JultikaUniversity of Oulu repository

The purpose of this thesis is to study a two-level system that is coupled non-linearly to a harmonic oscillator outside the Lamb-Dicke regime. These kind of non-linear couplings can be found between the vibrational and electronic degrees of freedom in a laser-irradiated trapped ions. In this work we study a superconducting circuit with similar properties. The circuit consist of LC-circuit that is coupled to a single-Cooper-pair transistor. The single-Cooper-pair transistor behaves as an artificial ion and the LC-circuit behaves as the electric potential that is used to trap the ion. When the coupling is weak the interaction can be approximated to be linear. The differences between the absorption spectrum of the linear approximation and the whole system is known as generalized Bloch-Siegert shift. In this thesis we will study the Bloch-Siegert shift as the coupling strength is increased. The non-linear coupling terms cause resonance shifts and additional resonance in the absorption spectrum.