A Topological Parametric Phonon Oscillator

Abstract

Topological bosonic systems have recently aroused intense interests in exploring exotic phenomena that have no counterparts in electronic systems. The squeezed bosonic interaction in these systems is particularly interesting, because it can modify the vacuum fluctuations of topological states, drive them into instabilities, and lead to topological parametric oscillators. However, these phenomena remain experimentally elusive because of limited nonlinearities in most existing topological bosonic systems. Here, a topological parametric phonon oscillator is experimentally realized based on a nonlinear nano-electromechanical Dirac-vortex cavity with strong squeezed interaction. Specifically, the Dirac-vortex cavity is parametrically driven to provide phase-sensitive amplification for topological phonons, leading to the observation of coherent parametric phonon oscillation above the threshold. Additionally, it is confirmed that the random frequency variation caused by fabrication disorders can be suppressed effectively by increasing the cavity size while the free spectral range reduces at a much slower rate, which benefits the realization of large-area single-mode lasers. These results represent an important advance in experimental investigations of topological physics with large bosonic nonlinearities and parametric gain.

Keywords: mechanical metamaterials; nano‐electromechanics; parametric oscillation; strong squeezed interaction; topological Dirac‐vortex states.

Figure 1

Nano‐electromechanical Dirac‐vortex state with squeezed interaction. a) Conceptual illustration of a nano‐electromechanical Dirac‐vortex parametric phonon oscillator. The device is based on a 2D array of suspended silicon nitride membranes, which encompasses a topological phase winding process rendered by the swirling cones. The heavily doped silicon substrate serves as the electrical ground, and the aluminum layer deposited on silicon nitride is connected to the signal electrode. Applying a combination of a d.c. bias voltage and an a.c. parametric pump voltage across the signal and ground electrodes leads to a strong squeezed interaction for the topological phonons, which enables a Dirac‐vortex parametric phonon oscillator. b) Scanning electron microscope image of the nano‐electromechanical crystal (before aluminum deposition) showing a unit cell of the hexagonal lattice (black dashed hexagon). Each unit cell contains two groups of suspended membranes (colored in blue and red), whose geometries are determined by the positions of the etched holes (r ₁, r ₂) and (r ₃, r ₄). c) Optical microscope image of the device (before aluminum deposition). It is color‐coded by the spatially varying parameters δ ₀(r)  =  δ _maxtanh(|r|/R ₀) and θ(r)  =  arg(r) with R ₀/l ₀ = 0.5; S, signal electrode; G, ground electrode. d) Simulated intensity profile of the Dirac‐vortex state. e) Measured mechanical intensity spectra of the device in panel (c). The gray‐shaded regions correspond to the bulk continuum. The Dirac‐vortex state lies in the bulk bandgap region with a resonant frequency ω ₀/2π = 45.238 MHz and a quality factor ≈2,874. f) Measured vibration quadrature of the Dirac‐vortex state without (V _pump = 0) and with (V _pump = 2.62 V, pump frequency = 2ω ₀) squeezed interaction.

Figure 2

Experimental demonstration of parametric phonon oscillation from a Dirac‐vortex state under a coherent pump. a) Measured mechanical power spectral density (PSD) of the device with R ₀/l ₀ = 0.5 under the pump voltage V _pump = 11 V, showing that the oscillation frequency is always half of the pump frequency. b) Measured peak PSD of the device with R ₀/l ₀ = 0.5 as a function of the pump frequency and pump voltage V _pump. c) Peak PSD as a function of V _pump with the pump frequency fixed at 90.44 MHz [along the white dashed line c in panel (b)]. d) Peak PSD as a function of the pump frequency with the pump voltage V _pump = 8.9 V [along the white dashed line d in panel (b)]. The bistable region is marked in gray. In panels (c) and (d), the dark blue circles and orange solid lines represent the measured data and theoretical fits, respectively.

Figure 3

Experimental demonstration of parametric phonon oscillation from a Dirac‐vortex state under an incoherent white‐noise pump. a) Measured mechanical PSD of the device with R ₀/l ₀ = 0.5 under different pump voltages. b) Measured normalized PSD under pump voltages V _pump = 2.16, 4.25, and 6.35 V. The solid black lines are Lorentzian fits of the measured data. c) Measured intensity modal profile of the Dirac‐vortex state above the oscillation threshold. d) Measured peak PSD of the device as a function of V _pump, showing a threshold pump voltage of 2.75 V. e) Fitted linewidth of the measured PSD as a function of V _pump. The error bars represent standard deviation during the linewidth fitting. f) Measured peak frequency as a function of V _pump. In panels (d)–(f), the region below the oscillation threshold is marked in pink, and the region of linewidth broadening is marked in light blue.

Figure 4

Experimental results of Dirac‐vortex parametric phonon oscillators with different modal areas. a) Optical microscope image of a fabricated device. The modal area S of the Dirac‐vortex state is controlled by the parameter R ₀. b) Measured mechanical intensity spectra of the devices with different R ₀ values, showing the existence of the Dirac‐vortex state in the bulk bandgap region of all these devices. c) Measured resonant frequencies of the Dirac‐vortex states with different R ₀/l ₀. The orange dots plot the frequency fluctuation δω measured from multiple devices with an identical design. We obtained δω/ω ₀ = 0.16%, 0.12%, 0.1%, and 0.045% for R ₀/l ₀ = 0.5, 1, 2, and 4, respectively. d) Measured mechanical PSD of the devices with R ₀/l ₀ = 1, 2, and 4 under an incoherent white‐noise parametric pump with a frequency bandwidth of 20 kHz. e) Measured oscillation intensity modal profiles |f ₀(r)|² (purple solid lines) along the x direction as indicated by the white dashed arrow in panel (a) from the devices with different R ₀ values. The orange dashed lines plot the fitted envelope function |g ₀(|r|)|². f) Measured resonant frequencies (purple dots) of the Dirac‐vortex state and its sidebands and the fitted modal area S (orange stars) of the Dirac‐vortex state from the devices with R ₀/l ₀ = 0.5, 1, 2, and 4.