Dynamics of Coherent States

A system of harmonic oscillators in the presence of interaction, and with an arbitrary number of degrees of freedom, is considered. The most general form of the Hamiltonian is derived under the restriction that the states which are initially coherent remain coherent at all times. The equation of motion for the annihilation operator, obtained by using this Hamiltonian, is solved, and the frequency spectrum of the annihilation operator is discussed. By giving specific examples it is shown that, in general, the annihilation operators (or their eigenvalues) contain positive as well as negative frequency components and hence are not analytic signals. Some special cases are also considered where the annihilation operators are analytic signals.