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Abstract
The pursuit of room temperature quantum optomechanics with tethered nanomechanical resonators faces stringent challenges owing to extraneous mechanical degrees of freedom. An important example is thermal intermodulation noise (TIN), a form of excess optical noise produced by mixing of thermal noise peaks. While TIN can be decoupled from the phase of the optical field, it remains indirectly coupled via radiation pressure, implying a hidden source of backaction that might overwhelm shot noise. Here we report observation of TIN backaction in a high-cooperativity, room temperature cavity optomechanical system consisting of an acoustic-frequency ${{\rm Si}_3}{{\rm N}_4}$ trampoline coupled to a Fabry–Perot cavity. The backaction we observe exceeds thermal noise by 20 dB and radiation pressure shot noise by 40 dB, despite the thermal motion being 10 times smaller than the cavity linewidth. Our results suggest that mitigating TIN may be critical to reaching the quantum regime from room temperature in a variety of contemporary optomechanical systems.
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1. INTRODUCTION
Room temperature quantum experiments are a longstanding pursuit of cavity optomechanics [1–3], spurred by the promise of fieldable quantum technologies [4–6] and simple platforms for fundamental physics tests [7,8]. Recently, ground state cooling has been achieved from room temperature using levitated nanoparticles [9,10]. Ponderomotive squeezing has also been achieved at room temperature, using both levitated nanoparticles [11,12] and an optically stiffened cantilever [13]. Despite this progress, however, including the recent development of ultracoherent nanomechanical resonators [14,15], room temperature quantum optomechanics with rigidly tethered nanomechanical resonators (e.g., strings and membranes) remains elusive, limited to signatures of weak optomechanical quantum correlations [2,3] and cooling to occupations of greater than 10 [16–18]. Overcoming this hurdle is important because tethered nanomechanical resonators are readily functionalized and integrated with chip-scale electronics [19], features that form the basis for optomechanical quantum technologies [4,6].
A key obstacle to room temperature quantum optomechanics is thermal intermodulation noise (TIN) [20], a form of excess optical noise produced in cavity optomechanical systems (COMSs) due to the mixing of thermomechanical noise peaks. TIN is especially pronounced in high-cooperativity tethered COMSs [16,20–23], which commonly employ nanomechanical resonators with free spectral ranges orders of magnitude smaller than the cavity linewidth [14,24]. In conjunction with the cavity’s transduction nonlinearity [25], this high mode density can give rise to spectrally broadband TIN orders of magnitude in excess of shot noise [20]—a severe impediment to protocols that rely on the observability of quantum backaction, such as ground state cooling [26] and squeezing [27].
Previous reports of TIN focus on its distortion of cavity-based measurement and methods to reduce it [20]. These include reducing optomechanical coupling and cavity finesse, operating at a “magic detuning” where the leading (quadratic) transduction nonlinearity vanishes [20], multimode cooling [28], and enhancing mechanical $Q$. Proposals for cavity-free quantum optomechanics with ultracoherent nanomechanical resonators represent an extreme solution [29–31]. A promising compromise exploits wide-bandgap phononic crystal nanoresonators in conjunction with broadband dynamical backaction cooling [17]. Another key insight is that the phase of a resonant cavity probe is insensitive to TIN, allowing efficient feedback cooling even in the presence of nonlinear thermal noise transduction [17,20].
With this paper, we wish to highlight a complementary form of TIN—stochastic radiation pressure backaction (TINBA)—that is resilient to some of the above methods and poses an additional obstacle to room temperature quantum optomechanics. While previous studies have observed the effect of nonlinear thermal noise transduction (hereafter “thermal nonlinearity”) on dynamical backaction—as an inhomogeneous broadening of the optical spring and damping effects [21]—TINBA, a type of classical intensity noise heating, is subtler, as it appears only indirectly in cavity-based measurements, and can dominate quantum backaction (QBA) even when the thermal nonlinearity is small. As a demonstration, we study TIN in a high-cooperativity, room temperature COMS based on an acoustic-frequency ${{\rm Si}_3}{{\rm N}_4}$ trampoline coupled to a Fabry–Perot cavity (a popularly proposed system for room temperature quantum experiments [1]). Despite the thermal motion of the trampoline being 10 times smaller than the cavity linewidth, we observe TINBA as high as 20 dB in excess of thermal noise and an estimated 40 dB in excess of QBA. We show that this noise can be precisely modeled, despite its apparent complexity, and explore tradeoffs to mitigating it via its strong dependence on temperature, finesse, and detuning.
