PHYSICAL REVIEW X 10, 041046 (2020) Featured in Physics

Fundamental Thermal Limits for Optical Microcavities

† Christopher Panuski,1,* Dirk Englund,1 and Ryan Hamerly 1,2, 1Research Laboratory of Electronics, MIT, Cambridge, Massachusetts 02139, USA 2NTT Research, Inc. Physics and Informatics Laboratories, 940 Stewart Drive, Sunnyvale, California 94085, USA

(Received 19 May 2020; revised 15 September 2020; accepted 21 October 2020; published 7 December 2020)

We present a joint theoretical and experimental analysis of thermorefractive noise in high-quality-factor (Q), small-mode-volume (V) optical microcavities. Analogous to well-studied stability limits imposed by in macroscopic Fabry-Perot resonators, we show that microcavity thermorefractive noise gives rise to a mode-volume-dependent maximum effective quality factor. State-of-the-art fabricated microcavities are found to be within one order of magnitude of this bound. By measuring the first thermodynamically limited frequency noise spectra of wavelength-scale high-Q=V silicon photonic pffiffiffiffiffiffi cavities, we confirm the assumptions of our theory, demonstrate a broadband sub-μK= Hz temperature sensitivity, and unveil a new technique for discerning subwavelength changes in microcavity mode volumes. To illustrate the immediate implications of these results, we show that thermorefractive noise limits the optimal performance of recently proposed room-temperature, all-optical qubits using cavity- enhanced bulk material nonlinearities. Looking forward, we propose and analyze coherent thermo-optic noise cancellation as one potential avenue toward violating these bounds, thereby enabling continued development in quantum optical measurement, precision sensing, and low-noise integrated photonics.

DOI: 10.1103/PhysRevX.10.041046 Subject Areas: Photonics, Statistical Physics

I. INTRODUCTION fundamental limits of macroscopic cavities in the presence of thermal fluctuations [10–12]. Room-temperature, high-quality-factor (Q) optical Here, we consider the opposite case: optical microcavities cavities enable the investigation of new physical phenom- [Fig. 1(b)] [13], whose small mode volumes (V˜ ∼ 1) ena by enhancing light- interaction [1], shaping facilitate single- label-free sensing [14], high- electromagnetic modes [2], and modifying the vacuum repetition-rate frequency combs [15,16] for frequency density of states [3]. However, these advantages synthesis [17], spectroscopy [18], and astronomy [19], come with an often forgotten cost—interaction with a enhanced coupling for atom-photon interfaces [20–22], thermally equilibrated confining medium inherently injects and low-energy (down to single-photon level) nonlinear noise into the optical mode in accordance with the fluc- interactions [23]. These microcavity-enhanced nonlinear- tuation-dissipation theorem (FDT) [4]. Macroscopic reso- ities even reveal new directions in physical science [24]: nators [Fig. 1(a)], such as those implemented in gravitational experimental demonstrations of non-Hermitian phenomena wave interferometers, minimize this interaction by support- [25], topological enhancement [26,27], synchronization ˜ ≫ 1 ˜ ing a large mode volume V in vacuum, where V ¼ [28], and chaotic dynamics [29] are just a few recent λ 3 λ V=ð =nÞ for the volume V of a -wavelength optical mode examples. However, shrinking mode confinement toward confined in a refractive index n. The surprising realization the near-diffraction-limited volumes offered by microcav- that the sensitivity of these kilometer-scale cavities can still ities significantly amplifies fundamental temperature fluc- be limited by Brownian motion in few-micron-thick mirror tuations hδT2i∝1=V [30]. These small-volume temperature e coatings [5,6] has spurred interest in low-noise mirror fluctuations are a classic problem in statistical mechanics coatings [7], grating-based mirrors [8,9], and the [31], and their exact nature was heavily debated [32–34] until initial measurements were reported in the 1990s [35]. *[email protected] Temperature noise has since been studied in diverse contexts † [email protected] including high-energy collisions [36], molecular dynamics [37], spin ensembles [38], and state-of-the-art electron- [39] Published by the American Physical Society under the terms of or graphene-based nanocalorimeters [40]. In optics, the the Creative Commons Attribution 4.0 International license. “ Further distribution of this work must maintain attribution to associated refractive index fluctuations, so-called thermor- the author(s) and the published article’s title, journal citation, efractive noise” (TRN), have been studied extensively in and DOI. fiber-based lasers and interferometers [41–44]. In recent

2160-3308=20=10(4)=041046(28) 041046-1 Published by the American Physical Society PANUSKI, ENGLUND, and HAMERLY PHYS. REV. X 10, 041046 (2020)

(a) magnitude of this bound. We verify our theory by meas- uring TRN as the dominant noise source in high-Q=V PhC cavities and demonstrate the ability to distinguish between subwavelength mode volumes (V<˜ 1) using fundamental noise spectra. To our knowledge, these are the first spectrally resolved measurements of a near-diffraction- limited optical mode operating at the thermal noise limit. We believe that our devices’ unique combination of (b) micron-scale spatial localization with a broadband temper- ature sensitivity comparable to state-of-the-art room-tem- perature optical thermometers will enable new directions in thermal physics and nonequilibrium thermodynamics [55,56]. As an example of the immediate impact of our formalism, we analyze the implications for an outstanding goal in quantum photonics: all-optical qubits using cavity- enhanced bulk material nonlinearities [57]. Since thermal noise is found to limit the qubit coherence, we propose and FIG. 1. Comparison of thermorefractive noise (TRN) in macro- analyze coherent thermo-optic noise cancellation as one scopic resonators (a) and microcavities (b). Mode-averaged potential avenue toward continued developments in low- temperature fluctuations δT¯ in large cavities induce refractive Q=V index noise δn¯ (and, thus, path length changes δL¯ ) due to the noise, high- microcavities. Together, these results ’ α reveal the importance of thermal noise in state-of-the-art mirrors nonzero thermo-optic coefficient TR ¼ dn=dT. The large mode volume V reduces δT¯ , yielding a narrow-band optical resonators, inform design choices to minimize its resonant frequency noise spectrum SωωðωÞ and an rms resonant impact on device performance, and motivate new research δω ≪ Γ frequency fluctuation rms , the cavity half-linewidth. This directions to violate the proposed bounds. nondominant thermal noise inhomogeneously broadens the intra- ω cavity field spectrum Saað Þ. Decreasing V increases both the II. FORMALISM δω Γ magnitude rms and bandwidth T of TRN, while increasing the resonator quality factor Q ¼ ω0=2Γ causes both quantities to As schematically illustrated in Fig. 1, fundamental exceed Γ. TRN, therefore, becomes a dominant source of stochastic temperature fluctuations δTðr;⃗ tÞ within a cavity homogeneous broadening in wavelength-scale high-Q=V micro- confining medium of refractive index n and thermo-optic cavities, leading to thermal dephasing and reduced resonant α coefficient TR ¼ dn=dT drive a mode-averaged refractive excitation efficiency of the cavity field aðtÞ. δ¯ α δ ¯ index change nðtÞ¼ TR TðtÞ. For an optical mode completely confined in dielectric, the resulting resonance ¯ shift δωðtÞ¼−ω0α δTðtÞ=n follows from first-order years, TRN has emerged as a principal source of resonant TR perturbation theory [58]. For now, we neglect temper- frequency noise in various dielectric microcavity geometries ature-induced deformations of the cavity, as the thermo- including microspheres [45], whispering-gallery-mode res- elastic coefficient of common dielectrics is typically 2 onators [46,47], ring resonators [48], and photonic crystal α orders of magnitude smaller than TR [46]. Resonant (PhC) cavities [49]. TRN has also recently been shown to enhancement of these mechanical effects over a narrow limit the stability of integrated lasers [50,51] and micro- bandwidth is possible; however, we are primarily interested cavity frequency combs [52]. in the broadband noise performance. To date, microcavity TRN has been considered only in a In the presence of TRN, the steady-state rotating-frame perturbative regime, where thep resultingffiffiffiffi rms resonant δω ∝ 1 intracavity field amplitude is frequency fluctuation rms = V is much less than Z R 2eΓ ω ffiffiffi t t the loaded cavity linewidth ¼ 0=Q. For sufficiently p −ðiΔþΓÞðt−t0Þþi dt00δωðt00Þ ˜ Γ˜ 0 t0 aðtÞ¼ sin dt e ð1Þ high Q and small V, this assumption becomes invalid. −∞ Continued improvements in microcavity performance— Γ yielding Q>107, V˜ ∼ 1 through fabrication advances for a loaded amplitude decay rate and critically coupled 5 −3 static drive s˜ detuned by Δ from the cavity resonance. [53] and Q ∼ 10 , V˜ ∼ 10 using novel subwavelength in The associated statistical moments can be computed using dielectric features [54]—thus raises a simple question: when the moment-generating properties of the characteristic will fundamental thermal noise limit the performance of R i t δωðt00Þdt00 high-Q=V microcavities? functional he t0 i, which in the case of zero-mean Here, we answer this open question by deriving general requires only the autocorrelation δω δω τ ω α 2 δ ¯ δ ¯ τ bounds for optical microcavity performance in the presence h ðtÞ ðt þ Þi¼ð 0 TR=nÞ h TðtÞ Tðt þ Þi [59,60]. of TRN and find that current photonic crystal and whisper- The latter autocorrelation of temperature fluctuations can ing-gallery-mode (WGM) devices are within one order of be computed from the heat equation

041046-2 FUNDAMENTAL THERMAL NOISE LIMITS FOR OPTICAL … PHYS. REV. X 10, 041046 (2020)  Z  τ ∂ 2 δTðr;⃗ tÞþD ∇ δTðr;⃗ tÞ¼F ðr;⃗ tÞð2Þ FðτÞ¼exp − dτ0ðτ − τ0ÞhδωðtÞδωðt þ τÞi ∂t T T  0   δω 2 rms −Γ τ in a medium of thermal diffusivity D driven by a Langevin 1 − Γ τ − T T ¼ exp Γ ð T e Þ ; ð7Þ forcing term FTðr;⃗ tÞ which satisfies the FDT. As we T illustrate for slab PhC cavities, Eq. (2) can be solved analytically for specific geometries; however, for general- where we implement the previously described character- ity, we follow the approach of Ref. [61] and enforce a istic functional properties with the frequency noise statis- single-mode decay approximation by introducing a phe- tics of Eq. (5). For sufficiently large VT (corresponding to a Γ nomenological thermal decay rate slow T) and small Q as assumed in previous analyses, the cavity resonance is quasistatic over the photon decay period R ∇ ϵ ⃗ ⃗⃗ 2 2 3⃗ and shifts by much less than a cavity linewidth over time. In f ½ ðrÞjEðrÞj g d r Γ τ ≪ 1 Γ ¼ D R ð3Þ this perturbative limit [Fig. 1(a)], T over the cavity T T 2 ⃗ 4 3 −Γ τ ϵðr⃗Þ jEðr⃗Þj d r⃗ ringdown time. We can therefore expand e T in Eq. (7) to −Γτ −δω2 τ2 2 second order, yielding ha˜ðtÞa˜ ðt þ τÞi ∼ e e rms = . evaluated for the envelope of intracavity energy The associated line shape, given by the Γ density. This form of T is chosen for consistency with of ha˜ðtÞa˜ ðt þ τÞi via the Wiener-Khinchin theorem, is δ ¯ 2 2 h T i¼kBT0=cVVT, the well-known statistical mechanics then the original Lorentzian with a small inhomogeneous result for temperature fluctuations in a volume VT of Gaussian broadening due to TRN. Simply put, TRN limits specific heat capacity cV at thermal equilibrium with a the cavity stability without substantially altering the intra- bath temperature T0 [31]. Averaging Eq. (2) over the cavity dynamics. δω optical mode profile, we then find However, as the thermal mode volume VT shrinks, rms and ΓT increase until they eventually exceed Γ [Fig. 1(b)]. d ω δT¯ ðtÞþΓ δT¯ ðtÞ¼F¯ ðtÞ; ð4Þ In this high-Q=V limit, the resonant frequency 0ðtÞ dt T T directly tracks the temperature noise over the relevant timescales [i.e., the frequency noise in Eq. (5) is effectively leading to the solution δ correlated], leading to a homogeneous broadening   2 2 of the resonance that dephases the intracavity field. ω0 kBT0 −Γ τ Γ τ ≫ 1 hδωðtÞδωðt þ τÞi ¼ α e T j j; ð5Þ Inserting the associated limit T into Eq. (7),we TR − Γ δω2 Γ τ |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}n cVVT find ha˜ðtÞa˜ ðt þ τÞi ∼ e ð þ rms= T Þ , corresponding to the 2 2 δω2 2Γ 2δω Γ ≈ 2δω Γ rms broadened linewidth þ rms= T rms= T. Our analysis focuses on the transition to this high-Q=V where the thermal mode volume limit. Specifically, we derive general solutions to Eq. (1) R for arbitrary TRN powers and bandwidths to calculate 2 3 2 “ ” ½ ϵðr⃗ÞjE⃗ðr⃗Þj d r⃗ mode-volume-dependent maximum effective cavity VT ¼ R ð6Þ quality factors Q that describe the fundamental limits ϵ ⃗2 ⃗⃗ 4 3⃗ eff ðrÞ jEðrÞj d r of microcavity stability and coherence. Whereas the simple free evolution model presented in this section reveals the is the common Kerr nonlinear mode volume found by essential physics of the extreme low- and high-Q=V cases, solving Eq. (2) in a homogeneous medium [45,62]. For a our latter generalized solutions are broadly applicable to three-dimensional mode with a Gaussian-shaped energy any highly confined dielectric microcavity, including density distribution, VR T is larger than the standard Purcell microspheres [45], micropillars [63], ring resonators [48], mode volume V ¼ ϵjE⃗j2d3r=⃗ maxfϵjE⃗j2g by a factor pffiffiffi microtoroids [64], PhC cavities [49], microdisks [65], of 2 2. vertical Fabry-Perot cavities [66], and more. Combining Eqs. (1) and (5), we can solve for the statistical moments of the driven cavity amplitude a˜ðtÞ III. TRN MEASUREMENT as a function of the primary parameters V and Γ . The T T Before pursuing these goals, we first experimentally results are discussed in Secs. IV and V based on the verify the fundamental assumptions of our TRN model by complete derivations in Appendix A. Here, however, we measuring the noise spectrum of high-Q=V PhC cavities. consider a simplified example of free cavity evolution, As shown in Fig. 2, our setup uses a Mach-Zehnder which demonstrates the basic techniques and the cavity interferometer to measure the of a cavity reflection behavior in two limiting regimes. In this case,R the amplitude tþτ signal via balanced homodyne detection. A variable beam −i δωðt0Þdt0 −Γτ autocorrelation ha˜ðtÞa˜ ðt þ τÞi ∼ he t ie ¼ splitter separates the emission from an amplified tunable FðτÞe−Γτ can be simplified by evaluating the dephasing infrared laser into local oscillator (LO) and cavity input function paths, which are passively balanced to minimize laser

