langle n\rangle$, thermal noise spectra are fit to Eq. (6a) with $g$ as a free parameter, assuming ${n_{{\rm th}}} = 1.56 \times {10^8}$, ${n_{{\rm imp}}} = 0.013$, ${\Gamma _0} = 2\pi \cdot 1.5\; {\rm mHz}$, and $\phi = - 0.15$ [Eq. (9)]. To facilitate fitting, in Fig. 4, we focus on high gain settings for which the loaded damping rate $(1 + g){\Gamma _0} \gt 1\; {\rm Hz}$. The model accurately reproduces the noise spectra until the damped peak coincides with the noise floor, for which the inferred gain is $g = 1.4 \times {10^5}$, corresponding to $\langle n\rangle = 3.0 \times {10^3}$. At higher gain, the noise floor exhibits typical “squashing” behavior [12,34], and the inferred $\langle n\rangle$ begins to increase.

 

Fig. 4. Radiation pressure feedback cooling. Upper plot: feedback cooling curve for parameters described in the main text. Colored points correspond to models overlaying experimental data. Lower plot: experimental measurements (colored) overlaid with models (dashed curves) using Eq. (9). The solid black curve is a model for $g = 0$ (no feedback).

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5. SUMMARY AND OUTLOOK

We have demonstrated measurement-based feedback cooling of a 40 kHz ${{\rm Si}_3}{{\rm N}_4}$ trampoline resonator from room temperature ($1.6 \times {10^8}$ phonons) to an effective temperature of 5 mK ($3 \times {10^3}$ phonons) using a simple two-path interferometer. The main limitation of our experiment is technical noise at the level of $10 \; {\rm fm}/\sqrt {{\rm Hz}}$. Absent this noise, the apparent 10% efficiency of our interferometer would in principle enable cooling to $\langle n\rangle \sim 100$ with a probe power of several megawatts. Operating at 4 K, assuming no increase in mechanical Q and no photothermal heating, would enable cooling to $\langle n\rangle \sim 10$, for which motional sideband asymmetry could be readily measured.

We speculate that a combination of monolithic interferometer design and moderate cryogenics could give access to $\langle n\rangle \sim 1$ without an optical cavity for state-of-the-art ${{\rm Si}_3}{{\rm N}_4}$ thin-film resonators. Particularly compelling are “soft-clamped” nanobeams [2], which have demonstrated megahertz modes with quality factors approaching ${10^9}$ and zero-point spectral densities exceeding $1 \;{\rm pm}/\sqrt {{\rm Hz}}$, and can be read out with high efficiency by evanescent coupling to optical waveguide. For soft-clamped resonators, an important challenge is the large density of states and low thermal conductance of the phononic crystal shield, which reduces power handling capacity and can introduce extraneous thermal noise. Clamp-optimized trampolines [3,4,30] might offer a simpler route, since the fundamental mode is well isolated and can also have ${Q_0}{ \gt 10^8}$ at millimeter dimensions [2]. In the future, resonators made of strained crystalline thin films promise ${Q_0}{ \gt 10^9}$ and increased thermal conductivity at 4 K [35]. A recent proposal for soft-clamping fundamental modes using a “fractal clamp” might push this performance even further [36].

Funding

National Science Foundation (ECCS-1725571).

Acknowledgment

C.M.P. gratefully acknowledges support from an Amherst College Fellowship. The reactive ion etcher used in this study was acquired by an NSF MRI grant ECCS-1725571.

Disclosures

The authors declare no conflicts of interest.

REFERENCES AND NOTE

1. Y. Tsaturyan, A. Barg, E. S. Polzik, and A. Schliesser, “Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution,” Nat. Nanotechnol. 12, 776–783 (2017). [CrossRef]  

2. A. H. Ghadimi, S. A. Fedorov, N. J. Engelsen, M. J. Bereyhi, R. Schilling, D. J. Wilson, and T. J. Kippenberg, “Elastic strain engineering for ultralow mechanical dissipation,” Science 360, 764–768 (2018). [CrossRef]  

3. R. A. Norte, J. P. Moura, and S. Gröblacher, “Mechanical resonators for quantum optomechanics experiments at room temperature,” Phys. Rev. Lett. 116, 147202 (2016). [CrossRef]  

4. C. Reinhardt, T. Müller, A. Bourassa, and J. C. Sankey, “Ultralow-noise sin trampoline resonators for sensing and optomechanics,” Phys. Rev. X 6, 021001 (2016). [CrossRef]  

5. R. Fischer, D. P. McNally, C. Reetz, G. G. Assumpcao, T. Knief, Y. Lin, and C. A. Regal, “Spin detection with a micromechanical trampoline: towards magnetic resonance microscopy harnessing cavity optomechanics,” New J. Phys. 21, 043049 (2019). [CrossRef]  

6. A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and O. Painter, “A high-resolution microchip optomechanical accelerometer,” Nat. Photonics 6, 768–772 (2012). [CrossRef]  

7. S. Nimmrichter, K. Hornberger, and K. Hammerer, “Optomechanical sensing of spontaneous wave-function collapse,” Phys. Rev. Lett. 113, 020405 (2014). [CrossRef]  

8. D. Carney, A. Hook, Z. Liu, J. M. Taylor, and Y. Zhao, “Ultralight dark matter detection with mechanical quantum sensors,” arXiv preprint arXiv:1908.04797 (2019).

9. Z.-Q. Yin, A. A. Geraci, and T. Li, “Optomechanics of levitated dielectric particles,” Int. J. Mod. Phys. B 27, 1330018 (2013). [CrossRef]  

10. A. Barg, Y. Tsaturyan, E. Belhage, W. H. Nielsen, C. B. Møller, and A. Schliesser, “Measuring and imaging nanomechanical motion with laser light,” Appl. Phys. B 123, 8 (2017). [CrossRef]  

11. E. Hebestreit, R. Reimann, M. Frimmer, and L. Novotny, “Measuring the internal temperature of a levitated nanoparticle in high vacuum,” Phys. Rev. A 97, 043803 (2018). [CrossRef]  

12. D. Wilson, V. Sudhir, N. Piro, R. Schilling, A. Ghadimi, and T. Kippenberg, “Measurement-based control of a mechanical oscillator at its thermal decoherence rate,” Nature 524, 325–329 (2015). [CrossRef]  

13. J. Milatz, J. Van Zolingen, and B. Van Iperen, “The reduction in the Brownian motion of electrometers,” Physica 19, 195–202 (1953). [CrossRef]  

14. J. Mertz, O. Marti, and J. Mlynek, “Regulation of a microcantilever response by force feedback,” App. Phys. Lett. 62, 2344–2346 (1993). [CrossRef]  

15. J.-M. Courty, A. Heidmann, and M. Pinard, “Quantum limits of cold damping with optomechanical coupling,” Eur. Phys. J. D 17, 399–408 (2001). [CrossRef]  

16. D. Vitali, S. Mancini, L. Ribichini, and P. Tombesi, “Macroscopic mechanical oscillators at the quantum limit through optomechanical cooling,” J. Opt. Soc. Am. B 20, 1054–1065 (2003). [CrossRef]  

17. F. Tebbenjohanns, M. Frimmer, V. Jain, D. Windey, and L. Novotny, “Motional sideband asymmetry of a nanoparticle optically levitated in free space,” Phys. Rev. Lett. 124, 013603 (2020). [CrossRef]  

18. M. Rossi, D. Mason, J. Chen, Y. Tsaturyan, and A. Schliesser, “Measurement-based quantum control of mechanical motion,” Nature 563, 53–58 (2018). [CrossRef]  

19. I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007). [CrossRef]  

20. F. Marquardt, J. P. Chen, A. A. Clerk, and S. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007). [CrossRef]  

21. C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008). [CrossRef]  

22. K. Jacobs, H. I. Nurdin, F. W. Strauch, and M. James, “Comparing resolved-sideband cooling and measurement-based feedback cooling on an equal footing: analytical results in the regime of ground-state cooling,” Phys. Rev. A 91, 043812 (2015). [CrossRef]  

23. P. Bushev, D. Rotter, A. Wilson, F. Dubin, C. Becher, J. Eschner, R. Blatt, V. Steixner, P. Rabl, and P. Zoller, “Feedback cooling of a single trapped ion,” Phys. Rev. Lett. 96, 043003 (2006). [CrossRef]  

24. M. Poggio, C. Degen, H. Mamin, and D. Rugar, “Feedback cooling of a cantilever’s fundamental mode below 5 mK,” Phys. Rev. Lett. 99, 017201 (2007). [CrossRef]  

25. A. Hopkins, K. Jacobs, S. Habib, and K. Schwab, “Feedback cooling of a nanomechanical resonator,” Phys. Rev. B 68, 235328 (2003). [CrossRef]  

26. A. Vinante, A. Kirste, A. Den Haan, O. Usenko, G. Wijts, E. Jeffrey, P. Sonin, D. Bouwmeester, and T. Oosterkamp, “High sensitivity squid-detection and feedback-cooling of an ultrasoft microcantilever,” App. Phys. Lett. 101, 123101 (2012). [CrossRef]  

27. A. H. Ghadimi, D. J. Wilson, and T. J. Kippenberg, “Radiation and internal loss engineering of high-stress silicon nitride nanobeams,” Nano Lett. 17, 3501–3505 (2017). [CrossRef]  

28. L. G. Villanueva and S. Schmid, “Evidence of surface loss as ubiquitous limiting damping mechanism in sin micro- and nanomechanical resonators,” Phys. Rev. Lett. 113, 227201 (2014). [CrossRef]  

29. We follow the method of Reinhardt et al. [4] using a simplified Teflon sample holder for wet etching. Fabrication begins with standard two-sided photolithography followed by dry etch of ${{\rm Si}_3}{{\rm N}_4}$, using plasma Ar+SF$_{6}$, to define the trampoline on one side and a square window on the other. The chip is then wet etched from both sides using a 45% percent KOH solution at 65 C for 21 hours. After the trampoline is released, the gradual dilution method described in [4] is used in lieu of a critical point dryer, in order to suspend the trampoline.

