# Why does Jaynes Cummings not describe an atom in free space?

I'm a bit confused about the application of the Jaynes Cummings model and what exactly is meant by "one single mode":

Usually, it is said that the Jaynes Cummings model describes a single atom in a high Q cavity. The atom then only interacts with a single mode of the light field and the Hamiltonian is written as:

$$H=H_0+H_I=\hbar \omega\sigma_+\sigma_-+\hbar\omega_La^{\dagger}a+\hbar g(\sigma_+a+\sigma_- a^{\dagger})$$

Question 1: Why does this Hamiltonian only describe cavity QED and not e.g. an atom in free space?

Question 2: Is it correct that this description is not valid for interaction with a laser field, since the coherent state is not an eigenstate of $a^{\dagger}a$, i.e. the above Hamiltonian describes interaction with a photon Fock state?

I would appreciate some help, I'm a bit confused about these different models...