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(a) (c) (d)

(b)

Multimode fit Multimode fit Single-mode fit Single-mode fit

FIG. 2. Calibrated measurement of TRN in high-Q=V silicon PhC cavities. A shot-noise-limited, balanced homodyne detector (a) is locked to the phase quadrature of the cavity reflection signal and records the spectrum of resonant frequency fluctuations. The finite ˜ difference time domain (FDTD) simulated mode profiles, thermal mode volume VT [Eq. (6)], and thermal decay rate ΓT [Eq. (3)] of the L3 and L4=3 devices tested are shown in (b). The radii of the green holes are increased by up to 5% to form superimposed gratings — which improve vertical coupling efficiency. The measured of cavity resonant frequency and temperature noise Sff ðfÞ — 3 4 3 and STTðfÞ, respectively (red) for L (c) and L = (d) cavities are compared to noise from a specular reflection off the sample surface, finite-element method (FEM) simulations of cavity TRN, as well as single- and multimode fits. The listed multimode fit parameters agree with the predicted values in (b). Inset reflection spectra of each device reveal quality factors on the order of 105, which compare favorably with the FDTD computed values (1.7 × 105 and 1.6 × 105 for the L4=3 and L3, respectively). Micrographs of the fabricated designs with enlarged holes relative to the optimal designs in (b) are also inset. frequency noise coupling. A λ=2 plate rotates the input from either cavity’s reflection (orange). The calibration tone signal by 45° relative to the dominant cavity is visible at 200 MHz, and we attribute the resonance at polarization axis such that the cavity reflection can be approximately 15 MHz to optomechanical coupling from isolated from any specular reflection from the sample using the fundamental flexural mode of the suspended membrane a polarizing beam splitter (PBS) [67]. The sample stage is [72]. In the corrected cavity noise curve (red), we subtract temperature controlled to better than 10 mK using a Peltier the LO and account for attenuation due to the plate and feedback temperature controller. A balanced, finite cavity linewidth. As expected, the wavelength-scale shot-noise-limited photodetector measures the homodyne mode volumes yield a spectral density of resonant frequency signal from the recombined cavity reflection and LO, and fluctuations Sff ðfÞ with nearly 2 orders of magnitude larger the result is recorded on an electronic amplitude and bandwidth compared to previous results in (ESA). By actively locking to the phase quadrature of the microspheres [45] and ring resonators [48]. The corrected homodyne signal with a piezocontrolled mirror, TRN- L4=3 noise spectra in Fig. 3 also confirm that the meas- induced cavity frequency noise is detected as frequency- urement is invariant across the range of acceptable input resolved voltage noise. To calibrate the spectrum, we inject powers, which is limited below by the homodyne locking a known with an electro-optic modulator stability and above by the onset of nonlinear effects leading (EOM) whose modulation efficiency is measured by side- to excess noise. Calibration data, an extended description of band fitting [68]. the setup, and an analysis of other possible noise sources are Figure 2 shows the resulting measurements for two provided in Appendix B. released (air-clad) silicon PhC cavities: the common L3 The measured noise spectra show excellent agreement cavity [69] and the recently proposed “L4=3” cavity [70]. with numerical simulations based on a modified version Fabricated cavities yield high quality factors (up to Q ≈ of the fluctuation-dissipation theorem for thermorefractive 400 000 at λ0 ≈ 1550 nm) with efficient vertical coupling noise [73], which is further described in Sec. VI. We also [71]. The variation of Purcell mode volume—V˜ ¼ develop an analytic noise model based on a multimode ð0.95; 0.32Þ for simulated L3 and L4=3 cavities, respec- solution to Eq. (2) in a thin slab, which is similarly tively—also allows us to confirm the expected volume well fitted to the data and yields the fit parameters ˜ L3 ΓL3 2π ˜ L4=3 ΓL4=3 2π 3 4 0 3 28 1 dependence of TRN. Whereas a direct reflection from the fVT ; T = ;VT ; T = g¼f . . ; MHz; sample surface (green trace) adds little additional noise to 1.4 0.1;80 3 MHzg that compare favorably with the the LO background (blue), we observe broadband noise expected values (f3.9; 29 MHz; 1.5; 84 MHzg) from

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magnitude less area and offer 3 orders of magnitude larger bandwidth. We expect this unique combination of micron- scale spatial resolution and broadband, thermodynamically limited readout to enable new directions in thermal physics.

IV. MICROCAVITY Q=V LIMITS Having experimentally verified the TRN model, we can now use it to estimate the fundamental performance limits of room-temperature microcavities. As described in Sec. II, 2Γ TRN broadens the cavity linewidth to eff according to the cavity mode volume VT and thermal decay rate ΓT. The ω 2Γ associated quality factor Qeff ¼ 0= eff is therefore bound to a maximum value indicative of the resonance stability, which, in turn, determines the fidelity of inte- grated optical frequency references or synthesizers [17], the FIG. 3. Comparison of corrected L4=3 cavity frequency noise minimum resolvable resonance shift in microcavity sensors ˜ spectra at various input powers Pin normalized to the maximum [78], and the spectral purity of microcavity lasers [79].We pvalueffiffiffiffiffiffiffiffiffiffiffiffi for the dataset. The rms fractional frequency fluctuation also specifically consider the quality factor to mode volume 2 hδω i=ω0 is computed from the integrated noise over the ratio Qeff=V, which is proportional to the peak intracavity plotted measurement bandwidth and plotted in the inset as a intensity and, therefore, of particular significance for cavity ˜ function of Pin. No significant power scaling or deviation from nonlinear optics [80] and enhanced sensitivity to pointlike the mean value (black dashed line) is observed, indicating that defects [81]. ω ˜ 2 2 ˜ 2 noise contributions from nonlinear effects can be neglected. The effective quality factor Qeff ¼ 0hjaðtÞj i= jsinj of interest in this case can alternatively be viewed as the ratio ˜ 2 Eqs. (6) (evaluated numerically from the finite element of intracavity energy hjaðtÞj i to energy input per cycle 2 ˜ 2 ω jsinj = 0 in a resonantly excited, critically coupled cavity. method-simulated mode profiles) and (3). In Eq. (3),we 2 assume a two-dimensional Gaussian mode and reduced Under the same conditions, solving Eq. (1) for hja˜ðtÞj i subject to the noise autocorrelation of Eq. (5) yields thermal diffusivity DT ¼ Dð1 − ϕÞ=ð1 þ ϕÞ for the pat- terned slab with porosity ϕ compared to the unpatterned Q ω0 thin film diffusivity D [74,75]. As predicted, the reduced eff exx−sγ s; x ; 8 ˜ ¼ 2Γ ˜ lð Þ ð Þ mode volume of the L4=3 cavity increases the bandwidth V TV and spectral density of thermal fluctuations. These obser- γ vations thereby illustrate a new technique for evaluating where l is the lower incomplete Gamma function, δω Γ 2 Γ Γ the mode volume of fabricated optical resonators using x ¼ð rms= TÞ , and s ¼ = T þ x. Intuitively, decreas- fundamental quantities as opposed to complex invasive ing the cavity linewidth 2Γ well below the broadened techniques, such as near-field scanning optical micros- linewidth has little impact: the prolonged energy storage copy [54]. offsets the reduced excitation rate of the rapidly shifting The noise spectra of the proposed single-mode approxi- resonance, leaving the intracavity energy unaltered. Qeff is mation [Eq. (5)] underestimate the measured noise of both maximized in this limiting case. The corresponding upper devices at low frequencies ω ≪ Γ but accurately approxi- bound of Eq. (8) at T ¼ 300 K is plotted for various T ˜ mate Sff in the range of frequencies of interest (near and material systems in Fig. 4 as a function of V assuming a above the cutoff frequency ΓT) and conserve the integrated three-dimensional Gaussian-shaped mode in a homo- δω2 geneous three- or two-dimensional confining medium. In frequency noise h rmsi. Our results indicate that TRN is the latter case, the decay rate Γ 3πD =V2=3 decreases the dominant noise source in high-Q=V resonators and pffiffiffi T ¼ T validate the suitability of a single-mode approximation by a factor of 3 2V1=3 to account for the restricted to describe the spectrum of frequency fluctuations in dimensionality of thermal diffusion. δω Γ ≪ Γ general microcavity geometries. To our knowledge, these In the joint limit ð RMS; Þ T, valid for sufficiently max measurements are the first demonstration of broadband, high-Q,low-V cavities, Eq. (8) simplifies to Qeff ¼ ω Γ 2δω2 wavelength-scale cavity readout at the thermodynamic 0 T= rms, thereby recovering the broadened linewidth 1 2 limit. The corresponding temperature sensitivity, S = ∼ 2δω2 =Γ derived in Sec. II. Qmax then scales as V1=3 in a pffiffiffiffiffiffi TT rms T eff 300 nK= Hz as shown in Fig. 2, is within one order of homogeneous medium, indicating that larger mode vol- magnitude of room-temperature records set by multimode umes reduce the integrated thermo-optic noise, as expected. 8 WGM thermometers [76,77]. Compared to those state-of- For this reason, recent ultrahigh-Q (Q>10 ) integrated the-art sensors, our PhC devices occupy 6 orders of resonators are specifically designed with V˜ ≫ 1 to limit

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have explored the feasibility of photonic microcavity-based quantum gates using strong photon-photon interactions mediated by bulk material nonlinearities. Driven by recent developments in high-Q=V microcavities [54,98] and thin film nonlinear optical materials, current experiments are approaching 1% [99] of this so-called “qubit limit of cavity nonlinear optics” [57] where single-photon nonlinearities outpace cavity losses. Strong emitter-based single-photon nonlinearities in high-Q=V cavities are also a promising route toward optically addressable qubits but rely on precise coupling to single atoms [100], ions [22], quantum dots [101], or defect centers [102]. We therefore focus our analysis on emitter-free, all-optical qubits using bulk non- linearities. These techniques promise room-temperature operation—the requisite hallmark for connecting distant nodes in future quantum networks [103]—by leveraging the relative immunity of optical to thermal noise. While this insensitivity is granted by Planck’s law, we show here FIG. 4. Thermal-noise-limited room-temperature quality factor that, through the thermorefractive effect, temperature fluc- max ˜ tuations can significantly impact light in a high-Q=V to mode volume ratios (Qeff =V) for the materials considered in Appendix C assuming a Gaussian-shaped mode at λ0 ¼ 1550 nm resonator. For coherent processes, TRN-induced dephasing admitting thermal diffusion in two or three dimensions (dash- of the field amplitude a˜ðtÞ must be considered in addition to dotted and lines, respectively). These limits are compared the previously discussed stability and intracavity energy our devices as well as other fabricated and proposed micro- limitations in Sec. IV. We therefore solve for ha˜ðtÞi in the cavities. Insets illustrate typical confinement geometries for the presence of TRN. This result corresponds to an effective range of V˜ listed (see Refs. [23,53,54,64,80,82–89]). ω ˜ 2 2 ˜ 2 quality factor Qeff ¼ 0jhaðtÞij = jsinj in a resonant, critically coupled cavity, yielding   TRN [65,90]. Alternatively, Fig. 4 illustrates the advantage 2 max ω0 Γ 2 −2 2 of reducing V to maximize Qeff =V. Further optimization Q Q e xx sγ s; x eff ¼ 2Γ ¼ Γ l ð Þð9Þ of subwavelength cavities [54,91] could therefore improve eff T the intensity enhancement achievable in room-temperature devices toward the open goal of microcavity-based quan- for the previously defined x and s. Equation (9) describes the tum nonlinear optics. ˜ coherence of the intracavity field and is synonymous with Surprisingly, the Qmax=V limits of several common 1 Γ 2 ω eff the dephasing time T2 ¼ = eff ¼ Qeff= 0 commonly materials lie within an order of magnitude. Yet this effect considered for quantum emitters [104]. is not fundamental: aluminum nitride, for example, is Γ We can compare eff to a nonlinear coupling rate g shown to outperform all other plotted materials by over between qubit basis states with the simple figure of merit an order of magnitude due to its simultaneously large 2Γ FOM ¼ g= eff, which intuitively corresponds to the thermal conductivity and small thermo-optic coefficient. number of qubit operations that can be completed prior While our review of high-Q=V cavities (Fig. 4) in various to decay or dephasing. For bulk χð3Þ and χð2Þ nonlinearities, materials shows that all fabricated cavities obey the the coupling rate g is a function of material parameters and projected bounds, silicon PhC slab cavities [53] and silica 3 mode volumes [105]. In the case of χð Þ interactions, g ¼ microtoroids [64] lie within an order of magnitude of the ω ˜ ˜ 0Kχ=VKerr for the Kerr interaction mode volume VKerr ¼ thermal noise limit. Furthermore, various simulated devices ˜ [84–87] exceed the limit; their practical realization thereby VT (as discussed in Sec. II) and coupling constant requires low-temperature operation—where TRN and the — 3 thermo-optic coefficient are both suppressed [92] or 3πℏc χð Þ novel noise-suppression techniques as discussed in Sec. VI. Kχ ¼ 4 ; ð10Þ 2nϵ0 λ0

