Single-Crystal Diamond Nanobeam Waveguide Optomechanics

Single-crystal diamond optomechanical devices have the potential to enable fundamental studies and technologies coupling mechanical vibrations to both light and electronic quantum systems. Here, we demonstrate a single-crystal diamond optomechanical system and show that it allows excitation of diamond mechanical resonances into self-oscillations with amplitude $>200\mathrm{nm}$. The resulting internal stress field is predicted to allow driving of electron spin transitions of diamond nitrogen-vacancy centers. The mechanical resonances have a quality factor $>7\times {10}^5$ and can be tuned via nonlinear frequency renormalization, while the optomechanical interface has a 150 nm bandwidth and $9.5\mathrm{fm}/\sqrt{\mathrm{Hz}}$ sensitivity. In combination, these features make this system a promising platform for interfacing light, nanomechanics, and electron spins.

Popular Summary

The superior optical, mechanical, and thermal properties of single-crystal diamond make it an attractive platform for technologies ranging from lasers to x-ray optics. Single-crystal diamond also has applications in optomechanics and quantum optics research. Intriguingly, single-crystal diamond is host to defects such as nitrogen-vacancy color centers with excellent quantum characteristics that act as sensors and quantum information storage elements. The electronic and nuclear states of these defects can be manipulated using optical and microwave photons and, as shown recently, phonons. On-chip optomechanical technology has the potential to provide control and measurement of phonons of individual devices using light, the preferred medium for transmitting quantum information. Here, we use a scalable fabrication technique to create a single-crystal diamond optomechanical system based on waveguide coupling to the nanobeams.

Optomechanical devices enhance optical forces, allowing low-power actuation and high-sensitivity readout of mechanical structures. They can simultaneously provide photon-phonon and phonon-spin coupling, enabling new hybrid approaches for optical control of spins and transferring quantum information between disparate systems. The nanobeams we employ double as optical waveguides, enabling sensitive optomechanical readout of their mechanical characteristics via phase-matched coherent coupling to an optical fiber. We are able to amplify the nanobeam thermal motion into self-oscillations greater than 200 nanometers in amplitude in cryogenic conditions (5 K). A comprehensive study of the nonlinear nanobeam dynamics reveals that the photothermal force in these devices is enhanced by the presence of compressive stress so that only 100 nW of absorbed optical power is needed to excite the self-oscillations.

These oscillations are predicted to enable strong phonon-spin coupling within the diamond nanobeam, paving the way for the future development of hybrid quantum devices that may be useful for sensing.

Figure 1

Diamond nanobeam waveguide-optomechanical system. (a) SEM images of a single-crystal diamond nanobeam waveguide. Dark high-contrast regions are due to variations in thickness of titanium deposited for imaging purposes. (b) Schematic of fiber taper evanescent waveguide coupling geometry and illustration of the waveguide-optomechanical coupling process. Nanobeam mechanical resonances, whose typical displacement profiles are shown, modulate the distance between the input waveguide, changing $\kappa (h)$. (c) Dispersion of $n_{\mathrm{eff}}$ of the waveguide modes of the fiber taper (diameter $1.1\mu \mathrm{m}$) and diamond nanobeam ($w\times d=460\times 250{\mathrm{nm}}^2$) when they are uncoupled (fiber taper shown by red dashed line, nanobeam shown by blue dashed line) and coupled with $h=300\mathrm{nm}$ (solid lines). Shaded region indicates the laser scan range used in the experiments. Insets: Mode profiles ($x\text{−}y$ plane, dominant electric field component $|E_y|$) of the symmetric ($+$) and antisymmetric ($-$) coupled modes.

Figure 2

Process flow for creating diamond nanobeams using quasi-isotropic reactive-ion undercut etching.

Figure 3

Efficient evanescent waveguide coupling. (a) Fiber taper transmission $\overline{T}(\lambda )$ versus wavelength for $h=200\mathrm{nm}$. Inset: Variation of $\overline{T}$ with nanobeam width $w$ and $\lambda$ for approximately constant $h$. (b) Dependence of fiber taper transmission on fiber height. Experimentally measured values for ${\overline{T}}_o$ (red points) are fit with the model described by Eq. (1) (red line). The corresponding $\kappa (h)$ extracted from the fit (blue line) and predicted from simulation (blue points) are also shown.

