Topological phases of driven-dissipative lattices: non-Hermitian physics and experimental implementations

We show that a system of coupled photonic or phononic modes with parametric couplings displays an exciting dissipative topological phase diagram. We define a winding number that identifies non-trivial topological phases in which the system behaves as a directional amplifier. Our theoretical framework relies on the singular value decomposition of the non-Hermitian coupling matrix. In our model, zero-singular modes are the analog of zero-energy states in topological insulator theory. Such zero-singular modes dominate the Green’s function of the system, leading to a directional amplification mechanism. Our work reveals a connection between topological insulator theory, non-Hermitian physics and directional amplification. Our ideas can be implemented with a variety of physical setups, including superconducting circuits, trapped ions and optomechanical systems. We present several schemes for such experimental implementation that rely on periodically driving boson local frequencies or, alternatively, by using Kerr-nonlinearities in a four-wave mixing scheme.