Position-Squared Coupling in a Tunable Photonic Crystal Optomechanical Cavity

We present the design, fabrication, and characterization of a planar silicon photonic crystal cavity in which large position-squared optomechanical coupling is realized. The device consists of a double-slotted photonic crystal structure in which motion of a central beam mode couples to two high-$Q$ optical modes localized around each slot. Electrostatic tuning of the structure is used to controllably hybridize the optical modes into supermodes that couple in a quadratic fashion to the motion of the beam. From independent measurements of the anticrossing of the optical modes and of the dynamic optical spring effect, a position-squared vacuum coupling rate as large as ${\tilde{g}}^{\prime }/2\pi =245\mathrm{Hz}$ is inferred between the optical supermodes and the fundamental in-plane mechanical resonance of the structure at ${\omega }_m/2\pi =8.7\mathrm{MHz}$, which in displacement units corresponds to a coupling coefficient of $g^{\prime }/2\pi =1\mathrm{THz}/{\mathrm{nm}}^2$. For larger supermode splittings, selective excitation of the individual optical supermodes is used to demonstrate optical trapping of the mechanical resonator with measured ${\tilde{g}}^{\prime }/2\pi =46\mathrm{Hz}$.

Popular Summary

Developing an optomechanical device capable of performing a continuous quantum nondemolition measurement of the energy stored in a mechanical resonator has been a long-sought-after goal. The stored energy in a mechanical resonator, proportional to its average squared displacement ($x^2$), can be used to infer the quantum jumps associated with individual photons or photons exiting or entering the optomechanical device. Despite significant technical advances made in recent years, the use of $x^2$ coupling for measuring or preparing nonclassical quantum states of a mesoscopic mechanical resonator remains an elusive goal. This fact stems from the small coupling rate to motion at the quantum level. For $x^2$ coupling, this rate scales as the square of the zero-point motion amplitude of the mechanical resonator, which for typical materials and mechanical objects, is roughly the diameter of a proton.

One method to greatly enhance $x^2$ coupling involves a multimoded cavity system in which two optical resonances, both coupled to mechanical motion, can be fine-tuned such that their mode splitting ($J$) is equal to that of the mechanical resonance frequency. In this work, we utilize a quasi-two-dimensional photonic crystal structure to create an optical cavity supporting a pair of tunable optical resonances that both couple to the motion of the structure. We measure an $x^2$-coupling coefficient as large as $1\mathrm{THz}/{\mathrm{nm}}^2$ to the fundamental mechanical resonance of the central beam at frequency 8.7 MHz. We also present additional measurements of the $x^2$ coupling through the dynamic and static optical spring effects. Compared with other systems, the corresponding vacuum $x^2$-coupling rate that we demonstrate is many orders of magnitude larger than what has been obtained in conventional Fabry-Pérot or fiber-gap membrane-in-the-middle systems.

We anticipate that the multimoded photonic crystal structures in this work will enable quantum nonlinear phononic and photonic systems to be realized and quantum nondemolition measurements of either phonon or photon number to be performed.

Figure 1

(a) Double-slotted photonic crystal cavity with optical cavity resonances ($a_1$, $a_2$) centered around the two slots, and three fundamental in-plane mechanical resonances corresponding to motion of the outer slabs ($b_1$, $b_2$) and the central nanobeam ($b_3$). Tuning the equilibrium position of the outer slabs $b_1$ and $b_2$, and consequently the slot size on either side of the central nanobeam, is achieved by pulling on the slabs (red arrows) through an electrostatic force proportional to the square of the voltage applied to capacitors on the outer edge of each slab. (b) Dispersion of the optical modes as a function of $x_3$, the in-plane displacement of the central nanobeam from its symmetric equilibrium position. Because of tunnel coupling at a rate $J$, the slot modes $a_1$ and $a_2$ hybridize into the even and odd supermodes $a_{+}$ and $a_{-}$, which have a parabolic dispersion near the central anticrossing point (${\omega }_1={\omega }_2$). (c) SEM image of a fabricated double-slotted photonic crystal device in the silicon-on-insulator material system. (d) Zoom-in SEM image showing the capacitor gap ($\sim 100\mathrm{nm}$) for the capacitor of one of the outer slabs. (e) Zoom-in SEM image showing some of the suspending tethers of the outer slabs which are of length $2.5\mu \mathrm{m}$ and width 155 nm. The central beam, which is much wider, is also shown in this image.

