diff --git a/main.tex b/main.tex index 4d63674..5e92eaa 100644 --- a/main.tex +++ b/main.tex @@ -1,327 +1,327 @@ \documentclass[aps,prx,a4paper,notitlepage,reprint,superscriptaddress]{revtex4-1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PACKAGE CONFIG. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{amsmath,amssymb,amsfonts} % standard AMS packages \usepackage{mathrsfs} % use \ma۳thscr{} for script letters in math \usepackage{mathtools} % for proper typesetting of := and =: \usepackage{color} \usepackage{chemformula} \usepackage{graphicx,float} \usepackage[colorlinks, linkcolor=red, citecolor=blue, urlcolor=red]{hyperref} \usepackage{cleveref} \usepackage{soul} \usepackage{changes} \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$_4\,$} \newcommand{\fnss}{\mathcal{F}} \newcommand{\mbrsize}[1]{#1\,mm$\times$#1\,mm} % 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{Optomechanical thermal intermodulation noise} \author{S. A. Fedorov} \thanks{These authors contributed equally} \email{sergey.fedorov@epfl.ch} \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} \email{nils.engelsen@epfl.ch} \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 in cavities limit the sensitivity of precision measurements ranging from detection of gravitational waves to stabilization of lasers for optical atomic clocks. They arise from fluctuations of length or refractive index caused by fundamental thermodynamic fluctuations of temperature, referred to as thermorefractive and thermoelastic noise. %\added{Thermal fluctuations limit the sensitivity of precision measurements ranging from laser interferometer gravitational wave observatories to optical atomic clocks. } %They are caused by fundamental thermodynamic temperature fluctuations that give rise to cavity frequency noise via thermal expansion and temperature dependent refractive indices of the mirror. %\added{Such thermal noise presently limits both the sensitivity of laser interferometer gravitational wave observatories, and frequency stability of reference cavities as employed in optical atomic clocks. } Here we report a \emph{broadband} noise process, thermal intermodulation noise, originating from the transduction of \added{thermal} frequency fluctuations by the nonlinearity in the cavity-laser detuning response. We study thermal intermodulation noise due to the Brownian motion of a thin $\ch{Si3N4}$ membrane resonator in an optomechanical cavity at room temperature, and show that it creates noise in the amplitude quadrature for resonant laser excitation. \added{We demonstrate that the noise scales with the quartic power the the ratio of optomechanical coupling rate and cavity linewidth, and study it's laser detuning dependence, which are found to be in excellent agreement with a developed theoretical model.} %it to be the dominant source of classical intracavity intensity fluctuations under resonant optical excitation. The noise process is particularly relevant to quantum optomechanics: Utilizing a phononic crystal membrane with a low mass, soft-clamped defect mode, that operates in the regime where quantum fluctuations of radiation pressure are expected to dominate (i.e. a nominal quantum cooperativity exceeding unity), the mixing products of the membrane modes create thermal intermodulation noise which exceeds the vacuum fluctuations by orders of magnitude, even within the bandgap, thereby preventing the observation of pondermotive squeezing. %Utilizing a phononic crystal membrane with a low effective mass, soft-clamped defect mode, we are able to operate an optomechanical system with a nominal quantum cooperativity of unity. In this regime, the mixing products of the membrane modes create thermal intermodulation noise which exceeds the vacuum fluctuations by orders of magnitude, even within the bandgap, thereby preventing the observation of pondermotive squeezing. The described noise process is broadly relevant to cavity-based measurements, and is especially pronounced when thermally induced frequency fluctuations are comparable to the optical linewidth. \end{abstract} \date{\today} \maketitle %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% BEGIN MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} Optical cavities are an enabling technology for precision metrology, with applications including gravitational wave detection \cite{ligo_collaboration_observation_gw_2016}, ultra stable lasers \cite{sterr_ultrastable_2009}, cavity QED \cite{ye_quantum_2008} to cavity optomechanics \cite{aspelmeyer_cavity_2014}. The ultimate limits in the frequency stability of such cavities are imposed by fundamental temperature fluctuations. The temperature dependent refractive index and expansion coefficient lead to thermo-refractive and thermoelastic fluctuations of the optical cavity pathlength that limit frequency stability\cite{braginsky_thermorefractive_2000,gorodetsky_thermal_noise_compensation_2008}. %At finite temperature, the optical resonance frequencies exhibits fundamental thermal fluctuations that limit frequency stability and which originate from Brownian motion of mirror surfaces, substrates %thermorefractive and thermoelastic fluctuations \cite{braginsky_thermorefractive_2000,gorodetsky_thermal_noise_compensation_2008}. % \added{Historically, understanding and minimizing such thermal noises has played a key role in design of laser interferometer gravitational wave detectors, where it motivated changes in the choice of mirror substrates[Ref VB 1999 Phys Lett A]. Moreover, thermal noise have motivated developing mirrors with lower noise, based on crystalline coatings [Ref. Cole nphoton.2013.174], and cryogenic operation [https://doi.org/10.1364/OPTICA.6.000240]. Fundamental thermodynamical noises are in particular prominent in optical micro-cavities[Grudinin, Gorodetsky Josa B 2004], which due to their lower mode volume exhibit larger fluctuations, which can limit for instance their ability to measure mechanical displacement in the near field [Anetsberger Nat. Phys.].