2. THEORY: QBA VERSUS TINBA
As illustrated in Fig. 1, TINBA arises due to a combination of transduction nonlinearity and radiation pressure in cavity-based readout of a multimode mechanical resonator. We here consider the observability of QBA in the presence of TINBA, focusing on a single mechanical mode with displacement coordinate $x$ and frequency ${\omega _{\text{m}}}$. As a figure of merit, we take the quantum cooperativity
Fig. 1. (a) Multimode cavity optomechanical system (COMS) with intracavity field amplitude $a$ and mechanical resonance frequencies ${\omega _i}$. (b) Nonlinear transduction of displacement $x$ into intensity (red) and linear transduction into phase (blue). Radiation pressure (yellow) couples intensity and phase. (c) Displacement and intensity ($I$) noise spectral density for resonantly probed COMS. Dashed lines: single-mode COMS with intensity dominated by shot noise (red) and total displacement (black) by thermal noise (green) and QBA (orange). Solid lines: multimode COMS with shot noise overwhelmed by TIN and QBA overwhelmed by TINBA. (d) Noise budget versus optical power for shaded region in (c).
To build a model for ${C_q}$, we first consider a single-mode COMS characterized by an optomechanical coupling:
where ${\omega _{\text{c}}}$ is the cavity resonance frequency, and $G$ (the “frequency pull parameter”) characterizes the optomechanical coupling strength. In the small displacement limit, the coupled Langevin equations describing this system are [33] where Eq. (3a) describes the displacement of a mechanical oscillator with mass $m$, resonance frequency ${\omega _{\text{m}}}$, and damping rate ${\Gamma _{\text{m}}}$, driven by a thermal force ${F_{{\rm th}}}$ and a radiation pressure backaction force ${F_{{\rm BA}}} = \hbar G|{a^2}|$; and Eq. (3b) describes the complex amplitude $a$ of the cavity field with energy decay rate $\kappa$, driven at rate ${\kappa _{{\rm in}}}$ by an input field with amplitude ${s_{{\rm in}}}$ and frequency ${\omega _0}$, and normalized so that $|a{|^2} = {n_{\text{c}}}$ is the intracavity photon number, and $|{s_{{\rm in}}}{|^2}$ is the input photon flux.Linearizing Eq. (3) about small fluctuations in $a$ yields
We hereafter specialize to the fast cavity limit ${\omega _{ \text{m}}} \ll \kappa$, in which most room temperature quantum optomechanics experiments with tethered mechanical resonators operate, including ours. In this case ${C_q} \gt 1$ requires
Turning to TIN, we first re-emphasize that Eq. (3) considers only a single mechanical mode whose displacement is perturbatively small, $Gx \ll \kappa$. In practice, however, operating in the fast cavity limit using tethered nanomechanical resonators usually implies that many mechanical modes are simultaneously coupled to the cavity:
Moreover, for a single mode designed for high cooperativity [Eq. (8)], low stiffness $m\omega _{\text{m}}^2$ and high temperature can readily lead to a root-mean-squared thermal displacement ${x_{{\rm th}}}$ exceeding the nonlinear transduction thresholdTo analyze TINBA in the fast cavity limit, it suffices to consider the steady-state dependence of ${n_{\text{c}}}$ on detuning, expanded to second order in deviations from the mean value $\Delta$. For convenience, following [20], we define the relative detuning $\nu = 2\Delta /\kappa$ and its deviation $\delta \nu$, yielding
Three features of TINBA bear emphasis. First, unlike QBA or photothermal heating (which both scale as ${S_x} \propto {n_{ \text{c}}}$), TINBA scales quadratically with ${n_{\text{c}}}$. Second, unlike the optical spring, TINBA does not vanish on resonance ($\nu = 0$). In fact, it is maximal in this case, simplifying to
