  We study a particular form of interaction Hamiltonian between qubits and
quantum harmonic oscillators, whose closed system dynamics results in qubit
controlled displacement operations. We show how this interaction is realizable
in many setups, including nanomechanical systems, ion traps, cavity QED and
circuit QED, and in each context we provide quantitative estimates for the
relevant parameters. The dynamics of the system is investigated through a
master equation, including typical decoherence mechanisms resulting from the
coupling of the qubit and oscillator to a thermal Markovian environment. We
show how to solve the master equation by adopting a phase-space representation
for the oscillator, and derive analytical and approximate solutions for many
special cases of interest. Finally, our techniques are applied to a relevant
example by studying the dynamics of qubit-oscillator entanglement and the
preparation of oscillator states with negative Wigner function.
