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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% BEGIN DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{document}
 \title{Optomechanical thermal intermodulation noise}
 	
 \author{S. A. Fedorov}
 \thanks{These authors contributed equally}
 \affiliation{Institute of Physics (IPHYS), {\'E}cole Polytechnique F{\'e}d{\'e}rale de Lausanne, 1015 Lausanne, Switzerland}
 	
 \author{A. Beccari}
 \thanks{These authors contributed equally}
 \affiliation{Institute of Physics (IPHYS), {\'E}cole Polytechnique F{\'e}d{\'e}rale de Lausanne, 1015 Lausanne, Switzerland}
 
 \author{A. Arabmoheghi}
 \affiliation{Institute of Physics (IPHYS), {\'E}cole Polytechnique F{\'e}d{\'e}rale de Lausanne, 1015 Lausanne, Switzerland}
 
 %\author{\\M. J. Bereyhi}
 %\affiliation{Institute of Physics (IPHYS), {\'E}cole Polytechnique F{\'e}d{\'e}rale de Lausanne, 1015 Lausanne, Switzerland}
 
 \author{D. J. Wilson}
 \affiliation{College of Optical Sciences, University of Arizona, Tucson, Arizona 85721, USA}
 
 \author{N. J. Engelsen}
 \affiliation{Institute of Physics (IPHYS), {\'E}cole Polytechnique F{\'e}d{\'e}rale de Lausanne, 1015 Lausanne, Switzerland}
 
 \author{T. J. Kippenberg}
 \email{tobias.kippenberg@epfl.ch}
 \affiliation{Institute of Physics (IPHYS), {\'E}cole Polytechnique F{\'e}d{\'e}rale de Lausanne, 1015 Lausanne, Switzerland}
 	
 \begin{abstract}
 Thermal fluctuations give rise to a number of noise processes in optical interferometers, limiting the sensitivity of precision measurements ranging from the detection of gravitational waves to the stabilization of lasers for optical atomic clocks. In optical cavities, thermal fluctuations of length and refractive index result in cavity frequency noise, which can linearly couple to the optical field. Here we describe a different kind of noise process, thermal intermodulation noise, produced from the cavity frequency fluctuations by the inherent nonlinearity of optical susceptibility in laser-cavity detuning. We study thermal intermodulation noise due to the Brownian motion of membrane resonators in membrane-in-the-middle optomechanical cavities at room temperature, and show it to be the dominant source of classical intracavity intensity fluctuations under nearly-resonant optical excitation. We are able to operate at nominal quantum cooperativity equal to one an optomechanical cavity with optical finesse $\mathcal{F}=1.5\times 10^4$ and a low effective mass soft clamped membrane mode with $Q=4.\times10^7$ as a mechanical oscillator. In this regime, the magnitude of thermal intermodulation noise created by the mixing products of all membrane modes exceed the vacuum fluctuations by tens of decibel, preventing the observation of pondermotive squeezing. The described noise process is broadly relevant to optical cavities, in particular for which thermal frequency fluctuations are not negligible compared to the cavity linewidth. 
 \end{abstract}
 	
 \date{\today}
 \maketitle
 	 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% BEGIN MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Introduction} 
 Optical cavities are ubiquitous in physical experiments. Their applications include precision interferometric position measurements, an extraordinary example of which is direct gravitational wave detection\cite{advanced_ligo_2015,ligo_collaboration_observation_gw_2016}, stable frequency references\textcolor{red}{[ref]}, and quantum experiments, including cavity quantum electrodynamics\textcolor{red}{[ref]} and optiomechanics\cite{aspelmeyer_cavity_2014}. Optical cavities have finite temperature and therefore their frequencies exhibit fundamental thermal fluctuations due to the Brownian motion of mirror surfaces, thermorefractive and thermoelastic fluctuations\cite{braginsky_thermorefractive_2000,gorodetsky_thermal_noise_compensation_2008} and other processes that modulate the effective cavity length. These fluctuations predominantly manifest as excess phase noise in an optical field resonant with the cavity. At the same time, the nonlinearity of cavity discrimination curve creates intensity noise in the resonant field, which is especially pronounced when the magnitude of frequency fluctuations is comparable to the optical linewidth. This effect is known as intermodulation noise as it mixes different harmonics of the frequency noise. Technical intermodulation noise is known to limit the stability of frequency standards\cite{audoin_intermodulation_1991} and cavity-stabilized lasers\cite{ferguson_laser-noise-induced_1990,bahoura_ultimate_2003}. Here we report and study thermal intermodulation noise (TIN) of fundamental origin. 
 
 The transduction of optical path difference into measured signal in optical interferometers is periodic with the period equal to wavelength, $\lambda$, and therefore inherently nonlinear. Similarly, an optical cavity transduces the fluctuations of round-trip optical path, $\delta x$, to the modulation of intracavity field linearly only as far as the accumulated phase shift, $\delta \phi$, given by
 \begin{equation}
 \delta \phi =\fnss \delta x/\lambda,
 \end{equation}
 is much smaller than one. Therefore high optical finesse not only increases the resolution of a cavity as an optical path sensor but also limits its dynamic range to $\lambda/\fnss$\cite{miao_standard_2009,khalili_preparing_2010}. This is a particularly important consideration in experiments in which, on one hand, high finesse is desirable to increase the strength of light-matter interaction, and, on the other hand, stringent constraints exist on the tolerable level of extraneous noise in both quadratures of the optical field. Experiments on quantum cavity optomechanics are among such.
 
 Quantum cavity optomechanics studies aspects of interaction between optical field and mechanical motion such as position measurements and feedback control in presence of measurement-backaction\cite{sudhir_appearance_2017,wilson_measurement-based_2015}, the preparation of mechanical ground\cite{chan_laser_2011,qiu_laser_2019,rossi_measurement-based_2018}, single-phonon\cite{hong_hanbury_2017} and entangled\cite{riedinger_remote_2018} states and ponderomotive squeezing\cite{safavi-naeini_squeezed_2013,purdy_strong_2013}. In a handful of recent experiments, some quantum optomechanical effects were demonstrated at room temperature\cite{purdy_observation_2016,sudhir_quantum_2017,cripe_measurement_2019,yap_broadband_2019,aggarwal_room_2018}, limited due to high thermal noise levels. Most of these experiments\cite{cripe_measurement_2019,yap_broadband_2019,aggarwal_room_2018} operated in an exotic regime when the radiation pressure spring exceeded the natural frequency of the mechanical oscillator by two orders of magnitude. An alternative platform which is considered promising for reaching the quantum regime of optomechanical interaction at room temperature is membrane in the middle system\cite{thompson_strong_2008,wilson_cavity_2009}. It is predicted that quantum-backaction dominated regime is reacheable at microwatt input optical powers with the help of recently developed high-stress \SiN membrane resonators hosting high-$Q$ and low mass soft-clamped modes\cite{tsaturyan_ultracoherent_2017,reetz_analysis_2019}. Yet, concomitant with this approach is a dense spectrum of membrane modes that equally couple to the optical field and produce large thermal frequency fluctuations. 
 
