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...