  Squeezed, nonclassical states are an integral tool of quantum metrology due
to their ability to push the sensitivity of a measurement apparatus beyond the
limits of classical states. While their creation in light has become a standard
technique, the production of squeezed states of the collective excitations in
gases of ultracold atoms, the phonons of a Bose-Einstein condensate (BEC), is a
comparably recent problem. This task is continuously gaining relevance with a
growing number of proposals for BEC-based quantum metrological devices and the
possibility to apply them in the detection of gravitational waves. The
objective of this thesis is to find whether the recently described effect of an
oscillating external potential on a uniform BEC can be exploited to generate
two-mode squeezed phonon states, given present day technology. This question
brings together elements of a range of fields beyond cold atoms, such as
general relativity and Efimov physics. To answer it, the full transformation
caused by the oscillating potential on an initially thermal phononic state is
considered, allowing to find an upper bound for the magnitude of this
perturbation as well as to quantify the quality of the final state with respect
to its use in metrology. These findings are then applied to existing
experiments to judge the feasibility of the squeezing scheme and while the
results indicate that they are not well suited for it, a setup is proposed that
allows for its efficient implementation and seems within experimental reach. In
view of the vast parameter space leaving room for optimization, the considered
mechanism could find applications not only in the gravitational wave detector
that originally motivated this work, but more generally in the field of quantum
metrology based on ultracold atoms.
