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...

Photonic cavities have emerged as an indispensable tool to control and manipulate harmonic motion in opto/ electromechanical systems 1. Invariably, in these systems a high-quality-factor photonic mode is parametrically coupled to a high-quality-factor mechanical oscillation mode 2-12. This entails the demanding challenges of either combining two physically distinct systems, or else optimizing the same nanostructure for both mechanical and optical properties 4-11. In contrast to these approaches, here we show that the cavity can be realized by the second oscillation mode of the same mechanical oscillator 13,14. A piezoelectric pump 15,16 generates strain-induced parametric coupling between the first and the second mode at a rate that can exceed their intrinsic relaxation rate. This leads to a mechanically induced transparency in the second mode which plays the role of the phonon cavity 17,18 , the emergence of parametric normal-mode splitting 19,20 and the ability to cool the first mode 2-11. Thus, the mechanical oscillator can now be completely manipulated by a phonon cavity 21. The dynamical backaction of a photonic cavity that is parametrically coupled to a mechanical oscillation mode has recently led to the realization of a quantized macroscopic mechanical system 10,11,22-24 , a long-standing goal in solid-state physics 25. The backaction of the mechanical motion on the cavity has also resulted in the emergence of opto/electromechanically induced transparency, which has great potential for communications technology and quantum information science 17,18,20,26-28. The success of this approach is leveraged on the requirement that the mechanical oscillation completes many cycles before the cavity relaxes, thus enhancing the effectiveness of the dynamical backaction, or in other words the coupled system is operated in the resolved sideband regime 1,22,23. The parametric coupling in these systems arises from the harmonic motion of the mechanical element modifying the cavity's resonance frequency by means of a change in either the cavity's length or capacitance. This has resulted in exquisitely engineered devices in which the mechanical motion can be manipulated by the parametrically coupled photonic cavity 1. In contrast to a photonic cavity, a phonon cavity operated in the resolved sideband regime should also be able to host dynamical backaction onto the mechanical element. One approach to this goal could be realized by physically coupling an additional mechanical oscillator to the system. Here we show that a more natural method is to simply couple two different colour mechanical oscillation modes in the same mechanical system, where the first mode represents the mechanical oscillation of interest and the second mode affords a phonon cavity. The key to this approach is a geometric intermodal coupling, where the motion of the first (second) mode creates tension that causes a shift in the frequency of the second (first) mode, that can be parametrically manipulated 29,30 , which enables phonon-cavity electromechanics ...

An electromechanical resonator is developed in which mechanical nonlinearities can be dynamically engineered to emulate the nondegenerate parametric down-conversion interaction. In this configuration, phonons are simultaneously generated in pairs in two macroscopic vibration modes, resulting in the amplification of their motion. In parallel, two-mode thermal squeezed states are also created, which exhibit fluctuations below the thermal motion of their constituent modes as well as harboring correlations between the modes that become almost perfect as their amplification is increased. The existence of correlations between two massive phonon ensembles paves the way towards an entangled macroscopic mechanical system at the single phonon level.

We propose an active mechanism for coupling the quantized mode of a nanomechanical resonator to the persistent current in the loop of superconducting Josephson junction (or phase slip) flux qubit. This coupling is independently controlled by an external coupling magnetic field. The whole system forms a novel solid-state cavity QED architecture in strong coupling limit. This architecture can be used to demonstrate quantum optics phenomena and coherently manipulate the qubit for quantum information processing. The coupling mechanism is applicable for more generalized situations where the superconducting Josephson junction system is a multi-level system. We also address the practical issues concerning experimental realization.