30. P. Sadeghi, M. Tanzer, S. L. Christensen, and S. Schmid, “Influence of clamp-widening on the quality factor of nanomechanical silicon nitride resonators,” J. Appl. Phys. 126, 165108 (2019). [CrossRef]  

31. D. Wilson, C. Regal, S. Papp, and H. Kimble, “Cavity optomechanics with stoichiometric sin films,” Phys. Rev. Lett. 103, 207204 (2009). [CrossRef]  

32. S. Sridaran and S. A. Bhave, “Electrostatic actuation of silicon optomechanical resonators,” Opt. Express 19, 9020–9026 (2011). [CrossRef]  

33. Q. P. Unterreithmeier, E. M. Weig, and J. P. Kotthaus, “Universal transduction scheme for nanomechanical systems based on dielectric forces,” Nature 458, 1001–1004 (2009). [CrossRef]  

34. M. Poggio, C. Degen, H. Mamin, and D. Rugar, “Feedback cooling of a cantilever’s fundamental mode below 5 mK,” Phys. Rev. Lett. 99, 017201 (2007). [CrossRef]  

35. E. Romero, V. M. Valenzuela, A. R. Kermany, F. Iacopi, and W. P. Bowen, “Engineering the dissipation of crystalline micromechanical resonators,” Phys. Rev. Applied 13, 04407 (2020). [CrossRef]  

36. S. A. Fedorov, A. Beccari, N. J. Engelsen, and T. J. Kippenberg, “Fractal-like mechanical resonators with a soft-clamped fundamental mode,” Phys. Rev. Lett. 124, 025502 (2020). [CrossRef]  

References

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  1. Y. Tsaturyan, A. Barg, E. S. Polzik, and A. Schliesser, “Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution,” Nat. Nanotechnol. 12, 776–783 (2017).
    [Crossref]
  2. A. H. Ghadimi, S. A. Fedorov, N. J. Engelsen, M. J. Bereyhi, R. Schilling, D. J. Wilson, and T. J. Kippenberg, “Elastic strain engineering for ultralow mechanical dissipation,” Science 360, 764–768 (2018).
    [Crossref]
  3. R. A. Norte, J. P. Moura, and S. Gröblacher, “Mechanical resonators for quantum optomechanics experiments at room temperature,” Phys. Rev. Lett. 116, 147202 (2016).
    [Crossref]
  4. C. Reinhardt, T. Müller, A. Bourassa, and J. C. Sankey, “Ultralow-noise sin trampoline resonators for sensing and optomechanics,” Phys. Rev. X 6, 021001 (2016).
    [Crossref]
  5. R. Fischer, D. P. McNally, C. Reetz, G. G. Assumpcao, T. Knief, Y. Lin, and C. A. Regal, “Spin detection with a micromechanical trampoline: towards magnetic resonance microscopy harnessing cavity optomechanics,” New J. Phys. 21, 043049 (2019).
    [Crossref]
  6. A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and O. Painter, “A high-resolution microchip optomechanical accelerometer,” Nat. Photonics 6, 768–772 (2012).
    [Crossref]
  7. S. Nimmrichter, K. Hornberger, and K. Hammerer, “Optomechanical sensing of spontaneous wave-function collapse,” Phys. Rev. Lett. 113, 020405 (2014).
    [Crossref]
  8. D. Carney, A. Hook, Z. Liu, J. M. Taylor, and Y. Zhao, “Ultralight dark matter detection with mechanical quantum sensors,” arXiv preprint arXiv:1908.04797 (2019).
  9. Z.-Q. Yin, A. A. Geraci, and T. Li, “Optomechanics of levitated dielectric particles,” Int. J. Mod. Phys. B 27, 1330018 (2013).
    [Crossref]
  10. A. Barg, Y. Tsaturyan, E. Belhage, W. H. Nielsen, C. B. Møller, and A. Schliesser, “Measuring and imaging nanomechanical motion with laser light,” Appl. Phys. B 123, 8 (2017).
    [Crossref]
  11. E. Hebestreit, R. Reimann, M. Frimmer, and L. Novotny, “Measuring the internal temperature of a levitated nanoparticle in high vacuum,” Phys. Rev. A 97, 043803 (2018).
    [Crossref]
  12. D. Wilson, V. Sudhir, N. Piro, R. Schilling, A. Ghadimi, and T. Kippenberg, “Measurement-based control of a mechanical oscillator at its thermal decoherence rate,” Nature 524, 325–329 (2015).
    [Crossref]
  13. J. Milatz, J. Van Zolingen, and B. Van Iperen, “The reduction in the Brownian motion of electrometers,” Physica 19, 195–202 (1953).
    [Crossref]
  14. J. Mertz, O. Marti, and J. Mlynek, “Regulation of a microcantilever response by force feedback,” App. Phys. Lett. 62, 2344–2346 (1993).
    [Crossref]
  15. J.-M. Courty, A. Heidmann, and M. Pinard, “Quantum limits of cold damping with optomechanical coupling,” Eur. Phys. J. D 17, 399–408 (2001).
    [Crossref]
  16. D. Vitali, S. Mancini, L. Ribichini, and P. Tombesi, “Macroscopic mechanical oscillators at the quantum limit through optomechanical cooling,” J. Opt. Soc. Am. B 20, 1054–1065 (2003).
    [Crossref]
  17. F. Tebbenjohanns, M. Frimmer, V. Jain, D. Windey, and L. Novotny, “Motional sideband asymmetry of a nanoparticle optically levitated in free space,” Phys. Rev. Lett. 124, 013603 (2020).
    [Crossref]
  18. M. Rossi, D. Mason, J. Chen, Y. Tsaturyan, and A. Schliesser, “Measurement-based quantum control of mechanical motion,” Nature 563, 53–58 (2018).
    [Crossref]
  19. I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
    [Crossref]
  20. F. Marquardt, J. P. Chen, A. A. Clerk, and S. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
    [Crossref]
  21. C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
    [Crossref]
  22. K. Jacobs, H. I. Nurdin, F. W. Strauch, and M. James, “Comparing resolved-sideband cooling and measurement-based feedback cooling on an equal footing: analytical results in the regime of ground-state cooling,” Phys. Rev. A 91, 043812 (2015).
    [Crossref]
  23. P. Bushev, D. Rotter, A. Wilson, F. Dubin, C. Becher, J. Eschner, R. Blatt, V. Steixner, P. Rabl, and P. Zoller, “Feedback cooling of a single trapped ion,” Phys. Rev. Lett. 96, 043003 (2006).
    [Crossref]
  24. M. Poggio, C. Degen, H. Mamin, and D. Rugar, “Feedback cooling of a cantilever’s fundamental mode below 5 mK,” Phys. Rev. Lett. 99, 017201 (2007).
    [Crossref]
  25. A. Hopkins, K. Jacobs, S. Habib, and K. Schwab, “Feedback cooling of a nanomechanical resonator,” Phys. Rev. B 68, 235328 (2003).
    [Crossref]
  26. A. Vinante, A. Kirste, A. Den Haan, O. Usenko, G. Wijts, E. Jeffrey, P. Sonin, D. Bouwmeester, and T. Oosterkamp, “High sensitivity squid-detection and feedback-cooling of an ultrasoft microcantilever,” App. Phys. Lett. 101, 123101 (2012).
    [Crossref]
  27. A. H. Ghadimi, D. J. Wilson, and T. J. Kippenberg, “Radiation and internal loss engineering of high-stress silicon nitride nanobeams,” Nano Lett. 17, 3501–3505 (2017).
    [Crossref]
  28. L. G. Villanueva and S. Schmid, “Evidence of surface loss as ubiquitous limiting damping mechanism in sin micro- and nanomechanical resonators,” Phys. Rev. Lett. 113, 227201 (2014).
    [Crossref]
  29. We follow the method of Reinhardt et al. [4] using a simplified Teflon sample holder for wet etching. Fabrication begins with standard two-sided photolithography followed by dry etch of ${{\rm Si}_3}{{\rm N}_4}$Si3N4, using plasma Ar+SF$_{6}$6, to define the trampoline on one side and a square window on the other. The chip is then wet etched from both sides using a 45% percent KOH solution at 65 C for 21 hours. After the trampoline is released, the gradual dilution method described in [4] is used in lieu of a critical point dryer, in order to suspend the trampoline.
  30. P. Sadeghi, M. Tanzer, S. L. Christensen, and S. Schmid, “Influence of clamp-widening on the quality factor of nanomechanical silicon nitride resonators,” J. Appl. Phys. 126, 165108 (2019).
    [Crossref]
  31. D. Wilson, C. Regal, S. Papp, and H. Kimble, “Cavity optomechanics with stoichiometric sin films,” Phys. Rev. Lett. 103, 207204 (2009).
    [Crossref]
  32. S. Sridaran and S. A. Bhave, “Electrostatic actuation of silicon optomechanical resonators,” Opt. Express 19, 9020–9026 (2011).
    [Crossref]
  33. Q. P. Unterreithmeier, E. M. Weig, and J. P. Kotthaus, “Universal transduction scheme for nanomechanical systems based on dielectric forces,” Nature 458, 1001–1004 (2009).
    [Crossref]
  34. M. Poggio, C. Degen, H. Mamin, and D. Rugar, “Feedback cooling of a cantilever’s fundamental mode below 5 mK,” Phys. Rev. Lett. 99, 017201 (2007).
    [Crossref]
  35. E. Romero, V. M. Valenzuela, A. R. Kermany, F. Iacopi, and W. P. Bowen, “Engineering the dissipation of crystalline micromechanical resonators,” Phys. Rev. Applied 13, 04407 (2020).
    [Crossref]
  36. S. A. Fedorov, A. Beccari, N. J. Engelsen, and T. J. Kippenberg, “Fractal-like mechanical resonators with a soft-clamped fundamental mode,” Phys. Rev. Lett. 124, 025502 (2020).
    [Crossref]

