Deep thermalization in Gaussian continuous-variable quantum systems

We uncover emergent universality arising in the equilibration dynamics of multimode continuous-variable systems. Specifically, we study the ensemble of pure states supported on a small subsystem of a few modes, generated by Gaussian measurements on the remaining modes of a globally pure bosonic Gaussian state. We find that beginning from highly entangled, complex global states, such as random Gaussian states and product squeezed states coupled via a deep array of linear optical elements, the induced ensemble attains a universal form, independent of the choice of measurement basis: it is composed of unsqueezed coherent states whose displacements are distributed normally and isotropically, with variance depending on only the particle-number density of the system. We further show that the emergence of such a universal form is consistent with a generalized maximum entropy principle, which endows the limiting ensemble, which we call the "Gaussian Scrooge distribution", with a special quantum information-theoretic property of having minimal accessible information. Our results represent a conceptual generalization of the recently introduced notion of "deep thermalization" in discrete-variable quantum many-body systems -- a novel form of equilibration going beyond thermalization of local observables -- to the realm of continuous-variable quantum systems. Moreover, it demonstrates how quantum information-theoretic perspectives can unveil new physical phenomena and principles in quantum dynamics and statistical mechanics.