The study of the quantum properties of mesoscopic mechanical systems is a rapidly evolving field that has been propelled by recent advances in the development of cavity optomechanical devices [1]. Nanophotonic cavity optomechanical structures [2] allow colocalization of photons and femtogram-to-picogram mechanical excitations, and have enabled demonstrations of ultrasensitive displacement and force detection [3 –7], ground-state cooling [8], and optical squeezing [9]. The development of cavity optomechanical systems with large nonlinear photon–phonon coupling has been motivated by quantum nondemolition (QND) measurement of phonon number [10] and shot noise [11], as well as mechanical quantum-state preparation [12], the study of photon–photon interactions [13], mechanical squeezing and cooling [14 –16], and phonon–photon entanglement [17].

Recent progress in developing optomechanical systems with large nonlinear optomechanical coupling has been driven by studies of membrane-in-the-middle (MiM) [18 –21] and whispering gallery mode [12,22,23] cavities. Demonstrations of massively enhanced quadratic coupling [19,21,22] have exploited avoided crossings between nearly degenerate optical modes, and have revealed rich multimode dynamics [21]. To surpass bandwidth limits [13,24] and parasitic linear coupling [25] imposed by closely spaced optical modes, it is desirable to develop devices that combine strong nonlinear coupling and large optical mode spacing. This can be achieved in short, low-mass, high-finesse optical cavities. In this Letter, we present such a nanocavity optomechanical system, which couples modes possessing low optical loss and terahertz free spectral range to mechanical resonances with femtogram mass, 300 kHz–220 MHz frequency, and large zero point fluctuation amplitude. This device has vanishing linear and large nonlinear optomechanical coupling, with a quadratic optomechanical coupling coefficient $g^{(2)}\approx 2\pi \times 400\mathrm{MHz}/{\mathrm{nm}}^2$ and a single-photon to two-phonon coupling rate of $\mathrm{\Delta }{\omega }_0=2\pi \times 16\mathrm{Hz}$.

The strength of photon–phonon interactions in nanocavity–optomechanical systems is determined by the modification of the optical mode dynamics via deformations to the nanocavity dielectric environment from excitations of mechanical resonances. In systems with dominantly dispersive optomechanical coupling, this dependence is expressed to second order in mechanical resonance amplitude $x$ as ${\omega }_o(x)={\omega }_0+g^{(1)}x+\frac{1}{2}{g}^{(2)}{x}^2$, where ${\omega }_o$ is the cavity resonance frequency, and $g^{(1)}=\delta {\omega }_o/\delta x$, $g^{(2)}={\delta }^2{\omega }_o/\delta x^2$ are the first- and second-order optomechanical coupling coefficients. In nanophotonic devices, $x$ parameterizes a spatially varying modification to the local dielectric constant, $\mathrm{\Delta }ϵ(\mathbf{r};x)$, whose distribution depends on the mechanical resonance shape and is responsible for modifying the frequencies of the nanocavity optical resonances.

Insight into nonlinear optomechanical coupling in nanocavities is revealed by the dependence of $\delta {\omega }^{(2)}$ on the overlap between $\mathrm{\Delta }ϵ$ and the optical modes of the nanocavity [26,27]:

The optomechanical device studied here, illustrated in Fig. 1, is a photonic crystal “paddle nanocavity” that combines operating principles of MiM cavities [10,29] and photonic crystal nanobeam optomechanical devices [2]. The device is designed to be fabricated from silicon-on-insulator (refractive index $n_{\mathrm{Si}}=3.48$, thickness $t=220\mathrm{nm}$), and to support modes near $\lambda \sim 1550\mathrm{nm}$. A “paddle” element is suspended within the optical mode of the nanocavity defined by two photonic crystal nanobeam mirrors. The width of the gap ($d=50\mathrm{nm}$) separating the mirrors from the paddle is chosen for smooth variation in the local effective index of the structure [30], and the paddle length ($L=958\mathrm{nm}$) is set to $\approx 1.5\lambda /n_{\mathrm{eff}}$ [31]. This allows the nanocavity to support high optical quality factor ($Q_o$) modes. The length ($l_s$) and width ($w_s$) of the paddle supports can be adjusted to tailor its mechanical properties, although $l_s\ge 200\mathrm{nm}$ and $w_s\le 200\mathrm{nm}$ are required to not degrade $Q_o$. We consider three support geometries, labeled $p1–p3$ (see Fig. 2(c) for dimension). All of these dimensions are realizable experimentally [7].