} These fluctuations predominantly manifest as excess phase noise in an optical field resonant with the cavity. \added{To estimate this phase noise, currently employed models generally assume that the transduction from effective pathlength fluctuations to the optical quadrature of the field is linear.} However, \deleted{At the same time,} the nonlinearity of the cavity response with respect to the cavity-laser detuning %of the cavity-laser detuning response creates noise in the amplitude can create noise in the amplitude quadrature of the resonant laser field. This effect is known as \emph{intermodulation noise}, as it mixes different Fourier components of the frequency noise\hl{[Ref missing]}. \added{In frequency locking using modulation sidebands}, such technical intermodulation noise \added{in the error signal} 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}. \hl{TJK: it is important to differentiate how TIN manifests itself in these experiments, vs. ours} Here we report and study \emph{\added{optomechanical} thermal intermodulation noise} (TIN) which has a fundamental thermodynamic origin. \added{Specifically, we show that this noise exists in optomechanical systems, arising from thermal motion of multiple mechanical modes coupled to the cavity field, and presents a key empedement to observing radiation pressure quantum noise at room temperature.} The transduction of optical path difference into measured signal in optical interferometers is periodic in the wavelength, $\lambda$, and therefore inherently nonlinear. Correspondingly, an optical cavity transduces the fluctuations of optical path length, $\delta l$, \added{into a linear phase shift (for a resonant laser field) only as long as the fluctuations in phase,} %to the modulation of intracavity field linearly only as far as the fluctuations of phase shift, \begin{equation} \delta \phi =\fnss \delta l/\lambda, \end{equation} accumulated over the cavity storage time, are much smaller than one. \added{As a consequence} high optical finesse ($\fnss $) not only increases the resolution of a cavity as an optical path length sensor but also limits its dynamic range to $\lambda/\fnss$ \cite{miao_standard_2009,khalili_preparing_2010}. This is an important consideration in experiments in which, on the 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 operate in this regime. Quantum cavity optomechanics studies aspects of the interaction between optical field and mechanical motion such as position measurements and feedback control in the presence of measurement backaction \cite{sudhir_appearance_2017,wilson_measurement-based_2015}, the preparation of the mechanical ground state\cite{chan_laser_2011,qiu_laser_2019,rossi_measurement-based_2018}, single-phonon states \cite{hong_hanbury_2017} and entangled states \cite{riedinger_remote_2018}, 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}. Most of these experiments \cite{cripe_measurement_2019,yap_broadband_2019,aggarwal_room_2018} were performed in an exotic regime where the radiation pressure spring exceeded the natural frequency of the mechanical oscillator by two orders of magnitude. The membrane-in-the-middle (MIM) system is considered a promising alternative platform for accessing quantum optomechanical experiments at room temperature \cite{thompson_strong_2008,wilson_cavity_2009}. Recent advances in high-stress \SiN membrane resonators hosting high-$Q$ and low mass soft-clamped modes \cite{tsaturyan_ultracoherent_2017,reetz_analysis_2019} have made it theoretically possible to reach the radiation pressure quantum backaction dominated regime at ambient temperatures for microwatt optical input powers. %It is predicted that the 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, which also couple to the optical field and produce large frequency fluctuations, proportional to the temperature, $T$. \added{We show, that} the nonlinear thermal intermodulation noise associated with these fluctuations is proportional to $T^2$, and therefore particularly strong at room temperature, \added{and limits accessing the quantum backaction dominated regime.} The optical transduction nonlinearity which creates thermal intermodulation noise was reported in optomechanical systems previously \cite{brawley_nonlinear_2016,leijssen_nonlinear_2017}. To the lowest order in the displacement divided by dynamic range, it manifests as the measurement of mechanical displacement squared. Such measurements potentially have enticing applications in quantum optomechanics: they can be used for the observation of phononic jumps \cite{gangat_phonon_2011}, phononic shot noise \cite{clerk_quantum_2010}, and the creation of mechanical squeezed states \cite{nunnenkamp_cooling_and_squeezing_2010} if the effects of linear measurement backaction are kept small \cite{martin_measurement_2007,brawley_nonlinear_2016}. Experiments that demonstrated quadratic optomechanical position measurements using position-squared coupling to the cavity frequency \cite{paraiso_position-squared_2015} remain deep in the classical regime due to small coupling rates. Optical cavity transduction can produce large, effective quadratic nonlinearity \cite{brawley_nonlinear_2016}, but it is inevitably accompanied by linear quantum backaction. \added{Our work shows that a further consequence of the quadratic transduction nonlinearity is a new noise source, that of optomechanical thermal intermodulation noise}. \hl{TJK: Note, this last paragraph was not connected with the present paper. It should be for the reader.