State preparation and tomography in 3D multimode circuit QED*

Multimode cavity quantum electrodynamics, where a two level system interacts simultaneously with many cavity modes, provides a versatile framework for quantum information processing and quantum optics. One platform for realizing this involves coupling a transmon circuit to a multimode superconducting microwave cavity, offering a combination of high coherences and strong interactions. Our implementation of such a multimode processor uses the quantum flute: a seamless rectangular 3D multimode cavity with a tailored mode dispersion, ~2 ms decay times, and single-photon cooperativities ~1 billion for 9 distinct modes. We present ways to take advantage of the nonlinearity of a single transmon circuit for universal control of all the cavity modes. These include quantum optimal control pulses that utilize photon number dependent phase shifts, and Hilbert space engineering with photon blockade. We use these schemes to prepare Fock states of increasing photon number in any mode and multimode entangled states. We also present a novel, general scheme for multimode Wigner tomography that does not require engineering of the dispersive shift, or the use of higher transmon levels.