Quantum phonon optics: Coherent and squeezed atomic displacements

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.