We present an equivalent circuit approach to understand, simulate and design experimental setups, where superconducting coplanar resonators are applied to probe quantum systems. We employ finite-element electromagnetic simulations to construct an equivalent lumped circuit representation of the microwave cavity and we use a circuit model based on the Bloch equations to simulate the behavior of a two-state model quantum system. Interconnection of the two circuit modules gives a semiclassical description of the coupled quantum/classical dynamics and yields predictions that are directly comparable with measurements on realized circuits.
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