  A quantum two level system coupled to a harmonic oscillator represents a
ubiquitous physical system. New experiments in circuit QED and
nano-electromechanical systems (NEMS) achieve unprecedented coupling strength
at large detuning between qubit and oscillator, thus requiring a theoretical
treatment beyond the Jaynes Cummings model. Here we present a new method for
describing the qubit dynamics in this regime, based on an oscillator
correlation function expansion of a non-Markovian master equation in the
polaron frame. Our technique yields a new numerical method as well as a
succinct approximate expression for the qubit dynamics. We obtain a new
expression for the ac Stark shift and show that this enables practical and
precise qubit thermometry of an oscillator.
