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We present a system composed of two flux qubits and a transmission-lineresonator. Instead of using the rotating wave approximation (RWA), we analysethe system by the adiabatical approximation methods under two opposite extremeconditions. Basic properties of the system are calculated and compared underthese two different conditions. Energy-level spectrum of the system in theadiabatical displaced oscillator basis is shown, and the theoretical result iscompared with the numerical solution.

Adiabatic approximation in the ultrastrong-coupling regime of a system consisting of an oscillator and two qubits