} The manuscript is structured as follows. First, we introduce a theoretical model of thermal intermodulation noise. Next, we present measurements in the low-cooperativity regime which reveal an extraneous intensity noise source in a resonantly driven membrane-in-the-middle cavity. We show that the noise matches the magnitude expected from the developed TIN model and verify the scaling with the ratio of the optomechanical coupling rate over the optical linewidth. Finally, employing a PnC membrane with a low effective mass soft clamped mode we conduct measurements at the onset of the quantum backaction-dominated regime. We study the dependence of TIN on laser detuning, and find it to be in excellent agreement with our theoretical prediction. Moreover, we show that TIN is a significant limit to the observability of quantum backaction-imprecision correlations. \section{Theory of intermodulation noise} \label{sec:genTheor} We begin by presenting the theory of thermal intermodulation noise with the assumption that the cavity 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. - -Our setup is an optical cavity with two ports, which is driven by a laser coupled to port one. The output from port two is directly detected on a photodiode. In the classical regime, i.e. neglecting vacuum fluctuations, the complex amplitude of the intracavity optical field, $a$, and the output field $s_\t{out,2}$ can be found from the input-output relations +We consider (as in our experimental setup) an optical cavity with two ports, which is driven by a laser coupled to port one. The output from port two is directly detected on a photodiode. In the classical regime, i.e. neglecting vacuum fluctuations, the complex amplitude of the intracavity optical field, $a$, and the output field $s_\t{out,2}$ can be found from the input-output relations \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 laser detuning from the cavity resonance, modulated by the cavity frequency noise, and $\kappa_{1,2}$ are the coupling rates of ports one and two. In the fast cavity limit, when the optical field adiabatically follows $\Delta(t)$, the intracavity field is found as +where $s_\t{in,1}$ is the constant coherent drive amplitude, $\Delta(t)=\omega_L-\omega_c(t)$ is the laser detuning from the cavity resonance, modulated by the cavity frequency noise, and $\kappa_{1,2}$ are the external coupling rates of ports one and two \hl{[total kappa not defined] +}. 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 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(\nu)=\frac{1}{1-i\nu}. \end{equation} Expanding $L$ in \eqref{eq:aFull} over small detuning fluctuations $\delta\nu$ around the mean value $\nu_0$ up to 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} 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 typical thermal noises, 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} \langle\delta\nu(t)^2\delta\nu(t+\tau)\rangle=0, \end{equation} where $\langle ...\rangle$ is the time average, for an arbitrary time delay $\tau$. Next, we consider the photodetected signal, which, up to a conversion factor, equals the intensity of the output light and is found to be \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} \noindent Notice that $\delta\nu(t)$ and $\delta\nu(t)^2$ can be distinguished by their detuning dependence. The linearly transduced fluctuations vanish on resonance ($\nu=0$), where $\partial L/\partial \nu=0$. Similarly, when $\partial^2 L/\partial\nu^2=0$, the quadratic frequency fluctuations vanish, and thus also the thermal intermodulation noise. We will denote these ``magic'' detunings by $\nu_0$, which is given by \begin{equation} \nu_0=\pm 1/\sqrt{3} \end{equation} \noindent In the following experiments, we will utilize this detuning dependence to independently characterize the spectra of $\delta\nu(t)$ and $\delta\nu(t)^2$. -The total spectrum \cite{specta_notations} of the detected signal, $I(t)$, is an incoherent sum the linear term, +The total spectrum \cite{specta_notations} of the detected signal, $I(t)$, is an incoherent sum of the linear term, \begin{equation}\label{eq:Snu} S_{\nu\nu}[\omega]=\int_{-\infty}^{\infty}\langle \delta\nu(t) \delta\nu(t+\tau) \rangle e^{i\omega \tau}d\tau, \end{equation} -and the quadratic term, which for Gaussian noise can be found using the Wick's theorem \cite{gardiner_handbook_1985} +and the quadratic term, which for Gaussian noise can be found using Wick's theorem \cite{gardiner_handbook_1985} \begin{equation} \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 \begin{multline}\label{eq:Snu2} 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 $\delta[\omega]$ is the Dirac delta function. \begin{figure}[t] \centering \includegraphics[width=\columnwidth]{fig_intro} -\caption{\small{a) Transduction of the oscillator's motion to the phase (upper panel) and amplitude (lower panel) quadratures of resonant intracavity light. b) Spectra of linear (upper panel) and quadratic (lower panel) position fluctuations of a multimode resonator. c) Experimental setup. PM: Phase modulator. AM: Amplitude modulator. ESA: Electronic spectrum analyzer.}} +\caption{\added{Physical mechanism of optomechanical thermal intermodulation noise.}\small{a) Transduction of the oscillator's motion to the phase (upper panel) and amplitude (lower panel) quadratures of resonant intracavity light. b) Spectra of linear (upper panel) and quadratic (lower panel) position fluctuations of a multimode resonator, which can lead to the emergence of a wideband noise process. c) Experimental setup in which TIN is studied consisting of a membrane-in-the-middle optomechanical system. PM: Phase modulator. AM: Amplitude modulator. ESA: Electronic spectrum analyzer.