  Temporal modulation of the quantum vacuum through fast motion of a neutral
body or fast changes of its optical properties is known to promote virtual into
real photons, the so-called dynamical Casimir effect. Empowering modulation
protocols with spatial control could enable to shape the spectral, spatial,
spin, and entanglement properties of the emitted photon pairs. Space-time
quantum metasurfaces have been proposed as a platform to realize this physics
via modulation of their optical properties. Here, we report the mechanical
analog of this phenomenon by considering systems whose lattice structure
undergoes modulation in space and in time. We develop a microscopic theory that
applies both to moving mirrors with modulated surface profile and atomic array
meta-mirrors with perturbed lattice configuration. Spatio-temporal modulation
enables motion-induced generation of steered frequency-path entangled photon
pairs in co- and cross-polarized states, as well as vortex photon pairs
featuring frequency-angular momentum entanglement. The proposed space-time
dynamical Casimir effect can be interpreted as an induced dynamical asymmetry
in the quantum vacuum.
