Shadows and photon rings of a quantum black hole

Recently, a black hole model in loop quantum gravity has been proposed by Lewandowski, Ma, Yang and Zhang (Phys. Rev. Lett. 130, 101501 (2023)). The metric tensor of the quantum black hole (QBH) is a suitably modified Schwarzschild one. In this paper, we calculate the radius of the circular null geodesic (light ring) and obtain the linear approximation of it with respect to the quantum correction parameter α: rl≃3M−α9M. We then assume the QBH is backlit by a large, distant plane of uniform, isotropic emission and calculate the radius of the black hole shadow and its linear approximation: rs=33M−α6(3M). We also consider the photon ring structures in the shadow when the impact parameter b of the photon approaches to a critical impact parameter bc, and obtain a formula for estimating the deflection angle, which is φdef=−2ωrl2log⁡(1−bc/b)+C˜(bc). We also numerically plot the images of shadows and photon rings of the QBH in three different illumination models and compare them with that of a Schwarzschild black hole. It is found that we could distinguish the quantum black hole with a Schwarzschild black hole via the shadow images in certain illumination models.