  In this paper we investigate coherent and squeezed quantum states of phonons.
The latter allow the possibility of modulating the quantum fluctuations of
atomic displacements below the zero-point quantum noise level of coherent
states. The expectation values and quantum fluctuations of both the atomic
displacement and the lattice amplitude operators are calculated in these
states---in some cases analytically. We also study the possibility of squeezing
quantum noise in the atomic displacement using a polariton-based approach.
