  The generalized rotating-wave approximation with counter-rotating
interactions has been applied to a biased qubit-oscillator system. Analytical
expressions are explicitly given for all eigenvalues and eigenstates. For a
flux qubit coupled to superconducting oscillators, spectra calculated by our
approach are in excellent agreement with experiment. Calculated energy levels
for a variety of biases also agree well with those obtained via exact
diagonalization for a wide range of coupling strengths. Dynamics of the qubit
has also been examined, and results lend further support to the validity of the
analytical approximation employed here. Our approach can be readily implemented
and applied to superconducting qubit-oscillator experiments conducted currently
and in the near future with a biased qubit and for all accessible coupling
strengths.
