How to derive the hamiltonian of a degenerate parametric amplification?

The hamiltonian is written as
$$H_{int} =hbarmu(a^{dagger 2}b+a^{2}b^{dagger}),$$
where $b$ and $a$ are the are the annihilation operators for the pump and signal modes, and $mu$ is a coupling strength between the pump field and the nonlinear crystal.
The paper I read didn’t give the derivation and I didn’t find it on the Internet, either.

1. How to derivate it? Is there a classical counterpart to this formula?
2. This is in the interaction picture. Can we get this hamiltonian in other pictures?
3. Could you recommend a note or book frendly to green hands on quantum optomechanics?

I’m new to this. Your detail answers will be very welcome.