corresponding to $S_{{\rm RIN}}^{{\rm TIN}}[\omega] = {S_{{\nu ^2}}}[\omega]$. Third, there exists a “magic” detuning $|\nu | = 1/\sqrt 3$ at which TINBA vanishes: corresponding to a ${\partial ^2}{n_{\text{c}}}/{\partial ^2}\nu = 0$. However, at this detuning, the optical spring is maximized, possibly leading to instability (${k_{{\rm opt}}} \approx - m\omega _{\text{m}}^2$) for large ${n_{\text{c}}}$.With these features in mind, we suggest three conditions for observing QBA (${C_q} \gtrsim 1$) in the presence of TIN, valid for negative detunings $(\nu \lt 0)$ in the fast cavity limit (see Supplement 1):
where ${Q_{\text{ m}}} = {\omega _{\text{ m}}}/{\Gamma _{\text{ m}}}$ is the mechanical quality factor, and $S_\nu ^{{\rm ZP}} = (16g_0^2/{\Gamma _{\text{m}}})/{\kappa ^2}$ is the normalized zero-point detuning spectral density, related to the zero-point displacement spectral density $S_x^{{\rm ZP}} = S_x^{{\rm th}}/(2{n_{{\rm th}}}) = 4x_{{\rm ZP}}^2/{\Gamma _{\text{m}}}$ by $S_\nu ^{{\rm ZP}} = 4{G^2}S_x^{{\rm ZP}}/{\kappa ^2}$. The first two conditions are independent of TIN and correspond to $S_F^{{\rm QBA}} \gt S_F^{{\rm th}}$ and ${k_{{\rm opt}}} \gt - m\omega _{\text{m}}^2$ (i.e. ${\omega _{{\rm eff}}} \gt 0$), respectively (see Supplement 1). The last condition implies that $S_F^{{\rm TIN}} \lesssim S_F^{{\rm th}}$ when ${C_q} = 1$, and is given by maximizing the relation (see Supplement 1)In the next section, we explore the requirements in Eq. (19) in a popular membrane-in-the-middle platform, and show that Eq. (19c) may not be met even if Eqs. (19a) and (19b) are.
3. TRAMPOLINE-IN-THE-MIDDLE SYSTEM
Our optomechanical system consists of a ${{\rm Si}_3}{{\rm N}_4}$ trampoline resonator coupled to a Fabry–Perot cavity in the membrane-in-the-middle configuration [38]—hereafter dubbed “trampoline-in-the-middle” (TIM). As shown in Fig. 2(a), the quasi-monolithic cavity is assembled by sandwiching the Si device chip between a concave (radius 10 cm) and a plano mirror, with the trampoline positioned nearer the plano mirror to minimize diffraction loss. [This is achieved by etching the plano mirror into a mesa-like structure (see Supplement 1), as shown in Fig. 2(a).] A relatively short cavity length of $L = 415\;{\unicode{x00B5}{\rm m}}$ is chosen, to reduce sensitivity to laser frequency noise (see Supplement 1). The bare (without trampoline) cavity finesse is as high as ${\cal F} = 3 \times {10^4}$ at wavelengths near the coating center, $\lambda = 850\;{\rm nm} $. To reduce thermal nonlinearity, ${\cal F}$ is reduced to $\approx 550$ by operating at $\lambda \approx 786\;{\rm nm} $, yielding a cavity decay rate of $\kappa = 2\pi \times 0.65\;{\rm GHz} $.
Fig. 2. Trampoline-in-the-middle (TIM) cavity optomechanical system. (a) Experimental setup: the TIM cavity is driven by a titanium sapphire laser (TiS) and probed in transmission with an avalanche photodiode (APD) and balanced homodyne detector (BHD). The BHD local oscillator (LO) is derived from the same TiS. (b) Displacement measurement using a resonant probe and the BHD tuned to the phase quadrature. Thermal noise models for the first five trampoline modes (a)–(d) are overlayed. (c) Ringdown measurements of the fundamental trampoline before (green) and after (red) mounting inside the TIM cavity. (d) Detuning sweep over cavity resonance, fit with a model that includes the thermal motion of the trampoline fundamental mode. (e) Measurement of the optical spring shift versus detuning, used to determine the intracavity photon number.