 From the perspective of linear optomechanics, high temperature only requires increasing the input optical power to the point at which the optomechanical coupling rate compensates for the mechanical decoherence and the quantum cooperativity, $C_q$, reaches unity,
 \begin{equation}
 C_q=\frac{4 g^2}{\kappa \Gamma_\t{th}}\sim 1.
 \end{equation}
 Here $g$ is the loaded optomechanical coupling rate, $\kappa$ is the optical linewidth and $\Gamma_\t{th}=\Gamma_m n_\t{th}$ is the mechanical thermal decoherence rate equal to the product of mechanical energy relaxation rate and phonon occupancy. If thermal intermodulation noise is taken into account, the effect of high temperature is more detrimental, as high absolute magnitude of Brownian motion produces strong extraneous classical noise. Unless the magnitude of intermodulation noise is smaller than the optical vacuum fluctuations, no quadrature of the optical field is quantum-limited. 
 
 The manuscript is structured as follows. In the beginning we introduce a theoretical model of thermal intermodulation noise due to the Brownian motion of mechanical resonator in an optomechanical cavity. Next we present measurements in low-cooperativity regime which reveal an extraneous intensity noise source in a resonantly driven membrane-in-the-middle cavity. We show the noise to match the expected from the model magnitude and scaling with optical linewidth. Finally, employing a PnC membrane with a low effective mass soft clamped mode we conduct measurements the regime $C_q\sim 1$, and study the dependence of TIN on laser detuning, and find it to be in excellent agreement with our theoretical prediction. Moreover, we show that the intermodulation noise poses a significant limitation for the observability of quantum backaction-imprecision correlations in such system.
 
+\textcolor{red}{[Mention in the introduction the relation to quadratic optomechanical transduction]}
+
 \begin{figure}[t]
 \centering
 \includegraphics[width=\columnwidth]{fig_intro}
 \caption{\small{a) Transduction of the oscillator motion to the phase (upper panel) and amplitude (lower panel) of resonant intracavity light. b) Spectra of linear (upper panel) and quadratic (lower panel) position fluctuations of a multimode system. c) Experimental setup.}}
 \label{fig:intro}
 \end{figure}
 
-\section{Classical thermomechanical intermodulation noise}
+\section{Theory of intermodulation noise}
+\label{sec:genTheor}
 
 \begin{figure*}[t]
 \centering
 \includegraphics[width=\textwidth]{fig_rect_mbr}
 \caption{\small{a) Noise from a MIM cavity with laser detuned from resonance and on resonance. 1 mm square membrane, $\kappa/2\pi=26.6$ MHz, $g_0/2\pi=330$ Hz. c) The low frequency part of data in a). b), d) and e) show measurements for MIM cavity with 2 mm square membrane.  b) dependence of the average RIN in $0.6-1.6$ MHz band. b) Power sweep on the resonance with wavelength 837.7 nm, band $\pm$ one standard deviation around the mean is shaded gray. e) Green points --- measured linewidths of different optical resonances of MIM cavity, the dashed line is a guide to eye. Orange line --- linewidth of an empty cavity with the same length.}}
 \label{fig:rectMbr}
 \end{figure*}
 
-The cavity-induced nonlinearity of optomechanical interaction was studied in the works\cite{brawley_nonlinear_2016,leijssen_nonlinear_2017,matsko_electromagnetic-continuum-induced_2018}. Nevertheless, possibly because this effect is not readily apparent from the optomechanic Hamiltonian, made it receive little attention in the literature. This is despite the observation that nonlinear transduction can produce signals, quadratic in mechanic displacement, that are orders of magnitude stronger than those from the conventionally considered $\partial^2 \omega_c/\partial x^2$ term\cite{brawley_nonlinear_2016}. Below we derive the classical dynamics of optical field in an optomechanical cavity taking into account terms that are quadratic in displacement. We show that in membrane in the middle cavity typical quadratic signals originating from the nonlinear transduction are $r\mathcal{F}$ larger than the signals due to the nonlinear optomechanical coupling, $\partial^2 \omega_c/\partial x^2$. 
+We begin by presenting the theory of thermal intermodulation noise in an optical cavity under the assumption that the frequency fluctuations are slow compared to the optical decay rate. We concentrate on the lowest-order, i.e. quadratic, nonlinearity of the cavity detuning transduction. 
 
