Scrambling and complexity in phase space*

In this talk, we will describe extensions of the study of scrambling and complexity to infinite-dimensional continuous variable (CV) systems. Unlike their discrete variable (DV) cousins, continuous variable systems exhibit two complementary domains of information scrambling: 1) scrambling in the phase space of a single mode and 2) scrambling across multiple modes. Moreover, for each of these domains, we identify two distinct "types" of scrambling; strict scrambling, where an initial operator localized in phase space spreads out and quasi-scrambling, where a local ensemble of operators distorts but the overall phase space volume remains fixed. To characterize these behaviors, we introduce a CV out-of-time-order correlator (OTOC) based upon displacement operators, which can be experimentally measured. By studying operator spreading and entanglement formation in a random local Gaussian circuit ensemble, we infer the dynamics of generic, chaotic, locally-interacting systems.Our work opens the door to experimentally probing phase space scrambling in CV systems, including cavity QED and quantum optics architectures.