Long-range and steady-state entanglement of driven-dissipative nitrogen vacancy centers using microwaves as a drive and synthetic antiferromagnet as a dissipator

The search for optimal schemes and dissipative environments for mediating long-range entanglement between two distant nitrogen-vacancy centers (NVCs) in diamond is the subject of ongoing vigorous efforts due to potential applications of such microscopic solid-state qubits in quantum sensing and quantum computing. However, stabilizing entanglement of NVCs into steady-state poses a significant challenge, typically requiring tuning the environment into a {\em nonequilibrium} state. Here we microscopically derive a Lindblad quantum master equation for a system of two driven-dissipative NVCs, where the drive is microwave radiation and dissipation is provided by a single magnetic bath that is kept in {\em equilibrium}. This equation allows us to predict precise conditions for long-range and steady-state entanglement of NVCs, while it also suggests synthetic antiferromagnet as an optimal choice for a dissipative environment. By using realistic parameters from available experiments, we estimate steady-state concurrence reaching $\mathcal{C}\simeq 0.28$ for two NVCs separated by $\sim 100 \: \mathrm{nm}$.