}} \label{fig:intro} \end{figure} \begin{figure*}[t] \centering \includegraphics[width=\textwidth]{fig_rect_mbr} -\caption{\small{a), b) and c) show measurements for a MIM cavity with 2 mm square membrane. a) Cavity reflection signal as the laser is scanned over two resonances, with low (left) and high (right) optomechanical coupling. b) Dependence of resonant RIN, averaged over $0.6-1.6$ MHz, on the input power. Parameters: $\kappa/2\pi=9.9$ MHz, $g_0/2\pi=84$ Hz for the fundamental mode. The interval of $\pm$ one standard deviation around the mean is shaded gray. c) Dependence of the average RIN in a $0.6-1.6$ MHz band on $g_0/\kappa$. d) Noise from a MIM cavity with the laser detuned from resonance (top) and on resonance (bottom) using a 1 mm square membrane, with $\kappa/2\pi=26.6$ MHz and $g_0/2\pi=330$ Hz for the fundamental mode. }} +\caption{\added{Observation of optomechanical thermal intermodulation noise}. \small{a), b) and c) show measurements for a \added{membrane-in-the middle} (MIM) cavity with 2 mm square membrane. a) Cavity reflection signal as the laser is scanned over two resonances, with low (left) and high (right) optomechanical coupling. b) Dependence of resonant RIN, averaged over $0.6-1.6$ MHz, on the input power. Parameters: $\kappa/2\pi=9.9$ MHz, $g_0/2\pi=84$ Hz for the fundamental mode. The interval of $\pm$ one standard deviation around the mean is shaded gray. c) Dependence of the average RIN in a $0.6-1.6$ MHz band on $g_0/\kappa$. d) Noise from a MIM cavity with the laser detuned from resonance (top) and on resonance (bottom) using a 1 mm square membrane, with $\kappa/2\pi=26.6$ MHz and $g_0/2\pi=330$ Hz for the fundamental mode. }} \label{fig:rectMbr} \end{figure*} % Wavelengths: % b), the the power dependence, is taken at \lambda=837.7 nm, % d), the example transmission noises with 1 mm membrane, are taken at \lambda=838.2 nm, \section{Thermal intermodulation noise} 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 resonator displacement, i.e. the sum of independent contributions $x_n$ of different mechanical modes (the effect of the finite cavity mode waist is treated in the SI). The spectrum of the Brownian frequency noise is then found to be \begin{equation}\label{eq:mbrFreqFluct} S_{\nu\nu}[\omega]=\left(\frac{2G}{\kappa}\right)^{2}\sum_n S_{xx,n}[\omega], \end{equation} -where $S_{xx,n}[\omega]$ are the displacement spectra of individual mechanical modes (see SI for more details). The thermomechanical frequency noise given by \eqref{eq:mbrFreqFluct} produces TIN which contains peaks at sums and differences of mechanical resonance frequencies and a broadband background due to off-resonant components of thermal noise, as illustrated in \figref{fig:intro}b. The magnitude of the intermodulation noise is related to the quadratic spectrum of the total mechanical displacement, $S_{xx}^{(2)}$, as +where $S_{xx,n}[\omega]$ are the (classical, two-sided) displacement spectra of individual mechanical modes (see SI for more details) \hl{[TJK: I suggest to ensure to comment that these are classical spectral densities that are symmetric, and two sided.]}. The thermomechanical frequency noise given by \eqref{eq:mbrFreqFluct} produces TIN which contains peaks at sums and differences of mechanical resonance frequencies and a importantly a \emph{broadband} background due to off-resonant components of thermal noise, as illustrated in \figref{fig:intro}b. The magnitude of the intermodulation noise is related to the quadratic spectrum of the total mechanical displacement, $S_{xx}^{(2)}$, as \begin{equation}\label{eq:intNoiseSxx2} S_{\nu\nu}^{(2)}=(2G/\kappa)^{4}S_{xx}^{(2)}. \end{equation} A reservation needs to be made: the theory presented in \secref{sec:genTheor} is only strictly applicable to an optomechanical cavity when the input power is sufficiently low, such that the driving of mechanical motion by radiation pressure fluctuations created by the intermodulation noise is negligible; otherwise the fluctuations of $x(t)$ and $\delta \nu(t)$ may deviate from purely Gaussian and correlations exist between $S_{\nu\nu}$ and $S_{\nu\nu}^{(2)}$. On a practical level, this reservation has minor significance for our experiment. Also, the presence of linear dynamical backaction of radiation pressure does not change the results of \secref{sec:genTheor} but does modify $S_{xx}$. -Thermal intermodulation noise can preclude the observation of linear quantum correlations, which are induced by the vacuum fluctuations of radiation pressure between the quadratures of light and manifest as ponderomotive squeezing \cite{purdy_strong_2013,safavi-naeini_squeezed_2013}, Raman sideband asymmetry \cite{sudhir_appearance_2017} and the cancellation of shot noise in force measurements \cite{kampel_improving_2017,sudhir_quantum_2017}. The observation of quantum correlations typically requires selecting a mechanical mode with high $Q$, a spectral neighbourhood free from other modes, and with a high coupling rate, $g_0$, given by +Thermal intermodulation noise can preclude the observation of linear quantum correlations, which are induced by the vacuum fluctuations of radiation pressure between the quadratures of light and manifest as ponderomotive squeezing \cite{purdy_strong_2013,safavi-naeini_squeezed_2013}, Raman sideband asymmetry \cite{sudhir_appearance_2017} and the cancellation of shot noise in force measurements \cite{kampel_improving_2017,sudhir_quantum_2017}. The observation of quantum correlations typically requires selecting a mechanical mode with high $Q$, a spectral neighbourhood free from other modes, and a high optomechanical vacuum coupling rate, $g_0$, given by \begin{equation} g_0=G x_\t{zpf}. \end{equation} -Here $x_\t{zpf}=\sqrt{\hbar /2 m_\t{eff} \Omega_m}$ is the magnitude of the zero point fluctuations, $\Omega_m$ is the mechanical resonance frequency and $ m_\t{eff}$ is the effective mass. The quantum regime of the linear optomechanical