${{\rm Si}_3}{{\rm N}_4}$ trampolines are popular candidates for room temperature quantum optomechanics experiments [1,24,29], owing to their high $Q$-stiffness ratio. We employ an 85-nm-thick trampoline with a $200 \times 200\; {{\unicode{x00B5}{\rm m}}^2}$ central pad, $1700 \times 4.2\; {{\unicode{x00B5}{\rm m}}^2}$ tethers, and tether fillets designed to optimize strain-induced dissipation dilution of the fundamental mode [39], yielding ${Q_{m}} = 2.6 \times {10^7}$ [29], ${\omega _{m}} \approx 2\pi \times 41\;{\rm kHz} $, and a COMSOL-simulated effective mass of $m = 12\; {\rm ng}$. Care was taken to mount the trampoline without introducing loss; nevertheless, two small dabs of adhesive reduced ${Q_{ \text{m}}}$ to $7.8 \times {10^6}$ (we speculate that this is due to hybridization with low-$Q$ chip modes [40], as evidenced by the satellite noise peaks in Fig. 4.) The resulting thermal force noise $S_F^{{\rm th}} \approx 8 \times {10^{- 17}}\; {\rm N}/\sqrt {{\rm Hz}}$ is in principle sufficient to observe QBA (${C_q} \sim 1$) with an input power of ${\sim}10\;{\rm mW} $ at ${\cal F} \sim {10^3}$. Challenging this prospect is the fact that the trampoline’s thermal motion, ${x_{{\rm th}}} = 0.07\;{\rm nm} $, is near the nonlinear transduction threshold [Eq. (10)] at ${\cal F} \sim {10^3}$. Moreover, $\kappa /{\omega _{\text{m}}} \sim {10^4}$ allows many higher-order trampoline modes to be coupled to the cavity field [Fig. 2(b)], satisfying the conditions for strong TIN.
Measurements characterizing the nonlinearity and cooperativity of our TIM system are shown in Figs. 2(d) and 2(e). A hallmark of thermal nonlinearity is modulation of the steady-state cavity response, as shown in Fig. 2(d). Here the cavity length is swept across resonance with strong optomechanical coupling, corresponding to the trampoline positioned between a node and an antinode of the intracavity standing wave [41]. Fitting the transmitted power to ${P_{{\rm out}}}(\nu (t)) \propto 1/(1 + (\nu + 8G{x_{{\rm th}}}\def\LDeqbreak{}\cos ({\omega _{\text{ m}}}\nu /\dot \nu)/\kappa ))^2)$ (where $\dot \nu$ is the constant slew rate of the detuning sweep) yields $G{x_{{\rm th}}}/\kappa \approx 0.04$ (see Supplement 1), corresponding to ${g_0} = G{x_{{\rm th}}}/\sqrt {2{n_{{\rm th}}}} \sim 2\pi \times 1\;{\rm kHz} $. A more careful estimate of ${g_0} = 2\pi \times 1.5\;{\rm kHz} $ was obtained using the frequency sideband calibration method [42], suggesting a vacuum cooperativity of ${C_0} \approx 3$. In the experiment below, for each input power ${P_{{\rm in}}}$, the intracavity photon number ${n_{ \text{c}}}$ is determined by recording the optical spring shift versus detuning and comparing to a model, $\Delta {\omega _{{\rm opt}}} \approx {k_{{\rm opt}}}/(2m{\omega _{\text{m}}}) \approx {\Gamma _{\text{m}}}{C_0}{n_{ \text{c}}}(\nu = 0)\nu /(1 + {\nu ^2}{)^2}$ [Fig. 2(e)]. For all measurements, the thermal nonlinearity is reduced by measurement-based feedback cooling of the fundamental mode. Full details and methods are presented in Supplement 1.
