I am applying to a European university for Ph.D. Candidates are encouraged to contact their potential advisor before submitting the application. I did that and had a discussion too. I have submitted my application now.
Now I am confused if I should email the professor that I have submitted my application? Can that somehow be seen as negative? Too eager/aggressive?
I would follow up with the supervisor - files can get lost in the shuffle, and it's always nice to know that a candidate has followed through on their promise of an application by actually applying. I would make it a short e-mail.
(For what it's worth, if a supervisor is going to get angry about a short, polite administrative update such as this, they are probably not going to be a good supervisor)
Short answer
Yes, send a succinct follow-up.
Long Answer
There's always this hesitancy among students, candidates, and applicants throughout the academic world (and likely all hiring processes) about this fine-line topic: how much contact is too much, not enough, and most importantly just right?
Well, I think it's safe to say that if any PhD advisor/supervisor considers a handful of professionally written, concise, and professionally-motived emails to be excessive, then that's a red flag about the (poor) degree of availability you should expect from them as one of their students.
I recommend every student applying to graduate school who is encouraged or required to work with a specific advisor/supervisor to engage this individual at least 3-4 times:
Initial outreach. Introduce yourself and indicate a desire to discuss the potential of joining their lab/program. Typically this is done via email and involves requesting to meet over the phone or in person (e.g. at an upcoming conference). Always indicate your availability to them instead of playing email tag!
However, cold-calls on phones prevent being ignored for long periods of time and allow you to each get a more personal feel for one another -- a big positive for the extroverted among us. (but be prepared to jump right into #2 below if you go the phone route).
I would never recommend leaving a phone message. they're scary and never hearing back will create a TON of awkward anxiety! Try the cold-call first and email as a follow-up -- this additional "effort" might reflect well (and will never reflect poorly).
"I tried calling your office the other day and must have just missed you. Could we schedule a time to meet....I'm available..."You might even consider reaching out to current grad students to info-gather and to generate recognition prior to reaching out to the PI/supervisor.
Substantial discussion. Aim to do this over the phone (or in person). Be able to seamlessly discuss their work, your work, and what you bring to the table (e.g., support existing efforts, expand current efforts in a related but novel way, bring new tech to the lab, etc.).
(-4). The follow-up(s). this is (or these are) such an important contact point to keep your name on the top of the individual's head and to indicate that "you're all in." YOU SHOULD ALWAYS FOLLOW-UP!
I always recommend to add a bit more depth to any follow-up beyond "I've applied" or "I'm still interested." While remaining succinct, follow-up with an additional achievement, update on your CV, etc. if you can (especially if any updates make you look better for the position or demonstrate your dedication/interest).
Depending on the timing of your application and the timeline for decisions, I'd recommend 2 follow-ups (only 1 if a short turnaround time is expected).
One shortly after applying (as you're interested in doing) to indicate interest and to keep your name fresh in their mind.
The second follow-up is more dependent on the timing of events:
If you submit your application long before the application window closes, you might follow-up again the day or two after the window closes.
If you apply in the fall for a position that won't be decided until the spring, I'd start off the spring semester with a message to keep you fresh on the supervisor's mind.
Perhaps both of these seem necessary in your circumstance. If they do, then I'd definitely be sure 1 or both of them provide an update from you about a recent submission, publication, presentation, etc that makes you look like an even better candidate!
You might also ask in your initial in-depth conversation with them when decisions are likely to be made (i.e., shortly after the application window closes, over winter break, 6 months form now, etc.). This might inform the best timing for a stretegic follow-up message.
Again, if a supervisor is annoyed by this 1 or 2 emails sent likely months apart, then you probably don't want to work with them anyway. Supervisors/advisors get 100s of emails from the same individuals every month, so getting 1-2 from you (that are professional and succinct) is likely not going to make the typical supervisor despise you. (the common lack of response to such a message is usually due to the 100s of other emails they get bumping yours out of sight/mind and not their displeasure in seeing your email).
Note that any follow-up on your behalf does not professionally necessitate a response from the supervisor. In fact, you'll often not get a response to your follow-ups. If the suspense is too much to bear, then you might consider including a concise, poignant, and well-formulated follow-up question in your follow-up message to try to encourage a response. If such a question seems pertinent to your decision to choose their program, then you following-up again due to any lack of response on their end will be professionally relevant and seem less uncomfortable for you.
(but then, perhaps you'd be inching ever closer to that fine line... :P)
