How does the displacement operator act on number states $|n\rangle$?

Question

The displacement operator generates the coherent state out of the vacuum.

$$\hat D(\alpha)|0\rangle = |\alpha\rangle$$

but I am wondering what the meaning of a displacement operator acting upon a number state with $n \neq 0$ is.

For example, is $\hat D(\alpha)|1\rangle$ meaningful?

I have not been able to find any mention of this online.

Answer 1

It is indeed meaningful; they're called, surprisingly enough, displaced number states, and they've appeared on this site e.g. here and here. It's hard to say anything else about them without knowing what exactly you want to know about them, though.