interaction begins when the quantum cooperativity, $C_q$, reaches unity, +Here $x_\t{zpf}=\sqrt{\hbar /2 m_\t{eff} \Omega_m}$ is the magnitude of the zero point fluctuations, $\Omega_m$ is the mechanical resonance frequency and $ m_\t{eff}$ is the effective mass. The quantum regime of the linear optomechanical interaction is reached when the quantum cooperativity, $C_q$, reaches unity, \begin{equation} C_q=C_0n_c/n_\t{th}\sim 1, \end{equation} -where $C_0=4 g_0^2/(\kappa \Gamma_m)$ is the single-photon cooperativity, $n_c$ is the mean intracavity photon number, $\kappa$ is the optical linewidth, $\Gamma_m$ is the mechanical energy relaxation rate and $n_\t{th}=\hbar\Omega_m/(k_B T)$ the mechanical oscillator phonon occupancy. In order to observe linear quantum correlations in this regime, the intermodulation noise must be negligible compared to the vacuum fluctuations, which occurs when the following condition is satisfied: +where $C_0=4 g_0^2/(\kappa \Gamma_m)$ is the single-photon cooperativity, $n_c$ is the mean intracavity photon number (\hl{Pls use $\bar{n}_c$ to denote average}), $\kappa$ is the optical linewidth, $\Gamma_m$ is the mechanical energy relaxation rate and $n_\t{th}=\hbar\Omega_m/(k_B T)$ the mean mechanical oscillator phonon occupancy. \added{In this regime quantum measurent backaction from radiation pressure quantum fluctuations dominate over the thermal noise for a given mechanical mode.} In order to observe linear quantum correlations in this regime, the intermodulation noise must be negligible compared to the vacuum fluctuations, which occurs when the following condition is satisfied: \begin{equation} C_q\left(\frac{g_0}{\kappa}\right)^2 \Gamma_m n_\t{th}\frac{S^{(2)}_{xx}[\omega]}{x_\t{zpf}^4}\ll 1. \end{equation} \noindent From this condition, we can learn that simply increasing the quantum cooperativity is not necessarily a successful strategy when limited by intermodulation noise. One can immediately observe that by reducing the mechanical dissipation and $g_0/\kappa$, one can keep the quantum cooperativity constant while lowering the intermodulation noise. However, a more involved approach of engineering the mode spectrum might be a more fruitful approach. Specifically, by reducing the mode density, for example by working with the fundamental mode, and attempting to engineer higher order modes such that their intermodulation products do not overlap with the fundamental mode frequency can mitigate the intermodulation noise. -The nonlinearity of the cavity-laser detuning response, producing TIN, modulates the optical field proportional to $x^2$ in a way analogous to, but not equivalent to, quadratic optomechanical coupling, $\partial^2 \omega_c/\partial x^2$. It was noticed that cavity transduction commonly results in effective quadratic coupling which is orders of magnitude stronger than the highest experimentally reported $\partial^2 \omega_c/\partial x^2$, in terms of the optical signal proportional to $x^2$ \cite{brawley_nonlinear_2016}. In the Supplementary Information, it is shown that the same is true in the MIM system. Here, the quadratic signal originating from nonlinear transduction, which creates the intermodulation noise, is larger than the signals due to nonlinear optomechanical coupling, $\partial^2 \omega_c/\partial x^2$, by a factor of $rF$, where $r$ is the membrane reflectivity. +The nonlinearity of the cavity-laser detuning response, producing TIN, modulates the optical field proportional to $x^2$ in a way analogous to, but not equivalent to, quadratic optomechanical coupling, $\partial^2 \omega_c/\partial x^2$. It was noticed that cavity transduction commonly results in effective quadratic coupling which is orders of magnitude stronger than the highest experimentally reported $\partial^2 \omega_c/\partial x^2$, in terms of the optical signal proportional to $x^2$ \cite{brawley_nonlinear_2016}. In the Supplementary Information, it is shown that the same is true in the MIM system. Here, the quadratic signal originating from nonlinear transduction, which creates the intermodulation noise, is larger than the signals due to nonlinear optomechanical coupling, $\partial^2 \omega_c/\partial x^2$, by a factor of $r \mathcal{F}$, where $r$ is the membrane reflectivity \added{and where $\mathcal{F}$ is the Finesse.} \section{Experimental observation of thermal intermodulation noise} \label{sec:rectMbr} TIN has a number of manifestations that are qualitatively different from other thermal noises in optical cavities. Namely, TIN is present in the amplitude quadrature of an optical field coupled to a cavity on resonance, and its magnitude depends very sensitively on the ratio of RMS cavity frequency fluctuations to the linewidth. In this section we present the observation of broadband classical intensity noise in the optical field resonant with membrane-in-the-middle optomechanical cavity at room temperature, and verify that this noise is due to the intermodulation of Brownian motion of membrane modes. Our experimental setup, shown in \figref{fig:intro}c, comprises a membrane-in-the-middle cavity, consisting of two high-reflectivity, dielectric mirrors with 100 ppm transmission and a 200 $\mu$m-thick silicon chip which is sandwiched directly between the mirrors and hosts a suspended high-stress stoichiometric \SiN membrane. The total length of the cavity is around 350 $\mu$m. The MIM cavity is situated in a vacuum chamber at room temperature and probed using a Ti:Sa or a tunable external cavity diode laser (ECDL) at a wavelength around 840 nm, close to the maximum reflectivity wavelength of the mirrors. The Ti:Sa laser was used in all the thermal noise measurements, whereas the diode laser was only used for the characterization of optical linewidths. The measurement signal was generated by direct