Eh, it would be best to apply for citizenship now and go after.
If that's not possible, you CAN study abroad and visit the US off and on, but what if you get a good job there? Won't you want to stay?
You'll need a re-entry permit that'll let you stay out for 2 years. You can apply for another one after that, which is usually granted. How long will the PhD take? 5 years?
Citizenship would be best.