Near-Field Integration of a SiN Nanobeam and a ${\mathrm{SiO}}_2$ Microcavity for Heisenberg-Limited Displacement Sensing

Placing a nanomechanical object in the evanescent near field of a high-$Q$ optical microcavity gives access to strong gradient forces and quantum-limited displacement readout, offering an attractive platform for both precision sensing technology and basic quantum optics research. Robustly implementing this platform is challenging, however, as it requires integrating optically smooth surfaces separated by $\lesssim \lambda /10$. Here we describe an exceptionally high-cooperativity, single-chip optonanomechanical transducer based on a high-stress ${\mathrm{Si}}_3{\mathrm{N}}_4$ nanobeam monolithically integrated into the evanescent near field of ${\mathrm{SiO}}_2$ microdisk cavity. Employing a vertical integration technique based on planarized sacrificial layers, we realize beam-disk gaps as little as 25 nm while maintaining mechanical $Qf>{10}^{12}\mathrm{Hz}$ and intrinsic optical $Q\sim {10}^7$. The combination of low loss, small gap, and parallel-plane geometry results in radio-frequency flexural modes with vacuum optomechanical coupling rates of 100 kHz, single-photon cooperativities in excess of unity, and large zero-point frequency (displacement) noise amplitudes of $10\mathrm{kHz}(\mathrm{fm})/\surd \mathrm{Hz}$. In conjunction with the high power-handling capacity of ${\mathrm{SiO}}_2$ and low extraneous substrate noise, the transducer performs particularly well as a sensor, with recent deployment in a 4-K cryostat realizing a displacement imprecision 40 dB below that at the standard quantum limit (SQL) and an imprecision-backaction product $<5\hslash$ [Wilson et al., Nature (London) 524, 325 (2015)]. In this report, we provide a comprehensive description of device design, fabrication, and characterization, with an emphasis on extending Heisenberg-limited readout to room temperature. Towards this end, we describe a room-temperature experiment in which a displacement imprecision 32 dB below that at the SQL and an imprecision-backaction product $<60\hslash$ is achieved. Our results extend the outlook for measurement-based quantum control of nanomechanical oscillators and suggest an alternative platform for functionally integrated “hybrid” quantum optomechanics.

Figure 1

(a) False-colored scanning electron micrograph of the device: a high-stress ${\mathrm{Si}}_3{\mathrm{N}}_4$ (red) nanomechanical beam integrated into the evanescent mode volume of a ${\mathrm{SiO}}_2$ (blue) microdisk. Disk and beam are integrated on a Si (gray) microchip. Subpanel (b) [(c)] highlights the lateral (vertical) positioning of the beam.

Figure 2

Survey of single-photon cooperativity ${\mathcal{C}}_0=4g_0^2/\kappa {\mathrm{\Gamma }}_m$ for various cavity optomechanical systems [19, 37, 38, 39, 40, 41, 42, 43] adapted with permission from Ref. [11]. Nonitalicized references are cited in Ref. [11]. Blue and red points correspond to cryogenic (typically $T<10\mathrm{K}$) and room-temperature experiments, respectively. Diagonal lines indicate the condition for ${\mathcal{C}}_0=n_{\mathrm{th}}\approx k_{B}T/\hslash {\mathrm{\Omega }}_m$ for various $T$. The reported result is highlighted with crosshairs.

Figure 3

$Q$ factor (red) and $Q\times \text{frequency}$ product (blue) of the first 11 odd-ordered out-of-plane flexural modes of a nanobeam with dimensions $\{l,w,t\}=\{60,0.6,0.05\}\mu \mathrm{m}$. Solid red curve is a fit to the $Q$-dilution model in Ref. [47], implying a limiting contribution from surface-related intrinsic loss.

Figure 4

(a), (b) SEM of a wedged microdisk; blue and gray indicate ${\mathrm{SiO}}_2$ and Si, respectively. (c) WGM quality factor $Q_o$ as a function of disk radius $r_d$ for stand-alone ${\mathrm{SiO}}_2$ microdisks of thickness $t_d\approx 700\mathrm{nm}$. TE and TM modes are not distinguished. Blue (red) points correspond to disks prepared with photolithography ($e$-beam lithography), which produce wedge angles of $\theta \approx 30°$ (11°). Horizontal lines represent constant cavity linewidth $\kappa =2\pi c/(\lambda Q_o)$, with $\lambda =780\mathrm{nm}$. Blue (red) dashed line is a guide to the eye for $Q\propto r_d$, corresponding to a fixed finesse of $\mathcal{F}=0.6(1.2)\times {10}^5$.

Figure 5

(a) Geometry of the nanobeam-microdisk system: $x$, $y$ represent the vertical (out-of-plane) and lateral (in-plane) position of the beam, respectively, with respect to the inner rim of the disk (thickness $t_d$, radius $r_d$). (b) Simulated optomechanical coupling versus beam position for the device shown in Fig. 1. The intensity profile of a TM-like WGM (computed using finite-element analysis) is shown in the background. Solid and dashed white lines denote the disk surface and the boundary within which the beam touches the disk surface for the coordinate system defined in (a). Contours indicate lines of constant $g_0$ for the 4.3-MHz fundamental out-of-plane beam mode. (c) Measured and simulated $g_0$ versus $y$ for the beam shown in Fig. 1. Black and blue data are for fundamental out-of-plane and in-plane beam modes, respectively (for details, see Sec. 4d). Black lines correspond to numerical solutions to Eq. (1) with a vertical offset of $x=25\mathrm{nm}$. Gray shading indicates the solution space for $x=20–30\mathrm{nm}$.

