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The generalized rotating-wave approximation with counter-rotatinginteractions has been applied to a biased qubit-oscillator system. Analyticalexpressions are explicitly given for all eigenvalues and eigenstates. For aflux qubit coupled to superconducting oscillators, spectra calculated by ourapproach are in excellent agreement with experiment. Calculated energy levelsfor a variety of biases also agree well with those obtained via exactdiagonalization for a wide range of coupling strengths. Dynamics of the qubithas also been examined, and results lend further support to the validity of theanalytical approximation employed here. Our approach can be readily implementedand applied to superconducting qubit-oscillator experiments conducted currentlyand in the near future with a biased qubit and for all accessible couplingstrengths.

Generalized rotating-wave approximation to biased qubit-oscillator systems