 

Fig. 1. Schematic of the photonic crystal paddle nanocavity (top-view). The paddle is separated from photonic crystal nanobeam mirrors by gaps $d=50\mathrm{nm}$, and has length $L=958\mathrm{nm}$. The elliptical hole horizontal and vertical semiaxes $(R_x,R_y)$ are tapered as shown.

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Figure 2(a) shows the first seven localized optical modes supported by the paddle nanocavity, calculated using finite element simulations (FEM). COMSOL software was used for all FEM simulations. The lowest-order mode ($M1$) has a resonance wavelength near 1550 nm (${\omega }_o/2\pi =191\mathrm{THz}$) and $Q_o>1.3\times {10}^4$. The mechanical resonances of the paddle nanocavity were also calculated using FEM simulations, and the displacement profiles of the four lowest mechanical frequency resonances are shown in Fig. 2(b). They are referred to here as “sliding” ($\mathcal{S}$), “bouncing” ($\mathcal{B}$), “rotational” ($\mathcal{R}$), and “torsional” ($\mathcal{T}$) resonances. As discussed below, we are particularly interested in the $\mathcal{S}$ resonance, whose frequency and effective mass [28] varies between $f_m=0.35–217\mathrm{MHz}$ and $m=314–589\mathrm{fg}$ for the support geometries $p1–p3$, as described in Fig. 2(b). Appropriate selection of geometry $p1–p3$ depends on the application, with $p1$ suited for sensitive actuation, $p2$ a compromise between ease of fabrication and sensitivity, and $p3$ for high-frequency operation and low thermal phonon occupation.

 

Fig. 2. (a) Properties of the localized optical modes of the paddle nanocavity (labeled M1–M7): electric field distribution ($E_y$ component), spatial symmetry in $x–y$ plane, optical frequency, and contribution $g_{{\omega }_1{\omega }_n}^{(2)}$ to the quadratic optomechanical coupling $g^{(2)}$ describing interaction between mode M1 and the $\mathcal{S}$ mechanical resonance shown in (b). (b) Displacement profiles and properties of the paddle nanocavity mechanical resonances. $m$ and $f_m$ are indicated for three support geometries, $p1–p3$, whose cross sections are given in (c).

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The spatial symmetry of the nanocavity results in vanishing $g^{(1)}$ for the mechanical resonances considered here. The intensity $E^2(x,y,z)$ of each nanocavity optical mode has even symmetry, denoted ${\sigma }_{x,y,z}=1$, while the mechanical resonances induce perturbation $\mathrm{\Delta }\epsilon$, which is odd in at least one direction, characterized by ${\sigma }_x=-1$ ($\mathcal{S}$, $\mathcal{R}$), ${\sigma }_y=-1$ ($\mathcal{R}$), or ${\sigma }_z=-1$ ($\mathcal{B}$, $\mathcal{T}$). As a result, $g^{(1)}\propto 〈E_{\omega }|\delta ϵ|E_{\omega }〉=0$. Similarly, the second-order self-overlap term in Eq. (1) is also zero. However, the electric field amplitude $E(x,y,z)$ may be even or odd, resulting in nonzero cross coupling $g_{{\omega }^{\prime },\omega }^{(2)}$ between optical modes with opposite ${\sigma }_{x,y,z}$. For example, displacement in the $\widehat{x}$ direction of $\mathcal{S}$ couples optical modes with opposite ${\sigma }_x$. In contrast, displacement in the $\widehat{z}$ direction by the $\mathcal{B}$ and $\mathcal{T}$ resonances does not induce cross coupling, as the localized optical modes all have even vertical symmetry (${\sigma }_z=1$). Here we focus on the nonlinear coupling between the $\mathcal{S}$ resonance and the $M1$ mode of the nanocavity.

