  This article discusses quantum fluctuation properties of a crystal lattice,
and in particular, phonon squeezed states. Squeezed states of phonons allow a
reduction in the quantum fluctuations of the atomic displacements to below the
zero-point quantum noise level of coherent phonon states. Here we discuss our
studies of both continuous-wave and impulsive second-order Raman scattering
mechanisms. The later approach was used to experimentally suppress (by one part
in a million only, which might be noise) fluctuations in phonons (and it has
not been clearly reproduced by other experimental groups). We calculate the
expectation values and fluctuations of both the atomic displacement and the
lattice amplitude operators, as well as the effects of the phonon squeezed
states on macroscopically measurable quantities, such as changes in the
dielectric constant. These results are compared with recent experiments.
Further information, including preprints and animations, are available in
http://www-personal.engin.umich.edu/\~nori/squeezed.html
