Nonlinear optomechanical paddle nanocavities

Nonlinear optomechanical coupling is the basis for many potential future experiments in quantum optomechanics (e.g., quantum non-demolition measurements, preparation of non-classical states), which to date have been difficult to realize due to small non-linearity in typical optomechanical devices. Here we introduce an optomechanical system combining strong nonlinear optomechanical coupling, low mass and large optical mode spacing. This nanoscale"paddle nanocavity"supports mechanical resonances with hundreds of fg mass which couple nonlinearly to optical modes with a quadratic optomechanical coupling coefficient $g^{(2)}>2\pi\times400$ MHz/nm$^2$, and a two phonon to single photon optomechanical coupling rate $\Delta \omega_0>2\pi\times 16$ Hz. This coupling relies on strong phonon-photon interactions in a structure whose optical mode spectrum is highly non--degenerate. Nonlinear optomechanical readout of thermally driven motion in these devices should be observable for T $>50 $ mK, and measurement of phonon shot noise is achievable. This shows that strong nonlinear effects can be realized without relying on coupling between nearly degenerate optical modes, thus avoiding parasitic linear coupling present in two mode systems.

Nonlinear optomechanical coupling is the basis for many potential future experiments in quantum optomechanics (e.g., quantum non-demolition measurements, preparation of non-classical states), which to date have been difficult to realize due to small non-linearity in typical optomechanical devices. Here we introduce an optomechanical system combining strong nonlinear optomechanical coupling, low mass and large optical mode spacing. This nanoscale "paddle nanocavity" supports mechanical resonances with hundreds of fg mass which couple nonlinearly to optical modes with a quadratic optomechanical coupling coefficient g (2) > 2π ×400 MHz/nm 2 , and a two phonon to single photon optomechanical coupling rate ∆ω0 > 2π × 16 Hz. This coupling relies on strong phononphoton interactions in a structure whose optical mode spectrum is highly non-degenerate. Nonlinear optomechanical readout of thermally driven motion in these devices should be observable for T > 50 mK, and measurement of phonon shot noise is achievable. This shows that strong nonlinear effects can be realized without relying on coupling between nearly degenerate optical modes, thus avoiding parasitic linear coupling present in two mode systems.
Recent progress in developing optomechanical systems with large nonlinear optomechanical coupling has been driven by studies of membrane-in-the-middle (MiM) [18][19][20][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 which combine strong nonlinear coupling and large optical mode spacing. This can be achieved in short, low-mass, high-finesse optical cavities. In this work we present such a nanocavity optomechanical system, which couples modes possessing low optical loss and THz 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 quadratic optomechanical coupling coefficient g (2) ≈ 2π × 400 MHz/nm 2 and single photon to two phonon coupling rate ∆ω 0 = 2π × 16 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 ω o (x) = ω 0 +g (1) x+ 1 2 g (2) x 2 , where ω o is the cavity resonance frequency, and g (1) = δω o /δx, g (2) = δ 2 ω o /δx 2 are the first and second order optomechanical coupling coefficients. In nanophotonic devices, x parameterizes a spatially varying modification to the local dielectric constant, ∆ (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 δω (2) on the overlap between ∆ and the optical modes of the nanocavity [26,27]: where the first term is a "self-term" and g (2) ω ,ω represents cross-couplings between the fundamental mode of inter- est (ω) and other modes supported by cavity (ω ): Here E ω denotes the electric field of a nanocavity mode at frequency ω, and the inner product is an overlap surface integral defined in Ref. [26] and developed in the context of optomechanics in Refs. [27,28] (see Supplementary information). In cavity optomechanical systems with no linear coupling (δω (1) = 0), the contribution in Eq. (1) from the self-overlap of the dielectric perturbation vanishes, and the quadratic coupling is determined entirely by mechanically induced cross-coupling between the nanocavity's optical modes. Enhancing this coupling can be realized in two ways. In the first approach, demonstrated in Refs. [18][19][20][21][22], the factor ω 2 /(ω 2 − ω 2 ) can be enhanced in a cavity with nearly-degenerate modes (ω ∼ ω ) which are coupled by a mechanical perturbation. An alternative approach which is desirable to avoid multimode dynamics [21] is to maximize the g (2) ω ,ω overlap terms. Here we investigate this route, and present a system with optical modes isolated by THz in frequency which possesses high quadratic optomechanical coupling owing to a strong overlap between optical and mechanical fields.
The optomechanical device studied here, illustrated in Fig. 1, is a photonic crystal "paddle nanocavity" which combines operating principles of MiM cavities [10,29] and photonic crystal nanobeam optomechanical devices [2]. The device is designed to be fabricated from siliconon-insulator (refractive index n Si = 3.48, thickness t = 220 nm), and to support modes near λ ∼ 1550 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 nm) separating the mirrors from the paddle is chosen for smooth variation in local effective-index of the structure [30], and the ω 1 ωn to the quadratic optomechanical coupling g (2) describing interaction between mode M1 and the S mechanical resonance shown in (b). (b) Displacement profiles and properties of the paddle nanocavity mechanical resonances. m and fm are indicated for three support geometries, p1 − p3, whose cross-sections are given in (c).
