Oh! You mean the Exodus from Egypt, not the Diaspora, which is self-evident.
Why would we need to prove the Exodus in order to prove the existence of the Jewish people? There is plenty of evidence for the latter, not the least of which being, living actual Jews.
As for why there is no archaeological evidence for it? Obviously, it didn't happen in the way it was written in Torah. Torah does not exist as a book of literal history. Its a book of faith. The stories are intended to reveal something about us and our relationship with the Divine. My guess is that we are looking for evidence either in the wrong place or in the wrong time. Egypt had considerable influence at certain times and was very much larger than it is today. It won territory, lost it, won it back. Numbers tend to be the least reliable parts of stories -- spans of time, numbers of people, ages of men. They tend to be largely symbolic. In all likelihood, the Exodus was much smaller, much shorter and much more local than is given in the story.