Switch PhD program: how to contact possible PhD advisors when already enrolled in PhD program? Say a student is enrolled in a PhD program in the US and is in the first or 2nd year and has not started his/her thesis work but has only done coursework. The student has always been in the same lab (no rotations). Say such student for whatever reason wants to switch to a PhD program in Europe.

How should such a student handle contacting PhD advisors and other institutions? Should the student mention in the first e-mail that he/she is already enrolled in a PhD program or can this be mentioned in the first phone call? Or should it be mentioned at all? How much would this situation decrease the chances of getting accepted into a PhD program in Europe? Or should the student first quit the current PhD program before applying to any other program?
Email, honesty and avoid burning bridges.

You can freely email (and otherwise) contact potential PhD advisors, of course remembering that they are always busy, so you need to be concise and have something of interest to offer.

You need to be honest about your current situation. If you are not and then the potential new advisor finds out about it, it will look bad and will likely ruin your chances.

Do not quit your current situation. That would look worse than wanting to change/improve your circumstances. And if you fail to find a new position, you'll be left with nothing.