Learning-based point cloud registration methods can handle clean point clouds well, while it is still challenging to generalize to noisy, partial, and density-varying point clouds. To this end, we propose a novel point cloud registration framework for these imperfect point clouds. By introducing a neural implicit representation, we replace the problem of rigid registration between point clouds with a registration problem between the point cloud and the neural implicit function. We then propose to alternately optimize the implicit function and the registration between the implicit function and point cloud. In this way, point cloud registration can be performed in a coarse-to-fine manner. By fully capitalizing on the capabilities of the neural implicit function without computing point correspondences, our method showcases remarkable robustness in the face of challenges such as noise, incompleteness, and density changes of point clouds.