-We first show how transduction nonlinearity emerges in a generic optomechanic system and evaluate the signal strength. We consider a two-port optomechanic cavity driven by laser from port $1$, and assume fast cavity limit, which corresponds to the cavity decay constant $\kappa$ being much larger than the frequency of mechanical fluctuations.
-The classical dynamics of optical field inside the cavity is described by the equation
-\begin{equation}
-\frac{da(t)}{dt}=\left(i\Delta(t)-\frac{\kappa}{2}\right)a(t)+\sqrt{\kappa_1}\, s_\t{1,in},
-\end{equation}
-where $\Delta=\omega_L-\omega_c$ is the laser detuning from the cavity resonance, modulated by the mechanical motion. In adiabatic limit, if the cavity field adiabatically follows the fluctuations of $\Delta(t)$, the intracavity field is found as
-\begin{equation}
-a(t)=\frac{2}{\sqrt{\kappa}}\sqrt{\eta_1} L(\delta(t))\, s_\t{1,in},
+Consider an optical cavity with two-ports which is driven by a laser coupled to port one and the output from port two of which is directly detected on a photodiode. In the classical regime, i.e. neglecting vacuum fluctuations, the intracavity optical field, $a$, and the output field $s_\t{out,2}$ are found from the equations
+\begin{align}
+&\frac{da(t)}{dt}=\left(i\Delta(t)-\frac{\kappa}{2}\right)a(t)+\sqrt{\kappa_1}\, s_\t{in,1},\\
+& s_\t{out,2}(t)=-\sqrt{\kappa_2}a(t).\label{eq:oi2}
+\end{align}
+where $s_\t{in,1}$ is the constant coherent drive amplitude, $\Delta(t)=\omega_L-\omega_c(t)$ is the detuning from the cavity resonance, modulated by the cavity frequency noise, and $\kappa_{1,2}$ are the coupling rates of the ports one and two. Observe that it follows from \eqref{eq:oi2} that the intensity of the detected light is directly proportional to the intracavity intensity. In the fast cavity limit, when the optical field adiabatically follows $\Delta(t)$, the intracavity field is found as
+\begin{equation}\label{eq:aFull}
+a(t)=2\sqrt{\frac{\eta_1}{\kappa}} L(\nu(t))\, s_\t{in,1},
 \end{equation}
-where we introduced the normalized detuning $\delta=2\Delta/\kappa$, the cavity decay ratio $\eta_{1,2}=\kappa_{1,2}/\kappa$  and Lorentzian susceptibility
+where we introduced for brevity the normalized detuning $\nu=2\Delta/\kappa$, the cavity decay ratios $\eta_{1,2}=\kappa_{1,2}/\kappa$  and Lorentzian susceptibility
 \begin{equation}
-L(\delta)=\frac{1}{1-i\delta}.
+L(\nu)=\frac{1}{1-i\nu}.
 \end{equation}
-Assuming the laser to be resonant with the cavity on average and expanding $L$ in small $\delta$ up to the second order we get
-\begin{equation}
-a(t)=\frac{2}{\sqrt{\kappa}}\sqrt{\eta_1}(1 -i\delta(t) -\delta(t)^2) s_\t{1,in}.
+Expanding $L$ in \eqref{eq:aFull} over small detuning fluctuations $\delta\nu$ around the mean value $\nu_0$ up to the second order we find the intracavity field as
+\begin{equation}\label{eq:aSq}
+a=2\sqrt{\frac{\eta_1}{\kappa}}L(\nu_0)(1+iL(\nu_0)\delta\nu -L(\nu_0)^2\delta\nu^2) s_\t{in,1}.
 \end{equation}
-The fluctuations of $\delta$ due to the mechanical displacement are given by
+According to \eqref{eq:aSq}, the intracavity field is modulated by the cavity frequency excursion, $\delta\nu$, and the frequency excursions squared, $\delta\nu^2$. If $\delta\nu(t)$ is a stationary Gaussian noise process, like a thermal noise, the linear and quadratic contributions are uncorrelated (despite clearly not being independent). This is due to the fact that odd-order correlations vanish for Gaussian noise,
 \begin{equation}
-\delta(t)\approx \frac{G}{\kappa} x(t)+\frac{G_2}{2\kappa} x(t)^2,
-\end{equation}
-where $G=-\partial \omega_c/\partial x$ and $G_2=-\partial^2 \omega_c/\partial^2 x$ are the linear and quadratic optomechanical coupling, respectively.
-So overall we have
-\begin{multline}
-a(t)\approx\frac{2}{\sqrt{\kappa}}\sqrt{\eta_1}(1 -i\delta(t) -\delta(t)^2) s_\t{1,in}\approx\\
-\frac{2}{\sqrt{\kappa}}\sqrt{\eta_1}\left(1 -i\frac{G}{\kappa} x(t) -\left(\left(\frac{G}{\kappa} \right)^2+i\frac{G_2}{2\kappa}\right) x(t)^2\right) s_\t{1,in}.
+\langle\delta\nu(t)^2\delta\nu(t+\tau)\rangle=0,
+\end{equation}  
+where $\langle ...\rangle$ is time-average, for arbitrary time delay $\tau$.
+
+Next, we consider the ptotodetected signal, which, up to an unimportant conversion factor, equals to the intensity of the output light and found as   
+\begin{multline}\label{eq:detectedNoise}
+I(t)=|s_\t{out,2}(t)|^2\propto\\
+|L(\nu_0)|^2 \left(1-\frac{2\nu_0}{1+\nu_0^2}\delta\nu(t)+\frac{3\nu_0^2-1}{(1+\nu_0^2)^2}\delta\nu(t)^2\right).
 \end{multline}
-It is instructive to compare the magnitudes of the two contributions to the prefactor of $x(t)^2$. The typical value for $G$ (assuming the membrane to be approximately in the cavity center) is 
-\begin{equation}
-G\sim 2r \frac{\omega_c}{L},
-\end{equation} 
-while the typical value for $G_2$ is\cite{thompson_strong_2008}
+Notice, that there exist ``magic" detunings, $\nu_0=\pm 1/\sqrt{3}$, at which quadratic frequency fluctuations do not contribute to the detected signal and the intermodulation noise vanishes to the leading order.
+
+The spectrum of the detected signal is an incoherent sum of linear term,
 \begin{equation}
-G_2\sim 4 \frac{r \omega_c^2}{Lc},
+S_{\nu\nu}[\omega]=\int_{-\infty}^{\infty}\langle \delta\nu(t) \delta\nu(t+\tau) \rangle e^{i\omega \tau}d\tau,
 \end{equation}
-so that the ratio of the two contributions can be evaluated as
+and quadratic term, which for Gaussian noise can be found using the Wick's theorem\cite{gardiner_handbook_1985}
 \begin{equation}
-\left(\frac{G}{\kappa} \right)^2/\left(\frac{G_2}{2\kappa}\right)\sim\mathcal{F}r.
+\langle \delta\nu(t)^2 \delta\nu(t+\tau)^2 \rangle=\langle\delta\nu(t)^2 \rangle^2+2\langle \delta\nu(t) \delta\nu(t+\tau) \rangle^2,
 \end{equation}
-As the cavity finesse $\mathcal{F}$ is typically large, on on the order of $10^3$ to $10^5$, and the membrane reflectivity $r$ is between $0.1$ and $0.5$, we conclude that linear optomechanical coupling needs to extremely well suppressed in order for the quadratic coupling $G_2$ to play any role. In the following we neglect $G_2$.
+as
+\begin{multline}\label{eq:SnuSq}
+S_{\nu\nu,2}[\omega]=\int_{-\infty}^{\infty}\langle \delta\nu(t)^2 \delta\nu(t+\tau)^2 \rangle e^{i\omega \tau}d\tau=\\
+2\pi\langle \delta\nu^2\rangle^2 \delta[\omega]+2\times\frac{1}{2\pi} \int_{-\infty}^{\infty} S_{\nu\nu}[\omega']S_{\nu\nu}[\omega-\omega']d\omega',
+\end{multline}
+where $\delta$ is delta-function.
+
+\section{Thermomechanical intermodulation noise}
 