paddle length (L = 958 nm) is set to ≈ 1.5λ/n 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 ≥ 200 nm and w s ≤ 200 nm is required to not degrade Q o . We consider three support geometries, labeled p1 − p3 (see 2 for dimentions). All of these dimensions are realizable experimentally [7]. Figure 2(a) shows the first seven localized optical modes supported by the paddle nanocavity, calculated using finite element simulations (FEM) [32]. The lowest order mode (M 1) has a resonance wavelength near 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" (S), "bouncing" (B), "rotational" (R) and "torsional" (T ) resonances. As discussed below, we are particularly interested in the S resonance, whose frequency and effective mass [28] varies between f m = 0.35 − 217 MHz and m = 314 − 589 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 sensitiv-ity, and p3 for high frequency operation and low thermal phonon occupation.
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 σ x,y,z = 1, while the mechanical resonances induce perturbation ∆ which is odd in at least one direction, characterized by σ (1) is also zero. However, the electric field amplitude E(x, y, z) may be even or odd, resulting in non-zero cross-coupling g (2) ω ,ω between optical modes with opposite σ x,y,z . For example, displacement in thê x direction of S couples optical modes with opposite σ x . In contrast, displacement in theẑ direction by the B and T resonances does not induce cross-coupling, as the localized optical modes all have even vertical symmetry (σ z = 1). Here we focus on the nonlinear coupling between the S resonance and the M 1 mode of the nanocavity.
To evaluate g (2) ω,ω , the mechanical and optical field profiles were input into Eq. (2). The resulting contributions g (2) ω1ωn of each localized mode to g (2) for optomechanical coupling between the S resonance and the M 1 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 ω o , which is small for the localized modes whose Q o > 10 2 , is also ignored. A total g (2) /2π ≈ −400 MHz/nm 2 is predicted, which matches with our direct FEM calculations (see Supplementary information). The corresponding single photon to two phonon coupling rate, ∆ω o depends on the support geometry. For the most flexible p1 geometry, ∆ω o ≡ |g (2) x 2 zpf | = 2π×16 Hz, where x zpf = /2mω m . This ∆ω 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 M 1 ↔ M 4 and M 1 ↔ M 7 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. (2) P (2ωm) as a function of detuning ∆, for varying quadratic coupling strengths g (2) .
Here ∆ = ω − ω o is the detuning between input photons and the nanocavity mode, and dT /d∆, d 2 T /d∆ 2 are the slope and curvature of the Lorentzian cavity resonance in T (∆). Eq. (4) shows that in general, both nonlinear transduction of linear optomechanics and linear transduction of nonlinear optomechanics contribute to the second order signal. The nonlinear mechanical displacement can be measured through photodetection of the waveguide optical output. For input power P i , the waveguide output optical power spectral density (PSD) due to transduction of x 2 (t) is S is the PSD of the x 2 mechanical motion of the mechanical resonance. To analyze the possibility of observing this signal, it is instructive to consider the scenario of a thermally-driven mechanical resonance. As shown in the Supplementary information and Refs. [15,33], S x 2 (ω) of a resonator in an phonon number thermal state is where Γ = ω m /Q m and Q m is the mechanical quality factor. Figure 3 shows S (2) P (ω) predicted from Eq. (5), for the S mode of a paddle nanocavity at room temperature (T b = 300K). The input optical field is set to P i = 100 µW, with detuning ∆ = κ/2 to maximize the nonlinear optomechanical coupling contribution. The predicted S (2) P (ω) is shown for p1 and p2 support geometries, assuming Q m = 10 3 , Q o = 1.4 × 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 (NEP=2.5 pW/ √ Hz). Resonances in S (2) P are evident at ω = 2ω m and ω = 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 2ω 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 kHz range. Nonlinear optomechanical coupling can be differentiated from nonlinear transduction by the ∆ dependence of the nonlinear signal. This is demonstrated in Fig. 3, which shows S (2) P (2ω m , ∆) with and without quadratic coupling, assuming that fabrication imperfections introduce nominal g (1) /2π = 50 MHz/nm. This demonstrates that at ∆ = κ/2, the nonlinear signal is dominantly from nonlinear optomechanical coupling.
Next we study the feasibility of QND phonon measurement using a paddle nanocavity. High ω 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 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 tot is the thermal lifetime quantifying the rate of decoherence due to bath phonons of the ground state cooled nanomechanical resonator, and S ωo 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 ∆ω o increases Σ (0) to 2.1 × 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 dn Σ (0) , where n d is the drive amplitude in units of phonon number, and n < 1 for a resonator in the quantum ground state. Using a p3 structure, SNR of above one is achievable assuming a drive amplitude of 62 pm (n d ≈ 7.8 × 10 6 ) and thermal bath phonon numbern = 1/4.
In conclusion, we have designed a single-mode nonlinear optomechanical nanocavity with THz mode spacing. The quadratic optomechanical coupling coefficient g (2) /2π = 400 MHz/nm 2 and single photon to two phonon coupling rate ∆ω 0 /2π = 16 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 a continuous QND measurements of phonon shot noise may be achievable for optimized device parameters. Supplementary Information for "Nonlinear optomechanical paddle nanocavities"

