  An effective scheme within two displaced bosonic operators with equal
positive and negative displacements is extended to study qubit-oscillator
systems analytically in an unified way. Many previous analytical treatments,
such as generalized rotating-wave approximation (GRWA) [Phys. Rev. Lett. 99,
173601 (2007)] and an expansion in the qubit tunneling matrix element in the
deep strong coupling regime [Phys. Rev. Lett. 105, 263603 (2010)] can be
recovered straightforwardly in the present scheme. Moreover, further improving
GRWA and extension to the finite-bias case are implemented easily. The
analytical expressions are then derived explicitly and uniquely, which work
well in a wide range of the coupling strengthes, detunings, and static bias
including the recent experimentally accessible parameters.
