Encircling exceptional points in a restricted parameter space

The eigenvalue spectrum of a non-Hermitian system has topological structure that is absent in Hermitian systems. This structure is evident when the system is parametrically tuned along a smooth path that returns to itself (i.e., control loop). Such a control loop causes the spectrum to trace out a braid and the specific braid is determined by how the control loop encircles degeneracies known as exceptional points. The relationship between control loops, degeneracies and braids has a natural description in the space spanned by the 2(N-1) parameters that provide full control over system’s spectrum (where N is the number of modes). In particular, for any N ≥ 2, the braids form the braid group B_N and homotopically equivalent control loops produce isotopically equivalent braids [1,2]. However, in cases where N is large, a visualization of control loops and degeneracies in the full control space is non-trivial. Moreover, in many applications, only a subset of these control parameters is accessible. In such a restricted subset of the control space, a theoretical description in terms of permutation matrices and representation theory [3] provides valuable insights. In this work, we use a cavity optomechanical system to experimentally validate this approach in a subset control space and show that the results map onto the generic description in the full control space.