Optimization of entanglement depends on whether a black hole is extremal

We consider two Unruh-DeWitt detectors interacting with a massless, minimally coupled scalar field in a $(1+1)$ dimensional Reissner-Nordstr\"om black hole spacetime. In particular, one of the detectors, corresponding to $Alice$, is moving along an outgoing null trajectory. While the other detector carried by $Bob$ is static. With this set-up, we investigate the entangling condition and the measure of the entanglement, concurrence, in the nonextremal and extremal scenarios. Our observations suggest, as expected, a qualitative similarity in characteristics of the entanglement between these two scenarios. However, we find quantitative differences between the nonextremal and extremal concurrences for a broad range of black hole charges. With moderately large detector transition energy, the extremal background always accounts for the larger entanglement than the nonextremal one. In contrast, with low detector transition energy, entanglement on the nonextremal background can be greater. Therefore, by adjusting the detector transition energy, one can perceive optimum entanglement from either the extremal or the nonextremal background.