V. IMPLICATIONS FOR ALL-OPTICAL QUBITS at wavelength λ0, where ℏ, c, and ϵ0 are Planck’s constant, These proposed thermal noise limits have practical impact the speed of light, and the vacuum permittivity, respec- for future devices. Chief among the applications driving the tively. In the strong coupling regime g=2Γ ≫ 1, the pursuit for high-Q=V cavities is quantum information. intensity-dependent refractive index leads to an anharmo- Within the past year alone, numerous proposals [93–97] nicity of Fock state energies that decouples the qubit

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FIG. 5. Performance of room-temperature all-optical qubits using bulk χð3Þ (left) or electric-field-induced χð2Þ (right) nonlinearities in silicon microcavities as a function of loaded cavity quality factor Q at λ0 ¼ 2.3 μm and the relevant normalized nonlinear mode volume ˜ ˜ ˜ ˜ ˜ ð3Þ ð2Þ V (VKerr ¼ VT and Vshg, assumed to be equal to VT , for χ and χ , respectively). The figure of merit (FOM)—the ratio of qubit coupling rate g to the composite decay and thermal dephasing rate 2Γeff ¼ ω0=Qeff —is largest for strong coupling (g=2Γ ≫ 1) and weak ˜ dephasing (Qeff ≈ Q). These competing characteristics yield an optimum quality factor Q ¼ Qopt for any V. Three-dimensional thermal diffusion in a homogeneous medium is assumed. basis (zero- and one-photon states) from higher-energy single-photon sources [105,107]. While silicon’s centro- states [57]. symmetric structure precludes an intrinsic χð2Þ nonlinearity, 2 Alternatively, bulk χð Þ nonlinearities can mediate cou- we assume the intrinsic χð3Þ nonlinearity can create an pling between doubly resonant first- and second-harmonic χð2Þ 3χð3Þ electric-field-induced ¼ Edc near the breakdown ω ˜ 1=2 qubit basis states. The coupling rate g ¼ 0Kϵ=Vshg then dc electric field Edc [108]. describes the frequency of Rabi oscillations between a An ideal qubit operates well within the strong coupling single photon in the second-harmonic mode and two regime with minimal dephasing. In the presence of TRN, photons at the fundamental frequency [93,106,107]. increasing Q=V improves the former at the cost of the latter, Assuming an equal amplitude decay rate Γ for both modes, leading to the observed mode-volume-dependent optimum ≈ ω Γ 2δω2 strong coupling again requires g=2Γ ≫ 1. Here, the appli- loaded quality factor Qopt 0 T= rms. Figure 5 also cable coupling constant and second-harmonic mode vol- illustrates a relative performance advantage for χð2Þ devices ume are defined as in silicon, as strong coupling can be achieved at lower sffiffiffiffiffiffiffiffiffi quality factors. For example, the peak FOMχð2Þ ∼ 10 is 3 πℏc χð2Þ orders of magnitude greater than FOMχð3Þ assuming Q ¼ Kϵ ¼ 3ϵ λ2 ; ð11Þ n 0 Q and V˜ ¼ V˜ ¼ 1. Although small V˜ —which as R R opt Kerr shg shg 2 ⃗ 2 3 2 ⃗ 2 3 2 illustrated in Eq. (12) involves maximizing the nonlinear ð n jE2ωj d r⃗Þð n jEωj d r⃗Þ V R 12 overlap function between two colocalized cavity modes— shg ¼ 3 ðiÞ ðjÞ ðkÞ 3 2 ð Þ j n E Eω Eω d r⃗j ˜ 2ω is generally more difficult to achieve than small VKerr [109], ˜ −1=2 ⃗ ⃗ FOMχð2Þ ∝ V also demonstrates more favorable scaling for the first- and second-harmonic modes Eω and E2ω, shg respectively, and the indices i, j, and k corresponding to at larger mode volumes. 2 field components associatedR with the dominant χð Þ tensor 3 element.R The integral ð…Þd r⃗is taken over all space, VI. BEYOND THE THERMAL NOISE LIMIT … 3⃗ while ð Þd r is restricted to the nonlinear material. This brief example manifests the practical limitations Extended derivations of Kχ, Kε, Vshg, and VKerr are imposed by TRN, which yield wavelength-scale ultrahigh- included in Appendix D. Q resonators [65], narrow-linewidth lasers [110], and high- The resulting figures of merit performance optical qubits thermodynamically forbidden. Breaking this fundamental “noise-volume trade-off” in Qeff Qeff FOMχð3Þ ¼ Kχ ; FOMχð2Þ ¼ Kϵ ð13Þ optical microcavities would lift our proposed Q=V limits V˜ ˜ 1=2 Kerr Vshg and could enable quantum-noise-limited readout of dif- fraction-limited optical modes, thereby unveiling new are plotted in Fig. 5 for Gaussian-shaped modes in silicon. possibilities in physics. Active cavity stabilization is one Similar figures of merit are applicable to high-fidelity candidate solution; however, the limited speed of control

041046-7 PANUSKI, ENGLUND, and HAMERLY PHYS. REV. X 10, 041046 (2020) loops and tuning mechanisms has restricted the noise (a) rejection bandwidth of previous demonstrations to a few hundred hertz [76]. Passive feedback noise rejection via photothermal backaction [61]—resonance-detuning- dependent laser heating that counteracts temperature fluctuations—has also been experimentally demonstrated (b) (d) Γ−1=2 [52], but the scales with T and therefore becomes ineffective for small cavities. Furthermore, neither feedback-based technique can suppress fluctuations in the high-Q=V regime, where the rate of resonant frequency fluctuations exceeds that of cavity leakage. To overcome these limitations, we propose and analyze coherent thermo-optic (TO) noise cancellation as an avenue toward the open goal of broadband thermal noise suppression. This technique stems from an analogous proposal for thermal noise reduction in mirror coatings (c) (e) [7,30,111–113]: when the uniform temperature of a thickness t coating is raised by δT, the increase in phase δϕ 2π δ λ 2π α δ λ accumulation ¼ t n= ¼ t TR T= from an increase in refractive index δn can be offset by the δ αL δ coating expansion t ¼ t TE T toward the incident beam αL provided the linear coefficient of thermal expansion TE α and TR have the same sign. In a microcavity, this relationship is reversed. Intuitively, we expect thermo- optic noise to be minimized when the normalized δω˜ δω ω thermoelastic (TE) frequency shift TR ¼ TE= ¼ −αL δ TE T of a freely expanding cavity equals that of FIG. 6. Athermal polymer (PMMA) microcavity design by δω˜ −α δ the thermorefractive (TR) effect, TR ¼ TR T=n, coherent thermo-optic noise cancellation. The Q-optimized base- 0 leading to the athermal condition line ladder-type nanobeam cavity (a) with fa;a ;wo;wi;wsg¼ f647nm;0.914a;3a;0.85w ;0g and thickness t ¼ 3a is formed by αL γ o TE − quadratically tapering the lattice constant a to a0 over the inner α ¼ ð14Þ “ ” η 0 4 TR n 12 periods with a constant rung duty cycle r ¼ wr=a ¼ . . This cavity is driven by anticorrelated volumetric and boundary for the energy confinement fraction γ ∈ ð0; 1 of the optical heat sources corresponding to TR and TE noise, respectively, in mode in the dielectric. Equation (14) can be satisfied by FEM-based fluctuation-dissipation simulations. The composite γ αL α ∝ ∇δ ⃗ 2 tuning provided TE and TR are of comparable magnitude TO frequency noise spectrum Sff ½ Tðr; fÞ is computed with opposite sign. Whereas we previously assumed from the resulting harmonic temperature profiles δTðr;⃗ fÞ (b). The αL α ≪ 1 TO noise level is bounded by the coherent (−) and incoherent (þ) j TE= TRj for common dielectrics, these two require- ments are remarkably well satisfied by a range of polymers, sum of TR and TE contributions (shaded region). The optimized a;a0;w ;w ;w ;t;η 697 ;0 874a;3a;0 85w ; αL α ≈ parameters f o i s rg¼f nm . . o where the Clausius-Mossotti relation dictates TE= TR 10 0 8 0 6 2 2 nm; . a; . g provide broadband TON cancellation (e) as −ðn þ 2Þðn − 1Þ [114]. In polymethyl-methacrylate visually depicted by reduced δTðr;⃗ fÞ at low frequencies (d). (PMMA, n ¼ 1.48), for example, γ ¼ 0.83 enables The color scale in (d) is equal to (b) to facilitate comparison. steady-state athermal operation. Our previously demon- strated high-Q=V polymer “ladder” cavity designs [115] i.e., the spatial average of temperature fluctuations schematically illustrated in Fig. 6(a) are therefore well suited δT r⃗ weighted by g r⃗, is calculated at frequency ω for TO noise cancellation. ð Þ TOð Þ i by driving the entropy S (conjugate to δT) with a But does steady-state athermal behavior imply complete δ ⃗ δ ⃗ harmonic, perturbative heat source QTOðr;tÞ¼T Sðr;tÞ¼ broadband TO noise cancellation? Unfortunately, the ⃗ ω answer is no, which we illustrate with the computational TQ0gTOðrÞcosð itÞ, where Q0 is a constant of proportion- form of the FDT used to simulate TRN in Sec. III [5,73].In ality. The resulting spectrum of fluctuations, this formalism, the total TO resonance shift 2kBT Sωω ω W ; Z ð iÞ¼πω2 2 diss ð16Þ i Q0 δω˜ ¼ ½g ðr⃗Þþg ðr⃗ÞδTðr⃗Þd3r;⃗ ð15Þ TO |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}TR TE at the equilibrium temperature T is then computed from the ⃗ gTOðrÞ time-averaged (h·i) power dissipation [73]

041046-8 FUNDAMENTAL THERMAL NOISE LIMITS FOR OPTICAL … PHYS. REV. X 10, 041046 (2020) Z κ confinement condition. However, the beam ends are W ¼ h½∇δTðr;⃗ tÞ2id3r⃗ ð17Þ diss T assumed to be fixed for support, which restricts lateral thermal expansion and leads to a dominant TR effect due to irreversible heat flow from the resulting harmonic as shown by the individual TR and TE noise spectra in temperature profile δTðr;⃗ tÞ in a material with thermal Fig. 6(c). If both heat sources are simultaneously simulated, conductivity κ. We obtain the functional form of the TR and the resulting TO noise resembles the coherent combination TE contributions to the composite TO weight function of the individual contributions; i.e., the noise amplitudes ⃗ ⃗ ⃗ gTOðrÞ¼gTRðrÞþgTEðrÞ by placing the associated first- subtract as desired. However, as depicted by the associated order perturbation theory results [58,116] into the form of harmonic temperature profiles in Fig. 6(b), the achievable Eq. (15): noise reduction is modest owing to the dominant amplitude Z of the TR source near the center, high-intensity region of δω˜ ≈ −α ⃗ 2 ⃗⃗ 2δ ⃗ 3⃗ the cavity mode. TR TRðrÞn jEðrÞj TðrÞd r; ð18Þ Ω |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} According to criterion (ii), it is evident that the TE source ⃗ gTRðrÞ must be amplified to improve the noise cancellation. Z   Freeing the cavity ends achieves this goal by enabling L ⃗ 2 α hðr⃗Þ jD⊥ðr⃗Þj δω˜ ≈ − TE E⃗ r⃗ 2 δT r⃗d2r:⃗ isotropic expansion but violates criterion (i): expansion TE 4 2 − 1 −1 j kð Þj þ 2 ð Þ ∂Ω |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}ðn Þ n along the beam length amplifies boundary displacements g ðr⃗Þ and the associated TE heat source away from the TR source TE maximum at the cavity center, thereby minimizing the ð19Þ spatial overlap between the two heat sources. As a result, TO noise suppression is again limited despite the nearly ⃗ ⃗ Here, Ek (D⊥) is the electric (electric displacement) field equal TR and TE noise contributions. parallel (perpendicular) to the cavity boundary ∂Ω, hðr⃗Þ is To simultaneously colocate the heat sources and magnify the dielectric thickness between neighboringR boundaries, the TE effect, we instead propose placing a thin air slot and we assume a normalized mode n2jE⃗j2d3r⃗¼ 1.We through the ladder rungs at the center of the cavity as shown also note that Eq. (19) is valid only for sufficiently low in Fig. 6(a). In the original design, expansion along the ⃗ 2 ⃗ ladder rung length displaces the side rail boundary, where frequencies ω ≪ DT= max fhðrÞg such that δTðrÞ is approximately uniform throughout the dielectric between the cavity field is weak. The modified design’s narrow slot neighboring boundaries. We therefore see that TRN is moves this displacement boundary to the center of the beam associated with an energy-density-dependent volumetric (the “half-rungs” expand to fill the gap), where the cavity δ ⃗ ⃗ field and TR heat source are maximized. We then thin the heat source QTRðrÞ¼TQ0gTRðrÞ, whereas TE noise is δ ⃗ beam to satisfy criterion (ii) within 1% and redesign the associated with a surface heat source QTEðrÞ¼ TQ0g ðr⃗Þ that depends on the field intensity and mechani- cavity defect region to maximize Q. The final optimized TE 4 cal displacement amplitude of the dielectric boundary. design supports a high-Q=V resonance (Q ¼ 6 × 10 , Given this formulation, we immediately see that the TO V˜ ¼ 0.75) that suppresses fundamental thermal noise by noise cannot be completely canceled at all frequencies due over an order of magnitude at low frequencies. The to the spatial mismatch between the heat sources, which temperature profiles in Fig. 6(e) graphically evidence this invariably leads to irreversible heat flow and the associated cancellation and suggest again that the achievable suppres- noise. However, our analysis reveals two intuitive design sion is limited by the imperfect coherence of the two principles to maximize the coherence between anticorre- spatially resolved heat sources. lated TR and TE sources, thereby minimizing the total TO This intuition-driven optimization is a first proof-of- noise: (i) The cavity boundaries should be placed near high- concept demonstration of broadband, coherent cancellation intensity regions of the cavity mode to maximize the spatial of fundamental thermodynamic fluctuations in a micro- overlap between TR and TE sources; (ii) we intuitively cavity. While the achievable noise reduction in our trial expect that the amplitude of the harmonic temperature structure is relatively limited, the design is based on simple profile, and thereby the TOR noise, should beR minimized if principles that can be readily extended to other designs, δ ⃗ 3 ⃗ δ ⃗ 2 ⃗ the total added heat QTO ¼ Ω QTRðrÞd rþ ∂Ω QTEðrÞd r materials, and optimization techniques. For example, the is zero. This expectation is true away from the sources at interdisciplinary thermo-optical design of high-Q=V,low- sufficiently low frequencies, where thermal diffusion effec- noise resonators requires structural modifications that are tively “masks” the exact source locations, and is actually well suited to recent developments in topological optimi- equivalent to the athermal condition of Eq. (14). zation of subwavelength integrated photonics [91,117,118]. Guided by these two simple principles, we optimized the The coherent thermo-optic cancellation scheme can also coherent TO noise cancellation of the PMMA ladder cavity be applied other materials beyond polymers, such as 5 ˜ in Fig. 6. Our baseline design yields Q ¼ 3 × 10 , V ¼ 2.6, CMOS-compatible titanium dioxide (TiO2) [119] or per- and γ ¼ 0.75—approximately 10% away from the athermal ovskites [120] with negative thermo-optic coefficients. Our

