Scrambling and complexity in phase space

The study of information scrambling in many-body systems has sharpened our understanding of quantum chaos. In this talk, we will address the question of scrambling and operator complexity in continuous variable (CV) systems. Unlike their discrete variable 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 of a many-body system. Moreover, for each of these domains, we identify two distinct "types" of scrambling; strong scrambling, where an initial operator localized in phase space spreads out and weak 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. Finally, we offer a number of results regarding the CV analog for unitary designs, enabling a complexity hierarchy for CV unitaries. Our work opens the door to experimentally probing phase space scrambling in CV systems, such as cavity QED architectures.