EVALUATION OF THE NONLINEAR OPTOMECHANICAL COUPLING COEFFICIENT
The matrix element used in the perturbation theory calculation of g (2) is a measure of the overlap of the nanocavity optical fields and the shifting dielectric boundaries of the mechanical resonance. It is discussed in detail in Refs. [S1-S3], and is given by where the integral is evaluated over the surface of the nanocavity, and e ω and d ⊥ ω are the components of the the optical mode electric and displacement fields parallel and perpendicular to the surface, respectively. The perturbation introduced by the mechanical resonance is described by the normalized displacement of the dielectric boundaries, q = Q(r)/|Q(r)| max where Q(r) is the vectorial displacement field. For the device studied here, the dielectric contrast is constant, and is described by ∆ = 1 − 2 and ∆( −1 ) = 1/ 2 − 1/ 1 , where 1 is the dielectric constant of the nanocavity, and 2 = 1 is the dielectric constant of the surrounding medium.

NONLINEAR OPTOMECHANICAL SIGNAL
Here we analyze the optical power spectrum generated by a thermally driven mechanical oscillator quadratically coupled to an optical nanocavity. As described by Eq. (5) in the main text, the optical energy spectrum of a quadratically coupled mechanical resonator in a cavity optomechanical system can be written in terms of the autocorrelation of displacement squared, Expressing the displacement in terms of annihilation and creation operators b and b † as x = x zpf (be iωmt + b † e −iωmt ) and substituting the displacement operator into Eq. (S2) yields S x 2 (ω) = 2πx 4 zpf (2(n + 1) 2 δ(ω − 2ω m ) + 2n 2 δ(ω + 2ω m ) + (8n(n + 1) + 1)δ(ω) .
wheren is the mean thermal phonon number and T b is the bath temperature. For large phonon numbersn 1, it is approximated byn = k b T b / ω m and the area under the nonlinear spectrum is given by which is in agreement with the moment relation for a thermal distribution x 4 = 3 x 2 2 [S4]. For low loss mechanical resonators (Γ ω m ) we can replace the delta functions with a Lorentzian δ(ω − ω m ) = 1 π Γ Γ 2 +(ω−ωm) 2 , resulting in the following formula for power spectral density, Assuming a large thermal phonon occupancy (n 1), for frequencies near the double mechanical frequency (ω ≈ 2ω m ) we obtain following normalized form (using Eq. (S4)) for nonlinear power spectral density One obtains a similar result from a classical analysis which assumes that during the mechanical decay time, ∆t ≈ 1/Γ, the thermal force acts as a delta function "kick". In this approximation, the spectral density of the thermal force is given by [S5] For a measurement time on the order of ∆t, From Eq. (S9) and Eq. (S2), and using the convolution properties of Fourier transforms, we find after imposing the normalization given by Eq. (S4). As illustrated in Fig. S1, the classical nonlinear signal described by Eq. (S10) matches the quantum result of Eq. (S5) whenn 1, in the neighbourhood of ω ∼ 2ω m . This analysis is in agreement with results in Ref. [S6].   (2) x 2 (solid line) with g (2) /2π = 400 MHz/nm 2 . Error bars determined by the lowest significant digit of ωo.

VALIDATING THE SECOND ORDER PERTURBATION THEORY
The accuracy of the second order perturbation theory, whose use has not been previously reported for nanophotonic cavity-optomechanical devices to the best of our knowledge, was tested by comparing its results with FEM calculations of ω o (x), Here x parameterizes the paddle displacement from the center position between the two mirrors of the simulated structure. This displacement closely approximates the motion of the S resonance which we are primarily interested in here.
This comparison is shown in Fig. S2, where we find good agreement for displacements |x| ≤ 2 nm, and deviation for larger displacements as the perturbation condition breaks down. This agreement confirms the validity of the assumptions underlying the second order perturbation theory. It also highlights the suitability of this method, as extracting g (2) from parameterized FEM simulations has considerable uncertainty due a 2 nm minimum mesh available with our computation tool.