2020 (3)

F. Tebbenjohanns, M. Frimmer, V. Jain, D. Windey, and L. Novotny, “Motional sideband asymmetry of a nanoparticle optically levitated in free space,” Phys. Rev. Lett. 124, 013603 (2020).
[Crossref]

E. Romero, V. M. Valenzuela, A. R. Kermany, F. Iacopi, and W. P. Bowen, “Engineering the dissipation of crystalline micromechanical resonators,” Phys. Rev. Applied 13, 04407 (2020).
[Crossref]

S. A. Fedorov, A. Beccari, N. J. Engelsen, and T. J. Kippenberg, “Fractal-like mechanical resonators with a soft-clamped fundamental mode,” Phys. Rev. Lett. 124, 025502 (2020).
[Crossref]

2019 (2)

P. Sadeghi, M. Tanzer, S. L. Christensen, and S. Schmid, “Influence of clamp-widening on the quality factor of nanomechanical silicon nitride resonators,” J. Appl. Phys. 126, 165108 (2019).
[Crossref]

R. Fischer, D. P. McNally, C. Reetz, G. G. Assumpcao, T. Knief, Y. Lin, and C. A. Regal, “Spin detection with a micromechanical trampoline: towards magnetic resonance microscopy harnessing cavity optomechanics,” New J. Phys. 21, 043049 (2019).
[Crossref]

2018 (3)

A. H. Ghadimi, S. A. Fedorov, N. J. Engelsen, M. J. Bereyhi, R. Schilling, D. J. Wilson, and T. J. Kippenberg, “Elastic strain engineering for ultralow mechanical dissipation,” Science 360, 764–768 (2018).
[Crossref]

M. Rossi, D. Mason, J. Chen, Y. Tsaturyan, and A. Schliesser, “Measurement-based quantum control of mechanical motion,” Nature 563, 53–58 (2018).
[Crossref]

E. Hebestreit, R. Reimann, M. Frimmer, and L. Novotny, “Measuring the internal temperature of a levitated nanoparticle in high vacuum,” Phys. Rev. A 97, 043803 (2018).
[Crossref]

2017 (3)

Y. Tsaturyan, A. Barg, E. S. Polzik, and A. Schliesser, “Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution,” Nat. Nanotechnol. 12, 776–783 (2017).
[Crossref]

A. Barg, Y. Tsaturyan, E. Belhage, W. H. Nielsen, C. B. Møller, and A. Schliesser, “Measuring and imaging nanomechanical motion with laser light,” Appl. Phys. B 123, 8 (2017).
[Crossref]

A. H. Ghadimi, D. J. Wilson, and T. J. Kippenberg, “Radiation and internal loss engineering of high-stress silicon nitride nanobeams,” Nano Lett. 17, 3501–3505 (2017).
[Crossref]

2016 (2)

R. A. Norte, J. P. Moura, and S. Gröblacher, “Mechanical resonators for quantum optomechanics experiments at room temperature,” Phys. Rev. Lett. 116, 147202 (2016).
[Crossref]

C. Reinhardt, T. Müller, A. Bourassa, and J. C. Sankey, “Ultralow-noise sin trampoline resonators for sensing and optomechanics,” Phys. Rev. X 6, 021001 (2016).
[Crossref]

2015 (2)

D. Wilson, V. Sudhir, N. Piro, R. Schilling, A. Ghadimi, and T. Kippenberg, “Measurement-based control of a mechanical oscillator at its thermal decoherence rate,” Nature 524, 325–329 (2015).
[Crossref]

K. Jacobs, H. I. Nurdin, F. W. Strauch, and M. James, “Comparing resolved-sideband cooling and measurement-based feedback cooling on an equal footing: analytical results in the regime of ground-state cooling,” Phys. Rev. A 91, 043812 (2015).
[Crossref]

2014 (2)

L. G. Villanueva and S. Schmid, “Evidence of surface loss as ubiquitous limiting damping mechanism in sin micro- and nanomechanical resonators,” Phys. Rev. Lett. 113, 227201 (2014).
[Crossref]

S. Nimmrichter, K. Hornberger, and K. Hammerer, “Optomechanical sensing of spontaneous wave-function collapse,” Phys. Rev. Lett. 113, 020405 (2014).
[Crossref]

2013 (1)

Z.-Q. Yin, A. A. Geraci, and T. Li, “Optomechanics of levitated dielectric particles,” Int. J. Mod. Phys. B 27, 1330018 (2013).
[Crossref]

2012 (2)

A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and O. Painter, “A high-resolution microchip optomechanical accelerometer,” Nat. Photonics 6, 768–772 (2012).
[Crossref]

A. Vinante, A. Kirste, A. Den Haan, O. Usenko, G. Wijts, E. Jeffrey, P. Sonin, D. Bouwmeester, and T. Oosterkamp, “High sensitivity squid-detection and feedback-cooling of an ultrasoft microcantilever,” App. Phys. Lett. 101, 123101 (2012).
[Crossref]

2011 (1)

2009 (2)

Q. P. Unterreithmeier, E. M. Weig, and J. P. Kotthaus, “Universal transduction scheme for nanomechanical systems based on dielectric forces,” Nature 458, 1001–1004 (2009).
[Crossref]

D. Wilson, C. Regal, S. Papp, and H. Kimble, “Cavity optomechanics with stoichiometric sin films,” Phys. Rev. Lett. 103, 207204 (2009).
[Crossref]

2008 (1)

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[Crossref]

2007 (4)

M. Poggio, C. Degen, H. Mamin, and D. Rugar, “Feedback cooling of a cantilever’s fundamental mode below 5 mK,” Phys. Rev. Lett. 99, 017201 (2007).
[Crossref]

M. Poggio, C. Degen, H. Mamin, and D. Rugar, “Feedback cooling of a cantilever’s fundamental mode below 5 mK,” Phys. Rev. Lett. 99, 017201 (2007).
[Crossref]

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[Crossref]

F. Marquardt, J. P. Chen, A. A. Clerk, and S. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[Crossref]

2006 (1)

P. Bushev, D. Rotter, A. Wilson, F. Dubin, C. Becher, J. Eschner, R. Blatt, V. Steixner, P. Rabl, and P. Zoller, “Feedback cooling of a single trapped ion,” Phys. Rev. Lett. 96, 043003 (2006).
[Crossref]

2003 (2)

2001 (1)

J.-M. Courty, A. Heidmann, and M. Pinard, “Quantum limits of cold damping with optomechanical coupling,” Eur. Phys. J. D 17, 399–408 (2001).
[Crossref]

1993 (1)

J. Mertz, O. Marti, and J. Mlynek, “Regulation of a microcantilever response by force feedback,” App. Phys. Lett. 62, 2344–2346 (1993).
[Crossref]

1953 (1)

J. Milatz, J. Van Zolingen, and B. Van Iperen, “The reduction in the Brownian motion of electrometers,” Physica 19, 195–202 (1953).
[Crossref]

Aspelmeyer, M.

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[Crossref]

Assumpcao, G. G.

R. Fischer, D. P. McNally, C. Reetz, G. G. Assumpcao, T. Knief, Y. Lin, and C. A. Regal, “Spin detection with a micromechanical trampoline: towards magnetic resonance microscopy harnessing cavity optomechanics,” New J. Phys. 21, 043049 (2019).
[Crossref]

Barg, A.

Y. Tsaturyan, A. Barg, E. S. Polzik, and A. Schliesser, “Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution,” Nat. Nanotechnol. 12, 776–783 (2017).
[Crossref]

A. Barg, Y. Tsaturyan, E. Belhage, W. H. Nielsen, C. B. Møller, and A. Schliesser, “Measuring and imaging nanomechanical motion with laser light,” Appl. Phys. B 123, 8 (2017).
[Crossref]

Beccari, A.

