# Hamiltonian tensor product 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?

• Yes, your interpretation is correct. Jun 18 at 21:34

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