041046-9 PANUSKI, ENGLUND, and HAMERLY PHYS. REV. X 10, 041046 (2020) preliminary results thus motivate an exciting new research In optical microcavities, V approaches diffraction-limited avenue into novel noise abatement schemes in pursuit of volumes leading to temperature fluctuations that signifi- ultrahigh-Q=V cavities beyond the thermal noise limit. cantly impact the resonance stability in materials with a temperature-dependent refractive index. VII. SUMMARY AND OUTLOOK Here, we derive the associated TRN spectrum in an optical microcavity under the single-mode approximation Ultimately, understanding the fundamental stability and described in the main text. Using this approximation, the coherence limits of optical microcavities relies on the intracavity field statistics are derived. In the typical pertur- proper characterization of thermo-optic noise. Toward this bative limit where the rms frequency fluctuation is much end, we have presented a general theory for thermorefrac- δω ≪ 2Γ smaller than the loaded cavity linewidth ( rms l), we tive noise in optical microcavities, discussed the resulting use perturbation theory to solve for the evolution of the practical limitations on future integrated photonic compo- cavity field aðtÞ and the associated noise spectrum SaaðtÞ. nents, and highlighted design choices that optimize device We also provide general solutions for the first and second performance in its presence. We experimentally verified our statistical moments of aðtÞ, which are used in the main text to model by measuring the dominant effect of temperature derive “effective” quality factors in the presence of thermal fluctuations in high-Q=V silicon PhC cavities, which noise. The solution for S ðtÞ in the limiting case of high-Q demonstrated the viability of optical microcavities as aa cavities—where the thermal decay rate Γ ≫ Γ and the high-spatial-resolution temperature probes operating at T l frequency noise can be assumed to be white—is also the fundamental thermal noise limit. Our results show that provided. Finally, we compare the single-mode noise nonperturbative TRN ultimately limits the achievable spectrum to that derived from a formal solution to heat quality factor in small-mode-volume cavities and that diffusion in an infinite two-dimensional slab, which we find experimental devices have neared this fundamental bound. to most accurately model the specific geometry of the Violating the observed trade-off between mode volume and photonic crystal microcavities in our experiments. thermo-optic noise stands as an exciting avenue for future A few notes on convention: (i) We derive two-sided investigation. We hope that our proposal and analysis of angular frequency noise spectra SωωðωÞ but plot one-sided one possible solution—coherent thermo-optic noise can- frequency spectra S ðfÞ¼2Sωωð2πfÞ=2π ¼ Sωωð2πfÞ=π cellation—toward low-noise, high-Q=V integrated optical ff for experimental measurements to conform with the devices further motivates this field of research. Ultimately, common conventions of the gravitational wave community; these advances will be necessary to achieve the perfor- (ii) temporal coupled mode theory decay rates Γ are mance required for future developments in optical quantum i amplitude decay rates; the associated quality factors are information processing, cavity optomechanics, precision therefore defined as Q ω0=2Γ . optical sensing, and beyond. i ¼ i

ACKNOWLEDGMENTS 1. Statistics of microcavity TRN The authors thank M. Dykman, G. Huang, and T. To first order, the change in resonant frequency under a δϵ ⃗ Kippenberg for useful discussions. C. P. was supported permittivity perturbation ðr; tÞ can be expressed as by the Hertz Foundation Elizabeth and Stephen Fantone R Family Fellowship and the MIT Jacobs Presidential δω t 1 δϵ r;⃗ t E⃗r⃗ 2d3r⃗ ð Þ − δϵ R ð Þj ð Þj Fellowship. R. H. was supported by an IC postdoctoral ¼ ⃗ 2 3 ω0 2 ϵjEðr⃗Þj d r⃗ fellowship at MIT, administered by Oak Ridge Institute for R Science and Education through U.S. DoE/ODNI. 1 nδnðr;⃗ tÞjE⃗ðr⃗Þj2d3r⃗ ≈− δn ; Experiments were supported in part by AFOSR Grant ⃗ 2 ðA2Þ Veff maxfϵjEðr⃗Þj g No. FA9550-16-1-0391, supervised by G. Pomrenke.

where we make the approximation δϵ ≈ 2n δn and intro- APPENDIX A: TRN THEORY duce the standard mode volume While the development of ultrahigh-performance optical resonators has only recently warranted its study within R ϵðr⃗ÞjE⃗ðr⃗Þj2d3r⃗ optical systems, stochastic temperature fluctuations are a V : A3 eff ¼ ⃗ 2 ð Þ fundamental concept in thermodynamics [31]. Assuming maxfϵðr⃗ÞjEðr⃗Þj g Boltzmann statistics within a finite volume V with specific heat capacity cV at temperature T, we find The change in refractive index δnðr;⃗ tÞ is directly propor- δ ⃗ 2 tional to temperature change Tðr; tÞ, with the thermo-optic 2 kBT α hδT i¼ : ðA1Þ coefficient TR ¼ dn=dT serving as the constant of pro- cVV portionality. We therefore find

041046-10 FUNDAMENTAL THERMAL NOISE LIMITS FOR OPTICAL … PHYS. REV. X 10, 041046 (2020) R Z Z δω 1 α δ ⃗ ⃗⃗ 2 3⃗ t tþτ ðtÞ δ TR Tðr; tÞnjEðrÞj d r δ ¯ δ ¯ τ 0 00 ¯ 0 ¯ 00 ¼ − n : ðA4Þ h TðtÞ Tðt þ Þi ¼ dt dt hFTðt ÞFTðt Þi ω ⃗ 2 −∞ −∞ 0 Veff maxfϵjEðr⃗Þj g −Γ − 0 −Γ τ− 00 × e T ðt t Þe T ðtþ t Þ: ðA12Þ Alternatively, Eq. (A2) can be evaluated for a uniform, ¯ ¯ mode-averaged temperature change δTðtÞ assuming com- The result requires the autocorrelation of FTðtÞ, which plete confinement of the mode within a homogeneous is readily evaluated using the mode-averaged form of medium. This approach yields Eq. (A10): R δω 1 2 3 ⃗ 2 2 ðtÞ ¯ 2D k T d r⃗f∇½ϵðr⃗ÞjE⃗ðr⃗Þj g ¼ − α δTðtÞ: ðA5Þ ¯ ¯ τ T B 0 δ τ ω TR hFTðtÞF ðt þ Þi ¼ 2 ð Þ 0 n T c V ϵ ⃗⃗ 2 2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}V eff ½maxf jEðrÞj g R ¯ ¯ ð0Þ Comparing Eqs. (A4) and (A5), we find FT FT R ¯ ¯ 0 δ τ : R ¼ FT FT ð Þ ð Þ ðA13Þ 1 δTðr;⃗ tÞϵðr⃗ÞjE⃗ðr⃗Þj2d3r⃗ δT¯ t : A6 ð Þ¼ ⃗ 2 ð Þ Veff maxfϵjEðr⃗Þj g Inserting this result into Eq. (A12) along with the change of variables t00 → t0 þ τ0 yields Equation (A5) can now be solved using the mode-averaged R ¯ ¯ 0 temperature change, whose evolution is derived from the ¯ ¯ FT FT ð Þ −Γ τ hδTðtÞδTðt þ τÞi ≈ e T j j: ðA14Þ heat equation (given a thermal diffusivity DT) 2ΓT

∂δTðr;⃗ tÞ Correspondence with Eq. (A1) therefore requires − D ∇2δT ¼ F ðr;⃗ tÞðA7Þ ∂t T T R 0 2 2 F¯ F¯ ð Þ kBT0 2kBT0ΓT T T ¼ ⇒ R ¯ ¯ ð0Þ¼ : ðA15Þ driven by a thermal Langevin source F ðr;⃗ tÞ with the FT FT T 2ΓT cVV cVV statistics [121] Comparing Eq. (A15) to Eq. (A13) then lends a calculable ⃗ ⃗ hFTðr1;t1ÞFTðr2;t2Þi form of the decay rate ΓT: 2 2 DTkBT0 ⃗ ⃗ R δ t1 − t2 ∇⃗ · ∇⃗ δ r⃗1 − r⃗2 A8 3 ⃗ ⃗ 2 2 ¼ ð Þ r1 r2 ½ ð Þ ð Þ d r⃗∇ ϵ r⃗ E r⃗ cV f ½ ð Þj ð Þj g ΓT ¼ DT R : ðA16Þ d3r⃗ϵðr⃗Þ2jE⃗ðr⃗Þj4 that satisfy the fluctuation-dissipation theorem. Averaging over the mode, we find the approximation Combining Eqs. (A14)–(A16) with the first-order per- ¯ turbation theory relationship δω0ðtÞ ≈−ðω0=nÞα δTðtÞ ¯ TR d½δTðtÞ lends the desired autocorrelation of the cavity resonance þ Γ δT¯ ðtÞ¼F¯ ðtÞ; ðA9Þ dt T T frequency

2 −Γ τ where Γ is introduced as a phenomenological thermal δω δω τ ≈ δω T j j T h ðtÞ ðt þ Þi rmse ; ðA17Þ decay rate whose form is chosen later to ensure the form of δ ¯ 2 h T i matches the canonical result from statistical mechan- for the rms resonant frequency fluctuation ics [Eq. (A1)]. In analog with δT¯ ðtÞ, the mode-averaged sffiffiffiffiffiffiffiffiffiffiffi thermal force F¯ ðtÞ is T ω α 2 δω 0 TR kBT0 R rms ¼ : ðA18Þ 2 3 n c V 1 F ðr;⃗ tÞϵðr⃗ÞjE⃗ðr⃗Þj d r⃗ V T F¯ t T : A10 Tð Þ¼ ⃗ 2 ð Þ Veff maxfϵjEðr⃗Þj g

The steady-state solution of Eq. (A9), 2. Derivation of the thermal mode volume Z The same correspondence to Eq. (A1) can be used to t ¯ ¯ −Γ − 0 solve for the thermal mode volume V [45]. For complete- δTðtÞ¼ Fðt0Þe T ðt t Þdt0; ðA11Þ T −∞ ness, we recapitulate this derivation. In an infinite homo- geneous medium, we can solve Eq. (A7) using Fourier can then be used to find the corresponding statistics of the modes instead of introducing the phenomenological param- temperature fluctuation at equilibrium (i.e., long t): eter ΓT, yielding