detection of the light transmitted through the cavity on an avalanche photodiode. The reflected light, separated using a circulator, was used for Pound-Drever-Hall (PDH) locking of the Ti:Sa frequency. The one-sided spectra \cite{specta_notations} of signals were detected in transmission and calibrated either as relative intensity noise (RIN) or as effective cavity detuning fluctuations, $S_\Delta$, with the help of calibration tones applied to the amplitude or phase quadratures of the laser, respectively. \subsection{MIM cavity with a uniform membrane} The characterization of TIN was performed using 20 nm-thick, square membranes with different side lengths as mechanical resonators. The insertion of a membrane into the cavity resulted in excess loss for most of the optical resonances. Nevertheless, for some resonances, the optical quality factors were reduced by only 10\% (for more data on the optical losses, see SI). The optomechanical cooperativity was kept low during the noise measurements to eliminate dynamical backaction of the light (damping or amplification of mechanical motion). For this purpose the residual pressure in the vacuum chamber was kept high, $0.22\pm 0.03$ mBar, such that the quality factors of the fundamental modes of the membranes were limited by gas damping to $Q\sim 10^3$. The reflection signals of two resonances of a MIM cavity with a \mbrsize{2} membrane are presented in \figref{fig:rectMbr}a. The resonances have similar optical linewidths but different optomechanical couplings, and the one with high coupling shows clear signatures of thermal noise. For the resonance with high coupling, the total RMS thermal frequency fluctuations are expected to be around 2 MHz, which is still well below the cavity linewidth, $\kappa/2\pi=16$ MHz. The thermal fluctuations imprinted on the optical field by the cavity are significant even when the laser is resonant with the cavity. Typical spectra of the detected noise are shown in \figref{fig:rectMbr}d for a cavity with a different, \mbrsize{1}, square membrane. When the laser is locked detuned from the cavity resonance (close to the ``magic" detuning, $\nu_0\approx -1/\sqrt{3}$), the transmission signal is dominated by the Brownian motion of membrane modes linearly transduced by the cavity. The magnitude of thermomechanical noise is gradually reduced at high frequencies due to the averaging of membrane mode profiles \cite{zhao_wilson_suppression_2012,wilson_thesis_2012} over the cavity waist (approx $35$ $\mu$m in our experiment), until it meets shot noise at around 15 MHz (verified by the optical power dependence, see SI). When the laser is locked on resonance, the output light also contains a large amount of thermal noise---at an input power of 5 $\mu$W the classical RIN exceeds the shot noise level by about 25 dB at MHz frequencies. Again, at high frequency the noise level approaches shot noise. An unambiguous proof of the intermodulation origin of the resonant intensity noise is obtained by examining the scaling of the noise level with $G/\kappa$. In thermal equilibrium the spectral density of frequency fluctuations, $\delta\nu(t)$, created by a particular membrane is proportional to $(G/\kappa)^2$, and therefore the spectral density of intermodulation noise is expected to be proportional to $(G/\kappa)^4$. We confirm this scaling by measuring the resonant intensity noise for different optical resonances of the same cavity with a \mbrsize{2} membrane and present in \figref{fig:rectMbr}b the noise magnitude, averaged over the frequency band from 0.6 to 1.6 MHz, as a function of $g_0/\kappa$. Here $g_0$ is that of the fundamental mechanical mode, measured using frequency noise calibration as described in Ref. \cite{gorodetksy_determination_2010}. By performing a sweep of the input laser power on one of the resonances of the same cavity we show (see \figref{fig:rectMbr}b) that the resonant intensity noise level is power-independent and therefore the noise is not related to radiation pressure effects. \begin{figure}[t] \centering \includegraphics[width=\columnwidth]{fig_theor} \caption{\small{Detuning fluctuation (top row) and relative intensity noise (bottom row) spectra produced by the modes of a 20-nm, \mbrsize{0.3}, rectangular, Si$_3$N$_4$ membrane. Red shows experimental data and blue is the theoretical prediction. }} \label{fig:theor} \end{figure} The TIN observed in our experiments agrees well with our model. By calculating the spectrum of total membrane fluctuations according to \eqref{eq:mbrFreqFluct} and applying the convolution formula from \eqref{eq:Snu2} (see SI for full details), we can accurately reproduce the observed noise. In \figref{fig:theor}, we compare the measured detuning and intensity noise spectra with the theoretical model. Here, we assume that the damping rate of all the membrane modes are identical as the experiment is operated in the gas-damping-dominated regime. While this model is not detailed enough to reproduce all the noise features, it accurately reproduces the overall magnitude and the broadband envelope of the intermodulation noise observed in the experiment. A comparison of the linear and quadratic displacements of rectangular membranes of different sizes is made in the SI, where we observe that the overall magnitude of the noise increases with increasing membrane size, owing to the increased mode density of larger structures. \begin{figure}[t] \centering \includegraphics[width=\columnwidth]{fig_soft_clamped} \caption{\small{Microscope images of PnC membranes (top) and ringdowns of their soft-clamped, localized modes (bottom). a) 3.6\,mm$\times$3.3\,mm$\times$40\,nm, with a localized mode at 853 kHz, b) 2\,mm$\times$2\,mm$\times$20\,nm membrane with a localized mode at 1.46 MHz.