Extended Data Fig. 1: Estimation of beamsplitter interaction strengths.

a, Mode splitting induced by a beamsplitter interaction observed in thermomechanical spectra. Each column corresponds to a beamsplitter interaction induced between a pair of resonators i ↔ j (left: 1 ↔ 2, middle: 2 ↔ 3, right: 1 ↔ 3) by a single drive laser modulation at frequency Δω_ij = ω_i − ω_j, where ω_i,j is the frequency of resonator i, j. Thermomechanical spectra (top row: resonator i, bottom row: resonator j) are recorded for increasing modulation depth c_m. The linear relationship \({J}_{{\rm{est}}}={c}_{{\rm{m}}}\sqrt{{\rm{\delta }}{\omega }_{i}{\rm{\delta }}{\omega }_{j}}/2\) is used to estimate the coupling strength J_est (top axis) from c_m, where δω_i,j is the optical spring shift of mode i, j. The estimated mode splitting (dashed) is slightly larger than observed, presumably due to frequency-dependent transduction (at d.c. and frequency Δω_ij) in the measurement of c_m. The difference is quantified by extracting Lorentzian peak frequencies from the spectra and subsequently fitting those linearly against modulation depth, and results in an observed mode splitting slope that is 78%, 90% and 90% of the estimated slope, respectively. The average estimation offset of 86% is applied to all (beamsplitter and squeezing) interaction strength calculations in our experiments. b, Time evolution of the coherent amplitude (in units of their zero-point fluctuations x_zpf) of a pair of resonators (1, blue and 2, red) coupled through a beamsplitter interaction (strength J/(2π) = 5 kHz). Resonator 1 is initially (time t < 0) driven to a high-amplitude steady state by a coherent drive laser modulation. At t = 0, the drive is switched off and the interaction is switched on. Rabi oscillations induced by the coupling interaction are observed, where energy is transferred back and forth between the resonators until the coherent energy in the resonators is dissipated. These dynamics illustrate the possibility for a transfer scheme in the strong-coupling regime where couplings are interrupted after a Rabi semi-cycle, that is, a time t_π = π/(2J). The energy transfer efficiency for this process can be calculated⁶⁷ to be ~64% for corresponding parameters and 70% for the coupling rates presented in Fig. 1.