041046-11 PANUSKI, ENGLUND, and HAMERLY PHYS. REV. X 10, 041046 (2020)   ω ⃗ ω 2 2 2Γ ⃗ Fð ; kÞ ω 0 α kBT0 T δTðω; kÞ¼ : ðA19Þ Sωωð Þ¼ TR 2 2 : ðA25Þ ⃗2 n cVVT Γ þ ω iω þ DTjkj T As noted in the main text, this approximate spectrum can be Taking the temperature mode average in Eq. (A6) and evaluated for any optical microcavity (photonic , inverse Fourier transforming yields microtoroids, microbottles, ring resonators, micropillars, Z Z microdisks, and so on) independent of its exact confining ⃗⃗ 2 1 3 ϵjEðrÞj − ω geometry. This feature allows us to derive general δT¯ ðtÞ¼ d r⃗ dωe i t 2π 4 ⃗ 2 ð Þ Veff maxfϵjEðr⃗Þj g noise limits as illustrated in the main text and derived in Z ⃗ Appendix A4. If a particular experimental system is of ⃗ Fðω; kÞ × d3ke⃗ ik·r⃗ : A20 interest, we can verify the accuracy of this approximation ⃗2 ð Þ iω þ DTjkj by solving the stochastic heat equation [Eq. (A7)] for that particular cavity geometry. Here, since we measure TRN in Using the frequency-space autocorrelation of the Langevin high-Q=Veff 2D slab photonic crystal cavities (see the main driving force [compare Eq. (A13)] [122], text and Appendix B), we demonstrate this evaluation for a Gaussian mode confined within an infinite two- ⃗ ⃗ dimensional slab. The heat equation in this case lends F ω1; k1 F ω2; k2 h ð Þ ð Þi logarithmically—as opposed to exponentially—decaying 2 2 4 kBT0DT ⃗2 ⃗ ⃗ temperature fluctuations in time. ¼ð2πÞ jkj δðk1 − k2Þδðω1 − ω2Þ; ðA21Þ c For a slab of thickness w lying atopP the xy plane, the V δ ⃗ ⃗ ϕ local temperature change Tðr; tÞ¼ n Tnðrk;tÞ nðzÞ can we can then solve for the autocorrelation of δT¯ : be expanded in terms of the out-of-plane eigenfunctions ϕnðzÞ¼cosðnπz=wÞ assuming insulating boundary con- ⃗ 2 −2 2 ditions on the top and bottom of the slab. The stochastic ½maxfϵjEðr⃗Þj g k T0 hδT¯ ðtÞδT¯ ðtþτÞi¼ B heat equation then simplifies to the form ð2πÞ3V2 c Z eff Z V 2 ∂ ⃗ 3 − ⃗2τ 3 2 ⃗⃗ Tnðrk;tÞ 2 2 ⃗ DT jkj ⃗ϵ ⃗ ⃗ ⃗ ik·r ∇ − π ⃗ × d ke d r ðrÞjEðrÞj e : ∂ ¼ DT½ ðn =wÞ Tnðrk;tÞ t Z 1 ðA22Þ þ ϕðzÞF ðr;⃗ tÞdz: ðA26Þ w n T τ 0 Equation (A22) must equal Eq. (A1) for ¼ , which If we assume a two-dimensional Gaussian mode profile reveals the final solution for the thermal mode volume V : T  1 − ⃗ 2 2σ2 jrkj = 0 ϵ ⃗ ⃗⃗ 2 2πσ2 e

041046-12 FUNDAMENTAL THERMAL NOISE LIMITS FOR OPTICAL … PHYS. REV. X 10, 041046 (2020)

Comparison of single-mode and multimode TRN spectra Normalized noise spectra Ratio

Multimode

˜ FIG. 7. Normalized noise spectra SωωðωÞ¼Iðω=ΓT Þ for single-mode [Eq. (A31)] and multimode [Eq. (A30)] TRN in an infinite slab of finite thickness.

Comparing Eqs. (A28) and (A29) to their three- normalized spectrum for comparison along with the ratio dimensional analogs [Eqs. (A7) and (A8)], we see that Imm=Ism. These results substantiate the claims in the main projecting onto the n ¼ 0 subspace reduces the finite- text: the single-mode approximation undershoots at low thickness slab to an infinite two-dimensional problem where frequency ω ≪ ΓT, slightly overshoots at intermediate −1=2 ω ∼ Γ FT is scaled by w . We can then apply the techniques of frequencies T, and converges to the multimode spec- Appendix A2(expansion in Fourier normal modes) to solve trum at high frequencies ω ≫ ΓT. We further note that the for the spectrum of temperature (and, therefore, resonant error of the multimode spectrum increases at low frequen- frequency) fluctuations. Without inverse Fourier transform- cies for any finite volume system: in our experiment, for ing frequency, the autocorrelation of Eq. (A20) gives example, the multimode estimate does not account for low- frequency heat transfer through the underlying oxide around mm Sωω ðωÞ the released membrane. Thus, in the range of frequencies   Z 2 2 2 of interest (i.e., near the thermal cutoff frequency ω0 k T DTk 2 α B 0 k ϵ 2 2 2 2 ¼ TR 2 2 2 j E ðkkÞj d kk ΓT ¼ DT=σ ), single-mode thermal decay is an appropriate n wcV ðDTk Þ þ ω   Z k simplifying assumption that allows the thermal noise spec- 2 2 ∞ ω0 k T x trum to be well approximated irrespective of the cavity’s α B 0 e−xdx ¼ TR 2π 2 ωσ2 2 exact geometry. n wcvDT |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}0 x þð =DTÞ 2 I ðωσ =DT Þ mm 4. Derivation of driven cavity dynamics ðA30Þ To determine the practical impact of thermorefractive σ 2 → with the change of variables ðkk Þ x.Notethatwetreat noise on microcavity dynamics, we now consider the case the effect of patterned holes in our experimental structures of a cavity driven by a monochromatic laser with frequency ω through a reduced thermal conductivity, and, therefore, L. Intuitively, we would expect that the large (relative to thermal diffusivity, as a function of the slab porosity (see the cavity linewidth), fast (relative to the cavity decay time) Appendix B for further detail). We can compare this result stochastic deviations of the resonance frequency in the with the single-mode approximation, which [by evaluating high-Q=Veff limit would restrict the maximum intensity in Eqs. (A16) and (A23) for the Gaussian mode profile in the cavity, as a narrow linewidth laser would no longer 2 always be on resonance with the fluctuating cavity reso- Eq. (A27)] gives the thermal mode volume VT ¼ 4πwσ , 2 nance. A mode-volume-dependent maximum “effective” decay rate ΓT ¼ DT=σ , and noise spectrum quality factor describing the stored energy should result. A   ω 2 2 1 similar effective quality factor could be derived if the sm 0 kBT0 SωωðωÞ¼ α : ðA31Þ coherence of the intracavity field—rather than the stored n TR 2πwc D 1 þðωσ2=D Þ2 v T |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}T energy alone—is also of interest. ωσ2 Ismð =DT Þ To prove these suppositions, we solve the driven tem- R poral coupled mode theory relation ∞ As expected, the integral −∞ Sωωdω=2π of either spectra 2 2 2 2 2 pffiffiffiffiffiffiffiffi yields hδω i¼ðω0α =nÞ hδT i¼ðω0α =nÞ k T0=c V daðtÞ TR TR B V T ¼½iω0ðtÞ − Γ aðtÞþ 2Γ s ðtÞ; ðA32Þ in correspondence with Eq. (A1). Figure 7 plots each dt l c in

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2 2 for the cavity field a t , where ω0 t ω0 δω t is the 2Γ ˜ 2Γ δω ð Þ ð Þ¼ þ ð Þ ω cjsinj T rms Saað Þ¼ 2 2 2 2 2 2 : ðA40Þ instantaneous resonant frequency, Γl ¼ ω0=2Ql is the Γ Δ Γ ω Γ ω Δ l þ ð T þ Þð l þð þ Þ Þ amplitude decay rate of a corresponding to a loaded quality Γ factor Ql, and c is the amplitude coupling rate of the drive The intracavity noise spectrum can therefore be approxi- ω ˜ i Lt field sinðtÞ¼sine þ c:c: detuned from resonance by mated as the product of two Lorentzians with spectral Δ ω − ω δω ¼ L 0. In the presence of TRN, ðtÞ is non- widths 2ΓT and 2Γl. When ΓT ≪ Γl, which often coincides δω ≪ Γ Markovian, zero-mean Gaussian noise with the autocorre- with the perturbative limit rms l for large mode lation given by Eq. (A17). Solving with an integrating −1=2 −2=3 volumes (δω ∝ V and ΓT ∝ V for a three-dimen- factor and introducing the slowly varying cavity amplitude rms T T − ω sional Gaussian mode), the resonant frequency fluctuations ˜ i Lt aðtÞ¼aðtÞe in a reference frame corotating with the are small and occur over timescales much longer than that ω drive frequency L, we find of intracavity photon decay. Thus, TRN leads to a weak Z R inhomogeneous broadening of the resonant mode that can pffiffiffiffiffiffiffiffi t t − Δ Γ − 0 i dt00δωðt00Þ ˜ 2Γ ˜ 0 ði þ lÞðt t Þ t0 aðtÞ¼ csin dt e e : ðA33Þ often be neglected for common applications of low-Q=Veff −∞ optical cavities. Gravitational wave interferometry [6,122] and ultrastable optical frequency references [47,123] are Since the steady-state solution is desired, we assume that two notable exceptions that have led to significant interest the integration starts at t −∞ such that the system has no ¼ in perturbative TRN. “memory” of the initial conditions. Using Eq. (A33),we can compute ha˜ðtÞi and hja˜ðtÞj2i, the mean cavity field ˜ amplitude and stored energy, respectively. In certain limit- b. General derivation for haðtÞi ing cases, the noise spectrum SaaðωÞ of the intracavity field Our work focuses on the transition to nonperturbative can also be derived. TRN in high-Q=Veff microcavities, where we are interested in general solutions for the statistical moments of Eq. (A33) ˜ a. Cavity spectrum in the perturbative limit in the presence of TRN. Specifically, haðtÞi provides insight into thermal-noise-induced dephasing while ha˜ 2ðtÞi lends a One of these limiting cases is the perturbative regime bound on the maximum allowable energy storage. commonly studied in the literature for low-Q=Veff cavities, The expected intracavity field amplitude wherein δω ≪ Γ . In this case, a˜ðtÞ and δωðtÞ— rms l Z R described by [cf. Eq. (A9)] pffiffiffiffiffiffiffiffi t t − Δ Γ − 0 dt00iδωðt00Þ ˜ 2Γ ˜ 0 ði þ lÞðt t Þ t0 haðtÞi ¼ csin dt e he i dδωðtÞ pffiffiffiffiffiffiffiffi −∞ ¼ −Γ δωðtÞþδω 2Γ WðtÞðA34Þ dt T rms T ðA41Þ

0 0 for a WðtÞ with hWðtÞffiffiffiffiffiffiffiffiWðt Þi ¼ δðt − t Þ— follows directly from Eq. (A33), where the average on the δω p2Γ can be expanded in orders of rms T. The zeroth- and right-hand side has a similar form to the characteristic first-order evolution equations (with subscripts 0 and 1, functional [60] respectively) are R R R ei kðtÞfðtÞdtP½fðtÞDfðtÞ ˜ pffiffiffiffiffiffiffiffi Φ i kðtÞfðtÞdt R da0ðtÞ ½kðtÞ ¼ he i¼ D ; ¼fi½δω0ðtÞ − Δ − Γ ga˜ 0ðtÞþ 2Γ s˜ ; ðA35Þ P½fðtÞ fðtÞ dt l c in ðA42Þ da˜ 1ðtÞ R ¼fi½δω0ðtÞ − Δ− Γ ga1ðtÞþiδω1ðtÞa˜ 0ðtÞ; ðA36Þ i kðtÞfðtÞdt dt l a normalized average of e along the paths fðtÞ with respective probabilities P½fðtÞ. For the special case of dδω0ðtÞ Gaussian noise, the moment-generating properties of the ¼ −Γ δω0ðtÞ; ðA37Þ dt T characteristic functional allow Eq. (A42) to be simplified to pffiffiffiffiffiffiffiffi R R R dδω1ðtÞ i kðtÞMðtÞdt −1=2 dt dt0kðtÞkðt0ÞhfðtÞfðt0Þi ¼ −Γ δω1ðtÞþδω 2Γ WðtÞ: ðA38Þ Φ½kðtÞ ¼ e e ; ðA43Þ dt T rms T Solving in the frequency domain yields which is characterized by two parameters only: the mean path MðtÞ and the autocorrelation of the noise fðtÞ, ffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffi p2Γ ˜ δω p2Γ ω hfðtÞfðt þ t0Þi. Comparing Eq. (A41) to Eq. (A42), we find ˜ ω csin i rms TWð Þ a1ð Þ¼Γ Δ Γ − ω Γ ω Δ ; ðA39Þ  l þ i ð T i Þ½ l þ ið þ Þ 1 t0