-Next we calculate the spectrum or photocurrent fluctuations in the setting relevant to our experiment, when the laser is coupled to the cavity from port one and the output of the second port is detected by a photodiode. For the output field we have
+In an optomechanical cavity the dominant source of cavity frequency fluctuations is the Brownian motion of mechanical modes coupled to the cavity,
+\begin{equation}\label{eq:dnuOpt}
+\delta\nu(t)= 2\frac{G}{\kappa} x(t),
+\end{equation}
+where $G=-\partial \omega_c/\partial x$ is the linear optomechanical coupling constant, and $x$ is the total membrane displacement, the sum of independent contributions $x_n$ of different mechanical modes. The spectrum of Brownian frequency noise is found as
+\begin{equation}\label{eq:mbrFreqFluct}
+S_{\nu\nu}[\omega]=G^2\sum_n S_{xx,n}[\omega],
+\end{equation}   
+where $S_{xx,n}[\omega]$ are the displacement spectra of individual membrane modes (see SI for more details). Applied to \eqref{eq:mbrFreqFluct}, the convolution in \eqref{eq:SnuSq} produces noise peaks at sums and differences of mechanical resonance frequencies, together with broadband background due to off-resonant components of thermomechanical noise, which is illustrated in \figref{fig:intro}b. 
+
+\textcolor{red}{[Introduce quantum optomechanical parameters]}
+
+For the following discussion of quantum optomechanical effects it is useful to introduce vacuum optomechanical coupling rate of an individual mode
 \begin{equation}
-s_\t{2,out}(t)=-2\sqrt{\eta_1\eta_2}L(\delta(t))s_\t{1,in},
+g_0=G x_\t{zpf},
 \end{equation}
-and for the photocurrent in direct detection
-\begin{multline}\label{eq:detectedNoise}
-I(t)=|s_\t{2,out}(t)|^2\propto|L(\delta_0)|^2\times \\\left(1-\frac{2\delta_0}{1+\delta_0^2}\frac{G}{\kappa}x(t)+\frac{3\delta_0^2-1}{(1+\delta_0^2)^2}\left(\frac{G}{\kappa}\right)^2 x(t)^2\right),
-\end{multline}
-where $\delta_0=\Delta_0/\kappa$ is the average laser detuning from the resonance.
+where $x_\t{zpf}=\sqrt{\hbar /2 m_\t{eff} \Omega_m}$ is the magnitude of zero point fluctuations, $\Omega_m$ is the mechanical resonance frequency and $ m_\t{eff}$ is the effective mass.
+
+The theory of \secref{sec:genTheor} applies in this case at optical powers sufficiently low so that the radiation-pressure mediated driving of the mechanical modes by the intermodulation noise is negligible. If this condition is not satisfied, radiation-pressure correlations may make the spectrum more complex. \textcolor{red}{[This is because the mechanical state becomes non-Gaussian\cite{brawley_nonlinear_2016}]} 
+
+The nonlinearity of optical detuning transduction imprints on the optical field signal proportional to $x^2$ in way analogous, but not equivalent, to the nonlinear optomechanical coupling, $\partial^2 \omega_c/\partial x^2$. This effect was studied previously\cite{brawley_nonlinear_2016,leijssen_nonlinear_2017,matsko_electromagnetic-continuum-induced_2018}. Interestingly, it was observed\cite{brawley_nonlinear_2016} that the cavity transduction commonly results in nonlinearity that is orders of magnitude stronger than the highest experimentally reported $\partial^2 \omega_c/\partial x^2$, when compared by the magnitude of the optical signal proportional to $x^2$. In the Supplementary Information it is shown that the same is true for membrane in the middle cavity, in which typical quadratic signals originating from the nonlinear transduction and leading to intermodulation noise are by the factor of $r\mathcal{F}$ (where $r$ is membrane reflectivity) larger than the signals due to the nonlinear optomechanical coupling, $\partial^2 \omega_c/\partial x^2$. 
 
-The frequency spectrum of the quadratic component of the photocurrent signal can be found using Wick's theorem\cite{gardiner_handbook_1985}
-\begin{multline}\label{eq:conv}
-S_2[\omega]=\int_{-\infty}^{\infty}\langle x(t)^2 x(t+\tau)^2 \rangle e^{i\omega \tau}d\tau=\\
-2\pi\langle x^2\rangle^2 d[\omega]+2\times\frac{1}{2\pi} \int_{-\infty}^{\infty} S_1[\omega']S_1[\omega-\omega']d\omega',
-\end{multline}
-where $d[\omega]$ is delta-function. If the cavity transduces the motion of multiple mechanical modes, the convolution results the appearance of frequency components at sum and difference frequencies, as illustrated in \figref{fig:intro}b. The quadratic spectrum of such system is dominated by intermodulation noise.
 
 \section{Experimental observation of extraneous amplitude noise}
 
 A startling manifestation of the optical transduction nonlinearity is the emergence of thermal amplitude noise in the field, output from an optomechanical cavity under resonant optical excitation. In the conventional picture of linear optomechanics, intensity fluctuations of the light that goes out of the cavity under such conditions is shot noise limited or contains a small portion of thermal signal transduced by dissipative coupling. In reality, however, the intensity of light can contain of vast amount of thermal intermodulation noise. We study this noise systematically with rectangular membranes in the low-cooperativity regime.
 
 Our experimental setup consists of a membrane in the middle cavity, composed of two supermirrors with the transmission of 100 ppm and a 200 $u$m-thick silicon chip sandwiched between them that hosts a suspended high-stress silicon nitride membrane (see \figref{fig:intro}c). The membrane in the middle cavity is situated in a vacuum chamber at room temperature, it is probed using the light form a Ti:Sa or a tunable ECDL laser with wavelength around 840 nm. The Ti:Sa laser is used for noise measurements, whereas the diode laser is used only for the characterization of optical linewidths. The light reflected from the cavity is used for PDH locking, whereas the light exiting the cavity from the second port and detected in direct detection constitutes our measurement signal. 
 
 While reaching the quatum regime of optomechnaical interaction requires engineering high-$Q$ and low mass mechanical oscillators, we perform the characterization of thermal intermodulation noise in low-cooperativity regime using conventional 20 nm-thick square membranes. In order to eliminate the influence of dynamic backaction, while characterizing the noises we keep the residual pressure in the vacuum chamber high, in the range $0.22\pm 0.03$ mBar, so that the quality factor of the fundamental membrane mode is gas-damping limited to $Q\sim 10^3$. 
 
 We start by presenting in \figref{fig:rectMbr}a and c the observation of strong classical amplitude noise in the output from the cavity subjected to resonant optical excitation. For MIM cavity with 1mm $\times$ 1mm $\times$ 20nm membrane and the input power of 5 $\mu$W the classical amplitude noise rises above the shot noise level by about 25 dB at low frequencies. In \figref{fig:rectMbr}a, the noise level approaches shot noise at high frequencies due to averaging of the membrane mode profiles of the cavity waist (approx $25$ $\mu$m in our experiment).
 
 Next we prove that the observed amplitude noise originates from the nonlinearity of cavity transduction by performing noise measurements across different optical resonances of the same cavity and referring the amplitude noise level to the value of vacuum optomechanical coupling rate of the fundamental mode over the optical linewidth (see \figref{fig:rectMbr}b). We use a 2mm $\times$ 2mm $\times$ 20nm membrane for this measurement. While the power spectral density of linearly transduced thermal fluctuations is $\propto (g_0/\kappa)^2$, the spectral density of quadratically transduced noise is $\propto (g_0/\kappa)^4$, a trend that is perfectly consistent with the data in \figref{fig:rectMbr}b. By performing a sweep of the input laser power on one of the resonances (see \figref{fig:rectMbr}d) we show that the intermodulation noise level is power-independent and therefore the observed noise is not related to the optomechanical dynamic backaction, negligible for the operation in low vacuum.
 