S. A. Fedorov, A. Beccari, N. J. Engelsen, and T. J. Kippenberg, “Fractal-like mechanical resonators with a soft-clamped fundamental mode,” Phys. Rev. Lett. 124, 025502 (2020).
[Crossref]

Becher, C.

P. Bushev, D. Rotter, A. Wilson, F. Dubin, C. Becher, J. Eschner, R. Blatt, V. Steixner, P. Rabl, and P. Zoller, “Feedback cooling of a single trapped ion,” Phys. Rev. Lett. 96, 043003 (2006).
[Crossref]

Belhage, E.

A. Barg, Y. Tsaturyan, E. Belhage, W. H. Nielsen, C. B. Møller, and A. Schliesser, “Measuring and imaging nanomechanical motion with laser light,” Appl. Phys. B 123, 8 (2017).
[Crossref]

Bereyhi, M. J.

A. H. Ghadimi, S. A. Fedorov, N. J. Engelsen, M. J. Bereyhi, R. Schilling, D. J. Wilson, and T. J. Kippenberg, “Elastic strain engineering for ultralow mechanical dissipation,” Science 360, 764–768 (2018).
[Crossref]

Bhave, S. A.

Blasius, T. D.

A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and O. Painter, “A high-resolution microchip optomechanical accelerometer,” Nat. Photonics 6, 768–772 (2012).
[Crossref]

Blatt, R.

P. Bushev, D. Rotter, A. Wilson, F. Dubin, C. Becher, J. Eschner, R. Blatt, V. Steixner, P. Rabl, and P. Zoller, “Feedback cooling of a single trapped ion,” Phys. Rev. Lett. 96, 043003 (2006).
[Crossref]

Bourassa, A.

C. Reinhardt, T. Müller, A. Bourassa, and J. C. Sankey, “Ultralow-noise sin trampoline resonators for sensing and optomechanics,” Phys. Rev. X 6, 021001 (2016).
[Crossref]

Bouwmeester, D.

A. Vinante, A. Kirste, A. Den Haan, O. Usenko, G. Wijts, E. Jeffrey, P. Sonin, D. Bouwmeester, and T. Oosterkamp, “High sensitivity squid-detection and feedback-cooling of an ultrasoft microcantilever,” App. Phys. Lett. 101, 123101 (2012).
[Crossref]

Bowen, W. P.

E. Romero, V. M. Valenzuela, A. R. Kermany, F. Iacopi, and W. P. Bowen, “Engineering the dissipation of crystalline micromechanical resonators,” Phys. Rev. Applied 13, 04407 (2020).
[Crossref]

Bushev, P.

P. Bushev, D. Rotter, A. Wilson, F. Dubin, C. Becher, J. Eschner, R. Blatt, V. Steixner, P. Rabl, and P. Zoller, “Feedback cooling of a single trapped ion,” Phys. Rev. Lett. 96, 043003 (2006).
[Crossref]

Carney, D.

D. Carney, A. Hook, Z. Liu, J. M. Taylor, and Y. Zhao, “Ultralight dark matter detection with mechanical quantum sensors,” arXiv preprint arXiv:1908.04797 (2019).

Chen, J.

M. Rossi, D. Mason, J. Chen, Y. Tsaturyan, and A. Schliesser, “Measurement-based quantum control of mechanical motion,” Nature 563, 53–58 (2018).
[Crossref]

Chen, J. P.

F. Marquardt, J. P. Chen, A. A. Clerk, and S. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[Crossref]

Christensen, S. L.

P. Sadeghi, M. Tanzer, S. L. Christensen, and S. Schmid, “Influence of clamp-widening on the quality factor of nanomechanical silicon nitride resonators,” J. Appl. Phys. 126, 165108 (2019).
[Crossref]

Clerk, A. A.

F. Marquardt, J. P. Chen, A. A. Clerk, and S. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[Crossref]

Courty, J.-M.

J.-M. Courty, A. Heidmann, and M. Pinard, “Quantum limits of cold damping with optomechanical coupling,” Eur. Phys. J. D 17, 399–408 (2001).
[Crossref]

Degen, C.

M. Poggio, C. Degen, H. Mamin, and D. Rugar, “Feedback cooling of a cantilever’s fundamental mode below 5 mK,” Phys. Rev. Lett. 99, 017201 (2007).
[Crossref]

M. Poggio, C. Degen, H. Mamin, and D. Rugar, “Feedback cooling of a cantilever’s fundamental mode below 5 mK,” Phys. Rev. Lett. 99, 017201 (2007).
[Crossref]

Den Haan, A.

A. Vinante, A. Kirste, A. Den Haan, O. Usenko, G. Wijts, E. Jeffrey, P. Sonin, D. Bouwmeester, and T. Oosterkamp, “High sensitivity squid-detection and feedback-cooling of an ultrasoft microcantilever,” App. Phys. Lett. 101, 123101 (2012).
[Crossref]

Dubin, F.

P. Bushev, D. Rotter, A. Wilson, F. Dubin, C. Becher, J. Eschner, R. Blatt, V. Steixner, P. Rabl, and P. Zoller, “Feedback cooling of a single trapped ion,” Phys. Rev. Lett. 96, 043003 (2006).
[Crossref]

Engelsen, N. J.

S. A. Fedorov, A. Beccari, N. J. Engelsen, and T. J. Kippenberg, “Fractal-like mechanical resonators with a soft-clamped fundamental mode,” Phys. Rev. Lett. 124, 025502 (2020).
[Crossref]

A. H. Ghadimi, S. A. Fedorov, N. J. Engelsen, M. J. Bereyhi, R. Schilling, D. J. Wilson, and T. J. Kippenberg, “Elastic strain engineering for ultralow mechanical dissipation,” Science 360, 764–768 (2018).
[Crossref]

Eschner, J.

P. Bushev, D. Rotter, A. Wilson, F. Dubin, C. Becher, J. Eschner, R. Blatt, V. Steixner, P. Rabl, and P. Zoller, “Feedback cooling of a single trapped ion,” Phys. Rev. Lett. 96, 043003 (2006).
[Crossref]

Fedorov, S. A.

S. A. Fedorov, A. Beccari, N. J. Engelsen, and T. J. Kippenberg, “Fractal-like mechanical resonators with a soft-clamped fundamental mode,” Phys. Rev. Lett. 124, 025502 (2020).
[Crossref]

A. H. Ghadimi, S. A. Fedorov, N. J. Engelsen, M. J. Bereyhi, R. Schilling, D. J. Wilson, and T. J. Kippenberg, “Elastic strain engineering for ultralow mechanical dissipation,” Science 360, 764–768 (2018).
[Crossref]

Fischer, R.

R. Fischer, D. P. McNally, C. Reetz, G. G. Assumpcao, T. Knief, Y. Lin, and C. A. Regal, “Spin detection with a micromechanical trampoline: towards magnetic resonance microscopy harnessing cavity optomechanics,” New J. Phys. 21, 043049 (2019).
[Crossref]

Frimmer, M.

F. Tebbenjohanns, M. Frimmer, V. Jain, D. Windey, and L. Novotny, “Motional sideband asymmetry of a nanoparticle optically levitated in free space,” Phys. Rev. Lett. 124, 013603 (2020).
[Crossref]

E. Hebestreit, R. Reimann, M. Frimmer, and L. Novotny, “Measuring the internal temperature of a levitated nanoparticle in high vacuum,” Phys. Rev. A 97, 043803 (2018).
[Crossref]

Genes, C.

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[Crossref]

Geraci, A. A.

Z.-Q. Yin, A. A. Geraci, and T. Li, “Optomechanics of levitated dielectric particles,” Int. J. Mod. Phys. B 27, 1330018 (2013).
[Crossref]

Ghadimi, A.

D. Wilson, V. Sudhir, N. Piro, R. Schilling, A. Ghadimi, and T. Kippenberg, “Measurement-based control of a mechanical oscillator at its thermal decoherence rate,” Nature 524, 325–329 (2015).
[Crossref]

Ghadimi, A. H.

A. H. Ghadimi, S. A. Fedorov, N. J. Engelsen, M. J. Bereyhi, R. Schilling, D. J. Wilson, and T. J. Kippenberg, “Elastic strain engineering for ultralow mechanical dissipation,” Science 360, 764–768 (2018).
[Crossref]

A. H. Ghadimi, D. J. Wilson, and T. J. Kippenberg, “Radiation and internal loss engineering of high-stress silicon nitride nanobeams,” Nano Lett. 17, 3501–3505 (2017).
[Crossref]

Gigan, S.

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[Crossref]

Girvin, S.

F. Marquardt, J. P. Chen, A. A. Clerk, and S. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[Crossref]

Gröblacher, S.

R. A. Norte, J. P. Moura, and S. Gröblacher, “Mechanical resonators for quantum optomechanics experiments at room temperature,” Phys. Rev. Lett. 116, 147202 (2016).
[Crossref]

Habib, S.

A. Hopkins, K. Jacobs, S. Habib, and K. Schwab, “Feedback cooling of a nanomechanical resonator,” Phys. Rev. B 68, 235328 (2003).
[Crossref]

Hammerer, K.

S. Nimmrichter, K. Hornberger, and K. Hammerer, “Optomechanical sensing of spontaneous wave-function collapse,” Phys. Rev. Lett. 113, 020405 (2014).
[Crossref]

Hebestreit, E.