}} \label{fig:softClamped} \end{figure} \begin{figure*}[t] \centering \includegraphics[width=\textwidth]{fig_detuning_sweep} \caption{\small{ a) Blue---protocurrent noise spectrum detected with the cavity-laser detuning set to $2\Delta/\kappa\approx-0.3$, red---shot noise level. The shaded region shows the noise averaging band for the plot in b. The inset shows an optical cavity mode (imaged at $\lambda\approx780$ nm) overlapping with the PnC membrane defect. b) The variation of the relative intensity noise at bandgap frequencies with cavity-laser detuning. Red dots are experimental measurements, blue line---fit \eqref{eq:Siidelta}, orange line --- cavity phase noise inferred from the fit, shaded blue region---independently calibrated cavity noise, with uncertainty from the selection of the averaging band (see SI).}} \label{fig:detuningSweep} \end{figure*} Before presenting our investigation of TIN with soft-clamped membranes, we would like to address two potential confounding effects: laser frequency noise and dissipative coupling. Laser frequency fluctuations contribute to detuning fluctuations in the same way as cavity frequency fluctuations. However, the noise of the Ti:Sa laser used in our experiments was much lower than the thermomechanical frequency noise of the short MIM cavities, and was therefore neglected. Additionally, we did not observe any significant effect of the laser lock performance on the magnitude of TIN, which indicates that the up-conversion of detuning noise from low frequencies($<10$ kHz), where the laser noise is largest, contributes negligibly to the TIN in our cavities. As dissipative coupling leads to the modulation of optical linewidth by mechanical position, it could also potentially explain intensity noise in a resonant optical field. Although dissipative coupling is generally present in MIM cavities \cite{wilson_thesis_2012}, the magnitude of this noise would expected to be orders of magnitude below that measured in our experiments (see SI for more details). Moreover, dissipative coupling cannot explain the observed scaling of resonant RIN ($\propto (G/\kappa)^4$) and the absence of correlation between the RIN level and the excess optical loss added by the membrane. \subsection{MIM cavity with a phononic crystal membrane} % Comment: Raw numbers for the input and output powers, uncorrected for the transmission of the vacuum chamber: % 137 uW input -> 15 uW output (resonance with kappa/2 pi = 34 MHz, at \lambda = 849.8 nm) % 30 uW input -> 8.3 uW output (resonance with kappa/2 pi = 25 MHz, at \lambda = 840.8 nm) Localized (``soft-clamped") defect modes in stressed phononic crystal (PnC) resonators can have quality factors in excess of $10^8$ at room temperature due to enhanced dissipation dilution \cite{tsaturyan_ultracoherent_2017,ghadimi_strain_2017}. Owing to their high $Q$ and low effective mass, which result in low thermal force noise, $S_\t{FF,th}= 2k_B T m_\t{eff}\Gamma_m$ \cite{saulson_thermal_1990}, these modes are promising for quantum optomechanics experiments \cite{rossi_measurement-based_2018}. Soft-clamped modes with thermal noises as low as 55~aN/$\sqrt{\text{Hz}}$ have been demonstrated in PnC membranes at room temperature \cite{tsaturyan_ultracoherent_2017,reetz_analysis_2019}. In \figref{fig:softClamped}a and b we present \SiN PnC membranes with soft-clamped modes optimized for low effective mass and high $Q$. The phononic crystals are formed by the hexagonal pattern of circular holes introduced in Ref.~\cite{tsaturyan_ultracoherent_2017}, which creates a bandgap for flexural modes. The phononic crystal is terminated to the silicon frame at half the hole radii in order to prevent mode localization at the membrane edges---such modes have low $Q$ and can have frequencies within the phononic bandgap, contaminating the spectrum. \figref{fig:softClamped}a shows a microscope image of a resonator with a trampoline defect, featuring $m_\t{eff}=3.8$ ng and $Q=1.65\times 10^8$ at $0.853$ MHz, corresponding to a thermal force noise $S_\t{FF,th}=13$ aN/$\sqrt{\text{Hz}}$. Another resonator, shown in \figref{fig:softClamped}b, is a \mbrsize{2} phononic crystal membrane with a defect engineered to create a single mode localized in the middle of the phononic bandgap. The displayed sample has $Q=7.4\times 10^7$ at $1.46$ MHz and $m_\t{eff}=1.1$ ng, corresponding to $S_\t{FF,th}=34$ aN/$\sqrt{\text{Hz}}$. The membrane designs can be found in \cite{zenodo_repos}. The phononic bandgap spectrally isolates soft-clamped modes from the thermomechanical noise created by the rest of the membrane spectrum. Nevertheless, when a PnC membrane is incorporated in a MIM cavity the entire multitude of membrane modes contributes to the TIN at bandgap frequencies, as TIN is produced by a nonlinear process. In the following, we present measurements of the in-bandgap excess noise in a MIM cavity at room temperature and show that it is dominated by TIN at all detunings except for the immediate vicinity of the ``magic" detuning $\nu_0=-1/\sqrt{3}$. Around $\nu_0=-1/\sqrt{3}$ the cavity noise, characterized by measuring the noise of an empty cavity (see SI), is the dominant excess noise. The measurements were conducted using a 2mm square PnC membrane with the patterning shown in \figref{fig:softClamped}b, but made of 40 nm-thick \SiN. The membrane has a single soft-clamped mode with $Q=4.1\times 10^7$ at $1.55$ MHz. The quality factor was characterized immediately before inserting the membrane in the cavity assembly. The measurements presented in this section were made using the same setup described in \secref{sec:rectMbr} and shown in \figref{fig:intro}c, the only difference being that the vacuum pressure was kept below $5\times 10^{-7}$ mBar in order to eliminate gas damping. \figref{fig:detuningSweep}a shows the spectrum of light transmitted through a resonance of