In the play, Romeo and Juliet, does Juliet know that Romeo is beneath her window?
She doesn't initially. She talks out of her window, daydreaming aloud to herself (or perhaps talking to the audience), and Romeo talks to himself (and perhaps the audience), listening to her.
The irony is that she addresses Romeo not thinking that he is in fact there or indeed that he can hear her:
O Romeo, Romeo! wherefore art thou Romeo?
Deny thy father and refuse thy name!
Why (wherefore) are you Romeo? Why did you have to be called that? That's what she's asking. And at the end of her speech, she asks Romeo to get rid of his name
Romeo, doff thy name;
And for that name, which is no part of thee,
Take all myself.
She is then absolutely astonished when Romeo agrees:
I take thee at thy word.
Call me but love, and I'll be new baptiz'd;
Henceforth I never will be Romeo.
And Juliet confirms for us that she really doesn't know he's there: she doesn't even know that it's Romeo (despite the fact that he's just said his name!). She actually asks him
What man art thou that, thus bescreen'd in night,
So stumblest on my counsel?
Who are you? He's screened in night, and he's stumbled upon her secret talk to herself (her 'counsel'). So after that she does know he's there, and she talks to him. Before that, she didn't.
I don't believe there is any evidence in the text that she does, so I would say that it is up to the director - how he/she decides to stage it. Juliet seems to be thinking out loud when she asks the rhetorical question, "Wherefore art thou Romeo?" - Why do you have to be Romeo Montague, my father's enemy? Personally, if I were going to direct this play, I would have Juliet be surprised to find out that Romeo is down there.