E. Hebestreit, R. Reimann, M. Frimmer, and L. Novotny, “Measuring the internal temperature of a levitated nanoparticle in high vacuum,” Phys. Rev. A 97, 043803 (2018).
[Crossref]

Heidmann, A.

J.-M. Courty, A. Heidmann, and M. Pinard, “Quantum limits of cold damping with optomechanical coupling,” Eur. Phys. J. D 17, 399–408 (2001).
[Crossref]

Hook, A.

D. Carney, A. Hook, Z. Liu, J. M. Taylor, and Y. Zhao, “Ultralight dark matter detection with mechanical quantum sensors,” arXiv preprint arXiv:1908.04797 (2019).

Hopkins, A.

A. Hopkins, K. Jacobs, S. Habib, and K. Schwab, “Feedback cooling of a nanomechanical resonator,” Phys. Rev. B 68, 235328 (2003).
[Crossref]

Hornberger, K.

S. Nimmrichter, K. Hornberger, and K. Hammerer, “Optomechanical sensing of spontaneous wave-function collapse,” Phys. Rev. Lett. 113, 020405 (2014).
[Crossref]

Iacopi, F.

E. Romero, V. M. Valenzuela, A. R. Kermany, F. Iacopi, and W. P. Bowen, “Engineering the dissipation of crystalline micromechanical resonators,” Phys. Rev. Applied 13, 04407 (2020).
[Crossref]

Jacobs, K.

K. Jacobs, H. I. Nurdin, F. W. Strauch, and M. James, “Comparing resolved-sideband cooling and measurement-based feedback cooling on an equal footing: analytical results in the regime of ground-state cooling,” Phys. Rev. A 91, 043812 (2015).
[Crossref]

A. Hopkins, K. Jacobs, S. Habib, and K. Schwab, “Feedback cooling of a nanomechanical resonator,” Phys. Rev. B 68, 235328 (2003).
[Crossref]

Jain, V.

F. Tebbenjohanns, M. Frimmer, V. Jain, D. Windey, and L. Novotny, “Motional sideband asymmetry of a nanoparticle optically levitated in free space,” Phys. Rev. Lett. 124, 013603 (2020).
[Crossref]

James, M.

K. Jacobs, H. I. Nurdin, F. W. Strauch, and M. James, “Comparing resolved-sideband cooling and measurement-based feedback cooling on an equal footing: analytical results in the regime of ground-state cooling,” Phys. Rev. A 91, 043812 (2015).
[Crossref]

Jeffrey, E.

A. Vinante, A. Kirste, A. Den Haan, O. Usenko, G. Wijts, E. Jeffrey, P. Sonin, D. Bouwmeester, and T. Oosterkamp, “High sensitivity squid-detection and feedback-cooling of an ultrasoft microcantilever,” App. Phys. Lett. 101, 123101 (2012).
[Crossref]

Kermany, A. R.

E. Romero, V. M. Valenzuela, A. R. Kermany, F. Iacopi, and W. P. Bowen, “Engineering the dissipation of crystalline micromechanical resonators,” Phys. Rev. Applied 13, 04407 (2020).
[Crossref]

Kimble, H.

D. Wilson, C. Regal, S. Papp, and H. Kimble, “Cavity optomechanics with stoichiometric sin films,” Phys. Rev. Lett. 103, 207204 (2009).
[Crossref]

Kippenberg, T.

D. Wilson, V. Sudhir, N. Piro, R. Schilling, A. Ghadimi, and T. Kippenberg, “Measurement-based control of a mechanical oscillator at its thermal decoherence rate,” Nature 524, 325–329 (2015).
[Crossref]

Kippenberg, T. J.

S. A. Fedorov, A. Beccari, N. J. Engelsen, and T. J. Kippenberg, “Fractal-like mechanical resonators with a soft-clamped fundamental mode,” Phys. Rev. Lett. 124, 025502 (2020).
[Crossref]

A. H. Ghadimi, S. A. Fedorov, N. J. Engelsen, M. J. Bereyhi, R. Schilling, D. J. Wilson, and T. J. Kippenberg, “Elastic strain engineering for ultralow mechanical dissipation,” Science 360, 764–768 (2018).
[Crossref]

A. H. Ghadimi, D. J. Wilson, and T. J. Kippenberg, “Radiation and internal loss engineering of high-stress silicon nitride nanobeams,” Nano Lett. 17, 3501–3505 (2017).
[Crossref]

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[Crossref]

Kirste, A.

A. Vinante, A. Kirste, A. Den Haan, O. Usenko, G. Wijts, E. Jeffrey, P. Sonin, D. Bouwmeester, and T. Oosterkamp, “High sensitivity squid-detection and feedback-cooling of an ultrasoft microcantilever,” App. Phys. Lett. 101, 123101 (2012).
[Crossref]

Knief, T.

R. Fischer, D. P. McNally, C. Reetz, G. G. Assumpcao, T. Knief, Y. Lin, and C. A. Regal, “Spin detection with a micromechanical trampoline: towards magnetic resonance microscopy harnessing cavity optomechanics,” New J. Phys. 21, 043049 (2019).
[Crossref]

Kotthaus, J. P.

Q. P. Unterreithmeier, E. M. Weig, and J. P. Kotthaus, “Universal transduction scheme for nanomechanical systems based on dielectric forces,” Nature 458, 1001–1004 (2009).
[Crossref]

Krause, A. G.

A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and O. Painter, “A high-resolution microchip optomechanical accelerometer,” Nat. Photonics 6, 768–772 (2012).
[Crossref]

Li, T.

Z.-Q. Yin, A. A. Geraci, and T. Li, “Optomechanics of levitated dielectric particles,” Int. J. Mod. Phys. B 27, 1330018 (2013).
[Crossref]

Lin, Q.

A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and O. Painter, “A high-resolution microchip optomechanical accelerometer,” Nat. Photonics 6, 768–772 (2012).
[Crossref]

Lin, Y.

R. Fischer, D. P. McNally, C. Reetz, G. G. Assumpcao, T. Knief, Y. Lin, and C. A. Regal, “Spin detection with a micromechanical trampoline: towards magnetic resonance microscopy harnessing cavity optomechanics,” New J. Phys. 21, 043049 (2019).
[Crossref]

Liu, Z.

D. Carney, A. Hook, Z. Liu, J. M. Taylor, and Y. Zhao, “Ultralight dark matter detection with mechanical quantum sensors,” arXiv preprint arXiv:1908.04797 (2019).

Mamin, H.

M. Poggio, C. Degen, H. Mamin, and D. Rugar, “Feedback cooling of a cantilever’s fundamental mode below 5 mK,” Phys. Rev. Lett. 99, 017201 (2007).
[Crossref]

M. Poggio, C. Degen, H. Mamin, and D. Rugar, “Feedback cooling of a cantilever’s fundamental mode below 5 mK,” Phys. Rev. Lett. 99, 017201 (2007).
[Crossref]

Mancini, S.

Marquardt, F.

F. Marquardt, J. P. Chen, A. A. Clerk, and S. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[Crossref]

Marti, O.

J. Mertz, O. Marti, and J. Mlynek, “Regulation of a microcantilever response by force feedback,” App. Phys. Lett. 62, 2344–2346 (1993).
[Crossref]

Mason, D.

M. Rossi, D. Mason, J. Chen, Y. Tsaturyan, and A. Schliesser, “Measurement-based quantum control of mechanical motion,” Nature 563, 53–58 (2018).
[Crossref]

McNally, D. P.

R. Fischer, D. P. McNally, C. Reetz, G. G. Assumpcao, T. Knief, Y. Lin, and C. A. Regal, “Spin detection with a micromechanical trampoline: towards magnetic resonance microscopy harnessing cavity optomechanics,” New J. Phys. 21, 043049 (2019).
[Crossref]

Mertz, J.

J. Mertz, O. Marti, and J. Mlynek, “Regulation of a microcantilever response by force feedback,” App. Phys. Lett. 62, 2344–2346 (1993).
[Crossref]

Milatz, J.

J. Milatz, J. Van Zolingen, and B. Van Iperen, “The reduction in the Brownian motion of electrometers,” Physica 19, 195–202 (1953).
[Crossref]

Mlynek, J.

J. Mertz, O. Marti, and J. Mlynek, “Regulation of a microcantilever response by force feedback,” App. Phys. Lett. 62, 2344–2346 (1993).
[Crossref]

Møller, C. B.

A. Barg, Y. Tsaturyan, E. Belhage, W. H. Nielsen, C. B. Møller, and A. Schliesser, “Measuring and imaging nanomechanical motion with laser light,” Appl. Phys. B 123, 8 (2017).
[Crossref]

Moura, J. P.

R. A. Norte, J. P. Moura, and S. Gröblacher, “Mechanical resonators for quantum optomechanics experiments at room temperature,” Phys. Rev. Lett. 116, 147202 (2016).
[Crossref]

Müller, T.

C. Reinhardt, T. Müller, A. Bourassa, and J. C. Sankey, “Ultralow-noise sin trampoline resonators for sensing and optomechanics,” Phys. Rev. X 6, 021001 (2016).
[Crossref]

Nielsen, W. H.

A. Barg, Y. Tsaturyan, E. Belhage, W. H. Nielsen, C. B. Møller, and A. Schliesser, “Measuring and imaging nanomechanical motion with laser light,” Appl. Phys. B 123, 8 (2017).
[Crossref]

Nimmrichter, S.