Figure 4

Nanomechanical and optomechanical properties. (a) Measured $s_x(f)$ when the fiber taper is coupled to a nanobeam ($L\times w\times d=50\times 0.46\times 0.25\mu {\mathrm{m}}^3$) in ambient conditions, with $P_d\sim 100\mu \mathrm{W}$. The vertical axis and noise floor (dashed line) of $s_x^o=9.5\mathrm{fm}/\sqrt{\mathrm{Hz}}$ are calibrated from the fit to the $f=1.3\mathrm{MHz}$ mechanical resonance thermomechanical displacement spectrum (red solid line). (b) Top: $S_v(f_1{)}^{1/2}$ observed (points) and fit with a function $\propto |d\overline{T}/dh|$ (solid line). Bottom: Displacement sensitivity $s_x^o(h)$ of the fundamental out-of-plane resonance from measurement (blue points) and predicted from the measured $T(h)$ for varying input power (solid lines). The starred point corresponds to the sensitivity and operating condition of the measurement in (a). Measured $S_v(f)$ for a nanobeam with $L\times w\times d=60\times 0.75\times 0.3\mu {\mathrm{m}}^3$ (c) in vacuum (d) at 5 K. Insets: Fits to observed mechanical resonances, and corresponding $Q_m$. Unlabeled peaks in (a), (c), and (d) below 1 MHz are related to fiber vibrations.

Figure 5

Nanobeam self-oscillations. (a) Time-resolved $T(h(t))$ of a phase-matched fiber taper nanobeam ($L\times w\times d=80\times 0.50\times 0.25\mu {\mathrm{m}}^3$) system as $h$ is reduced in discrete steps (visible as sharp steps in $T$). Inset: Large-amplitude oscillation of the nanobeam position. (b) Spectrograph showing the power spectral density of the data in (a). (c) Predicted renormalized mechanical dissipation rate compared with measured values. (d) Scatter plot of shift in nanobeam resonance frequency versus oscillation amplitude squared for data in the inset regions from (b) at the onset of self-oscillation. The solid line is a linear guide to the eye.

Figure 6

(a) Nanobeam cross sections, created by focused ion beam (FIB) milling, of two nanobeams with different $w$ patterned on the same chip. Note that the FIB milling process rounds the nanobeam edges, resulting in some apparent curvature in the nanobeam profiles. (b) SEM image of typical nanobeams observed to self-oscillate. Dashed straight lines are guides to emphasize the direction of nanobeam buckling.

Figure 7

Propagation constants (${\beta }_{\pm }$) of the even and odd supermodes of the coupled nanobeam and fiber taper waveguides, as a function of waveguide separation $h$, for varying $\lambda$. Points indicated by open squares and circles where calculated using Lumerical MODE Solutions. Solid lines are fits using the coupled mode theory model described in Appendix pp3.

Figure 8

Waveguide coupling coefficients as a function of fiber taper and nanobeam waveguide separation $h$. Each data point is obtained by fitting ${\beta }_{\pm }(\lambda )$ with the coupled-mode theory model. Solid lines are single exponential fits to the data points.

Figure 9

Predicted optomechanical coupling coefficients for varying fiber height when (a) $L_c=7\mu \mathrm{m}$, as in the experiment, and (b) $L_c=21\mu \mathrm{m}$, for a nanobeam of length $L=L_c$. Also shown is the coupling rate between the nanobeam and the forward propagating fiber taper waveguide mode ${\gamma }_e(h)$. (c) Corresponding transmission functions for $L_c=7\mu \mathrm{m}$ and $L_c=21\mu \mathrm{m}$.

Figure 10

Predicted achievable phonon number via optomechanical cooling of a fiber taper coupled nanobeam incorporated into an oval racetrack resonator (${\gamma }_p=0$). Device parameters are ${\gamma }_i/2\pi =1\mathrm{GHz}$, $L_c=L=21\mu \mathrm{m}$, ${\omega }_m/2\pi =7.3\mathrm{MHz}$, $m=3.1\mathrm{pg}$, $Q_m={10}^6$, 5 K bath temperature. For the oval racetrack cavity, $2L=2L_c+2\pi R_c$, where $R_c=10\mu \mathrm{m}$ is the radius of curvature of the cavity end.

Figure 11

Optical gradient force as a function of $h$, when the fiber taper is aligned along the nanobeam axis, and offset 500 nm laterally, approximating the position in the self-oscillation measurements.

Figure 12

(a) Longitudinal cross section of nanobeam used in ansys finite element simulations, showing the “arched” clamping-point geometry. (b) Finite element simulations of resonance frequency and nanobeam deflection as a function of axial compressive stress. Dashed lines indicate value of $F_i$ where simulated ${\omega }_m$ approximately matches the experimentally observed value. (c) Simulated change in resonance frequency and nanobeam deflection for $P_{\mathrm{abs}}=1\mu \mathrm{W}$ of absorbed optical power. (d) Simulated change in resonance frequency from a $F_g=1\mathrm{pN}$ transverse load applied across the coupling region of the nanobeam (length $L_c$). Nanobeam dimensions are $L\times w\times d=80\times 0.48\times 0.25\mu {\mathrm{m}}^3$, as in the self-oscillation measurements.

Figure 13

Predicted ${\omega }_m^{\prime }(h)$ generated using the model in Eq. (g10) and the parameters in Table 1.