GoKawiilExperimental realization
The gyroscopic system outlined in Fig. 1a is realized using a 4-mm-diameter on-chip disc resonator exhibiting 12-fold symmetry. Extended Data Fig. 1a shows a top-view photograph, highlighting the resonator in yellow. The deformable structure of the resonator features ten concentric rings linked by radial spokes, with its central part bonded to a substrate. Fabricated from 100-μm-thick P-type single-crystal silicon, the resonator is housed in a vacuum environment to reduce air damping. The resonator and associated circuitry are integrated on a printed circuit board mounted on a temperature-controlled (25 ± 0.05 °C) angular rate table. This set-up allows accurate out-of-plane rotations with precise angular velocities (error < 0.001° s−1). Electric connections are maintained during rotation using slip rings, linking the rotating parts with external equipment.
The elastic deformations in the rings and spokes of the silicon disc resonator lead to several eigenstates. This research uses a pair of nearly degenerate in-plane wineglass SW modes with wavenumber n = 3 (that is, with 2 × n = 6 nodes or antinodes arranged circularly), designated as modes 1 and 2 in Extended Data Fig. 1b. Each SW mode is a combination of two in-plane whispering-gallery TW modes sharing the wavenumber n = 3, which travel in opposite directions, denoted as CW and counterclockwise (CCW) in Extended Data Fig. 1c. The relationship between SW and TW modes is detailed later.
The fixed nodes and antinodes of SW modes enable actuation, transduction and tuning using fixed capacitive electrodes, situated with uniform 10-μm gaps from the resonator. Electrodes are colour-coded by function in Extended Data Fig. 1a. Differential actuation is achieved by applying antiphase signals to two opposite antinodal electrodes. Modes 1 and 2 are driven simultaneously, with the drive signal of mode 2 phase-shifted by +π/2 relative to mode 1. The quadrature excitation of these modes creates a CW TW mode. Antinodal displacements of modes 1 and 2 are detected capacitively in a differential set-up using charge amplifiers.
The transduced displacement signals, q 1,2 , are processed by a lock-in amplifier (Zurich Instruments MFLI) for demodulation against the reference driving signal, yielding the amplitudes |q 1,2 | and phases θ 1,2 . The relative phase ϑ between θ 2 and θ 1 is also recorded. Open-loop measurements use the parametric sweeper block of the lock-in amplifier, sequentially adjusting the reference driving frequency ω d while monitoring the amplitudes and phases as functions of ω d , that is, |q 1,2 |(ω d ), θ 1,2 (ω d ). In closed-loop operation, a PLL with a PID controller keeps the phase of mode 1 at −π/2, by adjusting the oscillation frequency. This PLL-controlled frequency, achieving θ 1 = −π/2 phase-tracking, is denoted ω T .
Imperfections in the degeneracy of the fabricated disc resonator are corrected through a preparatory tuning process using a proven mode-matching technique51. This procedure uses two electrode sets, depicted as pink and green in Extended Data Fig. 1a. A DC voltage V c is implemented on one (for negative V c ) or two (for positive V c ) off-axis green electrodes between the principal axes of modes 1 and 2 to facilitate stiffness coupling. This results in an electrostatically tunable off-axis spring with stiffness \({{\Delta }}_{{\rm{c}}}={E}_{1,2}(2{V}_{0}{V}_{{\rm{c}}}-{V}_{{\rm{c}}}^{2})\), in which E 1 ≈ 10,066 N m−1 kg−1 V−2 or E 2 ≈ 21,068 N m−1 kg−1 V−2 correspond to tuning factors for one or two electrodes52. Here V 0 = 30 V is the static voltage applied to the resonator body. The off-axis tuning introduces a coupling with strength g = −Δ c /(2ω 0 ), with the negative sign owing to a roughly −15° azimuthal angle between the tuning electrodes and the principal axis of mode 1. Fabrication flaws may cause slight misalignment of the electrodes with the central axis of modes 1 and 2, resulting in minor non-degeneracy, which is corrected by readjusting the in-axis tuning voltage V f on the pink electrodes in Extended Data Fig. 1a. The DC voltages V 0 , V c and V f are provided by the precise IT2800 Source Measure Units.
Sensitivity measurements
In the small-range measurement, the system is carefully tuned at the balanced cusp singularities, X 1 : (V c ≈ −0.777 V, Ω ≈ 0° s−1) or X 2 : (V c ≈ 2.170 V, Ω ≈ 0° s−1), to enhance gyroscopic sensitivity. The angular velocity, Ω, is varied incrementally from −0.15° s−1 to 0.15° s−1 in 0.01° s−1 steps, with stabilization at each point for about 120 s to allow the system to settle. The steady-state outputs, PhT frequency ω T and relative phase ϑ are recorded for 5 s, producing 525 data points for each angular velocity, depicted by blue (for X 1 ) or red (for X 2 ) points in Extended Data Fig. 2a,b for ω T and ϑ, respectively.
The ω T frequency outputs in small-range measurements are strongly affected by resonant-frequency fluctuations. To mitigate these effects, a differential output configuration is used, measuring the difference between each frequency output and a reference output ω T (Ω r ) at a constant angular velocity Ω r = 1.5° s−1 after each valid output (black points in Extended Data Fig. 2a). These differential values are adjusted by ω T (Ω r ) − ω T (0) to shift the origin to zero, with ω T (0) being the expected frequency at Ω = 0° s−1. The offset frequency differences δω X1,2 at singularities X 1 and X 2 are indicated by blue and red points in Fig. 3a, respectively. Each point represents the average of 525 samples, with error bars showing the standard deviation.
By contrast, the phase outputs ϑ for small-range measurements exhibit much greater stability, as shown in Extended Data Fig. 2b. Thus, no differential operation is applied to the ϑ outputs, notwithstanding that reference outputs ϑ(Ω r ) have been recorded (black points in Extended Data Fig. 2b). The mean and standard deviation of 525 ϑ samples for each angular velocity input are depicted as points with error bars in Fig. 4b.