Quantum correlations in the steady state of light-emitter ensembles from perturbation theory

The coupling of a quantum system to an environment leads generally to decoherence, and it is detrimental to quantum correlations within the system itself. Yet some forms of quantum correlations can be robust to the presence of an environment - or may even be stabilized by it. Predicting (let alone understanding) them remains arduous, given that the steady state of an open quantum system can be very different from an equilibrium thermodynamic state; and its reconstruction requires generically the numerical solution of the Lindblad equation, which is extremely costly for numerics. Here we focus on the highly relevant situation of ensembles of light emitters undergoing spontaneous decay; and we show that, whenever their Hamiltonian is perturbed away from a U(1) symmetric form, steady-state quantum correlations can be reconstructed via pure-state perturbation theory. Our main result is that in systems of light emitters subject to single-emitter or two-emitter driving, the steady state perturbed away from the U(1) limit generically exhibits spin squeezing; and it has minimal uncertainty for the collective-spin components, revealing that squeezing represents the optimal resource for entanglement-assisted metrology using this state.