Measurement of the Eigenvalue Braiding in the Vicinity of a Triple Exceptional Point

When a non-Hermitian system's dynamical matrix (or "Hamiltonian") is tuned around a closed loop in the vicinity of an exceptional point (ΕΡ), the system's complex eigenvalues trace out a braid. We have realized such eigenvalue braiding in a three-mode mechanical system by using cavity optomechanical techniques to tune the Hamiltonian of three vibrational modes in a SiN membrane. By measuring the modes' complex eigenvalues we show that the system can be tuned to a triple exceptional point (ΕΡ₃), and that the eigenvalues exhibit the expected behavior in the vicinity of this ΕΡ₃. Specifically, we show that varying the Hamiltonian in closed loops results in eigenvalue braids that correspond to the generators of the braid group Β₃. We also show that for any given loop, the specific braid which it produces is determined by how that loop encloses the set of double degeneracies (i.e., ΕΡ₂'s). This highlights the central role played by ΕΡ₂'s in determining the eigenvalue topology, even in systems with more than two modes. In the following talk, we describe measurements of the locations of the ΕΡ₂'s in the neighborhood of the ΕΡ₃, and show that they form a trefoil knot.