4. OBSERVATION OF TIN BACKACTION
We now explore TIN in our TIM system and present evidence that TINBA overwhelms QBA at a sufficiently high intracavity photon number. To this end, the cavity is probed on resonance ($\nu = 0$) with a titanium-sapphire laser ($M$-Squared Solstis) pre-stabilized with a broadband electro-optic intensity noise eater (see Supplement 1). The output field ${s_{{\rm out}}} \propto \sqrt \kappa a$ is monitored with two detectors. A balanced homodyne detector records the phase of ${s_{{\rm out}}}$, which encodes the trampoline displacement, ${\rm Arg}[{s_{{\rm out}}}] \propto x$, with a shot-noise-limited imprecision of $S_x^{{\rm imp}} \ge S_x^{{\rm ZP}}/(8{C_0}{n_{\text{c}}})$. An avalanche photodiode (APD) records the output intensity $|{s_{{\rm out}}}{|^2} \propto {n_{\text{c}}}$ and is also used to lock the cavity using the Pound–Drever–Hall technique (see Supplement 1).
TIN couples directly to intracavity intensity and indirectly to mechanical displacement, via TINBA. We explore this in Fig. 3, by comparing the intensity and phase fluctuations of an optical field passed resonantly through the TIM cavity. An input power of ${P_{{\rm in}}} = 0.2\;{\rm mW}$ is used, corresponding to ${n_{\text{c}}} = 4\eta {P_{{\rm in}}}/(\hbar {\omega _{\text{c}}}\kappa) = 4 \times {10^5}$ intracavity photons (with $\eta = 0.4$ determined from the optical spring shift) and an ideal quantum cooperativity of ${C_q} = 7 \times {10^{- 3}}$(in principle large enough to observe optomechanical quantum correlations [2,3]). The phase noise spectrum is calibrated in displacement units by bootstrapping to a model for the fundamental thermomechanical noise peak (see Supplement 1), yielding the apparent displacement noise spectrum
Fig. 3. Thermal intermodulation noise (TIN) in TIM system. (a) Low-frequency part of the phase measurement in Fig. 2(b) (green), compared to models of the thermal motion of the first seven trampoline modes (dotted red), their incoherent sum (solid red), shot noise (blue), and total noise (black). (b) Simultaneously recorded intensity noise (green), normalized by the mean intensity (RIN). Red curve is a TIN model (see main text), blue is a shot noise model, and gray is the measurement of the input laser RIN, after the noise eater (see Supplement 1).
Fig. 4. Evidence for TINBA in a strong displacement measurement. (a) Displacement noise near the fundamental trampoline resonance ${\omega _{\text{ m}}}$, with models for various components overlaid and shading for integration region in (d). (b) Noise spectra at different powers versus offset frequency $\omega - {\omega _{\text{ m}}}$. Shot noise dominates at low power and large offset frequencies, scaling inversely with power. At smaller offset frequencies, noise gradually increases with power, due to TINBA. (c) Coherence between the noise in (a) and a simultaneously recorded intensity noise. Values near unity imply a high degree of correlation. Inset: phase of the coherence function. The $\pi$ phase change on resonance is due the mechanical susceptibility. (d) Noise integrated over the frequency range shaded in (a), overlaid with models for shot noise (blue), thermal noise (green), and TINBA (yellow). Gray shading is the uncertainty of the TINBA model due to statistical variation of the measured TIN. Points from (b) are denoted with their respective color.
We now turn our attention to the spurious peaks in the phase noise spectrum in Fig. 3(a), which we argue is displacement produced by TINBA. To this end, in Fig. 4, we compare ${S_y}[\omega]$ at different probe powers, focusing on Fourier frequencies near the fundamental trampoline resonance. Qualitatively, as shown in Figs. 4(a) and 4(b), we observe an increase in the apparent displacement at larger powers, with a shape that is consistent with the measured intensity noise multiplied by the mechanical susceptibility. To confirm that this is TINBA, in Fig. 4(d) we plot ${S_y}[{\omega _{\text{m}}} + \delta]$ at an offset $\delta \gg {\Gamma _{\text{m}}}$ versus input power in the range ${P_{{\rm in}}} \in [10\;\unicode{x00B5}{\rm W},1.1\; {\rm mW}]$ (${n_{c}} \in [1.7 \times {10^4},2.2 \times {10^6}]$). The observed quadratic scaling with ${P_{{\rm in}}}$ is consistent with TINBA and distinct from QBA and photoabsorption heating. The absolute magnitude of the displacement moreover agrees quantitatively well with our TINBA model, ${S_x}[{\omega _{\text{m}}} + \delta] \approx {S_{{\rm RIN}}}[{\omega _{\text{m}}} + \delta]/(m\omega _{\text{ m}}^2\Gamma _{\text{m}}^2{)^2}$ (black line), allowing for statistical uncertainty (gray shading) due to fluctuations in the measured ${S_{{\rm RIN}}}[\omega]$.
