Level attraction in a microwave optomechanical circuit

Level repulsion-the opening of a gap between two degenerate modes due to coupling-is ubiquitous anywhere from solid-state theory to quantum chemistry. In contrast, if one mode has negative energy, the mode frequencies attract instead. They converge and develop imaginary components, leading to an instability; an exceptional point marks the transition. This only occurs if the dissipation rates of the two modes are comparable. Here we expose a theoretical framework for the general phenomenon and realize it experimentally through engineered dissipation in a multimode superconducting microwave optomechanical circuit. Level attraction is observed for a mechanical oscillator and a superconducting microwave cavity, while an auxiliary cavity is used for sideband cooling. Two exceptional points are demonstrated that could be exploited for their topological properties.