Linear circuit models for on-chip quantum electrodynamics
  We present equivalent circuits that model the interaction of microwave resonators and quantum systems. The circuit models are derived from a general interaction Hamiltonian. Quantitative agreement between the simulated resonator transmission frequency, qubit Lamb shift and experimental data will be shown. We demonstrate that simple circuit models, using only linear passive elements, can be very useful in understanding systems where a small quantum system is coupled to a classical microwave apparatus. 
Alp\'ar M\'aty\'as, Christian Jirauschek, Federico Peretti, Paolo   Lugli, and Gy\"orgy Csaba
quant-ph