S. Nimmrichter, K. Hornberger, and K. Hammerer, “Optomechanical sensing of spontaneous wave-function collapse,” Phys. Rev. Lett. 113, 020405 (2014).
[Crossref]

Nooshi, N.

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[Crossref]

Norte, R. A.

R. A. Norte, J. P. Moura, and S. Gröblacher, “Mechanical resonators for quantum optomechanics experiments at room temperature,” Phys. Rev. Lett. 116, 147202 (2016).
[Crossref]

Novotny, L.

F. Tebbenjohanns, M. Frimmer, V. Jain, D. Windey, and L. Novotny, “Motional sideband asymmetry of a nanoparticle optically levitated in free space,” Phys. Rev. Lett. 124, 013603 (2020).
[Crossref]

E. Hebestreit, R. Reimann, M. Frimmer, and L. Novotny, “Measuring the internal temperature of a levitated nanoparticle in high vacuum,” Phys. Rev. A 97, 043803 (2018).
[Crossref]

Nurdin, H. I.

K. Jacobs, H. I. Nurdin, F. W. Strauch, and M. James, “Comparing resolved-sideband cooling and measurement-based feedback cooling on an equal footing: analytical results in the regime of ground-state cooling,” Phys. Rev. A 91, 043812 (2015).
[Crossref]

Oosterkamp, T.

A. Vinante, A. Kirste, A. Den Haan, O. Usenko, G. Wijts, E. Jeffrey, P. Sonin, D. Bouwmeester, and T. Oosterkamp, “High sensitivity squid-detection and feedback-cooling of an ultrasoft microcantilever,” App. Phys. Lett. 101, 123101 (2012).
[Crossref]

Painter, O.

A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and O. Painter, “A high-resolution microchip optomechanical accelerometer,” Nat. Photonics 6, 768–772 (2012).
[Crossref]

Papp, S.

D. Wilson, C. Regal, S. Papp, and H. Kimble, “Cavity optomechanics with stoichiometric sin films,” Phys. Rev. Lett. 103, 207204 (2009).
[Crossref]

Pinard, M.

J.-M. Courty, A. Heidmann, and M. Pinard, “Quantum limits of cold damping with optomechanical coupling,” Eur. Phys. J. D 17, 399–408 (2001).
[Crossref]

Piro, N.

D. Wilson, V. Sudhir, N. Piro, R. Schilling, A. Ghadimi, and T. Kippenberg, “Measurement-based control of a mechanical oscillator at its thermal decoherence rate,” Nature 524, 325–329 (2015).
[Crossref]

Poggio, M.

M. Poggio, C. Degen, H. Mamin, and D. Rugar, “Feedback cooling of a cantilever’s fundamental mode below 5 mK,” Phys. Rev. Lett. 99, 017201 (2007).
[Crossref]

M. Poggio, C. Degen, H. Mamin, and D. Rugar, “Feedback cooling of a cantilever’s fundamental mode below 5 mK,” Phys. Rev. Lett. 99, 017201 (2007).
[Crossref]

Polzik, E. S.

Y. Tsaturyan, A. Barg, E. S. Polzik, and A. Schliesser, “Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution,” Nat. Nanotechnol. 12, 776–783 (2017).
[Crossref]

Rabl, P.

P. Bushev, D. Rotter, A. Wilson, F. Dubin, C. Becher, J. Eschner, R. Blatt, V. Steixner, P. Rabl, and P. Zoller, “Feedback cooling of a single trapped ion,” Phys. Rev. Lett. 96, 043003 (2006).
[Crossref]

Reetz, C.

R. Fischer, D. P. McNally, C. Reetz, G. G. Assumpcao, T. Knief, Y. Lin, and C. A. Regal, “Spin detection with a micromechanical trampoline: towards magnetic resonance microscopy harnessing cavity optomechanics,” New J. Phys. 21, 043049 (2019).
[Crossref]

Regal, C.

D. Wilson, C. Regal, S. Papp, and H. Kimble, “Cavity optomechanics with stoichiometric sin films,” Phys. Rev. Lett. 103, 207204 (2009).
[Crossref]

Regal, C. A.

R. Fischer, D. P. McNally, C. Reetz, G. G. Assumpcao, T. Knief, Y. Lin, and C. A. Regal, “Spin detection with a micromechanical trampoline: towards magnetic resonance microscopy harnessing cavity optomechanics,” New J. Phys. 21, 043049 (2019).
[Crossref]

Reimann, R.

E. Hebestreit, R. Reimann, M. Frimmer, and L. Novotny, “Measuring the internal temperature of a levitated nanoparticle in high vacuum,” Phys. Rev. A 97, 043803 (2018).
[Crossref]

Reinhardt, C.

C. Reinhardt, T. Müller, A. Bourassa, and J. C. Sankey, “Ultralow-noise sin trampoline resonators for sensing and optomechanics,” Phys. Rev. X 6, 021001 (2016).
[Crossref]

Ribichini, L.

Romero, E.

E. Romero, V. M. Valenzuela, A. R. Kermany, F. Iacopi, and W. P. Bowen, “Engineering the dissipation of crystalline micromechanical resonators,” Phys. Rev. Applied 13, 04407 (2020).
[Crossref]

Rossi, M.

M. Rossi, D. Mason, J. Chen, Y. Tsaturyan, and A. Schliesser, “Measurement-based quantum control of mechanical motion,” Nature 563, 53–58 (2018).
[Crossref]

Rotter, D.

P. Bushev, D. Rotter, A. Wilson, F. Dubin, C. Becher, J. Eschner, R. Blatt, V. Steixner, P. Rabl, and P. Zoller, “Feedback cooling of a single trapped ion,” Phys. Rev. Lett. 96, 043003 (2006).
[Crossref]

Rugar, D.

M. Poggio, C. Degen, H. Mamin, and D. Rugar, “Feedback cooling of a cantilever’s fundamental mode below 5 mK,” Phys. Rev. Lett. 99, 017201 (2007).
[Crossref]

M. Poggio, C. Degen, H. Mamin, and D. Rugar, “Feedback cooling of a cantilever’s fundamental mode below 5 mK,” Phys. Rev. Lett. 99, 017201 (2007).
[Crossref]

Sadeghi, P.

P. Sadeghi, M. Tanzer, S. L. Christensen, and S. Schmid, “Influence of clamp-widening on the quality factor of nanomechanical silicon nitride resonators,” J. Appl. Phys. 126, 165108 (2019).
[Crossref]

Sankey, J. C.

C. Reinhardt, T. Müller, A. Bourassa, and J. C. Sankey, “Ultralow-noise sin trampoline resonators for sensing and optomechanics,” Phys. Rev. X 6, 021001 (2016).
[Crossref]

Schilling, R.

A. H. Ghadimi, S. A. Fedorov, N. J. Engelsen, M. J. Bereyhi, R. Schilling, D. J. Wilson, and T. J. Kippenberg, “Elastic strain engineering for ultralow mechanical dissipation,” Science 360, 764–768 (2018).
[Crossref]

D. Wilson, V. Sudhir, N. Piro, R. Schilling, A. Ghadimi, and T. Kippenberg, “Measurement-based control of a mechanical oscillator at its thermal decoherence rate,” Nature 524, 325–329 (2015).
[Crossref]

Schliesser, A.

M. Rossi, D. Mason, J. Chen, Y. Tsaturyan, and A. Schliesser, “Measurement-based quantum control of mechanical motion,” Nature 563, 53–58 (2018).
[Crossref]

A. Barg, Y. Tsaturyan, E. Belhage, W. H. Nielsen, C. B. Møller, and A. Schliesser, “Measuring and imaging nanomechanical motion with laser light,” Appl. Phys. B 123, 8 (2017).
[Crossref]

Y. Tsaturyan, A. Barg, E. S. Polzik, and A. Schliesser, “Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution,” Nat. Nanotechnol. 12, 776–783 (2017).
[Crossref]

Schmid, S.

P. Sadeghi, M. Tanzer, S. L. Christensen, and S. Schmid, “Influence of clamp-widening on the quality factor of nanomechanical silicon nitride resonators,” J. Appl. Phys. 126, 165108 (2019).
[Crossref]

L. G. Villanueva and S. Schmid, “Evidence of surface loss as ubiquitous limiting damping mechanism in sin micro- and nanomechanical resonators,” Phys. Rev. Lett. 113, 227201 (2014).
[Crossref]

Schwab, K.

A. Hopkins, K. Jacobs, S. Habib, and K. Schwab, “Feedback cooling of a nanomechanical resonator,” Phys. Rev. B 68, 235328 (2003).
[Crossref]

Sonin, P.

A. Vinante, A. Kirste, A. Den Haan, O. Usenko, G. Wijts, E. Jeffrey, P. Sonin, D. Bouwmeester, and T. Oosterkamp, “High sensitivity squid-detection and feedback-cooling of an ultrasoft microcantilever,” App. Phys. Lett. 101, 123101 (2012).
[Crossref]

Sridaran, S.

Steixner, V.

P. Bushev, D. Rotter, A. Wilson, F. Dubin, C. Becher, J. Eschner, R. Blatt, V. Steixner, P. Rabl, and P. Zoller, “Feedback cooling of a single trapped ion,” Phys. Rev. Lett. 96, 043003 (2006).
[Crossref]

Strauch, F. W.

