  We study squeezed quantum states of phonons, which allow the possibility of
modulating the quantum fluctuations of atomic displacements below the
zero-point quantum noise level of coherent phonon states. We calculate the
corresponding expectation values and fluctuations of both the atomic
displacement and the lattice amplitude operators, and also investigate the
possibility of generating squeezed phonon states using a three-phonon
parametric amplification process based on phonon-phonon interactions.
Furthermore, we also propose a detection scheme based on reflectivity
measurements.