To evaluate $g_{\omega ,{\omega }^{\prime }}^{(2)}$, the mechanical and optical field profiles were input into Eq. (2). The resulting contributions $g_{{\omega }_1{\omega }_n}^{(2)}$ of each localized mode to $g^{(2)}$ for optomechanical coupling between the $\mathcal{S}$ resonance and the $M1$ mode are summarized in Fig. 2(a). Contributions from delocalized modes are neglected due to their large mode volume and low overlap. The imaginary part of ${\omega }_o$, which is small for the localized modes whose $Q_o>{10}^2$, is also ignored. A total $g^{(2)}/2\pi \approx -400\mathrm{MHz}/{\mathrm{nm}}^2$ is predicted, which matches with our direct FEM calculations (see Supplement). The corresponding single-photon to two-phonon coupling rate, $\mathrm{\Delta }{\omega }_o$, depends on the support geometry. For the most flexible $p1$ geometry, $\mathrm{\Delta }{\omega }_o\equiv |g^{(2)}{x}_{zpf}^2|=2\pi \times 16\mathrm{Hz}$, where $x_{zpf}=\sqrt{\hslash /2m{\omega }_m}$. This $\mathrm{\Delta }{\omega }_o$ is about four orders of magnitude higher than typical MiM systems [18,21], while the mode spacing is five orders of magnitude higher than other nonlinear optomechanical systems [13,18,22]. The dominant contributions to $g^{(2)}$ arise from cross coupling between modes $M1\leftrightarrow M4$ and $M1\leftrightarrow M7$ due to strong spatial overlap between their fields and the paddle–nanobeam gaps. Increasing $g^{(2)}$ through additional optimization, for example, by concentrating the optical field more strongly in the gap, should be possible.

Given $g^{(2)}$ of the paddle nanocavity, the optical response of the device can be predicted. In experimental applications of optomechanical nanocavities, photons are coupled into and out of the nanocavity using an external waveguide. Mechanical fluctuations, $x(t)$, are monitored via variations, $dT(t)$, of the waveguide transmission, $T$. In the sideband unresolved regime (${\omega }_m\ll {\omega }_o/Q_o$), optomechanical coupling results in a fluctuating waveguide output $dT=G_{1}x(t)+\frac{1}{2}{G}_{2}x{(t)}^2$, where

Figure 3(a) shows $S_P^{(2)}(\omega )$ predicted from Eq. (5), for the $\mathcal{S}$ mode of a paddle nanocavity at room temperature ($T_b=300\mathrm{K}$). The input optical field is set to $P_i=100\mathrm{\mu W}$, with detuning $\mathrm{\Delta }=\kappa /2$ to maximize the nonlinear optomechanical coupling contribution. The predicted $S_P^{(2)}(\omega )$ is shown for $p1$ and $p2$ support geometries, assuming $Q_m={10}^3$, $Q_o=1.4\times {10}^4$, and relatively weak fiber coupling $T_o=0.90$. Note that $Q_m$ is specified assuming the device is operating in moderate vacuum conditions [7], and can increase to ${10}^5$ in cryogenic vacuum conditions [9]. Also shown are estimated noise levels, assuming direct photodetection using a Newport 1811 photoreceiver ($\mathrm{NEP}=2.5\mathrm{pW}/\mathrm{Hz}$). Resonances in $S_P^{(2)}$ are evident at $\omega =2{\omega }_m$ and $\omega =0$, corresponding to energies of the two-phonon processes characteristic of $x^2$ optomechanical coupling. Figure 3 suggests that even for these relatively modest device parameters, the nonlinear signal at $2{\omega }_m$ is observable. This signal can be further enhanced with improved device performance. For example, if $Q_m={10}^4$, the nonlinear signal is visible for temperatures as low as 50 mK for the $p1$ geometry. Note that additional technical noise will increase as $f_m$ is further decreased into the kilohertz range.

