I have a bosonic mode associated to the usual operators $a$, $a^\dagger$. I'm interested in knowing the evolution of a coherent state $\vert \alpha \rangle = e^{\alpha a^\dagger - \alpha^\ast a}\vert 0\rangle$ under the action of the nonlinear (Kerr-type) operator $U=e^{-i \chi (a^\dagger a)^2}$. To summarise, I need to know $$ \vert \psi(\chi)\rangle=e^{-i \chi (a^\dagger a)^2} \vert\alpha\rangle \;, $$ in the coherent state basis.
I tried to expand the exponential, but the calculation seems to get cumbersome. What is the best approach to proceed? Is this problem discussed somewhere in the literature? Is there actually an analytical solution for the state $\vert\psi(\chi)\rangle$ (for arbitrary $\chi$)?
[EDIT] My attempt starts from the expression $\vert\alpha\rangle=e^{-\vert\alpha\vert^2/2}\sum_n \dfrac{\alpha^n}{\sqrt{n!}}\vert n\rangle$. From this, I evaluate $e^{-i \chi (a^\dagger a)^2} \vert n\rangle$, which is $e^{-i \chi n^2} \vert n\rangle$ (correct?). At this point, I would like to express $e^{-\vert\alpha\vert^2/2}\sum_n \dfrac{\alpha^n e^{-i \chi n^2}}{\sqrt{n!}}\vert n\rangle$ in the coherent state basis. Does this approach make sense?
