Topological amplification in photonic lattices and the classification of topological non-Hermitian phases

A proper characterization of non-trivial topological phases in dissipative systems is highly non-trivial. This is the case of photonic lattices, superconducting circuits or even vibronic systems like trapped ions, whose description typically involves the eigenstates of a non-Hermitian coupling matrix, H.

In my talk I will present a definition of topological invariants for driven dissipative lattices that relies on the singular value decomposition of non-Hermitian coupling matrices, H, or, equivalently, on the eigenstates of an effective Hamiltonian, H_eff = σ⁺ H + H^† σ^-. I will show that topological non-trival phases of H_eff correspond to steady-states in which the driven dissipative lattice behaves as a directional amplifier. Thus, our work leads to a connection between topological band theory and amplification in photonic lattices, which also leads to a physically motivated classification of topological phases non-Hermitian lattices. I will briefly sketch an implementation of our ideas with periodically driven superonducting circuits.

[1] D. Porras and S. Fernández-Lorenzo, Phys. Rev. Lett. 122, 143901 (2019)