As an additional consistency check, we measure the coherence between the phase and intensity noise, allowing us to rule out artifacts such as imperfect cancellation of intensity noise in the balanced homodyne receiver. We define the coherence between signals $a$ and $b$ as ${C_{\textit{ab}}}[\omega] = |{S_{\textit{ab}}}[\omega {]|^2}/({S_a}[\omega]{S_b}[\omega])$ [43], where ${S_{\textit{ab}}}[\omega]$ is the cross-spectrum of $a$ and $b$, so that ${C_{\textit{ab}}}[\omega] = 1$ for perfectly correlated or anti-correlated signals, and ${\rm Arg}[{S_{\textit{ab}}}]$ characterizes the relative phase of the correlated signal components. Figure 4(c) shows the coherence of the phase and intensity noise measurements with ${n_{ \text{c}}} = 2 \times {10^6}$, together with a model that predicts the coherence based on the measured TIN noise and the mechanical susceptibility. A high degree of coherence is observed over a ${\sim}100\;{\rm kHz} $ bandwidth surrounding the fundamental resonance. Moreover, near resonance, the argument of the coherence undergoes a $\pi$-phase shift, indicative of the response of the mechanical susceptibility, and consistent with TINBA-driven motion. Full details about models and measurements can be found in Supplement 1.
Fig. 5. Strategies for mitigating TINBA. (a) Estimated quantum cooperativity ${C_q}$ of TIM system at $T = 300\; {\rm K}$ (red) and 4 K (blue) assuming no TIN (dashed line) and parameters in the experiment (solid line, $S_{{\rm RIN}}^{{\rm TIN}} \approx {10^{- 11}}\; {{\rm Hz}^{- 1}}$, ${g_0} = 2\pi \times 1.5\;{\rm kHz} $, and $\kappa = 2\pi \times 650\;{\rm MHz}$). Cryogenic operation reduces thermal noise and TIN, increasing the maximum ${C_q}$ by $1/{T^2}$. (b) ${C_q}$ of TIM system as a function of linewidth and input power, for $T = 300\; {\rm K}$ and 4 K. The red line indicates the slice shown in (a). The dashed line indicates the maximum ${C_q}$ for a given $\kappa$ and ${P_{{\rm in}}}$. (c) Readout of the TIM cavity at the “magic detuning,” showcasing a decrease in TIN in the intensity noise (red), but an increase of TIN in the phase (displacement) noise (blue).