Figure 2

(a) Band structure diagram of the periodic (along $x$) double-slotted photonic crystal waveguide structure. Here, we show only photonic bands that are composed of modes with even vector symmetry around the “vertical” (${\sigma }_z$) mirror plane. The two waveguide bands of interest lie within the quasi-2D photonic band gap of the outer photonic crystal slabs and are shown as bold red and black curves. These waveguide bands are labeled “even” (bold black curve) and “odd” (bold red curve) due to the spatial symmetry of their mode shape for the dominant electric field polarization in the $y$ direction, $E_y$. The simulated structure is defined by the lattice constant between nearest-neighbor holes in the hexagonal lattice ($a_0=480\mathrm{nm}$), the thickness of the silicon slab ($d=220\mathrm{nm}$), the width of the two slots ($s=100\mathrm{nm}$), and the refractive index of the silicon layer ($n_{\mathrm{Si}}=3.42$). The hole radius in the outer slabs and the central nanobeam is $r=144\mathrm{nm}$. The gray shaded region represents a continuum of radiation modes which lie above the light cone for the air cladding which surrounds the undercut silicon slab structure. (b) Normalized $E_y$ field profile at the $X$ point of the odd waveguide supermode, shown for several unit cells along the $x$ guiding axis. (c) $E_y$ field profile of the even waveguide supermode. Waveguide simulations of (a)–(c) are performed using the plane-wave mode solver MPB [32, 33]. Normalized $E_y$ field profile of the corresponding localized cavity supermodes of (d) odd and (e) even spatial symmetry about the $y$ axis mirror plane. The lattice constant $a_0$ is decreased by 4.5% for the central five lattice constants between the dashed lines to localize the waveguide modes. Simulations of the full cavity modes are performed using the comsol finite-element method mode solver package [34].

Figure 3

Tuning of the slot widths of the double-slotted photonic crystal cavity showing (a) the mean wavelength shift and (b) the splitting $2J={\omega }_{+}-{\omega }_{-}$ of the even and odd cavity supermodes versus slot width $s=s_1=s_2$. (c),(d) Avoided crossing of the cavity supermodes obtained by tuning $s_1$ while keeping $s_2$ fixed at (c) $s_2=90\mathrm{nm}$ and (d) $s_2=95\mathrm{nm}$. The red and black data points correspond to the supermode branch with even and odd symmetry at the center of the anticrossing, respectively. Note that the upper and lower supermode branch switch symmetry between small slots ($s_2=90\mathrm{nm}$) and large slots ($s_2=95\mathrm{nm}$). For all simulations in (a)–(d) the parameters of the cavity structure are the same as in Fig. 2, except for the slot widths. The simulations are performed using the comsol FEM mode solver [34].

Figure 4

(a)–(c) Optical transmission measurements versus the wavelength of the probe laser showing the cavity mode anticrossing and tuning of the photon tunneling rate. In these measurements the probe laser wavelength (horizontal axis) is scanned across the optical cavity resonances as the voltage across the first capacitor $V_1$ is swept from low to high (vertical axis shows $V_1^2$ in $V^2$, proportional to slab displacement). The second capacitor is held fixed at (a) $V_2=1\mathrm{V}$, (b) $V_2=15\mathrm{V}$, and (c) $V_2=18\mathrm{V}$. The color scale indicates the fractional change in the optical transmission level $\mathrm{\Delta }T$, with blue corresponding to $\mathrm{\Delta }T=0$ and red corresponding to $\mathrm{\Delta }T\approx 0.25$. From the three anticrossing curves we measure a splitting $2J/2\pi$ equal to (a) 50 GHz, (b) 12 GHz, and (c) $-25\mathrm{GHz}$. (d)–(f) Corresponding simulations of the normalized optical transmission spectra for the slot width $s_1$ varied and the second slot width held fixed at (d) $s_2=93\mathrm{nm}$, (e) $s_2=95\mathrm{nm}$, and (f) $s_2=97\mathrm{nm}$. The dispersion and tunneling rate of the slot modes are taken from simulations similar to that found in Fig. 3. Panels (g) and (h) show the measured linewidths of the high-frequency upper (black) and low-frequency lower (red) optical resonance branches as a function of $V_1^2$, extracted from (a) and (c), respectively. The narrowing (broadening) is a characteristic of the odd (even) nature of the cavity supermode, indicating the inversion of the even and odd supermodes for the two voltage conditions $V_2=1\mathrm{V}$ and $V_2=18\mathrm{V}$. The lines are guides for the eye.