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00 and fðtÞ¼δωðt Þ. Since hω0ðtÞi ¼ 0, as expected from noiseless temporal coupled mode theory.  Z Z  Assuming critical coupling (Γc ¼ Γl=2) and resonant 1 t t 2 −Γ 00− 0 excitation (Δ ¼ 0), we find the “effective” quality factor Φ − 0 00δω T jt2 t2j ¼ exp dt2 dt2 rmse 2 t0 t0     ω ˜ 2 Γ 2 δω2 0jhaðtÞij l 2x −2s 2 rms −Γ − 0 Q ¼ ¼ Q e x γ ðs; xÞðA51Þ ¼ exp ½1 − Γ ðt − t0Þ − e T ðt t Þ : ðA44Þ eff 2 ˜ 2 l Γ l Γ2 T jsinj T T ω ˜ 2 by analogy to the noiseless result where Q¼ 0jhaðtÞijT¼0= The above form, combined with the change of variable 2 2 −Γ − 0 2 ˜ τ˜ ¼ðδω =Γ Þ e T ðt t Þ, simplifies to jsinj . rms T This result is used in the main text to describe dephasing ffiffiffiffiffiffiffiffi   p 2 − Δ−Γ − δω2 Γ Γ 2Γ ˜ δω f½ i l ð rms= T Þ= T g in the qubit limit of cavity nonlinear optics. For a given csin δω Γ 2 rms ha˜ðtÞi ¼ eð rms= T Þ opt ≈ Γ Γ mode volume, the optimum loaded quality factor Ql T T ω Γ 2δω2 δω ≪ Γ Z 0 T= rms (assuming rms T, which is valid for the δω =Γ 2 ð rms T Þ Δ Γ δω2 Γ Γ −1 −τ˜ × τ˜f½i þ lþð rms T Þ= T g e dτ˜; range of mode volumes plotted in Fig. 5 of the main text) 0 maximizes the resonant cavity amplitude: lower quality ðA45Þ factors incur excess loss, whereas higher quality factors allow the qubit to “explore” a larger region of the phase which is in the form of the lower incomplete Gamma space, thereby reducing the integrated cavity amplitude. function Intuitively, the resulting maximum amplitude jha˜ðtÞij Z increases with increasing mode volume due to the reduced x s−1 −τ˜ magnitude of temperature fluctuations. γlðs; xÞ¼ τ˜ e dτ˜: ðA46Þ 0 c. General derivation for ha˜2ðtÞi The final closed-form solution is, therefore, Solving for ha˜ðtÞ2i generally follows the same procedure pffiffiffiffiffiffiffiffi 2Γ s˜ and reveals a limit on the allowable intracavity optical ˜ c in x −sγ haðtÞi ¼ e x lðs; xÞ; ðA47Þ energy in the presence of TRN. Starting from Eq. (A33), ΓT   the amplitude correlation takes the form δω 2 rms Z ≡ t x ; ðA48Þ 2 − Δ Γ − 0 Γ ˜ ˜ 0 2Γ ˜ 0 ði þ lÞðt t Þ T haðtÞa ð Þi ¼ cjsinj dt e Z −∞ R R Γ iΔ 0 t 0 l þ 00 −ðiΔ−Γ Þt00 iδωðt2Þdt2− iδωðt2Þdt2 s ≡ þ x: ðA49Þ × dt e l he t0 t00 i: ΓT −∞ A52 To confirm this solution, we can evaluate the limiting case ð Þ δω → 0 → 0 of rms (x ), corresponding to a noiseless thermal reservoir when T → 0. Using the series expansion of Following the method of Ref. [59], the average can be expressed in the form of Eq. (A42): γlðs; xÞ in terms of s, x, and the complete Gamma function γ R R R fðzÞ, we find t 0 ∞ iδωðt2Þdt2− iδωðt2Þdt2 i kðt2Þδωðt2Þdt2 e t0 t00 e −∞ ; A53 pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi h i¼h i ð Þ 2Γ ˜ γ s 2Γ ˜ 1 ˜ csin fð Þ csin haðtÞiT¼0 ¼ ¼ by appropriately defining k t . As illustrated in Fig. 8, the ΓT γfðs þ 1Þ ΓT s 0 ð Þ x¼ 0 00 ffiffiffiffiffiffiffiffi dependence of k t2 upon t , t , and t differs in three sectors p2Γ ˜ ð Þ csin of the region of integration. Simple diagrams in each of the ¼ Γ Δ ðA50Þ l þ i three scenarios can be used to find a closed form for kðtÞ:

8 00 0 0 0 00 0 00 > sgnðt − t Þ;t< 0 and minðt ;t Þ ≤ t2 ≤ maxðt ;t Þ > < 0 0 0 1; ðt < 0 and 0 ≤ t2 ≤ tÞ orðt > 0 and t ≤ t2 ≤ tÞ; k t2 A54 ð Þ¼ 0 00 ð Þ > −1;t> 0 and t ≤ t2 ≤ 0; :> 0; else:

This definition allows us to rewrite the autocorrelation as

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R ðtÞ¼ha˜ðtÞa˜ ð0Þi aa Z Z  Z Z  t 0 1 ∞ ∞ 2 Δ−Γ − 0 Δ Γ 00 2 −Γ 0 − 00 2Γ ˜ 0 ði lÞðt t Þ 00 ði þ lÞt − 0 00 0 00 δω T jt2 t2j ¼ cjsinj dt e dt e exp dt2 dt2kðt2Þkðt2Þ rmse : ðA55Þ −∞ −∞ 2 −∞ −∞

Γ t0 While a general solution to the full autocorrelation in With a second substitution τ˜2 ¼ xe T , we find Eq. (A55) appears intractable, we can find hja˜j2i¼ ˜ 0 2 0  hjað Þj i by evaluating Eq. (A55) at t ¼ (thereby elimi- ex γ s0;x ˜ 0 2 2Γ ˜ 2 −s0 lð Þ nating integration region 3 in Fig. 8), which yields hjað Þj i1 ¼ cjsinj Γ x 2Γ Z Z T l 0 0 Z  2 2 Δ Γ 0 − Δ Γ 00 −2Γ =Γ ˜ 2Γ ˜ 0 ði þ lÞt 00 ð i þ lÞt x l T x 2Γ Γ −1 hjaj i¼ cjsinj dt e dt e − τ˜ð l= T Þ γ 0 τ˜ τ˜ −∞ −∞ 2 lðs ; 2Þd 2    ΓT 0 2    δω 00 0 rms 00 0 −Γ jt −t j x 0 −2Γl=ΓT 1 − Γ − − T e 0 γ ðs ;xÞ x Γ × exp Γ ð Tjt t j e Þ : 2Γ ˜ 2 −s l − T T ¼ cjsinj Γ x 2Γ Γ 2Γ  T l T  l  ðA56Þ τ˜ 2Γ Γ 2Γ 2¼x τ˜ l= T γ 0 τ˜ γ 0 l τ˜ × 2 lðs ; 2Þþ u s þ Γ ; 2 ; Note that the integral in region 1 or region 2 is nearly the T τ˜2¼0 same—exchanging t0 and t00 in either region returns the ðA59Þ integral for the other region but conjugates iΔ. Therefore, we focus on evaluating Eq. (A56) in region 1 ðt00 >t0Þ and γ 0 then generalize this result to the other region by taking where uðs ;xÞ is the upper incomplete Gamma function the complex conjugate. In region 1, the substitution τ˜ ¼ defined by ðδω =Γ Þ2 exp ½−Γ ðt00 − t0Þ yields rms T T Z Z Z ∞ 0 x x γ 0 τs0−1 −τ τ 2 2 2Γ 0 e − 0 0−1 −τ˜ uðs ;xÞ¼ e d : ðA60Þ ˜ 0 2Γ ˜ 0 lt s τ˜s τ˜ hjað Þj i1 ¼ cjsinj dt e x e d x Γ Γ 0 −∞ T xe T t ex 2Γ ˜ 2 −s0 ¼ cjsinj Γ x Evaluating the final terms, the result simplifies nicely to Z T 0 0 0 0 2Γlt 0 0 ΓT t     × dt e ½γlðs ;xÞ − γlðs ;xe Þ; ðA57Þ 2 js˜ j Γ 0 2Γ −∞ 2 in c x −½s þð2Γl=ΓT Þ 0 l hja˜ð0Þj i1 ¼ e x γl s þ ;x : ΓT Γl ΓT where x¼ðδω =Γ Þ2 and s0 ¼ðδω2 =Γ − Γ þ iΔÞ=Γ . rms T rms T l T ðA61Þ The first term can be directly evaluated, while the second can be simplified with integration by parts using the relationship To find the complete result, we simply add the second term 0 in Eq. (A56) to find ∂γ ðs ;xÞ 0 l ¼ xs −1e−x: ðA58Þ ∂x   ˜ 2 Γ 2 jsinj c x −s −s hja˜j i¼ e ½x γlðs; xÞþx γlðs ;xÞ; ðA62Þ ΓT Γl   δω 2 x ≡ rms ; ðA63Þ ΓT

Γ þ iΔ s ≡ l þ x: ðA64Þ ΓT

Note the similarity to Eqs. (A47)–(A49). Once again, we must ensure that our solution corresponds to the noiseless δω → 0 FIG. 8. The complete region of integration can be divided into result expected when rms . Using the series expan- three subspaces which yield different conditions for kðtÞ. sion of γlðs; xÞ, we find

041046-16 FUNDAMENTAL THERMAL NOISE LIMITS FOR OPTICAL … PHYS. REV. X 10, 041046 (2020)    pffiffiffiffiffiffiffiffi 2 s˜ Γ γ ðsj 0Þ γ ðs j 0Þ ˜ − Δ − Γ ˜ 2Γ ˜ ˜ 2 j inj c f x¼ f x¼ daSðtÞ¼½ð i lÞaSðtÞþ csindt hjaj iT¼0 ¼ þ sffiffiffiffiffiffi Γ Γ γ ðsj 0 þ 1Þ γ ðs j 0 þ 1Þ T  l  f x¼ f x¼ 2 ˜ 2 Γ Γ Γ þ δω a˜ ðtÞdWðtÞ; ðA69Þ jsinj c T T Γ rms S ¼ þ ; T ΓT Γl Γl þ iΔ Γl − iΔ 2Γ ˜ 2 2 cjsinj     hja˜j i 0 ¼ ðA65Þ δω2 pffiffiffiffiffiffiffiffi T¼ Δ2 þ Γ2 ˜ − Δ − Γ rms ˜ 2Γ ˜ l daIðtÞ¼ i l þ Γ aIðtÞþ csin dt sffiffiffiffiffiffi T ˜ as expected. Similar to the solution for haðtÞi, we define the 2 δω ˜ effective quality factor þ rmsaIðtÞdWðtÞðA70Þ ΓT 2 ω0hja˜j i ω0 Q exx−sγ s; x eff ¼ 2 ˜ 2 ¼ 2Γ lð ÞðA66Þ jsinj T in Stratonovich and Itô forms, respectively. Applying the Itô rule ½dWðtÞ2 ¼ dt to the latter, we can solve for the for resonant excitation (Δ ¼ 0) and critical coupling steady-state moments (Γc ¼ Γl=2). pffiffiffiffiffiffiffiffi As opposed to the nonmonotonic scaling of the mean 2Γ s˜ ˜ ˜ 2 a˜ t c in ; A71 field amplitude jhaðtÞij with Ql, the stored energy hjaðtÞj i h ð Þi ¼ Γ δω2 Γ Δ ð Þ ½ l þ rms= Tþi increases monotonically with increasing Ql. This behavior is intuitively described in the main text: continuing to increase Q decreases the cavity linewidth until Q is 2Γ 1 δω2 Γ Γ ˜ 2 l eff ˜ 2 cð þ rms= T lÞjsinj saturated by mode-volume-dependent thermal noise in the hjaðtÞj i¼ Γ δω2 Γ 2 Δ2 ; ðA72Þ ½ l þ rms= T þ high-Ql=Veff regime. Finally, we note that the maximum energy storage (although not necessarily the maximum intensity, which also depends on the mode volume) is which by comparison to Eqs. (A50) and (A65) immediately 2Γ → 2Γ 2δω2 Γ achieved with large mode volumes due to reduced thermo- reveals a thermal broadening l l þ rms= T of optic noise. the microcavity linewidth. We can also derive an equation of motion for the autocorrelation RaaðτÞ¼ha˜ðtÞa˜ ðt þ τÞi: d. Cavity spectrum in the limit    δω2 pffiffiffiffiffiffiffiffi d τ Δ − Γ rms τ 2Γ ˜ ˜ The complete field autocorrelation in Eq. (A55) sim- Raað Þ¼ i l þ Raað Þþ csinhaðtÞi: dτ ΓT plifies considerably in the high-Ql limit where ΓT ≫ Γl,as the cavity resonant frequency ω0ðtÞ can be assumed to ðA73Þ directly track the temperature noise over the relevant timescales. The frequency noise is then effectively Solving Eq. (A73) subject to the τ ¼ 0 conditions of delta correlated in time, and the aforementioned—albeit Eq. (A72), we find tedious—“integration by regions” technique can then be similarly applied to solve for the field noise spectrum 2Γ 2 ω cjsinj Saað Þ. A more intuitive approach to this solution is R ðτÞ¼ ω ’ aa ½Γ þ δω2 =Γ 2 þ Δ2 through adiabatic elimination of 0ðtÞ s dynamics follow- l rms T  δω2 ing the procedure in Ref. [124]. Converting the optical field rms Δ−Γ −δω2 Γ τ ði l rms= T Þ 1 and resonant frequency evolution equations [Eqs. (A32) × e þ ; ðA74Þ ΓTΓl and (A34)] into stochastic differential equations yields pffiffiffiffiffiffiffiffi corresponding to the optical noise spectrum ˜ ( δω − Δ − Γ ˜ 2Γ ˜ ) daðtÞ¼ fi½ ðtÞ lgaðtÞþ csin dt; ðA67Þ pffiffiffiffiffiffiffiffi Γ 2δω2 js j2=Γ δω −Γ δω δω 2Γ S ω c rms in T d ðtÞ¼ T ðtÞdt þ rms TdWðtÞðA68Þ aað Þ¼Γ Γ δω2 Γ 2 Δ2 l ½ l þ rms= T þ 2 Γ δω2 Γ for both Itô and Stratonovich forms, since the frequency ð l þ rms= TÞ ffiffiffiffiffiffiffiffi × : ðA75Þ δω p2Γ Γ δω2 Γ 2 ω Δ 2 noise is additive ( rms T is constant). In the limit ð l þ rms= TÞ þð þ Þ ΓT → ∞, we can adiabatically eliminate the resonant frequency dynamics, yielding a steady-state value δω t δω ≪ pffiffiffiffiffiffiffiffiffiffi ð Þ¼ Equation (A75) evaluated in the perturbative limit rms 2 Γ δω = T rmsdWðtÞ=dt. The cavity evolution can then be Γl coincides with the low-frequency (ω ≪ ΓT) limit of the simplified to previous perturbative spectrum [Eq. (A40)].

