Topological Amplification in Photonic Lattices

We characterize topological phases in photonic lattices by unveiling a formal equivalence between the singular value decomposition of the non-Hermitian coupling matrix and the diagonalization of an effective Hamiltonian. Our theory reveals a relation between topological insulators and directional amplifiers. We exemplify our ideas with an array of photonic cavities which can be mapped into an AIII topological insulator. We investigate stability properties and prove the existence of stable topologically nontrivial steady-state phases. Finally, we show numerically that the topological amplification process is robust against disorder in the lattice parameters.

Figure 1

Purple (orange) line is the absolute value of the $n=N$ right (left) singular vector of the 1D model (9) with $t_c=t_d=1$, $\varphi =\pi /3$, $N=50$. Dotted lines: ${\gamma }_p=0$ (topologically trivial case). Continuous lines: ${\gamma }_p=1$ (topologically nontrivial case). Inset: Singular values for ${\gamma }_p=0$, 1, 1.5, 2.0, showing the emergence of a zero singular value state at $n_E=N$.

Figure 2

Singular value gap, ${\mathrm{\Delta }}_s=s_{N-1}-s_N$, for the model defined by Eq. (9), $N=100$. Dashed lines separate topological insulator (TI) and normal (N) phases. The intersection, TI-S, is the region of stable nontrivial topological phases. Continuous lines separate stable (S) and unstable (U) phases. (a) $\varphi =\pi /2$, $t_c=1$ and different values of $t_d$ and ${\gamma }_p$. (b) $t_d=t_c=1$ and different values of $\varphi$ and ${\gamma }_p$.

Figure 3

Gap between the second and lowest singular value of $H$ given by Eq. (9) with $t_c=t_d={\gamma }_p=1$, as a function of disorder strength, averaged over $10^5/N$ realizations of disorder. Inset: Field amplitude at site $j=N$ induced by a coherent drive at $j=1$ with ${\epsilon }_1=1$ in a chain with $N=100$.

Figure 4

Scheme for a physical implementation of the 1D topological amplifier (9). White circles: main local photonic modes. Orange circles: auxiliary fast decaying modes for reservoir engineering.