Synthetic magnetism has been used to control charge neutral excitations for applications ranging from classical beam steering to quantum simulation. In optomechanics, radiation-pressure-induced parametric coupling between optical (photon) and mechanical (phonon) excitations may be used to break time-reversal symmetry, providing the prerequisite for synthetic magnetism. Here we design and fabricate a silicon optomechanical circuit with both optical and mechanical connectivity between two optomechanical cavities. Driving the two cavities with phase-correlated laser light results in a synthetic magnetic flux, which in combination with dissipative coupling to the mechanical bath, leads to nonreciprocal transport of photons with 35 dB of isolation. Additionally, optical pumping with blue-detuned light manifests as a particle non-conserving interaction between photons and phonons, resulting in directional optical amplification of 12 dB in the isolator through direction. These results indicate the feasibility of utilizing optomechanical circuits to create a more general class of nonreciprocal optical devices, and further, to enable novel topological phases for both light and sound on a microchip.

We investigate the synchronization of oscillators based on anharmonic nanoelectromechanical resonators. Our experimental implementation allows unprecedented observation and control of parameters governing the dynamics of synchronization. We find close quantitative agreement between experimental data and theory describing reactively coupled Duffing resonators with fully saturated feedback gain. In the synchronized state we demonstrate a significant reduction in the phase noise of the oscillators, which is key for sensor and clock applications. Our work establishes that oscillator networks constructed from nanomechanical resonators form an ideal laboratory to study synchronization-given their high-quality factors, small footprint, and ease of cointegration with modern electronic signal processing technologies. Synchronization is a ubiquitous phenomenon both in the physical and biological sciences. It has been observed to occur over a wide range of scales-from the ecological [1], with oscillation periods of years, to the microscale [2], with oscillation periods of milliseconds. Although synchronization has been extensively studied theoretically [3][4][5], relatively few experimental systems have been realized that provide detailed insight into the underlying dynamics. Here we show that oscillators based on nanoelectromechanical systems (NEMS) can readily enable the resolution of such details, while providing many unique advantages for experimental studies of nonlinear dynamics [6][7][8]. In addition, nanomechanical systems might prove useful for exploring quantum synchronization [9,10].Nanomechanical oscillators also have been exploited for a variety of applications [11][12][13]. In particular, nanoscale mechanics exhibits enhanced nonlinearity [14,15] and tunability [16,17], which has been used to suppress feedback noise [18,19] and create new types of electromechanical oscillators [20][21][22]. These oscillators may find application as mass [23], gas [24,25], or force sensors [26], without the need of an external frequency source.Building frequency sources from arrays of NEMS may yield enhanced applicability, but is challenging. For example, statistical deviations in batch fabrication inevitably lead to undesirable array dispersion [24]. If an array has appreciable frequency dispersion, global sensor responsivity gets reduced. However, if the elements of the array are made into a self-sustained oscillators and synchronized with one another, then the array responsivity will recover due to a reduction in phase noise [3]. Since NEMS have numerous applications, and are useful in studying nonlinear dynamics, we set an important milestone by demonstrating synchronization in nanomechanical systems.There are previous reports of synchronization in microor nanomechanical systems. However, these do not, in fact, demonstrate the phenomenon as conventionally defined [3] -that is, the phase locking of weakly coupled selfsustained oscillators. Shim et al.[27] reported synchronization of the driven excitations in coupled resonat...

I. DESIGN OF MECHANICAL CAVITY AND WAVEGUIDE COUPLINGIn the optomechanical cavity-waveguide coupled devices, we can change the cavity-waveguide coupling (i.e. γ e ) on purpose. This is achieved by the design of a low-Q mechanical cavity mode and varying the number of mirror cells . As shown in Fig. S-1b, the blue curves are the mechanical band structure of the mirror unit cell (blue rectangle in Fig. S-1a). We design the cavity such that the mechanical cavity frequency (red dashed line in Fig. S-1b) overlaps with the band of mirror unit cell, such that the mechanical cavity mode can tunnel through the mirror cells into waveguide. Meanwhile, the optical cavity frequency (red dashed line in Fig. S-1c) lies within the optical band gap of mirror unit cell, such that the optical cavity mode keeps high-Q. By varying the number of unit cells, we find the simulated radiation mechanical coupling rate into waveguide (γ e /2π) oscillates between a few MHz to as high as 30 MHz, due to the interference within the mirror unit cells. II. SIMULATION OF PHONON PULSE PROPAGATIONIn this section, we show propagation and bouncing of phonon pulses in the cavity-waveguide system (Fig. 3a) can be well simulated by a group of coupled mode equations using input-output formalism. The dynamics captured by the coupled mode equations is a phonon pulse travelling in a waveguide terminated by two cavities with bare mechanical frequency ω mL,R and waveguide coupling rate γ eL,R . We approximate ω mL,R to be the frequency of cavity-dominated modes L 1 and R 1 in the simulation. Since the response time of the optical cavity is much shorter than that of the mechanical cavity, we can exclude the dynamics of optical modes from these equations. Thus, the coupled mode equations can be written as follows,where α 0L and α +L are the amplitudes of optical pump and its red sideband in the left cavity, τ is the duration of excitation pulse, ω s is the frequency of pulse, Θ(t) is the Heaviside step function, γ is the effective decay rate of the excited mechanical mode, α ≈ γ/v g is the waveguide loss rate, and t w = 1/(is the single trip time the pulse spent in the waveguide.From the mechanical spectrum we find γ = 2π × 2.1 MHz for L 1 mode (the main coherently-driven mode) during the pulse measurement; and by fitting the pulse tails detected in each cavity we find γ eL = 2π ×34.7 MHz and γ eR = 2π × 25.5 MHz. Using these parameters, |b L | and |b R | can be numerically calculated from the coupled mode equations and the proportional voltage signals are shown in Fig. 3a. The simulated result captures the main features of the measured pulse data. In particular, the pulse splitting observed from cavity R is due to the fact that the pulse frequency is not in resonant with cavity R and thus experiences destructive interference inside this cavity.The phonon transfer efficiency from cavity L to cavity R is about e −γ/(2fFSR) ≈ 67%. The phonon transfer efficiency from cavity to waveguide for cavity L and R is γ eL(R) /(γ eL(R) + γ) ≈ 94%(92%).We summarize the measu...

Synchronization of oscillators, a phenomenon found in a wide variety of natural and engineered systems, is typically understood through a reduction to a first-order phase model with simplified dynamics. Here, by exploiting the precision and flexibility of nanoelectromechanical systems, we examined the dynamics of a ring of quasi-sinusoidal oscillators at and beyond first order. Beyond first order, we found exotic states of synchronization with highly complex dynamics, including weak chimeras, decoupled states, traveling waves, and inhomogeneous synchronized states. Through theory and experiment, we show that these exotic states rely on complex interactions emerging out of networks with simple linear nearest-neighbor coupling. This work provides insight into the dynamical richness of complex systems with weak nonlinearities and local interactions.