Hamiltonian tensor product question

Question

I am reading a book that models a two-level atom interacting with an EM environment as $$H = H_S + H_E + V$$ where $H_S$ is the free Hamiltonian of the two-level system, $H_E$ is the Hamiltonian for the EM environment, and $V$ is the interaction.

The book states that \begin{equation} H_S = \frac{\omega _a}{2}\sigma_z \\ H_E = \sum_j \omega_j b_{j}^{\dagger}b_j \\ V = \sum_j g_j(\sigma_+ b_j + \sigma_- b_{j}^{\dagger}) \end{equation}

I think it is implied that $H_S = H_S \otimes I_E$ and $H_E = I_S \otimes H_E$, but what is $V$ here? How is it split between the two spaces? Something like $\sigma_+ \otimes b_j$ in the sum?

Answer 1

Basically, V is a product operator. The term $\sigma_+ b_j$ is, in reality: $$ \sigma_+ \otimes b_j$$ that acting on a state $\mid \sigma, j \rangle = \mid \sigma\rangle \otimes \mid b_j \rangle$ is just

$$ \left(\sigma_+ \mid \sigma\rangle \right)\otimes\left( b_j\mid b_j \rangle\right) $$

Each operator acts in its own subspace. Your interpretation is extremelly correct.