Harmonic oscillators might be one of the most fundamental entities described by physics. Yet they stay relevant in recent research. The topological properties associated with exceptional points that can occur when two modes interact have generated much interest in recent years. Harmonic oscillators are also at the heart of new quantum technological applications: the long lifetime of high-Q resonators make them advantageous as quantum memories, and they are employed for narrowband processing of quantum signals, as in Josephson parametric amplifiers.

The goal of this thesis is to explore different fundamental regimes of two coupled harmonic oscillators using cavity optomechanics as the experimen- tal platform. With consistent progress in attaining ever increasing Q factors, mechanical and electromagnetic resonators realize near-ideal harmonic oscillators. By parametrically modulating the nonlinear optomechanical interaction between them, an effective linear coupling is achieved, which is tunable in strength and in the relative frequencies of the two modes. Thus cavity optomechanics provides a framework with excellent control over system parameters for the study of two coupled harmonic modes. The specific optomechanical implementation employed are superconducting circuits with the vibrating top plate of a capacitor as the mechanical element. Multimode optomechanical circuits are realized, with two microwave modes interacting with one or two mechanical oscillators. The supplementary modes serve either as intermediaries in the relation of the two modes of interest, or as auxiliaries used to control a parameter of the system.

Three main experimental results are achieved. First, an auxiliary microwave mode allows the engineering of the effective dissipation rate of a mechanical oscillator. The latter then acts as a reservoir for the main microwave mode with which it interacts. The microwave mode susceptibility can be tuned, resulting in an instability akin to that of a maser and in resonant amplification of incoming microwave signals with an added noise close to the quantum minimum. Second, we study the conditions for a nonreciprocal interaction between two microwave modes, when the information flows in one direction but not in the other. The two modes interact through two mechanical oscillators, leading to frequency conversion between the two cavities. Dissipation in the mechanical modes is essential to the scheme in two ways: it provides a reciprocal phase necessary for the interference and eliminates the unwanted signals. Third, level attraction between a microwave and a mechanical mode is demonstrated, where the eigenfrequencies of the system are drawn closer as the result of interaction, rather distancing themselves as in the more usual case of level repulsion. The phenomenon is theoretically connected to exceptional points, and a general classification of the possible regimes of interaction between two harmonic modes is exposed, including level repulsion and attraction as special cases.