K. Jacobs, H. I. Nurdin, F. W. Strauch, and M. James, “Comparing resolved-sideband cooling and measurement-based feedback cooling on an equal footing: analytical results in the regime of ground-state cooling,” Phys. Rev. A 91, 043812 (2015).
[Crossref]

Sudhir, V.

D. Wilson, V. Sudhir, N. Piro, R. Schilling, A. Ghadimi, and T. Kippenberg, “Measurement-based control of a mechanical oscillator at its thermal decoherence rate,” Nature 524, 325–329 (2015).
[Crossref]

Tanzer, M.

P. Sadeghi, M. Tanzer, S. L. Christensen, and S. Schmid, “Influence of clamp-widening on the quality factor of nanomechanical silicon nitride resonators,” J. Appl. Phys. 126, 165108 (2019).
[Crossref]

Taylor, J. M.

D. Carney, A. Hook, Z. Liu, J. M. Taylor, and Y. Zhao, “Ultralight dark matter detection with mechanical quantum sensors,” arXiv preprint arXiv:1908.04797 (2019).

Tebbenjohanns, F.

F. Tebbenjohanns, M. Frimmer, V. Jain, D. Windey, and L. Novotny, “Motional sideband asymmetry of a nanoparticle optically levitated in free space,” Phys. Rev. Lett. 124, 013603 (2020).
[Crossref]

Tombesi, P.

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[Crossref]

D. Vitali, S. Mancini, L. Ribichini, and P. Tombesi, “Macroscopic mechanical oscillators at the quantum limit through optomechanical cooling,” J. Opt. Soc. Am. B 20, 1054–1065 (2003).
[Crossref]

Tsaturyan, Y.

M. Rossi, D. Mason, J. Chen, Y. Tsaturyan, and A. Schliesser, “Measurement-based quantum control of mechanical motion,” Nature 563, 53–58 (2018).
[Crossref]

A. Barg, Y. Tsaturyan, E. Belhage, W. H. Nielsen, C. B. Møller, and A. Schliesser, “Measuring and imaging nanomechanical motion with laser light,” Appl. Phys. B 123, 8 (2017).
[Crossref]

Y. Tsaturyan, A. Barg, E. S. Polzik, and A. Schliesser, “Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution,” Nat. Nanotechnol. 12, 776–783 (2017).
[Crossref]

Unterreithmeier, Q. P.

Q. P. Unterreithmeier, E. M. Weig, and J. P. Kotthaus, “Universal transduction scheme for nanomechanical systems based on dielectric forces,” Nature 458, 1001–1004 (2009).
[Crossref]

Usenko, O.

A. Vinante, A. Kirste, A. Den Haan, O. Usenko, G. Wijts, E. Jeffrey, P. Sonin, D. Bouwmeester, and T. Oosterkamp, “High sensitivity squid-detection and feedback-cooling of an ultrasoft microcantilever,” App. Phys. Lett. 101, 123101 (2012).
[Crossref]

Valenzuela, V. M.

E. Romero, V. M. Valenzuela, A. R. Kermany, F. Iacopi, and W. P. Bowen, “Engineering the dissipation of crystalline micromechanical resonators,” Phys. Rev. Applied 13, 04407 (2020).
[Crossref]

Van Iperen, B.

J. Milatz, J. Van Zolingen, and B. Van Iperen, “The reduction in the Brownian motion of electrometers,” Physica 19, 195–202 (1953).
[Crossref]

Van Zolingen, J.

J. Milatz, J. Van Zolingen, and B. Van Iperen, “The reduction in the Brownian motion of electrometers,” Physica 19, 195–202 (1953).
[Crossref]

Villanueva, L. G.

L. G. Villanueva and S. Schmid, “Evidence of surface loss as ubiquitous limiting damping mechanism in sin micro- and nanomechanical resonators,” Phys. Rev. Lett. 113, 227201 (2014).
[Crossref]

Vinante, A.

A. Vinante, A. Kirste, A. Den Haan, O. Usenko, G. Wijts, E. Jeffrey, P. Sonin, D. Bouwmeester, and T. Oosterkamp, “High sensitivity squid-detection and feedback-cooling of an ultrasoft microcantilever,” App. Phys. Lett. 101, 123101 (2012).
[Crossref]

Vitali, D.

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[Crossref]

D. Vitali, S. Mancini, L. Ribichini, and P. Tombesi, “Macroscopic mechanical oscillators at the quantum limit through optomechanical cooling,” J. Opt. Soc. Am. B 20, 1054–1065 (2003).
[Crossref]

Weig, E. M.

Q. P. Unterreithmeier, E. M. Weig, and J. P. Kotthaus, “Universal transduction scheme for nanomechanical systems based on dielectric forces,” Nature 458, 1001–1004 (2009).
[Crossref]

Wijts, G.

A. Vinante, A. Kirste, A. Den Haan, O. Usenko, G. Wijts, E. Jeffrey, P. Sonin, D. Bouwmeester, and T. Oosterkamp, “High sensitivity squid-detection and feedback-cooling of an ultrasoft microcantilever,” App. Phys. Lett. 101, 123101 (2012).
[Crossref]

Wilson, A.

P. Bushev, D. Rotter, A. Wilson, F. Dubin, C. Becher, J. Eschner, R. Blatt, V. Steixner, P. Rabl, and P. Zoller, “Feedback cooling of a single trapped ion,” Phys. Rev. Lett. 96, 043003 (2006).
[Crossref]

Wilson, D.

D. Wilson, V. Sudhir, N. Piro, R. Schilling, A. Ghadimi, and T. Kippenberg, “Measurement-based control of a mechanical oscillator at its thermal decoherence rate,” Nature 524, 325–329 (2015).
[Crossref]

D. Wilson, C. Regal, S. Papp, and H. Kimble, “Cavity optomechanics with stoichiometric sin films,” Phys. Rev. Lett. 103, 207204 (2009).
[Crossref]

Wilson, D. J.

A. H. Ghadimi, S. A. Fedorov, N. J. Engelsen, M. J. Bereyhi, R. Schilling, D. J. Wilson, and T. J. Kippenberg, “Elastic strain engineering for ultralow mechanical dissipation,” Science 360, 764–768 (2018).
[Crossref]

A. H. Ghadimi, D. J. Wilson, and T. J. Kippenberg, “Radiation and internal loss engineering of high-stress silicon nitride nanobeams,” Nano Lett. 17, 3501–3505 (2017).
[Crossref]

Wilson-Rae, I.

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[Crossref]

Windey, D.

F. Tebbenjohanns, M. Frimmer, V. Jain, D. Windey, and L. Novotny, “Motional sideband asymmetry of a nanoparticle optically levitated in free space,” Phys. Rev. Lett. 124, 013603 (2020).
[Crossref]

Winger, M.

A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and O. Painter, “A high-resolution microchip optomechanical accelerometer,” Nat. Photonics 6, 768–772 (2012).
[Crossref]

Yin, Z.-Q.

Z.-Q. Yin, A. A. Geraci, and T. Li, “Optomechanics of levitated dielectric particles,” Int. J. Mod. Phys. B 27, 1330018 (2013).
[Crossref]

Zhao, Y.

D. Carney, A. Hook, Z. Liu, J. M. Taylor, and Y. Zhao, “Ultralight dark matter detection with mechanical quantum sensors,” arXiv preprint arXiv:1908.04797 (2019).

Zoller, P.

P. Bushev, D. Rotter, A. Wilson, F. Dubin, C. Becher, J. Eschner, R. Blatt, V. Steixner, P. Rabl, and P. Zoller, “Feedback cooling of a single trapped ion,” Phys. Rev. Lett. 96, 043003 (2006).
[Crossref]

Zwerger, W.

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[Crossref]

App. Phys. Lett. (2)

J. Mertz, O. Marti, and J. Mlynek, “Regulation of a microcantilever response by force feedback,” App. Phys. Lett. 62, 2344–2346 (1993).
[Crossref]

A. Vinante, A. Kirste, A. Den Haan, O. Usenko, G. Wijts, E. Jeffrey, P. Sonin, D. Bouwmeester, and T. Oosterkamp, “High sensitivity squid-detection and feedback-cooling of an ultrasoft microcantilever,” App. Phys. Lett. 101, 123101 (2012).
[Crossref]

Appl. Phys. B (1)

A. Barg, Y. Tsaturyan, E. Belhage, W. H. Nielsen, C. B. Møller, and A. Schliesser, “Measuring and imaging nanomechanical motion with laser light,” Appl. Phys. B 123, 8 (2017).
[Crossref]

Eur. Phys. J. D (1)

J.-M. Courty, A. Heidmann, and M. Pinard, “Quantum limits of cold damping with optomechanical coupling,” Eur. Phys. J. D 17, 399–408 (2001).
[Crossref]

Int. J. Mod. Phys. B (1)

Z.-Q. Yin, A. A. Geraci, and T. Li, “Optomechanics of levitated dielectric particles,” Int. J. Mod. Phys. B 27, 1330018 (2013).
[Crossref]

J. Appl. Phys. (1)

P. Sadeghi, M. Tanzer, S. L. Christensen, and S. Schmid, “Influence of clamp-widening on the quality factor of nanomechanical silicon nitride resonators,” J. Appl. Phys. 126, 165108 (2019).
[Crossref]