5. IMPLICATIONS FOR QUANTUM OPTOMECHANICS EXPERIMENTS
TIN backaction currently limits the quantum cooperativity of our TIM system. For example, at the highest intracavity photon number in Fig. 4, ${n_{\text{c}}} = 2 \times {10^6}$, $S_x^{{\rm TIN}}/S_x^{{\rm th}} \approx {10^2}$, and $S_x^{{\rm TIN}}/S_x^{{\rm QBA}} = (S_x^{{\rm TIN}}/S_x^{{\rm th}})/C_q^0 \approx 2500$, implying that ${C_q} \approx\def\LDeqbreak{} 4 \times {10^{- 4}}$ instead of the ideal value, $C_q^0 = {C_0}{n_{\text{c}}}/{n_{{\rm th}}} = 0.04$. As implied by Eq. (19c), this could have been anticipated by comparing the measured TIN to the a priori zero-point frequency noise $S_\nu ^{{\rm ZP}} = 4{C_0}/\kappa \approx 3 \times {10^{- 9}}$ scaled by the thermal occupation ${n_{{\rm th}}} \approx 1.5 \times {10^8}$. In our case, $S_{{\rm RIN}}^{{\rm TIN}} \approx {10^{- 11}} {{\rm Hz}^{- 1}}$, yielding ${C_q} \lesssim {10^{- 3}}$ according to Eq. (20):
In the case of TINBA, which originates from thermal nonlinearity [Eq. (10)], inroads can be made by leveraging the dependence of the nonlinearity on $G$, $\kappa$, and $\nu$. For example, the fact that $S_F^{{\rm TIN}} \propto {(G/\kappa)^4}$ and $S_F^{{\rm QBA}} \propto {(G/\kappa)^2}$ suggests that TINBA can be mitigated by using a higher $\kappa$ (lower-finesse ${\cal F}$) cavity. In Fig. 5(b) we consider this strategy for our TIM system using the above experimental parameters and $\nu = 0$. Evidently, ${C_q} \sim 1$ is possible with 100-fold lower ${\cal F}$; however, it would require a proportionately larger laser power. This is problematic because of photothermal heating and increased demands on classical laser intensity noise suppression. Figures 5(a) and 5(b) show the same computation at $T = 4\; {\rm K}$, revealing that power demands are relaxed in proportion to $T$, since $S_F^{{\rm TIN}} \propto {T^2}$. This observation reaffirms the demands of room temperature quantum optomechanics and, conversely, the advantages of cryogenic pre-cooling.
Finally, we re-emphasize the strong detuning dependence of TIN and TINBA. Operating at the magic-detuning $\nu = 1/\sqrt 3$, as shown in Fig. 5(c), can eliminate TIN in the optical intensity; however, as evident in the blue data, the phase response of the cavity becomes nonlinear, potentially preventing the observation of quantum correlations generated via the optomechanical interaction. Moreover, in the regime of strong QBA, the associated optical spring (maximal at $\nu = 1/\sqrt 3$ in the fast cavity limit) can be substantial. To avoid instability (${\omega _{{\rm eff}}} = 0$), one strategy is to use a dual-wavelength probe with $\nu = \pm 1/\sqrt 3$, but this does not resolve the phase nonlinearity issue. Another promising strategy—not considered in our theoretical analysis—is to exploit optical damping at $\nu \ne 0$ to realize multimode cooling [17]. The success of this strategy will depend on the details of the system and may benefit by operating in an intermediate regime between the fast and slow cavity limits.
6. CONCLUSION
We have explored the effect of thermal intermodulation noise (TIN) on the observability of radiation pressure quantum backaction (QBA) in a room temperature cavity optomechanical system and argued that TIN backaction (TINBA) can overwhelm QBA under realistic conditions. As an illustration, we studied the effects of TINBA in a high-cooperativity trampoline-in-the-middle cavity optomechanical system and found that it overwhelmed thermal noise and QBA by several orders of magnitude. The conditions embodied by our TIM system—transduction nonlinearity, large thermal motion, and a multimode mechanical resonator—can be found in a wide variety of room temperature quantum optomechanics experiments based on tethered nanomechanical resonators, including an emerging class of systems based on ultracoherent phononic crystal nanomembranes and strings [16,17]. Anticipating and mitigating TINBA in these systems may therefore be a key step to operating them in the quantum regime. In addition to increasing $Q$, a program combining multimode coherent [17] or measurement-based [28] feedback cooling, dual-probes at the “magic detuning” [44], and/or engineering of the effective mass and frequency spectrum [45] may be advantageous.
Funding
National Science Foundation (ECCS-1725571, ECCS-1945832).
Acknowledgment
The authors thank Charles Condos for assistance in refining the experimental apparatus and Ute Drechsler for fabricating the mesa mirrors. CMP acknowledges support from the ARCS Foundation, an Amherst College Fellowship, and a Grant in Aid of Research from Sigma Xi. ARA acknowledges support from a CNRS-UArizona iGlobes fellowship. The reactive ion etcher used for this study was funded by NSF.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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