Coupled mechanical oscillations were first observed in paired pendulum clocks in the mid-seventeenth century and were extensively studied for their novel sympathetic oscillation dynamics [1, 2]. In this era of nanotechnologies, coupled oscillations have again emerged as subjects of interest when realized in nanomechanical resonators for both practical applications and fundamental studies [3][4][5][6][7][8][9][10][11]. However, a key obstacle to the further development of this architecture is the ability to coherently manipulate the coupled oscillations. This limitation arises as a consequence of the usually weak coupling between the constituent nanomechanical elements. Here, we report parametrically coupled mechanical resonators in which the coupling strength can be dynamically adjusted by modulating (pumping) the stress in the mechanical elements via a piezoelectric transducer. The parametric control enables the coupling rate between the two resonators to be made so strong that it exceeds their intrinsic energy dissipation rate by more than a factor of four. This ultra-strong coupling can be exploited to coherently transfer phonon populations, namely phonon Rabi oscillations [12,13], between the mechanical resonators via two coupled vibration modes, realizing superposition states of the two modes and their time-domain control. More unexpectedly, the nature of the parametric coupling can also be tuned from a linear first-order interaction to a non-linear higher-order process in which more than one pump phonon mediates the coherent oscillations. This demonstration of multipump phonon mixing echoes multi-wave photon mixing [14] and suggests that concepts from nonlinear optics can also be applied to mechanical systems. Ultimately, the parametric pumping is not only useful for controlling classical oscillations [15] but can also be extended to the quantum regime [12,13,[16][17][18], opening up the prospect of entangling two distinct macroscopic mechanical objects [19,20].The dynamic parametric coupling is developed in GaAs-based paired mechanical beams shown in Fig. 1a, in which the piezoelectric effect is exploited to mediate all-electrical displacement transduction [21]. The frequency response of beam 1 measured by harmonically driving it while the parametric pump is deactivated displays two coupled vibration modes (Fig. 1b), where mode 1 (ω 1 = 2π × 293.93 kHz) is dominated by the vibration of beam 1 while mode 2 (ω 2 = 2π × 294.37 kHz) is dominated by the vibration of beam 2. The amplitude of mode 2 is much smaller than that of mode 1 reflecting the energy exchange due to the structural coupling via the overhang is inefficient because of the eigenfrequency difference between the two beams. This difference can be compensated by activating the parametric pump, which is induced by piezoelectrically modulating the spring constant of beam 1 with the pump frequency ω p at around the frequency difference between the two modes, ∆ω ≡ ω 2 − ω 1 (Fig. 1c).The dynamics of this system can then be expressed by the following e...

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Transport in InAs/GaSb heterostructures with different InAs layer thicknesses is studied using a six-terminal Hall bar geometry with a 2-µm edge channel length. For a sample with a 12-nm-thick InAs layer, non-local resistance measurements with various current/voltage contact configurations reveal that the transport is dominated by edge channels with negligible bulk contribution. Systematic non-local measurements allow us to extract the resistance of individual edge channels, revealing sharp resistance fluctuations indicative of inelastic scattering. Our results show that the InAs/GaSb system can be tailored to have conducting edge channels while keeping a gap in the bulk region and provide a way of studying 2D topological insulators even when quantized transport is absent.

Nanoelectromechanical systems (NEMS), utilizing localized mechanical vibrations, have found application in sensors, signal processors and in the study of macroscopic quantum mechanics. The integration of multiple mechanical elements via electrical or optical means remains a challenge in the realization of NEMS circuits. Here, we develop a phonon waveguide using a one-dimensional array of suspended membranes that offers purely mechanical means to integrate isolated NEMS resonators. We demonstrate that the phonon waveguide can support and guide mechanical vibrations and that the periodic membrane arrangement also creates a phonon bandgap that enables control of the phonon propagation velocity. Furthermore, embedding a phonon cavity into the phonon waveguide allows mobile mechanical vibrations to be dynamically switched or transferred from the waveguide to the cavity, thereby illustrating the viability of waveguide-resonator coupling. These highly functional traits of the phonon waveguide architecture exhibit all the components necessary to permit the realization of all-phononic NEMS circuits.