 

Fig. 3. (a) $S_P^{(2)}(\omega )$ generated by thermal motion of the $\mathcal{S}$ mode of a paddle nanocavity for $p1$ and $p2$ support geometry, assuming room temperature operation, $\mathrm{\Delta }=\kappa /2$, $P_i=100\mathrm{\mu W}$, and $Q_m={10}^3$. (b) $S_P^{(2)}(2{\omega }_m)$ as a function of detuning $\mathrm{\Delta }$, for varying quadratic coupling strengths $g^{(2)}$.

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Nonlinear optomechanical coupling can be differentiated from nonlinear transduction by the $\mathrm{\Delta }$ dependence of the nonlinear signal. This is demonstrated in Fig. 3(b), which shows $S_P^{(2)}(2{\omega }_m,\mathrm{\Delta })$ with and without quadratic coupling, assuming that fabrication imperfections introduce nominal $g^{(1)}/2\pi =50\mathrm{MHz}/\mathrm{nm}$. This demonstrates that at $\mathrm{\Delta }=\kappa /2$, the nonlinear signal is dominantly from nonlinear optomechanical coupling.

Next we study the feasibility of QND phonon measurement using a paddle nanocavity. High ${\omega }_m$ is advantageous for ground-state cooling, which is required for QND measurements. The large optical mode spacing of the paddle nanocavity allows this without introducing Zener tunneling effects [13] or parasitic linear coupling and resulting backaction [25]. Cryogenic temperature of 10 mK could directly cool the $\mathcal{S}$ resonance of the $p3$ structure to its quantum ground state. For feasible optical and mechanical quality factors $Q_o={10}^6$ [8,30] and $Q_m={10}^5$ [9], the signal-to-noise ratio (SNR) introduced in Ref. [10] of a quantum jump measurement in such a device is ${\mathrm{\Sigma }}^{(0)}={\tau }_{\text{tot}}^{(0)}\mathrm{\Delta }{\omega }_0^2/S_{{\omega }_o}=6.4\times {10}^{-8}$. Here ${\tau }_{\text{tot}}^{(0)}$ is the thermal lifetime quantifying the rate of decoherence due to bath phonons of the ground-state-cooled nanomechanical resonator, and $S_{{\omega }_o}$ is the shot-noise-limited sensitivity of an ideal Pound–Drever–Hall detector. Introducing laser cooling would potentially allow preparation of the $p1$ device to its quantum ground state, where the larger $x_{zpf}$ and $\mathrm{\Delta }{\omega }_o$ increase ${\mathrm{\Sigma }}^{(0)}$ to $2.1\times {10}^{-5}$. However, this would require development of sideband unresolved nonlinear optomechanical cooling [15].

A more feasible approach for observing discreteness of the paddle nanocavity mechanical energy is a QND measurement of phonon shot noise [11]. The SNR of such a measurement scales with the magnitude of an applied drive, which enhances the signal by $S=8n_d\overline{n}{\mathrm{\Sigma }}^{(0)}$, where $n_d$ is the drive amplitude in units of phonon number, and $\overline{n}<1$ for a resonator in the quantum ground state. Using a $p3$ structure, SNR of more than 1 is achievable assuming a drive amplitude of 62 pm ($n_d\approx 7.8\times {10}^6$) and thermal bath phonon number $\overline{n}=1/4$.

In conclusion, we have designed a single-mode nonlinear optomechanical nanocavity with terahertz mode spacing. The quadratic optomechanical coupling coefficient $g^{(2)}/2\pi =400\mathrm{MHz}/{\mathrm{nm}}^2$ and single-photon to two-phonon coupling rate $\mathrm{\Delta }{\omega }_0/2\pi =16\mathrm{Hz}$ of this system are among the largest single-mode quadratic optomechanical couplings predicted to date. Observing a thermal nonlinear signal from this structure is possible in realistic conditions, and continuous QND measurements of phonon shot noise may be achievable for optimized device parameters. We acknowledge support from: