PHYSICAL REVIEW X 10, 041046 (2020) Featured in Physics
Fundamental Thermal Noise 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 Brownian motion 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 crystal 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-matter interaction [1], shaping facilitate single-molecule label-free sensing [14], high- electromagnetic modes [2], and modifying the vacuum repetition-rate frequency combs [15,16] for frequency photon 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 Gaussian noise 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 Fourier transform 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 phase 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 spectral density 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 polarization 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 shot noise 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 spectrum analyzer 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 phase noise 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 photons 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 solid 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 noise reduction 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 ðr⃗Þ δTðr⃗Þd3r;⃗ 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;⃗ tÞ 2id3r⃗ 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 crystals, 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 ϕ ðzÞF 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