The superposition of quantum states is one of the hallmarks of quantum physics, and clear demonstrations of superposition have been achieved in a number of quantum systems. However, mechanical systems have remained a challenge, with only indirect demonstrations of mechanical state superpositions, in spite of the intellectual appeal and technical utility such a capability would bring 1,2 . This is due in part to the highly linear response of most mechanical systems, making quantum operation difficult, as well as their characteristically low frequencies, making it difficult to reach the quantum ground state [3][4][5][6][7][8] . In this work, we demonstrate full quantum control of the mechanical state of a macroscopic mechanical resonator. We strongly couple a surface acoustic wave 9 resonator to a superconducting qubit, using the qubit to control and measure quantum states in the mechanical resonator. Most notably, we generate a quantum superposition of the zero and one phonon states and map this and other states using Wigner tomography 10-15 . This precise, programmable quantum control is essential to a range of applications of surface acoustic waves in the quantum limit, including using surface acoustic waves to couple disparate quantum systems 16,17 .Linear resonant systems are traditionally challenging to control at the level of single quanta, as they are always in the correspondence limit 18 . The recent advent of engineered quantum devices in the form of qubits has enabled full quantum control over some linear systems, in particular electromagnetic resonators 14,15 . A number of experiments have demonstrated that qubits may provide similar control over mechanical degrees of freedom, including qubits coupled to bulk acoustic modes 3,8 , surface acoustic waves 19,20 , and flexural modes in suspended beams [21][22][23][24] . Of particular note are experiments in which a superconducting qubit is coupled via a piezoelectric material to a microwave-frequency bulk acoustic mode 25 , where the ground state can be achieved at moderate cryogenic temperatures, and demonstrations include controlled vacuum Rabi swaps between the qubit and the mechanical mode 3,8 . However, the level of quantum control and measurement has been limited by the difficulty of engineer-ing a single mechanical mode with sufficient coupling and quantum state lifetime. More advanced operations, such as synthesizing arbitrary acoustic quantum states and measuring those states using Wigner tomography, remain a challenge. Here we report a significant advance in the level of quantum control of a mechanical device, where we couple a superconducting qubit to a microwave-frequency surface acoustic wave resonance, demonstrating groundstate operation, vacuum Rabi swaps between the qubit and the acoustic mode, and the synthesis of mechanical Fock states as well as a Fock state superposition. We map out the Wigner function for these mechanical states using qubit-based Wigner tomography. We note that a similar achievement has recently been reported with an ex...

Phonons, and in particular surface acoustic wave phonons, have been proposed as a means to coherently couple distant solid-state quantum systems. Recent experiments have shown that superconducting qubits can control and detect individual phonons in a resonant structure, enabling the coherent generation and measurement of complex stationary phonon states. Here, we report the deterministic emission and capture of itinerant surface acoustic wave phonons, enabling the quantum entanglement of two superconducting qubits. Using a 2 mmlong acoustic quantum communication channel, equivalent to a 500 ns delay line, we demonstrate the emission and re-capture of a phonon by one qubit; quantum state transfer between two qubits with a 67% efficiency; and, by partial transfer of a phonon between two qubits, generation of an entangled Bell pair with a fidelity of FB = 84 ± 1%.

We describe the principles of design, fabrication, and operation of a piezoelectric optomechanical crystal with which we demonstrate bi-directional conversion of energy between microwave and optical frequencies. The optomechanical crystal has an optical mode at 1523 nm co-located with a mechanical breathing mode at 3.8 GHz, with a measured optomechanical coupling strength g om /2p of 115 kHz. The breathing mode is driven and detected by curved interdigitated transducers that couple to a Lamb mode in suspended membranes on either end of the optomechanical crystal, allowing the external piezoelectric modulation of the optical signal as well as the converse, the detection of microwave electrical signals generated by a modulated optical signal. We compare measurements to theory where appropriate.