Dissipative Pairing Interactions: Quantum Instabilities, Topological Light, and Volume-Law Entanglement

We analyze an unusual class of bosonic dynamical instabilities that arise from dissipative (or non-Hermitian) pairing interactions. We show that, surprisingly, a completely stable dissipative pairing interaction can be combined with simple hopping or beam-splitter interactions (also stable) to generate instabilities. Further, we find that the dissipative steady state in such a situation remains completely pure up until the instability threshold (in clear distinction from standard parametric instabilities). These pairing-induced instabilities also exhibit an extremely pronounced sensitivity to wave function localization. This provides a simple yet powerful method for selectively populating and entangling edge modes of photonic (or more general bosonic) lattices having a topological band structure. The underlying dissipative pairing interaction is experimentally resource friendly, requiring the addition of a single additional localized interaction to an existing lattice, and is compatible with a number of existing platforms, including superconducting circuits.

Figure 1

Stability diagram for a minimal three-mode bosonic system (see inset) with loss on mode $\widehat{a}$ (rate $\kappa$), gain on mode $\widehat{c}$ (rate ${\eta }^2\kappa$), and tunnel couplings $J_1$, $J_2$ [cf. Eq. (3)]. In the absence of dissipative pairing, the system is dynamically unstable above the dashed line. Adding dissipative pairing $i\eta \kappa (\widehat{a}\widehat{c}+\mathrm{H}.\mathrm{c}.)$ shifts the onset of instability to the solid line, see Eq. (4). Remarkably, this boundary is independent of $\overline{J}/\kappa \eta$, where $\overline{J}=\sqrt{J_1^2+J_2^2}$. The dissipative steady state remains pure (with a high density) as one approaches instability, see main text. Red lines in each plot are the same cut of parameter space, $\overline{J}/\kappa \eta =1$ and $J_1/J_2=0.75$. Solid lines show hopping, dashed line shows the dissipative pairing interaction.

Figure 2

Steady state correlation functions for a 99-site SSH chain with $\delta =-0.65$. There is a single jump operator of the form of Eq. (6) with $\overline{0}=4$ and $\overline{1}=0$, and $\eta =0.999{\eta }_c\sim 0.045$. The squeezing correlation functions show a pure, single-mode squeezed state exponentially localized to the edge. Inset: Schematic of the dissipatively stabilized SSH chain. A single jump operator generates a dissipative pairing interaction, selectively exciting the edge mode.

Figure 3

(a) A $24\times 24$ site Hofstadter lattice, which has uniform hopping and a quarter flux per plaquette $\mathrm{\Phi }=\frac{1}{4}{\mathrm{\Phi }}_0$. There is a single dissipator of the form of Eq. (6), with $\overline{1}=(11,23)$ and $\overline{0}=(12,20)$, and with $\eta =0.999{\eta }_c\sim 0.0007$. The color corresponds to local steady state photon number, which is exponentially localized to the edges of the lattice. (b) The same system, now showing steady state squeezing correlations between the randomly chosen edge site (18,23) and the rest of the lattice. Every edge site has exponentially enhanced squeezing with every other edge site on the same sublattice.