  Time-varying optical media, whose dielectric properties are actively
modulated in time, introduce a host of novel effects in the classical
propagation of light, and are of intense current interest. In the quantum
domain, time-dependent media can be used to convert vacuum fluctuations
(virtual photons) into pairs of real photons. We refer to these processes
broadly as ``dynamical vacuum effects'' (DVEs). Despite interest for their
potential applications as sources of quantum light, DVEs are generally very
weak, providing many opportunities for enhancement through modern techniques in
nanophotonics, such as using media which support excitations such as plasmon
and phonon polaritons. Here, we present a theory of DVEs in arbitrary
nanostructured, dispersive, and dissipative systems. A key element of our
framework is the simultaneous incorporation of time-modulation and
``dispersion'' through time-translation-breaking linear response theory. We
propose a highly efficient scheme for generating entangled surface polaritons
based on time-modulation of the optical phonon frequency of a polar insulator.
We show that the high density of states, especially in hyperbolic polaritonic
media, may enable high-efficiency generation of entangled phonon-polariton
pairs. More broadly, our theoretical framework enables the study of quantum
light-matter interactions in time-varying media, such as spontaneous emission,
and energy level shifts.
