Recent experimental realizations of uniform confining potentials for ultracold atoms make it possible to create quantum acoustic resonators and explore nonequilibrium dynamics of quantum field theories. These systems offer a promising platform for studying the dynamical Casimir effect, since they allow one to achieve relativistic, i.e., near sonic, velocities of the boundaries. In comparison to previously studied optical and classical hydrodynamic systems, ultracold atoms allow one to realize a

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... broader class of dynamical experiments combining both classical driving and vacuum squeezing. In this paper, we discuss theoretically two types of experiments with interacting one-dimensional condensates with moving boundaries. Our analysis is based on the Luttinger liquid model which utilizes the emergent conformal symmetry of the low-energy sector of the Lieb-Liniger model. The first system we consider is a variable-length interferometer with two Y-shaped junctions connected back to back. We demonstrate that dynamics of the relative phase between the two arms of the interferometer can be analyzed using the formalism developed by Moore in the problem of electromagnetic vacuum squeezing in a cavity with moving mirrors. The second system we discuss is a single condensate in a box potential with periodically moving walls. This system exhibits classical excitation of the mode resonant with the drive as well as nonlinear generation of off-resonant modes. In addition, we find strong parametric multimode squeezing between modes whose energy difference matches integer multiples of the drive frequency.