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 [1–4]. 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 ${\Omega _0}$,

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

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) [17,23–26]. To this end, agnostic to the measurement scheme, consider a real-time estimate $y$ of the oscillator displacement $x$ obscured by imprecision noise ${x_{{\rm imp}}}$:

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 [27–29], 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

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

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].