  We consider a simple model of a Josephson junction phase qubit coupled to a
solid-state nanoelectromechanical resonator. This and many related
qubit-resonator models are analogous to an atom in an electromagnetic cavity.
When the systems are weakly coupled and nearly resonant, the dynamics is
accurately described by the rotating-wave approximation (RWA) or the
Jaynes-Cummings model of quantum optics. However, the desire to develop faster
quantum-information-processing protocols necessitates approximate, yet analytic
descriptions that are valid for more strongly coupled qubit-resonator systems.
Here we present a simple theoretical technique, using a basis of dressed
states, to perturbatively account for the leading-order corrections to the RWA.
By comparison with exact numerical results, we demonstrate that the method is
accurate for moderately strong coupling, and provides a useful theoretical tool
for describing fast quantum information processing. The method applies to any
quantum two-level system linearly coupled to a harmonic oscillator or
single-mode boson field.
