How to approach potential PhD advisors

What is the correct way for a new PhD student to approach an interested faculty member to discuss becoming their student?

Is it better to come with a few thesis project ideas prepared to pitch? Or is it better to come in expecting to have the faculty member to pitch them to the student?

Typically, the student has a year or two after joining a group to formulate an actual research proposal - but at the beginning of the program, when approaching potential advisers, there is only limited time, and limited knowledge of the lab's current focus and planned future projects.

Of course, the answer depends very much on the actual faculty member in question. But in what ways does it depend? Are there any aspects to the decision that remain invariant?

I am in the physical sciences (biology) but to make the question useful to more users, perhaps it may be better if we could have answers covering the peculiarities and commonalities of this question for different fields.

Answer

(This is for Computer Science and Medical Informatics, I suspect that most STEM fields are similar.)

Is it better to come with a few thesis project ideas prepared to pitch? Or is it better to come in expecting to have the faculty member to pitch them to the student?

In my experience, if a faculty member is looking for a PhD student, they already have one or more projects in mind.

I think one reason for this is funding, and what faculty need to do to get it: The process of coming up with a grant proposal, submitting it, and getting a response takes months, and some grants can only be applied for once a year. That means that when you, the student, are joining the lab, there already is funding for you from a grant, and some general research goals are outlined in the grant proposal that your research needs to fall within. Some grants are more general than others, and there are also "training grants" out there the goal of which is to train you in a field (almost no constraints on the research topic).

The other reason is that faculty also have research interests.

I seriously doubt that you will need to pitch an idea out of the blue. It's possible that the faculty member will give you a general idea of what the direction of the research will be, since it would be under the constraints of their own interests and under the constraints of whatever grant is funding you, and then you would need to propose something more specific within those constraints. It's also possible that they will have something much more specific in mind and it will just be a matter of saying whether you're willing to do that research.

[A]t the beginning of the program, when approaching potential advisers, there is only limited time, and limited knowledge of the lab's current focus and planned future projects.

That's why you need to read up on the potential advisor's interests, look at their publications, look at the publications coming out of the lab, and figure out what the research focus and direction of the lab seems to be.