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FIG. 9. Schematic of the setup built to measure TRN in photonic crystal cavities. An amplified (PriTel PMFA) continuous wave laser (Santec TSL-710) is separated into a local oscillator and cavity signal by a polarizing beam splitter (PBS). The LO line is passively path- length matched to the cavity signal using a tunable retroreflector delay line. The cavity signal is combined with a linearly polarized (LP) white light source (LED) using a dichroic mirror (DM) and is reflected from a PhC cavity rotated 45° from the incident polarization (adjustable with a half-wave plate, λ=2), which allows the cavity signal to be isolated from the specular reflection using a PBS. A quarter-wave plate (λ=4) allows the specular reflection to be extracted for comparison to the cavity-only reflection. The reflected illumination light is separated and imaged onto a silicon CCD. The cavity signal can be directed with flip mirrors toward an IR camera for imaging, an IR avalanche photodetector (ThorLabs PDB410C 10 MHz InGaAs APD) to collect low-noise reflection spectra, or toward the balanced homodyne detector. For the latter, a balanced photodetector (ThorLabs PDB480C-AC 1.6 GHz InGaAs p-i-n photodetector) measures the homodyne signal from the recombined cavity reflection and local oscillator, and the result is recorded on an electronic spectrum analyzer (ESA; Agilent N9010A EXA Signal Analyzer). The dc signal extracted from a low-pass filter (LPF) is used as the feedback signal for a digital PID controller which stabilizes the signal-LO phase difference by actuating a piezoactuated mirror. An EOM provides a known phase noise which can be used to calibrate the frequency noise of the PhC cavity. The sample stage is temperature stabilized to ΔT<0.01 K using a Peltier plate and a feedback temperature controller.

APPENDIX B: EXPERIMENTAL TRN IN (HF) acid. The designs are adapted from Refs. [69,70].As PHOTONIC CRYSTAL CAVITIES shown in Fig. 2 of the main text, superimposed gratings are added to improve vertical coupling efficiency. The gratings The single-mode thermal decay approximation made in are formed via periodic hole radii perturbations in the range Eq. (A9) implies the decay rate of Eq. (A16) and the Δr 0 → 0.05r at a period equal to twice the lattice spectral density of cavity resonant frequency in Eq. (A25). ¼ constant a. Although devices with quality factors as large This result is commonly used as a simplifying assumption as 400 000 are measured for small values of Δr, the results for temperature fluctuations [35,61]; however, it is not Δ 0 05 immediately clear that the single-mode approximation presented in the main text use r ¼ . r, which signifi- holds in the case of small-mode-volume optical micro- cantly improves collection efficiency into our fiber-coupled cavities, where the characteristic length scales of the near detector. diffraction-limited optical mode (∼λ=n) can approach that of the phonon mean free path [75]. In the absence of any 2. Experimental setup experimental data in the literature to verify the assumption, A more detailed version of the experimental setup we construct an experiment to measure thermorefractive depicted in the main text is provided in Fig. 9. The setup noise in high-Ql=Veff silicon photonic crystal cavities. consists of a typical polarized light microscope, where the The experiment also allows us to compare measured signal reflected from a PhC cavity is measured with TRN with the spectra derived from our multimode theory balanced homodyne detection. The homodyne detector is (Appendix A3). balanced by zeroing the dc component of the homodyne signal with a digital PID feedback controller connected to a 1. Photonic crystal cavity sample details piezoactuated mirror. In this configuration, the homodyne The L3 and L4=3 photonic crystal cavities are fabricated voltage signal by Applied Nanotools foundry via electron-beam pattern- ∼ ˜ ˜ δϕ ing and dry etching of 220-nm-thick undoped silicon-on- vh jaLOjjacavityj cavityðtÞðB1Þ insulator wafers with a 2-μm-thick buried oxide layer. To ˜ suspend the devices, the buried oxide is subsequently for a local oscillator signal aLO is directly proportional to ˜ released via a 60 s timed wet etch in 49% hydrofluoric the cavity amplitude jacavityj and to phase fluctuations

041046-18 FUNDAMENTAL THERMAL NOISE LIMITS FOR OPTICAL … PHYS. REV. X 10, 041046 (2020) pffiffiffiffiffiffiffiffiffiffiffiffi δϕ ˜ ≈ η 1 ϕ ω cavityðtÞ resulting from the stochastic resonant frequency. aLO HPin½ þ i mðVpÞ cosð mtÞ: ðB8Þ An electronic spectrum analyzer is used to measure the power spectral density Svv of this homodyne voltage signal. Similarly, assuming ωm ≪ Γl (as is the case in our experi- ment), the cavity response yields the output signal 3. Phase noise calibration pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Γ − Γ a˜ ≈ 1 − η P c l The resonant frequency Sωω can out ð HÞ in Γ then be determined from S using the absolute calibration   l  vv ω technique discussed in Refs. [68,125]. For example, con- 1 ϕ ω m ω × þ i mðVpÞ cosð mtÞþ Γ sinð mtÞ : ðB9Þ sider a Mach-Zehnder interferometer with input power Pin l and splitting ratio ηh, which creates the in-phase local oscillator and cavity input signals The homodyne signal pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Γ − Γ ˜ η ˜ 1 − η c l aLO ¼ HPin; ain ¼ ð HÞPin: ðB2Þ v t ≈ 2G η 1 − η P ω ϕ V ω t hð Þ c Hð HÞ in Γ2 m mð Þ sinð m Þ l Assuming resonant drive (Δ ¼ ω − ω0 ¼ 0) for a cavity L ðB10Þ with input power coupling rate Γc, total loss rate Γl, and a perturbative resonant frequency noise δωðtÞ, the output corresponds to a power spectral density cavity signal is then   2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi δϕ 2Γ − Γ 2Γ − Γ m ≈ 4 2η 1 − η c l 2 ω2 ˜ ≈ 1 − η c l Svv Gc Hð HÞ 2 P mSϕϕjω¼ω ; aout ð HÞPin ; ðB3Þ Γ in m Γl þ iδωðtÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}l

Kexpt yielding a homodyne detection voltage ðB11Þ v ðtÞ ≈ 2G ja˜ jja˜ jδϕ ðtÞ h c LO out cavity which, similar to Sδω, is directly proportional to K . K pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Γ − Γ vv expt expt ≈−2G η 1 − η P c l δω t can therefore be eliminated to yield an absolute calibration c Hð HÞ in Γ2 ð ÞðB4Þ l for the resonant frequency noise spectral density:

δω 2 for a detector conversion gain Gc. The final frequency noise ω Sϕϕ Svv m jω¼ωm δω spectral density Sωω ≈ ≈ Svv : ðB12Þ δϕm Kexpt Svv jω ω   ¼ m 2 δω 2 2Γc − Γl 2 S ≈ 4G η 1 − η P Sωω K Sωω vv c Hð HÞ Γ2 in ¼ expt This result can be simplified by evaluating the phase |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}l spectral density Kexpt Z ∞ 2 −iωτ ðB5Þ SϕϕðωÞ¼ ϕmðVpÞhcosðωmtÞ cos½ωmðt þ τÞie dτ −∞   is therefore a function of various experimental constants ϕ2 V 1 1 mð pÞ δ ω − ω δ ω ω and cavity coupling parameters. ¼ 2 2 ð mÞþ2 ð þ mÞ : ðB13Þ However, the value of Kexpt can be exactly determined by δϕ ϕ ω injecting a known phase noise ðtÞ¼ mðVpÞ cosð mtÞ The spectrum analyzer convolves the δ functions with the into the interferometer with an electro-optic modulator intermediate frequency filter function FðωÞ, which is ω driven with an electrical tone with frequency m and peak normalized such that Fð0Þ¼1=ENBW [68], where the voltage Vp. Under the same experimental conditions, the effective noise bandwidth ENBW ¼ ηFRBW for a resolu- local oscillator and cavity input signals are, respectively, tion bandwidth RBW and a filter-shape-dependent ηF ≈ 1. pffiffiffiffiffiffiffiffiffiffiffiffi Therefore, the measured noise spectral density evaluated at ϕ ω ˜ η i mðVpÞ cosð mtÞ ω aLO ¼ HPine ; ðB6Þ the modulation frequency m becomes pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϕ ω ϕ2 ϕ2 a˜ ¼ ð1 − η ÞP ei mðVpÞ cosð mtÞ: ðB7Þ mðVpÞ mðVpÞ in H in Sϕϕðω ¼ ω Þ¼ FðωÞδðω − ω Þ¼ : m 4 m 4 · ENBW ϕ π With a small enough modulation depth mðVpÞ¼ Vp=Vπ ðB14Þ (and, therefore, a small enough drive voltage Vp relative to the half-wave voltage Vπ), the local oscillator can be Using this result, the calibration term in Eq. (B12) can be approximated to first order as simplified to a final form

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(a) (b)

FIG. 10. Measurement of phase modulator modulation depth ϕmðVpÞ at λ ¼ 1550 nm. A balanced homodyne measurement is performed on the output of a Mach-Zehnder interferometer with the EOM in one arm, yielding spectra similar to that of (a). The sideband amplitudes are fitted to find the modulation depth at each peak drive voltage Vp, and a linear fit is applied to find the modulation efficiency (b). The measured value ϕm=Vp ¼ 0.82 0.01 rad=V corresponds to a half-wave voltage Vπ ¼ 3.83 V.

ω2 ϕ2 V δω ω ac-coupled spectrum analyzer) via a least-squares regres- ω m mð pÞ Svv ð Þ Sωωð Þ¼ δϕ ðB15Þ ϕ 4η m sion yields mðVpÞ for any peak drive voltage Vp. F · RBW Svv ðωmÞ Figure 10(a) illustrates the result for λ ¼ 1550 nm and 2 24 that agrees with Eq. (20) of Ref. [68]. Vp ¼ . V, where the Bessel functions evaluated at ϕm ≈ 0.6π rad (red points) are well fitted to the measured a. Electo-optic phase modulator calibration (blue curve) peak amplitudes. After repeating the experi- ment for multiple values of Vp, a linear fit [Fig. 10(b)]gives Equation (B15) demonstrates that the calibrated fre- the modulation efficiency quency noise can be readily obtained by comparing δω the recorded rf power spectral density Svv ðωÞ to the δϕ ϕ m ω η m 0 82 0 01 calibration PSD Svv ð mÞ (which corresponds to a known mod ¼ ¼ . . rad=V; ðB17Þ V λ λ phase spectral density) for a given calibration frequency p 0¼ cal ωm=2π and spectrum analyzer RBW. The ENBW correc- η tion factor F is a function of various spectrum analyzer corresponding to a half-wave voltage Vπ ¼ 3.75 V (roughly settings (see Ref. [126], for example) and is therefore in line with the manufacturer-quoted value of 3.17 V) at the measured by comparing the noise marker amplitude (dBm/ pffiffiffiffiffiffi calibration wavelength. Hz) to the measured PSD divided by the RBW. This Note that the dc phase of the fiber interferometer in this technique yields ηF ≈ 1.057, which is approximately equal experiment is allowed to drift, while the measurement is to the value given in Ref. [126] assuming typical spectrum averaged on a timescale much longer than that of the drift— analyzer settings. a standard technique [127] which affects only the total The only remaining unknown parameter required for power of the homodyne signal, not the relative magnitude calibration is the peak-voltage-dependent modulation depth of the sidebands. ϕmðVpÞ of the phase modulator, which can be determined with a sideband fitting technique as shown in Fig. 10.An b. Balanced homodyne detector characterization EOM is embedded in one arm of an unbalanced Mach- Zehnder interferometer, yielding a homodyne signal Using the measured EOM modulation efficiency, the X calibrated thermorefractive noise measurements in Fig. 2 of the main text are obtained by measuring the cavity vh ∝ JnðϕmÞ cosðnωmtÞðB16Þ n reflection with the stabilized homodyne detector in Fig. 9. We confirm that the balanced photodetection is shot noise for a modulation frequency ωm. The power spectrum limited (with 10 dB of shot noise clearance) for frequencies observed on the spectrum analyzer therefore consists of greater than approximately 100 kHz and balance the a periodic sequence of spectrum analyzer filter functions interferometer arms to well within 1 mm—over an order Fðω − nωmÞ at frequencies ωn ¼ nωm with powers propor- of magnitude shorter than the expected cavity delay 2 tional to JnðϕmÞ. Fitting the sideband powers (relative to (approximately centimeters). This balance is achieved by the n ¼ 1 sideband, as the n ¼ 0 peak is inaccessible on the tuning a retroreflector-based delay line while observing