 \begin{figure}[t]
 \centering
 \includegraphics[width=\columnwidth]{fig_theor}
-\caption{\small{Cavity-waist averaged position (top row) and position squared (bottom row) noises produced by the modes of 20-nm Si$_3$N$_4$ rectangular membranes of different sizes. Red is experimental data and blue is theoretical prediction. \textcolor{red}{(Warning: two-sided theory spectra might be plotted, also the data for 2 mm membrane should be replaced) }}}
+\caption{\small{Cavity-waist averaged position (top row) and position squared (bottom row) noises produced by the modes of 20-nm Si$_3$N$_4$ rectangular membranes of different sizes. Red is experimental data and blue is theoretical prediction. \textcolor{red}{(Warning: two-sided theory spectra might be plotted, also the data for 2 mm membrane should be replaced) [also need to remove the cavity delay correction]}}}
 \label{fig:theor}
 \end{figure}
 
-The intermodulation noise observed in our experiment is well reproduced by a theoretical model with no fitting parameters. The frequency noise produced by the thermal membrane motion in a MIM cavity can be found as
-\begin{equation}\label{eq:mbrFreqFluct}
-S_{\Delta}[\omega]=G^2\sum_n A^2_n S_{x,n}[\omega],
-\end{equation}   
-where $S_{x,n}[\omega]$ are the displacement noises of individual membrane modes and $A_n$ are the factors characterizing the signal low-pass filtering due to the geometric overlap of mechanical mode profiles with the optical mode and due to the finite cavity response time. In our experiment the effect of finite cavity response time is small compared to the effect of geometric overlap, but we still take it into account. By using \eqref{eq:detectedNoise} and \eqref{eq:conv} we can find the spectrum of quadratically tansduced fluctuations and the resulting amplitude noise given the spectrum of cavity frequency fluctuations from \eqref{eq:mbrFreqFluct}. We present a comparison between experimentally measured linear and quadratic displacement noise spectra and the ones calculated using the theoretical model in \figref{fig:theor}. The inputs for the model in this case are experimentally measured $g_0$ of the fundamental membrane mode, the optical linewidth $\kappa$, the membrane size and its quality factors (for simplicity, assumed to be the same for all the modes). While our model is not detailed enough to reproduce precisely all the noise features, it well reproduces the overall level and the broadband envelope of the intermodulation noise that were observed in experiment.  
+The intermodulation noise observed in our experiment is well reproduced by a theoretical model with no fitting parameters. 
+
+In our experiment the effect of finite cavity response time is small compared to the effect of geometric overlap, but we still take it into account. By using \eqref{eq:detectedNoise} and \eqref{eq:conv} we can find the spectrum of quadratically tansduced fluctuations and the resulting amplitude noise given the spectrum of cavity frequency fluctuations from \eqref{eq:mbrFreqFluct}. We present a comparison between experimentally measured linear and quadratic displacement noise spectra and the ones calculated using the theoretical model in \figref{fig:theor}. The inputs for the model in this case are experimentally measured $g_0$ of the fundamental membrane mode, the optical linewidth $\kappa$, the membrane size and its quality factors (for simplicity, assumed to be the same for all the modes). While our model is not detailed enough to reproduce precisely all the noise features, it well reproduces the overall level and the broadband envelope of the intermodulation noise that were observed in experiment.  
 
 
 \begin{figure*}[t]
 \centering
 \includegraphics[width=\textwidth]{fig_soft_clamped}
 \caption{\small{a) Microscope image of a 3.6mm $\times$3.3mm $\times$ 20nm, with a low $m_\t{eff}$ localized mode frequency of approximately 800 kHz, b) displacement spectrum of the membrane in a\textcolor{red}{(replace data for this to be true)} c) ringdown measurement of quality factor of the membrane in a.
 d) Microscope image of a 2mm $\times$ 2mm $\times$ 20nm membrane hosting a soft clamped mode. e) Blue---protocurrent noise spectrum detected with laser detuned from the cavity resonance, red---shot noise level. f) The variation of the relative intensity noise of the light output from MIM cavity at bandgap frequencies with laser-cavity detuning. Blue dots are experimental points, dashed line - single-parameter model fit.}}
 \label{fig:softClamped}
 \end{figure*}
 
+\textcolor{red}{[Mention that the laser locking does not affect the intermodulation noise]}
+
+\textcolor{red}{[Add discussion that the potential dissipative coupling can only produce a much weaker effect on the intermodulation noise]}
+
+\textcolor{red}{[Mention that Sxx are compared in Fig3 because these noises are intrinsic to membranes]}
+
 
 \section{Intermodulation noise in an optomechanical cavity with a phononic crystal membrane}
 
 The quantum regime of optomechanic interaction, when the radiation pressure shot noise matches the thermal force noise, is reached at the input laser power given by
 \begin{equation}\label{eq:pcrit}
 P_\t{in}=\frac{\pi c}{32 \hbar} \frac{\lambda}{\mathcal{F}^2}\frac{S_\t{FF, th}}{4r^2},
 \end{equation}
 where it is assumed that the membrane is positioned along the cavity to maximize the linear optomechanical coupling, $\lambda$ is the optical wavelength, $\mathcal{F}$ is cavity finesse, $r$ is membrane reflectivity and $S_\t{FF, th}$ is the thermal force noise spectral density given by\cite{saulson_thermal_1990}
 \begin{equation}
 S_\t{FF,th}=2k_B T m_\t{eff}\Gamma_m.
 \end{equation}
 The reduction of thermal noise is essential for reaching the quantum backaction-dominated regime. Recently thermal noise down to 55 aN/$\sqrt{\text{Hz}}$ was demonstrated at room temperature  for soft-clamped modes localized in stressed phononic crystal membrane nanoresonators\cite{tsaturyan_ultracoherent_2017,reetz_analysis_2019}, owing to the simultaneous enhancement of quality factor and the reduction of effective mass. Although similar force noise levels are attainable with trampoline resonators\cite{reinhardt_ultralow-noise_2016}, the advantage of soft-clamped localized modes is their high frequency, on the order of MHz, which makes them less affected by classical laser noises. Even lower thermal noise, down to 10 aN/$\sqrt{\text{Hz}}$, was demonstrated for soft-clamped modes in nanobeam\cite{ghadimi_strain_2017} resonators, but nanobeams are not straightforward to combine with Fabry-Perot cavities.
 
 The integration of a membrane resonator with a Fabry-Perot optical cavity generally involves tradeoffs for the attainable thermal noise. Practical constraints need have to be satisfied include maintaining a good overlap between the mechanical mode and the optical cavity waist and ensuring that the mechanical mode of interest is spectrally well isolated from other membrane modes. 
 
 In \figref{fig:softClamped}a and b we present designs of PnC membranes with defects optimized to create low effective mass and high-$Q$ soft-clamped modes. The phononic crystal is made by a hexagon pattern of circular holes, which was introduced in Ref.~\cite{tsaturyan_ultracoherent_2017} and makes the simplest arrangement that creates a complete phononic bandgap for the flexural modes. The phononic crystal is terminated to the frame at half the hole radii, which is necessary to avoid the appearance of modes, localized at the membrane edges---such modes have frequencies withing the phononic bandgap and can contaminate the mechanical spectrum.  
 
