  The study of information scrambling in many-body systems has sharpened our
understanding of quantum chaos, complexity and gravity. Here, we extend the
framework for exploring information scrambling to infinite dimensional
continuous variable (CV) systems. Unlike their discrete variable cousins,
continuous variable systems exhibit two complementary domains of information
scrambling: i) scrambling in the phase space of a single mode and ii)
scrambling across multiple modes of a many-body system. Moreover, for each of
these domains, we identify two distinct `types' of scrambling; genuine
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 correlation (OTOC) function based upon displacement
operators and offer a number of results regarding the CV analog for unitary
designs. Finally, we investigate operator spreading and entanglement growth in
random local Gaussian circuits; to explain the observed behavior, we propose a
simple hydrodynamical model that relates the butterfly velocity, the growth
exponent and the diffusion constant. Experimental realizations of continuous
variable scrambling as well as its characterization using CV OTOCs will be
discussed.