Figure 6

Fabrication process flow: blue, red, green, and (light) gray indicate ${\mathrm{SiO}}_2$, ${\mathrm{Si}}_3{\mathrm{N}}_4$, ${\mathrm{Al}}_2{\mathrm{O}}_3$, and (poly-)Si, respectively. Steps (a)–(h) are detailed in Sec. 3.

Figure 7

Defining the vertical gap between the disk and the nanobeam. (a) Top view of patterned ${\mathrm{SiO}}_2$ prior to selective etch of the microdisk. Photoresist protects the sacrificial structures, while a window is exposed around the microdisk. (b) Top view after selective etch of the microdisk and removal of the photoresist. The altered color of the microdisk indicates thinning.

Figure 8

Defining the nanobeam and the “mesa.” (a) Top view of sample after etching of ${\mathrm{Si}}_3{\mathrm{N}}_4$ (pink and purple). Surrounding ${\mathrm{SiO}}_2$ structures, including the microdisk, appear green. (b) Image of the mesa photomask.

Figure 9

(a) Overview of the experimental apparatus described in Sec. 4a. (b) Representative optical $Q$ measurement. WGM loss rates ($\kappa$) and mode splitting ($\gamma$) are inferred from the cavity transmission profile (red) generated by sweeping the diode laser frequency while monitoring the transmitted power. The sweep is calibrated by simultaneously monitoring transmission through a fiber loop cavity (blue). (c) Representative thermomechanical noise measurement. ${\mathrm{\Omega }}_m$, ${\mathrm{\Gamma }}_m$, and $g_0$ are inferred from the center frequency, linewidth, and area beneath the thermal-noise peak (pink), respectively. The latter is calibrated by normalizing to the area beneath a FM tone (blue).

Figure 10

Optical spring measurement. (a) Thermal-noise spectrum of the fundamental beam mode as a function of laser detuning. Blue and red spectra indicate blue ($\mathrm{\Delta }>0$) and red ($\mathrm{\Delta }<0$) detuning, respectively. Lighter shades indicate smaller detuning. Blue spectra are vertically offset. (b) Plot of optical spring shift $\mathrm{\Delta }{\mathrm{\Omega }}_m$ versus normalized detuning, $\mathrm{\Delta }/\kappa$. Dashed black lines are a fit to Eq. (4) using $g_0$ as a free parameter.

Figure 11

(a) Measured vacuum optomechanical coupling rate ($g_0$) and single-photon cooperativity (${\mathcal{C}}_0$, assuming ${\mathrm{\Gamma }}_m=2\pi \times 20\mathrm{Hz}$) versus lateral beam position ($y$) for TM (solid circles) and TE (open circles) cavity modes. (b) Corresponding intrinsic cavity decay rate (${\kappa }_0$). (c) Measured $g_0$ versus beam width ($w$) for two disk thicknesses ($t_d$). (d) Measured $g_0$ versus mode frequency, ${\mathrm{\Omega }}_m^{(n)}\approx n{\mathrm{\Omega }}_m^{(0)}$. Red dots correspond to odd harmonics ($n=1,3,5\dots$). Solid and dashed lines are model curves [Eq. 5] for a sampling length of $l_{\mathrm{eff}}=9.6$ and $l_{\mathrm{eff}}=0$, respectively.

Figure 12

(a) Nanobeam displacement noise measured by balanced homodyne detection of the microdisk output field, for various input powers. Noise spectra are expressed in units relative to the cavity frequency noise produced by one phonon of fundamental out-of-plane vibration $2S_{\omega }^{\mathrm{ZP}}({\mathrm{\Omega }}_m)=(2\pi \times 10\mathrm{kHz}/\sqrt{\mathrm{Hz}}{)}^2$, where ${\mathrm{\Omega }}_m=2\pi \times 4.4\mathrm{MHz}$. At large powers, the fundamental noise peak is shifted and broadened by optical spring softening and damping, respectively. The peak at 4.9 MHz is due to thermal motion of the fundamental in-plane mode. The gray curve is a model for the intrinsic thermal motion of the fundamental out-of-plane and in-plane modes [Eq. (6)]. (b) Measured phonon-equivalent displacement $n_{\mathrm{tot}}=S_{\omega }({\mathrm{\Omega }}_m)/2S_{\omega }^{\mathrm{ZP}}({\mathrm{\Omega }}_m)$, displacement imprecision $n_{\mathrm{imp}}=S_{\omega }^{\mathrm{imp}}({\mathrm{\Omega }}_m)/2S_{\omega }^{\mathrm{ZP}}({\mathrm{\Omega }}_m)$, and their geometric mean versus intracavity photon number $n_c$ weighted by single-photon cooperativity ${\mathcal{C}}_0$. Dashed lines indicate ideal values for $n_{\mathrm{tot}}=n_{\mathrm{th}}+n_{\mathrm{ba}}+n_{\mathrm{imp}}$ (orange), $n_{\mathrm{ba}}={\mathcal{C}}_0{n}_c$ (red), and $n_{\mathrm{imp}}=1/16{\mathcal{C}}_0{n}_c$ (blue), using $n_{\mathrm{th}}=1.4\times {10}^6$ and ${\mathcal{C}}_0=0.45$. Green arrow indicates proximity to the uncertainty limit, $4\sqrt{n_{\mathrm{imp}}{n}_{\mathrm{tot}}}\ge 1$.

Figure 13

Variation on a theme: suspending the nanobeam from tethers enables higher $g_0$ by reducing mass without changing optomechanical mode overlap. Here the central beam coincides with the effective sampling length of the optical mode.