membrane-in-the-middle cavity with $g_0/2\pi=0.9$ kHz for the soft-clamped mode, $\kappa/2\pi=34$ MHz (estimated roundtrip excess optical loss is 300 ppm) and $C_0=2.5$. The input power in the measurement was 100 uW after correcting for spatial mode matching, which corresponds to a nominal $C_Q\sim1$. The shot noise level was calibrated in a separate measurement by directing an independent laser beam on the detector. The noise at bandgap frequencies is dominated by TIN, which exceeds the shot noise by four orders of magnitude. The spectrum also shows a dispersive feature in the middle of the bandgap, which is a signature of classical correlations due to the intracavity TIN exciting the localized mechanical mode. We next present in \figref{fig:detuningSweep}b the dependence of the bandgap noise level on the laser detuning, measured on a different optical resonance of the same MIM cavity and at lower input power. In this measurement $g_0/2\pi=360$ Hz for the localized mode, $\kappa/2\pi=24.8$ MHz (estimated 150 ppm excess loss per roundtrip) and the input power was 30 $\mu$W. The bandgap noise was averaged over a 35 kHz band indicated in \figref{fig:detuningSweep}a. The detuning of the laser from the cavity resonance was controlled by and inferred from the locking offset. For detunings greater than $2\Delta/\kappa\approx 0.5$, where the PDH error flips sign, side of the line locking was used instead of PDH. With the laser drive detuned from the resonance, both linear, $\delta\nu(t)$, and quadratic, $\delta\nu(t)^2$, cavity frequency fluctuations contribute to the detected signal, as described in \eqref{eq:detectedNoise}. $S_{\nu\nu}$ is dominated by cavity noise at bandgap frequencies, while $S_{\nu\nu}^{(2)}$ consists of the intermodulation products of all the membrane modes. Moreover, radiation pressure cooling of membrane modes must be taken into account. Then, the overall detuning dependence of intensity noise, $S_{II}$, is given by (see SI for a detailed derivation) \begin{equation}\label{eq:Siidelta} S_{II}\propto \frac{4 \nu_0^2}{(1+\nu_0^2)^2} C_1+\frac{1}{|\nu_0|}\frac{(3\nu_0^2-1)^2}{1+\nu_0^2} C_2, \end{equation} where $C_1$ and $C_2$ are proportional to $S_{\nu\nu}$ and $S_{\nu\nu}^{(2)}$, respectively. $C_1$ and $C_2$ are taken as free parameters in fitting the data in \figref{fig:detuningSweep}b, but the inferred value of $C_1$ is consistent with the cavity noise level calibrated independently (see SI). As can be seen from \figref{fig:detuningSweep}b, our model reproduces the observed variation of output noise with detuning very well. While at small detunings from resonance, TIN is the dominant contribution to the overall noise, around the ``magic" detuning, TIN is suppressed and the output noise is limited by the cavity noise. Notice that the intensity of the detected light in our measurement is proportional to the intensity of the intracavity field. Therefore, the suppresion of TIN in the output necessarily implies the suppression of the corresponding radiation pressure noise, which can lead to classical heating of the mechanical oscillator and thereby limit the true quantum cooperativity. \section{Conclusions and outlook} To summarize, we have presented the observation and characterization of a previously unreported broadband thermal noise in optical cavities, TIN, which originates from the quadratic transduction of thermal cavity frequency noise. Although produced by the cavity frequency noise, TIN is not correlated with it (neglecting radiation pressure effects) and therefore in many ways behaves as an independent noise. The key qualitative feature of TIN is that it creates intensity fluctuations in an optical field resonant with the cavity. The TIN magnitude grows quadratically in the ratio of RMS thermal frequency fluctuations by the optical linewidth, and therefore it strongly affects high-finesse optical cavities with large frequency fluctuations, such as optomechanical membrane-in-the-middle cavities at room temperature. Thermal intermodulation noise in optomechanical experiments can be avoided by using cavities with low finesse (equivalently, low $g_0/\kappa$), and by coupling them to mechanical resonators with lower total thermal fluctuations, i.e. which have fewer mechanical modes, higher frequency, and higher $Q$ for all modes. The latter consideration could make the fundamental modes of mechanical resonators (e.g. low-mass trampolines \cite{reinhardt_ultralow-noise_2016}) seem preferable compared to high-$Q$ but high-order PnC defect soft-clamped modes. In this context, a newly proposed method of exploiting self-similar structures as mechanical resonators with soft-clamped fundamental modes \cite{fedorov_fractal-like_2020} could potentially be fruitful for overcoming TIN. Another way of reducing the TIN is laser cooling of mechanical motion, either by dynamical backaction of a red-detuned beam or by active feedback. In this case, however, all mechanical modes that contribute to the total cavity noise must be efficiently cooled, which could be technically challenging. The raw measurement data, analysis scripts and membrane designs are available in \cite{zenodo_repos}. \section{Acknowledgements} The authors thank Ryan Schilling for fabrication advice. All samples were fabricated and grown in the Center of MicroNanoTechnology (CMi) at EPFL. This work was supported by the Swiss National Science Foundation under grant no. 182103 and the Defense Advanced Research Projects Agency (DARPA), Defense Sciences Office (DSO), under contract no. D19AP00016 (QUORT). A.B. acknowledges support from the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement no. 722923 (OMT). N.J.E. acknowledges support from the Swiss National Science Foundation under grant no. 185870 (Ambizione). \bibliography{references} \end{document}