 \figref{fig:softClamped}a shows the microscope image of a resonator with trampoline defect, featuring a particularly low $m_\t{eff}=1.9$ ng at $\Omega_m/2\pi = 0.853$ kHz and $Q=1.65\times 10^8$, corresponding to the force noise of $S_\t{FF,th}=15$ aN/$\sqrt{\text{Hz}}$. Another design, shown in \figref{fig:softClamped}d, is a 2 mm by 2 mm by 40 nm phononic crystal membrane with defect in the center that was engineered to create a single mode localized in the middle of phononic bandgap. This design features $Q=4.1\times 10^7$ at $1.5$ MHz and the effective mass of 2.2 ng, which results in predicted $S_\t{FF,th}=200$ aN/$\sqrt{\text{Hz}}$. The overall membrane size in this case of the second design is kept small enough so that no other modes, including in-plane ones, fall into the phononic bandgap. The complete membrane designs is available from Zenodo repository\cite{zenodo_repos}. 
 
 Using the membrane shown in \figref{fig:softClamped}d we were able to reproducibly assemble membrane-in-the-middle cavities with single-photon cooperativity $C_0=0.1-1$ (when operated in high vacuum, around $4\times 10^{-7}$ mBar in our case) and round trip loss lower than 200 ppm. According to the estimate provided by \eqref{eq:pcrit}, in such cavities the quantum backaction-dominated regime is expected to be reached at the input powers of a few hundreds of $\mu$W. Our experiment, however, shows that at these powers the optical amplitude noise in such cavities is far from being limited by the vacuum fluctuations of light due to the intermodulation noise. The latter is big challenge for the exploration of quantum aspects of radiation pressure interaction at room temperature.  
 
 \figref{fig:softClamped}e shows the spectrum of light output from a membrane in the middle cavity with length around 350 $\mu$m, $g_0/2\pi=360$ Hz, $\kappa/2\pi=24.8$ MHz and the intrinsic loss rate around 100 ppm, close to the output coupling rate from the two cavity mirrors having the transmission of 100 ppm each. The laser was detuned to the red from the cavity resonance in this measurement, and the spectrum of output fluctuations contains both the contribution of thermomechanical noise linearly transduced by the cavity detuning and the intermodulation noise due to the nonlinearity in $G/\kappa$. In particular, at frequencies within the phononic bandgap the noise level is dominated by the intermodulation noise, which rises almost 40 dB above the level of vacuum fluctuations (calibrated separately by directing an auxiliary laser beam of the the same power on the detector). The intermodulation origin of the noise in the bandgap can be proven by considering the variation of the noise level with laser detuning presented in 
 \figref{fig:softClamped}f. The laser power in this measurement was kept fixed to 30 $\mu$W, the cavity resonance wavelength is $840.1$ nm.
 
 We can understand the data in \figref{fig:softClamped}f using the general formula for the photocurrent produced in the detection of outgoing light (\eqref{eq:detectedNoise}). Linear and quadratic position fluctuations are transduced differently by the cavity, but almost within the entire range of the detunings the quadratically transduced fluctuations dominate. The exception is the vicinity of the detuning $\Delta=\kappa/(2\sqrt{3})$ at which the quadratic transduction by the cavity is compensated by the quadratic transduction by the nonlinearity of photodetection (see SI for discussion). At this detuning the in-bandgap noise level is consistent with the mirror noise. The overall variation of noise with the detuning can be described by the formula 
 \begin{equation}\label{eq:Siidelta}
-S_\t{RIN}\propto \frac{4 \delta_0^2}{(1+\delta_0^2)^2}S_1+\frac{1}{\delta_0}\frac{(3\delta_0^2-1)^2}{1+\delta_0^2}S_{2},
+S_\t{RIN}\propto \frac{4 \nu_0^2}{(1+\nu_0^2)^2}S_1+\frac{1}{\nu_0}\frac{(3\nu_0^2-1)^2}{1+\nu_0^2}S_{2},
 \end{equation}
 where $S_1$ is the contribution of mirror noise, which is independently calibrated, and $S_{2}$ is the contribution of quadratic noise that we use as a fitting parameter for the dashed curve in \figref{fig:softClamped}f. Aside from the cavity transduction, \eqref{eq:Siidelta} takes into account the laser cooling of mechanical modes by dynamic backaction (assuming that the optical damping is much larger than the intrinsic linewidth, see SI for details). As can be seen from \figref{fig:softClamped}f, \eqref{eq:Siidelta} very well reproduces the experimental data.
 
+\section{Conclusions and outlook}
+
+The suppression of intermodulation noise can be done by engineering mechanical resonators with lower multimode thermal noise and fewer modes, or by using optomechanical cavities with lower $g_0/\kappa$.
 
+As a potential way to suppress the intermodulation noise we may suggest engineering the optical susceptibility in a way that the quadratic transduction vanishes, for example, using double resonance.
 
 \section{Acknowledgements}
 This work was supported by...
 
 
 \bibliography{references}
 
 \end{document}
 
 
 
 
 