Figure 5

(a) Fine-tuning scan around the anticrossing point of the measured dispersion curve of Fig. 4. Black circles are the resonance frequencies obtained from fitting a double Lorentzian curve to each measured spectrum. (b) Plot of the linewidth versus $\mathrm{\Delta }{V}_1^2$ around the anticrossing point from the double-Lorentzian fit to the measured spectra of (a). Again, black (red) data points correspond to the upper (lower) frequency optical supermode. (c) Plot of the resonance frequency splitting versus $\mathrm{\Delta }{V}_1^2$. The solid black curves correspond to the fit curves for the 95% confidence interval of the fit value of the tunneling rate, $2J/2\pi =12.2\pm 1.1\mathrm{GHz}$.

Figure 6

rf photocurrent noise spectrum for the optically transmitted light past the double-slotted photonic crystal cavity. Here, the applied voltage $V_2=1\mathrm{V}$ is held fixed and $V_1$ is swept from just below to just above the anticrossing point of Fig. 4. In these measurements the probe laser power is $10\mu \mathrm{W}$ at the input to the cavity, the probe laser frequency is set on the blue side of the upper frequency supermode resonance, ${\mathrm{\Delta }}_L\approx \kappa /2\sqrt{3}$, and the fiber taper is placed in the near field of the photonic crystal cavity resulting in an on-resonance dip in transmission of approximately $\mathrm{\Delta }T=15\%$. The vertical axis in this plot is converted to a change in the slot width $\delta s_1$ using the numerically simulated value of ${\alpha }_{\mathrm{cap}}=0.025\mathrm{nm}/{\mathrm{V}}^2$. The color indicates the magnitude of the rf noise power spectral density (PSD) in $\mathrm{dBm}/\mathrm{Hz}$, where the color scale from 0 to 14 MHz is shown on the left of the scale bar and the color scale from 14 to 20 MHz is shown on the right of the scale bar (a different scale is used to highlight the noise out at $2{\omega }_{b_3}$).

Figure 7

(a) Dynamic optical spring effect measured by exciting the upper frequency supermode resonance far from the anticrossing point ($\sim a_2$ mode) ($V_1=V_2=1\mathrm{V}$, $P_{\mathrm{in}}=10\mu \mathrm{W}$, $\kappa /2\pi =12.5\mathrm{GHz}$, $\mathrm{\Delta }T\approx 10\%$). (b) Static optical spring shift of the $b_3$ resonance frequency versus laser detuning ${\mathrm{\Delta }}_L$ from the upper ($\sim a_{+}$) supermode resonance near the anticrossing point ($V_1=10.8\mathrm{V}$, $V_2=1\mathrm{V}$, $P_{\mathrm{in}}=50\mu \mathrm{W}$, $\kappa /2\pi =26\mathrm{GHz}$, $\mathrm{\Delta }T\approx 25\%$). In both (a) and (b) $V_2$ is fixed at 1 V [see Fig. 4] and the measured data (circles) correspond to a Lorentzian fit to the resonance frequency of the optically transduced thermal noise peak at ${\omega }_{b_3}$. In (a) the red curve is a fit to the data using a dynamical optical spring model [38] with linear optomechanical coupling coefficient ${\tilde{g}}_{a_2,b_3}/2\pi =1.72\mathrm{MHz}$. In (b) the red curve is a fit to the data using a static spring model [cf. Eqs. (8) and (9)] with $x^2$-coupling coefficient ${\tilde{g}}_{b_3}^{\prime }/2\pi =46\mathrm{Hz}$. In both the spring models of (a) and (b) the intracavity photon number versus detuning $n({\mathrm{\Delta }}_L)$ is calibrated from the known input laser power, cavity linewidth, and on-resonance transmission contrast.