It's not like there were trumpets sounding the arrival of modern physics. I don't think there is a unique answer to this question. I'm not even sure there is really a crystal clear definition of what modern physics is.
If we take modern physics to be "quantum mechanics + special relativity + statistical mechanics," then an arbitrary line in the sand could be 1905, Einstein's [*annum mirabilis*](https://en.wikipedia.org/wiki/Annus_mirabilis_papers), when he published ground breaking papers on quantum mechanics, special relativity, and statistical mechanics. Of course, these previous papers built on developments in quantum mechanics, special relativity, and statistical mechanics, (at least, early versions of those subjects), so there is plenty of room to argue for a different date.