I am in the process of writing my PhD thesis, and I would like to extend the state-of-the-art part into a review article. Although I am confident that I have investigated and understood the existing literature thoroughly, I think the overall quality and visibility of the review article would increase if I find an expert to collaborate with on the paper. My PhD supervisor is not an expert in the field and I expect his willingness to actively contribute to be near zero.
I have already identified a rising expert (junior PI) that I would like to contact. A strong plus is also that he seems to be an expert on the area of the article where I have no published results, whereas I have some published research in the other of the two subareas of the review article. However, I am not sure how and when to contact this person (I have not met him in person).
One approach would be to (nearly) finish the paper and then send him a direct email, asking whether he would like to contribute, and send him the paper after he agreed to handle the draft confidentially so he can decide if he wants to contribute. I would also mention that I am willing to do the ground work but expect him to comment on the draft and help improving the overall quality of the paper.
Another approach would be to upload a finished (to the best of my abilities) paper on arXiv and than send him the mail asking for his willingness to contribute.
Is there a better approach?