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+\documentclass[aps,a4paper,notitlepage,aps,pra]{revtex4-1}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PACKAGE CONFIG. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%start the SI figur labeling with S
+\usepackage[english]{babel}
+%\addto\captionsenglish{\renewcommand{\figurename}{Figure}}
+\renewcommand{\thefigure}{S\arabic{figure}}
+\renewcommand{\theequation}{S\arabic{equation}}
+
+\usepackage{amsmath,amssymb,amsfonts} % standard AMS packages
+\usepackage{changes}
+
+\usepackage{bm}	% bold symbols in math mode \bm{...}
+\renewcommand{\mathbf}{\bm}
+\usepackage{dsfont}	% proper mathbb format
+\renewcommand{\mathbb}{\mathds}	% redefine \mathbb
+
+\usepackage{mathrsfs} % use \mathscr{} for script letters in math
+
+\usepackage{mathtools} % for proper typesetting of := and =:
+\newcommand{\eqdef}{\vcentcolon=\,}
+\newcommand{\defeq}{=\vcentcolon\,}
+
+\usepackage{graphicx,float}
+\usepackage[colorlinks,
+	linkcolor=red,
+	citecolor=blue,
+	urlcolor=red]{hyperref}
+	
+\usepackage{chemformula}
+\usepackage{siunitx}
+
+\interfootnotelinepenalty=10000	% prevents footnotes from splitting across pages
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CUSTOM MACROS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+% macros for physics objects
+\newcommand{\ket}[1]{\vert{#1}\rangle}
+\newcommand{\bra}[1]{\langle{#1}\vert}
+\newcommand{\op}[1]{\hat{#1}}
+\newcommand{\tr}{\mathrm{Tr}}
+\newcommand{\avg}[1]{\left\langle{#1}\right\rangle}
+
+% macros for math objects
+\newcommand{\re}{\mathrm{Re}\,}
+\newcommand{\im}{\mathrm{Im}\,}
+\newcommand{\abs}[1]{\left\vert{#1}\right\vert}
+\newcommand{\symtext}[2]{\ensuremath{\stackrel{{#2}}{{#1}}}}
+
+% specific macros for this document
+\renewcommand{\t}[1]{\mathrm{#1}}
+\newcommand{\SiN}{Si$_3$N$_4\,$}
+
+% useful reference macros
+\newcommand{\figref}[1]{Fig.~\ref{#1}}
+\renewcommand{\eqref}[1]{Eq.~\ref{#1}}
+\newcommand{\secref}[1]{Sec.~\ref{#1}}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% BEGIN DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{document}
+
+\title{Supplementary information\\
+	``Optomechanical thermal intermodulation noise''}
+
+\author{S. A. Fedorov}
+\thanks{These authors contributed equally}
+\affiliation{Institute of Physics (IPHYS), {\'E}cole Polytechnique F{\'e}d{\'e}rale de Lausanne, 1015 Lausanne, Switzerland}
+	
+\author{A. Beccari}
+\thanks{These authors contributed equally}
+\affiliation{Institute of Physics (IPHYS), {\'E}cole Polytechnique F{\'e}d{\'e}rale de Lausanne, 1015 Lausanne, Switzerland}
+
+\author{A. Arabmoheghi}
+\affiliation{Institute of Physics (IPHYS), {\'E}cole Polytechnique F{\'e}d{\'e}rale de Lausanne, 1015 Lausanne, Switzerland}
+
+%\author{\\M. J. Bereyhi}
+%\affiliation{Institute of Physics (IPHYS), {\'E}cole Polytechnique F{\'e}d{\'e}rale de Lausanne, 1015 Lausanne, Switzerland}
+
+\author{D. J. Wilson}
+\affiliation{College of Optical Sciences, University of Arizona, Tucson, Arizona 85721, USA}
+
+\author{N. J. Engelsen}
+\affiliation{Institute of Physics (IPHYS), {\'E}cole Polytechnique F{\'e}d{\'e}rale de Lausanne, 1015 Lausanne, Switzerland}
+
+\author{T. J. Kippenberg}
+\email{tobias.kippenberg@epfl.ch}
+\affiliation{Institute of Physics (IPHYS), {\'E}cole Polytechnique F{\'e}d{\'e}rale de Lausanne, 1015 Lausanne, Switzerland}
+	
+\maketitle
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% BEGIN MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\tableofcontents
+\addtocontents{toc}{\protect\setcounter{tocdepth}{1}}
+
+\section{Membrane fabrication}
+
+Patterned and unpatterned membrane samples are fabricated on the same \SI{100}{\milli \meter} wafer. Stoichiometric, high stress \ch{Si_3N_4} is grown by low pressure chemical vapor deposition (LPCVD) on both sides of a \SI{200}{\micro \meter}-thick silicon wafer. The initial deposition stress is estimated a posteriori from the observation of membrane resonant frequencies, and varies in the range \SI{900}{}-\SI{1100}{MPa}, changing slightly with deposition run.
+
+The fabrication process relies on bulk wet etching of silicon in \ch{KOH} through the whole wafer thickness, to create openings for optical access to the membranes samples \cite{tsaturyan2017ultracoherent,reinhardt2016ultralow,norte2016mechanical,gartner2018integrated}. The extremely high selectivity of \ch{Si_3N_4} to \ch{Si} during \ch{KOH} etching allows the use of the backside nitride layer as a mask, to define the outline of the membranes on the frontside.
+
+Initially, the frontside nitride (\ch{Si_3N_4}) layer is patterned with h-line photolithography and \ch{CHF_3}/\ch{SF6}-based reactive ion etching (RIE) (steps 2-3 of figure \ref{fig:processflow}). The photoresist film is then stripped with a sequence of hot N-Methyl-2-pyrrolidone (NMP) and \ch{O_2} plasma; this procedure is carefully repeated after each etching step. The frontside nitride layer is then protected by spinning a thick layer of negative-tone photoresist (MicroChemicals AZ\textregistered 15nXT), prior to flipping the wafer and beginning the patterning of membrane windows on the backside nitride layer (steps 4-5). We noticed a reduction in the occurrence of local defects and increased overall membrane yield when the unreleased membranes on the frontside were protected from contact with hot plates, spin-coaters and plasma etchers chucks. The backside layer is then patterned with membrane windows, in a completely analogous way. The exposure step requires a wafer thickness-dependent rescaling of membrane windows, to account for the slope of slow-etching planes in \ch{KOH}, and careful alignment with frontside features.
+
+After stripping the photoresist, the wafer is installed in a PTFE holder for the first wet etching step in \ch{KOH} at $\approx \SI{75}{\celsius}$ (step 6). The holder clamps the wafer along its rim, sealing off the wafer frontside with a rubber O-ring, while exposing the backside to chemical etching by \ch{KOH}. This procedure is necessary to ensure that PnC membranes are suspended correctly: we noticed that releasing PnC samples by etching from both sides of the wafer produced a large number of defects in the phononic crystal, probably due to the particular dynamics of undercut and stress relaxation in the film. The wafer is etched until \SI{30}{}-\SI{40}{\micro\meter} of silicon remains, leaving the samples robust during the subsequent fabrication steps. The wafer is then removed from the KOH bath and the PTFE holder, rinsed and cleaned in concentrated \ch{HCl} at room temperature for 2 hours \cite{nielsen2004particle}.
+
+Subsequently, the wafer is coated with thick, protective photoresist and diced into $\SI{8.75}{mm}\times\SI{8.75}{mm}$ chips, and the remainder of the process is carried on chip-wise. Chips are cleaned again with hot solvents and \ch{O_2} plasma, and the membrane release is completed by exposing chips to \ch{KOH} from both sides (step 7). The temperature of the solution is lowered ($\approx \SI{55}{}-\SI{60}{\celsius}$), to mitigate the perturbation of fragile samples by buoyant \ch{N_2} bubbles, a byproduct of the etching reaction. After the undercut is complete, the samples are carefully rinsed, cleaned in \ch{HCl}, transferred to an ethanol bath and gently dried in a critical point dryer (CPD).