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TABLE I. Parameters used for calibrating the noise spectrum and computing or fitting ΓT and VT . Independent values n for L3 and L4=3 microcavities are listed as fnL3;nL4=3g for cavity-dependent parameters. The mode confinement factor γSi ∼ 1 confirms the validity of Eq. (B15), which assumes complete confinement of the mode in silicon. Parameter Symbol Value Source Temperature T 295.68 K Measured Si refractive index nSi 3.48 [128] αSi 1 8 10−4 −1 Si thermo-optic coefficient TR . × K [128] Si 1 64 3 Si specific heat cV . J=cm ·K [75] Si thermal conductivity κSi 70 W=m·K [75] Si 2 Si thermal diffusivity (thin film) D 0.43 cm =s κ=cV Lattice porosity ϕ f0.29; 0.26g Calculated Si 0 23 0 25 2 Si 1 − ϕ 1 ϕ Patterned thermal diffusivity DT f . ; . g cm =s D ð Þ=ð þ Þ [74] Resonant wavelength λ0 f1559.3; 1551.5g nm Measured Quality factor Ql f168 000; 163 000g Measured η 0 821 Phase modulator efficiency mod . rad=V Measured (1550 nm) ESA noise correction factor ηF 1.057 Measured γ 0 96 0 95 Mode confinement factor Si f . ; . g Simulated ˜ 0 95 0 32 Mode volume Veff f . ; . g Simulated ˜ 3 92 1 51 Thermal mode volume VT f . ; . g Simulated [Eq. (A23)] pulse delays from a picosecond fiber laser on both recombines, producing random local heating analogous interferometer paths. to fundamental thermorefractive noise. Considering this similarity, we can analyze the MPA photothermal shot 4. Summary of experimental parameters noise by redefining the statistics of the mode-averaged ¯ Table I summarizes the various experimental parameters temperature driving force FTðtÞ in Eq. (A9). The mean rate used to generate the data and fit parameters shown in the of intracavity k-photon absorption is [81] main text. Note that, as described in the caption of Fig. 2 of β the main text, the expected thermal diffusivity is based on r k Ik V ; h kPAi¼ ℏω pk kPA ðB18Þ thermal conductivity measurements in thin silicon films k 0 [75] and the hole lattice porosity ϕ [74]. The porosity pffiffiffi where I ¼ cja˜j2=2nV is the peak intensity of ϕ ¼ 3πr2=ð3 3a2=2 − 3πr2Þ—calculated as the ratio of pk eff the stored energy ja˜j2, β is the k-photon absorption hole area 3πr2 (assuming a hole radius r) to material area k coefficient, c is the speed of light, and V ¼ within a hexagonal unit cell of a lattice with lattice constant R kPA jE⃗ðr⃗Þj2kd3r= maxfjE⃗ðr⃗Þj2kg. Note that we assume a—reduces the thin film diffusivity to DT ≈ Dð1 − ϕÞ= dielectric ð1 þ ϕÞ [74]. This “restricted” diffusivity is used to that the heating produced by the photoexcited free carriers is local (i.e., no carrier diffusion). The variance calculate the expected decay rates in Fig. 2 of the main text. ¯ of FTðtÞ is then determined from the temperature change 5. Comparison of other noise sources expected from the variance of MPA events within an infinitesimally small time (a Poisson process), yielding Other stochastic processes can also produce resonant the autocorrelation frequency noise. Here, we consider two such sources: (i) multiphoton absorption leading to photothermal shot kℏω0 hF¯ ðtÞF¯ ðt0Þi ¼ β Ik V δðt − t0Þ: ðB19Þ noise from free carrier recombination and (ii) self-phase T;k T;k c2 V2 k pk kPA modulation. Both noise sources evaluated at their respec- V T — “ tive nonlinear thresholds as an estimate of the worst- Following the method of Appendix A1, we arrive at the ” — case maximum noise levels are found to be more than spectral density one order of magnitude weaker than TRN. Since the cavity   2 is measured well within the linear regime, we find that TRN ω0 kℏω0 1 kPA ω α β k dominates both other contributions combined, thus further Sωω ð Þ¼ TR 2 2 kIpkVkPA 2 2 ; ðB20Þ n c V Γ þ ω confirming our experimental measurements. V T T which can be evaluated for any intracavity stored energy. a. Multiphoton absorption Here, we consider Ipk at the nonlinear threshold, i.e., the Multiphoton absorption (MPA) within the resonator peak intensity for a linewidth resonance shift jhΔωij=2Γl ¼ α Δ 1 leads to a free carrier population that stochastically TRh TkPAiQl=n ¼ . The threshold intensity can

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1 2 FIG. 11. Approximate spectrum of microcavity noise sources for the experimental parameters in Table I. Note that S = is plotted as a pffiffiffiffiffiffi ff fractional stability (units 1= Hz to aid comparison with cavity stabilization literature). from two-photon absorption (2PA) and self-phase modulation (SPM) at their respective nonlinear threshold powers—which approximates the maximum noise level—are still smaller than TRN.

2 therefore be derived from the steady-state value of where ϵ0 is the free space permittivity, ja˜ðtÞj is the stored Eqs. (A9), which lends the average temperature change energy, and the Kerr mode volume VKerr is equal to the thermal mode volume VT [80]. When driven with a ℏω β k k 0hrkPAi kIpkVkPA classical source, the intracavity energy autocorrelation hΔT i¼ ¼ : ðB21Þ kPA c V Γ c Γ V V T T V T T 2Γ 2 0 2 c −Γ jt−t0j 2 δ a˜ t δ a˜ t e l ℏω0 s˜ Substituting this result into the spectral density equation h j ð Þj j ð Þj i¼ Γ2 hj inj iðB24Þ assuming two-photon absorption as the dominant process l (true for our silicon cavities driven at approximately can derived from temporal coupled mode theory assuming 1550 nm), we can simplify to the final result ˜ 2 Γ a constant pump power hjsinj i coupled at rate c to a cavity with composite amplitude decay rate Γ . The corresponding ω2α ℏω 2Γ l 2PA;threshold ω 0 TR 0 T resonant frequency autocorrelation can then be used to Sωω ð Þ¼ Γ2 ω2 : ðB22Þ n QlcVVT T þ compute the noise spectral density     This result is plotted in Fig. 11 assuming the 3χð3Þ 2 4Γ =Γ SPM ω c l ℏω3 ˜ 2 experimental parameters of our devices listed in Table I. Sωω ð Þ¼ 4 2 2 0hjsinj i: ðB25Þ 4ϵ0n V Γ þ ω 2PA;threshold TRN T l Comparing with Eq. (A31), we find Sωω =Sωω ¼ α ℏω ðn=Ql TRTÞð 0=kBTÞ, which accounts for the factor of Similar to the multiphoton absorption case, we evaluate approximately 10 weaker maximum photothermal shot Δω 2Γ 1 this result at the nonlinear threshold h Kerri= l ¼ for a noise in our devices as shown in Fig. 11. We operate with conservative estimate of the associated noise. The final an input power much lower than the nonlinear threshold result is Δω ≪ Γ power (such that h 2PAi l), so the experimental    photothermal shot noise is substantially weaker than the 3χð3Þ 2Γ SPM;threshold l 3 Sωω ω ℏω : maximum value calculated here. ð Þ¼ 4ϵ 4 Γ2 ω2 0 ðB26Þ 0n VTQl l þ b. Kerr self-phase modulation Even at the threshold power, Fig. 11 shows that the SPM When confined in a χð3Þ nonlinear material, Poissonian noise is over an order of magnitude weaker than TRN. fluctuations of the mean intracavity photon number impart self-phase modulational (SPM) noise on the resonant APPENDIX C: COMPARISON OF TRN IN frequency. From first-order perturbation theory, the Kerr VARIOUS MATERIALS ð3Þ ⃗ 2 3 index change δnðr⃗Þ¼3χ ϵðr⃗ÞjEðr⃗Þj =8ϵ0n results in a max Surprisingly, the Qeff =Veff limits shown in Fig. 4 of the resonant frequency shift main text for several common materials lie within an order   of magnitude. As shown in Table II, this observed invari- δωðtÞ 3χð3Þ − δ a˜ t 2; ance can be attributed to an inverse relationship between ω ¼ 4ϵ 4 j ð Þj ðB23Þ 0 Kerr 0n VKerr the thermo-optic coefficient and thermal diffusivity in

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TABLE II. Material properties used to calculate the thermal noise limits in Fig. 4 of the main text. Aluminum nitride is the only material listed with a favorable thermo-optic coefficient and thermal diffusivity.

−1 3 2 Material Index n TR coefficient αTR [K ] Density ρ [g=cm ] Heat capacity cV [J=g · K] Thermal diffusivity DT [cm =s] Si 3.48 1.8 × 10−4 2.32 0.7 0.8 GaAs 3.38 2.35 × 10−4 5.32 0.35 0.31 InP 3.16 2 × 10−4 4.81 0.31 0.37 −5 Si3N4 1.99 2.5 × 10 4.65 0.7 0.02 −5 −3 LiNbO3 2.21 3.2 × 10 5.32 0.63 7 × 10 AlN 2.19 3 × 10−5 3.23 0.6 1.47 common materials. Yet this relationship is not fundamental: a. Kerr (χ ð3Þ) interaction aluminum nitride, for example, is shown to outperform all In the χð3Þ case, we have a cavity with a single resonant other plotted materials by over an order of magnitude due to mode Eω. The polarization term due to the Kerr interaction its simultaneously large thermal conductivity and small ð3Þ −iωt 3 is P ¼ ϵ0χ ∶ðCωAωEωe þ c:c:Þ . This term gives rise thermo-optic coefficient. This realization demonstrates the _ − χ 2 importance of material choice when designing state-of-the- to the equation of motion Aω ¼ i jAωj Aω, where Q=V R art high- eff resonators. 2 ð3Þ 2 2 3 2 3ℏω χ 1 ð n jEωj d r⃗Þ χ − ;V ≡ R : ¼ 4 4ϵ Kerr 4 4 3 ⃗ ðD4Þ APPENDIX D: EFFECTS OF TRN ON n 0 VKerr n jEωj d r ALL-OPTICAL QUBITS Quantizing the field to satisfy the commutation relations ˆ ˆ † 1. Derivation of qubit coupling strengths ½Aω; Aω¼1, this equation of motion can be generated This section derives the figures of merit for qubit from the Hamiltonian: operation in nonlinear optical cavities. For more informa- 1 tion, see Refs. [57,95]. The procedure is to first derive the ˆ † ˆ † ˆ ˆ H ¼ χAωAωAωAω: ðD5Þ classical equations of motion for fields in nonlinear oscil- Kerr 2 lators and then to quantize them, deriving the Hamiltonian and the single-photon coupling strength. In classical cavity As an open quantum system, the field interacts with a bath through Lindblad dissipation terms, in this case electrodynamics, a cavity field can be expressed as a sum of pffiffiffiffiffiffi resonant modes L ¼ 2ΓAω, where Γ ¼ ω=2Q. The figure of merit for X strong coupling is −iωt E x;⃗ t Cω Aω t Eω x⃗e c:c: ; D1 ð Þ¼ ½ ð Þ ð Þ þ ð Þ 3 ω χ 3πℏc χð Þ Q ð3Þ : pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi FOMχ ¼ 2Γ ¼ 2 ϵ λ4 ˜ ðD6Þ |fflfflfflfflffl{zfflfflfflfflffl}n 0 VKerr where Cω ¼ ℏω=2ϵ0. The modes Eω satisfy the Helmholtz 2 2 2 Kχ equation ∇ × ð∇ × EωÞ¼ðn ω =c ÞEω.Thisformisa generalized eigenvalue equation,R and the resultingR solutions 2 3⃗ 2 3⃗ can be orthogonalized: n Eω0 Eωd r ¼ c Bω0 Bωd r ¼ δ b. Second-order (χ ð2Þ) interaction ω0ω. With this normalization, we find that theP electromag- 2 netic energy density in the cavity is U ¼ ω ℏωjAωj . In this case, we have two fields at frequencies ðω; 2ωÞ. ð2Þ −iωt Therefore, Aω is the normalized field operator, where The polarization term is P ¼ ϵ0χ ∶ðCωAωEωe þ 2 −2iωt 2 jAωj gives the number of photons in the mode Eω. C2ωA2ωE2ωe þ c:c:Þ , which gives rise to the equa- Nonlinear interactions can be treated as perturbations, tions because the nonlinearity is weak on the order of a single optical cycle. The Helmholtz equation acquires a nonlinear 1 _ 2 _ ð2Þ 2 ð3Þ 3 A2ω ¼ − ϵAω; Aω ¼ ϵA2ωAω; ðD7Þ polarization P ¼ ϵ0ðχ ∶E þ χ ∶E þÞ, which can 2 be integrated to give perturbations to the equations of motion for Aω [129]: where pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 ∂ 1 ∂ ϵ ω ℏω=ϵ0 n E ðP= 0Þ ϵ χð2Þ ∇ × ð∇ × EÞþ ¼ − ðD2Þ ¼ 1 2 ; ðD8Þ c2 ∂t2 c2 ∂t2 n3V = R shg R Z   2 2 3 2 2 3 2 ω ⃗ n E2ω d r⃗ n Eω d r⃗ dAω i Pðx; tÞ iωt 3 ð j R j Þð j j Þ ⇒ Eω x⃗ e d r:⃗ D3 V ¼ ðD9Þ ¼ 2 ð Þ ϵ ð Þ shg 3 3⃗2 dt Cω 0 ω j n E2ωEωEωd rj

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TABLE III. Silicon material properties assumed to calculate the qubit figures of merit at λ0 ¼ 2.3 μm and T ¼ 300 K.

Parameter Symbol Value Source 3 −13 2 χð Þ nonlinear index n2 1.2 × 10 cm =W [130], Sec. 11 3 −11 χð Þ FOM constant Kχ 8.7 × 10 Calculated [Eq. (D6)] χð2Þ (dc E-field induced) χð2Þ 40 pm=V [108] 2 −7 χð Þ FOM constant Kϵ 1.3 × 10 Calculated [Eq. (D13)] αSi 1 8 10−4 −1 Thermo-optic coefficient TR . × K [128] Thermal diffusivity DSi 0.8 cm2=s [75]

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