  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 new platform for studying the dynamical Casimir effect, since they
allow to achieve relativistic, i.e. near sonic, velocities of the boundaries.
In comparison to previously studied optical and classical hydrodynamic systems
ultracold atoms allow to realize a 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-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.