+
+\begin{figure}[t]
+\includegraphics[width=0.6\textwidth]{fig_process_flow.pdf}
+\centering
+\caption{Main steps of the fabrication process. Magenta - \ch{Si_3N_4}; gray - Si; green - photoresist.
+\label{fig:processflow}}
+\end{figure}
+
+\section{Quadratic mechanical displacement transduction by the optical cavity versus quadratic optomechanical coupling}
+
+Nonlinear cavity transduction can produce signals, quadratic in mechanic displacement, that are orders of magnitude stronger than those due $\partial^2 \omega_c/\partial x^2$ terms that were ever experimentally demonstrated\cite{brawley_nonlinear_2016}. Below we derive the classical dynamics of optical field in an optomechanical cavity taking into account terms that are quadratic in displacement. We show that in membrane in the middle cavity typical quadratic signals originating from the nonlinear transduction are $r\mathcal{F}$ larger than the signals due to the nonlinear optomechanical coupling, $\partial^2 \omega_c/\partial x^2$. 
+
+The fluctuations of $\nu$ due to the mechanical displacement are given by
+\begin{equation}
+\delta\nu(t)\approx 2\frac{G}{\kappa} x(t)+\frac{G_2}{\kappa} x(t)^2,
+\end{equation}
+where $G=-\partial \omega_c/\partial x$ and $G_2=-\partial^2 \omega_c/\partial^2 x$ are the linear and quadratic optomechanical coupling, respectively, and the total displacement $x$ is composed by partial contributions of different modes $x_n$
+\begin{equation}
+x(t) = \sum_n x_n(t).
+\end{equation}
+For resonant lase probe we can find the intracavity field as
+\begin{equation}\label{eq:aResDisp}
+a(t)\approx 2\sqrt{\frac{\eta_1}{\kappa}}(1 -i\nu(t) -\nu(t)^2) s_\t{in,1}=2\sqrt{\frac{\eta_1}{\kappa}}\left(1 -2i\frac{G}{\kappa} x(t) -\left(\left(2\frac{G}{\kappa} \right)^2+i\frac{G_2}{\kappa}\right) x(t)^2\right) s_\t{in,1}.
+\end{equation}
+It is instructive to compare the magnitudes of the two contributions to the prefactor of $x(t)^2$. The typical value for $G$ (assuming the membrane to be not very close to one of the mirrors) is 
+\begin{equation}
+G\sim 2r \frac{\omega_c}{l_c},
+\end{equation} 
+while the typical value for $G_2$ is\cite{thompson_strong_2008}
+\begin{equation}
+G_2\sim 4 \frac{r \omega_c^2}{c \, l_c},
+\end{equation}
+where $c$ is the speed of light, $r$ is the membrane reflectivity and $l_c$ is the cavity length. The ratio of the two contributions is evaluated as
+\begin{equation}
+\left. \left(2\frac{G}{\kappa} \right)^2\right/\left(\frac{G_2}{\kappa}\right)\sim\mathcal{F}r.
+\end{equation}
+As the cavity finesse $\mathcal{F}$ is typically large, on on the order of $10^3$ to $10^5$, and the membrane reflectivity $r$ is between $0.1$ and $0.5$, we conclude that linear optomechanical coupling needs to extremely well suppressed in order for the quadratic coupling $G_2$ to contribute. 
+
+\section{Dissipative coupling}
+In an optomechanical membrane-in-the-middle cavity dissipative coupling, $\partial \kappa/\partial x$, exists in addition to the dispersive coupling, $\partial \omega_c/\partial x$. Dissipative coupling modulates the optical decay rate, both external coupling and intrinsic loss, and potentially can produce intensity noise in a resonantly locked probe laser. However, for the parameters of our experiment the dissipative coupling is negligible.
+
+The noise due to dissipative coupling can be upper-bound as follows. The cavity linewidth cannot change by more than $\kappa$ as the membrane is translated by $\lambda$ inside the cavity, and therefore the dissipative coupling rate is limited by
+\begin{equation}
+\frac{\partial \kappa}{\partial x}\lesssim \frac{\kappa}{\lambda}= \frac{1}{\mathcal{F}}\frac{\omega_c}{2l_c}\sim \frac{G}{\mathcal{F}},
+\end{equation} 
+where in the last transition it was assumed that the membrane reflectivity is not very much smaller than one.
+
+Resonant intracavity field modulated by dissipative coupling only is given by
+\begin{equation}
+a(t)\approx 2\sqrt{\frac{\eta_1}{\kappa}}\left(1 - \frac{1}{2\kappa}\frac{\partial \kappa}{\partial x} x(t)\right) s_\t{in,1}.
+\end{equation}
+Comparing to \eqref{eq:aResDisp}, we find that the noise produced by dissipative coupling is negligible compared to the intermodulation noise if
+\begin{equation}
+\frac{G}{\kappa}x \gg \frac{1}{\mathcal{F}}.
+\end{equation}
+In all the experiments presented in this work this condition is satisfied, $Gx/\kappa$ ranges from \textcolor{red}{0.1 to 0.01 [verify exact numbers]}, whereas $1/\mathcal{F}$ is always less than $10^{-4}$.
+
+\section{Laser and mirror noises}
+Here we present the measurements of amplitude noise of TiSa laser plus the frequency noise noise of an empty Fabry-Perot cavity.
+
+\section{Details of the TIN calculations}
+
+\section{Model of TIN in presence of dynamical backaction cooling}
+
+\bibliography{supp_references}
+
+\end{document}
diff --git a/supplementary/fig_process_flow.pdf b/supplementary/fig_process_flow.pdf
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@@ -0,0 +1,54 @@
+@article{nielsen2004particle,
+  title={Particle precipitation in connection with KOH etching of silicon},
+  author={Nielsen, C Bergenstof and Christensen, Carsten and Pedersen, Casper and Thomsen, Erik Vilain},
+  journal={Journal of The Electrochemical Society},
+  volume={151},
+  number={5},
+  pages={G338--G342},
+  year={2004},
+  publisher={The Electrochemical Society}
+}
+
+@article{tsaturyan2017ultracoherent,
+  title={Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution},
+  author={Tsaturyan, Yeghishe and Barg, Andreas and Polzik, Eugene S and Schliesser, Albert},
+  journal={Nature Nanotechnology},
+  volume={12},
+  number={8},
+  pages={776},
+  year={2017},
+  publisher={Nature Publishing Group}
+}
+
+@article{reinhardt2016ultralow,
+  title={Ultralow-noise SiN trampoline resonators for sensing and optomechanics},
+  author={Reinhardt, Christoph and M{\"u}ller, Tina and Bourassa, Alexandre and Sankey, Jack C},
+  journal={Physical Review X},
+  volume={6},
+  number={2},
+  pages={021001},
+  year={2016},
+  publisher={APS}
+}
+
+@article{norte2016mechanical,
+  title={Mechanical resonators for quantum optomechanics experiments at room temperature},
+  author={Norte, Richard A and Moura, Joao P and Gr{\"o}blacher, Simon},
+  journal={Physical Review Letters},
+  volume={116},
+  number={14},
+  pages={147202},
+  year={2016},
+  publisher={APS}
+}
+
+@article{gartner2018integrated,
+  title={Integrated optomechanical arrays of two high reflectivity SiN membranes},
+  author={Gärtner, Claus and Moura, Jo{\~a}o P and Haaxman, Wouter and Norte, Richard A and Gröblacher, Simon},
+  journal={Nano Letters},
+  volume={18},
+  number={11},
+  pages={7171--7175},
+  year={2018},
+  publisher={ACS Publications}
+}
\ No newline at end of file