  Under certain conditions, the quantum delta-kicked harmonic oscillator
displays quantum resonances. We consider an atom-optical realization of the
delta-kicked harmonic oscillator, and present a theoretical discussion of the
quantum resonances that could be observed in such a system. Having outlined our
model of the physical system we derive the values at which quantum resonances
occur and relate these to potential experimental parameters. We discuss the
observable effects of the quantum resonances, using the results of numerical
simulations. We develop a physical explanation for the quantum resonances based
on symmetries shared between the classical phase space and the
quantum-mechanical time evolution operator. We explore the evolution of
coherent states in the system by reformulating the dynamics in terms of a
mapping over an infinite, two-dimensional set of coefficients, from which we
derive an analytic expression for the evolution of a coherent state at quantum
resonance.