J. Opt. Soc. Am. B (1)

Nano Lett. (1)

A. H. Ghadimi, D. J. Wilson, and T. J. Kippenberg, “Radiation and internal loss engineering of high-stress silicon nitride nanobeams,” Nano Lett. 17, 3501–3505 (2017).
[Crossref]

Nat. Nanotechnol. (1)

Y. Tsaturyan, A. Barg, E. S. Polzik, and A. Schliesser, “Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution,” Nat. Nanotechnol. 12, 776–783 (2017).
[Crossref]

Nat. Photonics (1)

A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and O. Painter, “A high-resolution microchip optomechanical accelerometer,” Nat. Photonics 6, 768–772 (2012).
[Crossref]

Nature (3)

D. Wilson, V. Sudhir, N. Piro, R. Schilling, A. Ghadimi, and T. Kippenberg, “Measurement-based control of a mechanical oscillator at its thermal decoherence rate,” Nature 524, 325–329 (2015).
[Crossref]

M. Rossi, D. Mason, J. Chen, Y. Tsaturyan, and A. Schliesser, “Measurement-based quantum control of mechanical motion,” Nature 563, 53–58 (2018).
[Crossref]

Q. P. Unterreithmeier, E. M. Weig, and J. P. Kotthaus, “Universal transduction scheme for nanomechanical systems based on dielectric forces,” Nature 458, 1001–1004 (2009).
[Crossref]

New J. Phys. (1)

R. Fischer, D. P. McNally, C. Reetz, G. G. Assumpcao, T. Knief, Y. Lin, and C. A. Regal, “Spin detection with a micromechanical trampoline: towards magnetic resonance microscopy harnessing cavity optomechanics,” New J. Phys. 21, 043049 (2019).
[Crossref]

Opt. Express (1)

Phys. Rev. A (3)

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[Crossref]

K. Jacobs, H. I. Nurdin, F. W. Strauch, and M. James, “Comparing resolved-sideband cooling and measurement-based feedback cooling on an equal footing: analytical results in the regime of ground-state cooling,” Phys. Rev. A 91, 043812 (2015).
[Crossref]

E. Hebestreit, R. Reimann, M. Frimmer, and L. Novotny, “Measuring the internal temperature of a levitated nanoparticle in high vacuum,” Phys. Rev. A 97, 043803 (2018).
[Crossref]

Phys. Rev. Applied (1)

E. Romero, V. M. Valenzuela, A. R. Kermany, F. Iacopi, and W. P. Bowen, “Engineering the dissipation of crystalline micromechanical resonators,” Phys. Rev. Applied 13, 04407 (2020).
[Crossref]

Phys. Rev. B (1)

A. Hopkins, K. Jacobs, S. Habib, and K. Schwab, “Feedback cooling of a nanomechanical resonator,” Phys. Rev. B 68, 235328 (2003).
[Crossref]

Phys. Rev. Lett. (11)

M. Poggio, C. Degen, H. Mamin, and D. Rugar, “Feedback cooling of a cantilever’s fundamental mode below 5 mK,” Phys. Rev. Lett. 99, 017201 (2007).
[Crossref]

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[Crossref]

F. Marquardt, J. P. Chen, A. A. Clerk, and S. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[Crossref]

D. Wilson, C. Regal, S. Papp, and H. Kimble, “Cavity optomechanics with stoichiometric sin films,” Phys. Rev. Lett. 103, 207204 (2009).
[Crossref]

P. Bushev, D. Rotter, A. Wilson, F. Dubin, C. Becher, J. Eschner, R. Blatt, V. Steixner, P. Rabl, and P. Zoller, “Feedback cooling of a single trapped ion,” Phys. Rev. Lett. 96, 043003 (2006).
[Crossref]

M. Poggio, C. Degen, H. Mamin, and D. Rugar, “Feedback cooling of a cantilever’s fundamental mode below 5 mK,” Phys. Rev. Lett. 99, 017201 (2007).
[Crossref]

L. G. Villanueva and S. Schmid, “Evidence of surface loss as ubiquitous limiting damping mechanism in sin micro- and nanomechanical resonators,” Phys. Rev. Lett. 113, 227201 (2014).
[Crossref]

F. Tebbenjohanns, M. Frimmer, V. Jain, D. Windey, and L. Novotny, “Motional sideband asymmetry of a nanoparticle optically levitated in free space,” Phys. Rev. Lett. 124, 013603 (2020).
[Crossref]

S. Nimmrichter, K. Hornberger, and K. Hammerer, “Optomechanical sensing of spontaneous wave-function collapse,” Phys. Rev. Lett. 113, 020405 (2014).
[Crossref]

R. A. Norte, J. P. Moura, and S. Gröblacher, “Mechanical resonators for quantum optomechanics experiments at room temperature,” Phys. Rev. Lett. 116, 147202 (2016).
[Crossref]

S. A. Fedorov, A. Beccari, N. J. Engelsen, and T. J. Kippenberg, “Fractal-like mechanical resonators with a soft-clamped fundamental mode,” Phys. Rev. Lett. 124, 025502 (2020).
[Crossref]

Phys. Rev. X (1)

C. Reinhardt, T. Müller, A. Bourassa, and J. C. Sankey, “Ultralow-noise sin trampoline resonators for sensing and optomechanics,” Phys. Rev. X 6, 021001 (2016).
[Crossref]

Physica (1)

J. Milatz, J. Van Zolingen, and B. Van Iperen, “The reduction in the Brownian motion of electrometers,” Physica 19, 195–202 (1953).
[Crossref]

Science (1)

A. H. Ghadimi, S. A. Fedorov, N. J. Engelsen, M. J. Bereyhi, R. Schilling, D. J. Wilson, and T. J. Kippenberg, “Elastic strain engineering for ultralow mechanical dissipation,” Science 360, 764–768 (2018).
[Crossref]

Other (2)

D. Carney, A. Hook, Z. Liu, J. M. Taylor, and Y. Zhao, “Ultralight dark matter detection with mechanical quantum sensors,” arXiv preprint arXiv:1908.04797 (2019).

We follow the method of Reinhardt et al. [4] using a simplified Teflon sample holder for wet etching. Fabrication begins with standard two-sided photolithography followed by dry etch of ${{\rm Si}_3}{{\rm N}_4}$Si3N4, using plasma Ar+SF$_{6}$6, to define the trampoline on one side and a square window on the other. The chip is then wet etched from both sides using a 45% percent KOH solution at 65 C for 21 hours. After the trampoline is released, the gradual dilution method described in [4] is used in lieu of a critical point dryer, in order to suspend the trampoline.

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Figures (4)

Fig. 1.
Fig. 1. ${{\rm Si}_3}{{\rm N}_4}$ trampoline resonator. (Top left) Camera image of a typical device. (Top right) Microscope image of the trampoline used in the experiment. (Bottom right) Finite element simulation of the fundamental 40 kHz vibrational mode. (Bottom left) Energy ringdown of the fundamental mode before (red) and after (blue) deposition of a dust particle onto a tether.
Fig. 2.
Fig. 2. Setup for probing the trampoline, consisting of a confocal microscope embedded in a balanced Michelson interferometer. Electronics for stabilizing the interferometer path length (PI = proportional integral controller, Newport LB1005) and for radiation pressure feedback cooling (see main text for details) are indicated in black. An image of the focused optical beam on the trampoline pad is shown at bottom left.
Fig. 3.
Fig. 3. Characterization of interferometer sensitivity. Upper plot: imprecision in noise quanta units versus power, compared to Eq. (8) (blue line). Lower plot: apparent displacement spectrum of the trampoline versus frequency for different optical powers. The dashed line is a model for ${S_\textit{xx}}$. The blue line is obtained by blocking the signal arm of the interferometer at highest power. (Inset: broadband spectrum for highest power).
Fig. 4.
Fig. 4. Radiation pressure feedback cooling. Upper plot: feedback cooling curve for parameters described in the main text. Colored points correspond to models overlaying experimental data. Lower plot: experimental measurements (colored) overlaid with models (dashed curves) using Eq. (9). The solid black curve is a model for $g = 0$ (no feedback).

Equations (10)

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Γ t h = k B T 0 Q 0 < Ω 0 ,
S xx i m p , g s = 4 x z p 2 Γ t h = 2 2 k B T 0 Q 0 m Ω 0 ,
n = k B T e f f Ω 0 S xx i m p S xx i m p , g s 3 × 10 3 .
y = x + x i m p .
m x ¨ + m Γ 0 x ˙ + m Ω 0 2 x = 2 k B T 0 m Γ 0 ξ ( t ) g m Γ 0 y ˙ ,
S xx [ Ω ] 2 S xx z p = | χ g [ Ω ] | 2 ( n t h + g 2 n i m p ) ,
S yy [ Ω ] 2 S xx z p = | χ g [ Ω ] | 2 ( n t h + ( 1 + g ) 2 | χ 0 [ Ω ] | 2 n i m p ) ,
n + 1 2 = x 2 2 x z p 2 = n t h + g 2 n i m p 1 + g 2 n t h n i m p .
S xx i m p , s h o t = c λ 16 π η R m P ,
δ F f b g m Γ 0 ( y ˙ + Ω 0 cot ( ϕ ) y ) ,

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