This study demonstrates a beamsplitter-type interaction between multiple phonon modes of a high-overtone bulk acoustic-wave resonator (HBAR) coupled to a superconducting transmon qubit. The interaction is mediated by a parametrically driven qubit and can be tailored to couple pairs or triplets of phononic modes. The researchers show that this interaction enables the Hong–Ou–Mandel effect between phonons, which is a key quantum interference phenomenon. The work provides a foundation for using phononic systems as quantum memories and platforms for quantum simulations. The study focuses on circuit quantum acoustodynamics (cQAD), where a superconducting qubit is coupled to gigahertz-frequency acoustic modes. This system has been used to demonstrate the generation and measurement of non-trivial quantum states and entanglement between mechanical modes. The HBAR's large effective mass and multimode properties make it an excellent platform for bosonic quantum simulations, encodings, and quantum metrology. The researchers engineered a phononic iSWAP gate, which allows for the direct exchange of quanta between mechanical modes. This was achieved through a beamsplitter interaction, a coupling mechanism that has been studied in various quantum systems. The phononic beamsplitter interaction can serve as a building block for quantum computing architectures and offer new possibilities for simulating complex quantum systems. The study shows that the beamsplitter interaction can be used to create entanglement between two acoustic overtone modes of the HBAR. Additionally, the researchers created an interference between three phononic modes and explored the multimode dynamics governing the system. They also demonstrated the Hong–Ou–Mandel effect between macroscopic mechanical modes, showing that phonons can exhibit quantum interference. The device used in the study is a cQAD system where a superconducting qubit is flip-chip bonded to an HBAR. The qubit is a three-dimensional transmon with specific parameters, and the HBAR has a specific longitudinal free spectral range. The qubit and HBAR are coupled through a piezoelectric transducer, mediating a Jaynes–Cummings interaction. The researchers applied two off-resonant drives on the qubit to create a nonlinear mixing element, which enabled the beamsplitter interaction. They used qubit spectroscopy to study the effects of the drives and observed multiple sidebands that mediate the desired beamsplitter coupling. They then performed time-domain experiments to demonstrate iSWAP and √i SWAP gates, using the latter to demonstrate entanglement between two acoustic overtone modes. The study also explored the multimode dynamics of the system and showed that the beamsplitter interaction can be used to exchange multiple excitations between the modes. The results demonstrate the potential of phononic systems for quantum information processing and quantum simulations. The work provides a fundamental building block for performing quantum-optics-type experiments with massive mechanical excitations and addresses a key challenge towards realizingThis study demonstrates a beamsplitter-type interaction between multiple phonon modes of a high-overtone bulk acoustic-wave resonator (HBAR) coupled to a superconducting transmon qubit. The interaction is mediated by a parametrically driven qubit and can be tailored to couple pairs or triplets of phononic modes. The researchers show that this interaction enables the Hong–Ou–Mandel effect between phonons, which is a key quantum interference phenomenon. The work provides a foundation for using phononic systems as quantum memories and platforms for quantum simulations. The study focuses on circuit quantum acoustodynamics (cQAD), where a superconducting qubit is coupled to gigahertz-frequency acoustic modes. This system has been used to demonstrate the generation and measurement of non-trivial quantum states and entanglement between mechanical modes. The HBAR's large effective mass and multimode properties make it an excellent platform for bosonic quantum simulations, encodings, and quantum metrology. The researchers engineered a phononic iSWAP gate, which allows for the direct exchange of quanta between mechanical modes. This was achieved through a beamsplitter interaction, a coupling mechanism that has been studied in various quantum systems. The phononic beamsplitter interaction can serve as a building block for quantum computing architectures and offer new possibilities for simulating complex quantum systems. The study shows that the beamsplitter interaction can be used to create entanglement between two acoustic overtone modes of the HBAR. Additionally, the researchers created an interference between three phononic modes and explored the multimode dynamics governing the system. They also demonstrated the Hong–Ou–Mandel effect between macroscopic mechanical modes, showing that phonons can exhibit quantum interference. The device used in the study is a cQAD system where a superconducting qubit is flip-chip bonded to an HBAR. The qubit is a three-dimensional transmon with specific parameters, and the HBAR has a specific longitudinal free spectral range. The qubit and HBAR are coupled through a piezoelectric transducer, mediating a Jaynes–Cummings interaction. The researchers applied two off-resonant drives on the qubit to create a nonlinear mixing element, which enabled the beamsplitter interaction. They used qubit spectroscopy to study the effects of the drives and observed multiple sidebands that mediate the desired beamsplitter coupling. They then performed time-domain experiments to demonstrate iSWAP and √i SWAP gates, using the latter to demonstrate entanglement between two acoustic overtone modes. The study also explored the multimode dynamics of the system and showed that the beamsplitter interaction can be used to exchange multiple excitations between the modes. The results demonstrate the potential of phononic systems for quantum information processing and quantum simulations. The work provides a fundamental building block for performing quantum-optics-type experiments with massive mechanical excitations and addresses a key challenge towards realizing