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Hosting non-classical states of light in three-dimensional microwave cavities has emerged as a promising paradigm for continuous-variable quantum information processing. Here we experimentally demonstrate high-fidelity generation of a range of Wigner-negative states useful for quantum computation, such as Schr\"{o}dinger-cat states, binomial states, Gottesman-Kitaev-Preskill (GKP) states, as well as cubic phase states. The latter states have been long sought after in quantum optics and were never achieved experimentally before. To do so, we use a sequence of interleaved selective number-dependent arbitrary phase (SNAP) gates and displacements. We optimize the state preparation in two steps. First we use a gradient-descent algorithm to optimize the parameters of the SNAP and displacement gates. Then we optimize the envelope of the pulses implementing the SNAP gates. Our results show that this way of creating highly non-classical states in a harmonic oscillator is robust to fluctuations of the system parameters such as the qubit frequency and the dispersive shift.

openaire: EC/H2020/732894/EU//HOT Funding Information: We acknowledge the facilities and technical support of Otaniemi research infrastructure for Micro and Nanotechnologies (OtaNano) that is part of the European Microkelvin Platform. This work is supported by the Academy of Finland (Contracts No. 307757 and No. 312057), by the European Research Council (615755-CAVITYQPD), by the Wihuri Foundation, and by the Aalto Centre for Quantum Engineering. The work is performed as part of the Academy of Finland Centre of Excellence program (Project No. 336810). We acknowledge funding from the European Union’s Horizon 2020 research and innovation program under Grant Agreement No. 732894 (FETPRO HOT). Publisher Copyright: © 2022 American Physical Society. Studies of micromechanical and acoustic modes in the quantum regime have shed light on quantum properties of massive objects. Integrating these systems into superconducting circuits shows great promise for applications as quantum memory elements, bosonic codes, or in frequency conversion. To this end, investigation of nonclassical properties of acoustic degrees of freedom is critical also for applications. Here, we investigate a strongly driven system consisting of a transmon qubit interacting with a high-overtone bulk acoustic resonator. We observe multiphonon transitions, which enable mapping the energy landscape in the coupled system. At a high driving amplitude comparable to the qubit-oscillator coupling, we observe a shift of the multiphonon spectral lines, reminiscent of Stark shift, which is explained using a simple model. Our work thus also provides insight in multiquanta transitions in other qubit-oscillator systems, not limited to acoustics or circuit quantum electrodynamics.

Mechanical resonators are a promising way for interfacing qubits in order to realize hybrid quantum systems that offer great possibilities for applications. Mechanical systems can have very long energy lifetimes, and they can be further interfaced to other systems. Moreover, integration of a mechanical oscillator with qubits creates a potential platform for the exploration of quantum physics in macroscopic mechanical degrees of freedom. The utilization of high overtone bulk acoustic resonators coupled to superconducting qubits is an intriguing platform towards these goals. These resonators exhibit a combination of high-frequency and high-quality factors. They can reach their quantum ground state at dilution refrigeration temperatures and they can be strongly coupled to superconducting qubits via their piezoelectric effect. In this paper, we demonstrate our system where bulk acoustic phonons of a high overtone resonator are coupled to a transmon qubit in a planar circuit architecture. We show that the bulk acoustic phonons are interacting with the qubit in a simple design architecture at the quantum level, representing further progress towards the quantum control of mechanical motion.