Tomás Ramos

Driven-dissipative quantum systems can exhibit robust transport and amplification in topological regimes, yet the connection between topology and the extent of correlations remains largely unexplored. In this work, we develop a general framework that links topological phases in driven-dissipative systems to bosonic correlations via the singular value decomposition (SVD). In essence, we claim that non-Hermitian topology in quadratic Liouvillians is directly encoded in steady-state correlations, providing an intrinsic characterization of topology without external probes. We show that topological amplification induces disorder-robust long-range order (LRO) in steady-state correlations at fixed frequency, establishing frequency-resolved correlations as direct signatures of non-Hermitian topological phases. We introduce a vector-valued topological invariant that captures the total number of singular-value gap closings across the frequency axis, extending the concept of adiabatic deformation from topological insulators to the case of topological phases of quadratic Liouvillians. Within this framework, we further demonstrate that the spatial structure of equal-time correlations encodes global topological information, manifested as a Gaussian spatial decay with distance in the topological phase, in contrast to the exponential decay characteristic of trivial phases. These findings open new avenues for quantum sensing and correlation engineering in non-Hermitian systems, with feasible implementations in platforms such as trapped ions and superconducting circuits.

We propose two setups for realizing a chiral quantum network, where two-level systems representing the nodes interact via directional emission into discrete waveguides, as introduced in T. Ramos et al. [Phys. Rev. A 93, 062104 (2016)]. The first implementation realizes a spin waveguide via Rydberg states in a chain of atoms, whereas the second one realizes a phonon waveguide via the localized vibrations of a string of trapped ions. For both architectures, we show that strong chirality can be obtained by a proper design of synthetic gauge fields in the couplings from the nodes to the waveguide. In the Rydberg case, this is achieved via intrinsic spin-orbit coupling in the dipole-dipole interactions, while for the trapped ions it is obtained by engineered sideband transitions. We take long-range couplings into account that appear naturally in these implementations, discuss useful experimental parameters, and analyze potential error sources. Finally, we describe effects that can be observed in these implementations within state-of-the-art technology, such as the driven-dissipative formation of entangled dimer states.