Coherent drive Hamiltonian

The coherent drive represents an electromagnetic field interacting with the light/atom in the cavity. Say you have a drive (electromagnetic field):

$$E\propto \epsilon b +\epsilon^*b^\dagger$$

and the light in the cavity has creation/annihilation operators $a, a^\dagger$. Then, the interaction between the drive and the light in the cavity would be represented as:

$$H_d \propto (\epsilon b +\epsilon^*b^\dagger)(a+a^\dagger)$$

The reason for such an interaction is that photons would be added (pumped into) the cavity by a laser. You can look into this more deeply by reading about cavity optomechanics. In the interaction picture, it can be seen that the operators of the light in the cavity evolve as:

$$a(t)\propto a(0) e^{-iwt}$$

where $w$ is the frequency of the light in the cavity. Also, in the interaction picture, the light of the coherent drive evolves as:

$$E(t)\propto\epsilon b e^{-iw_dt}+\epsilon^*b^\dagger e^{iw_dt}$$

where $w_d$ is the frequency of the coherent drive. Due to this, the interaction term in the interaction picture Hamiltonian looks like:

$$H_d^{I}\propto (\epsilon b e^{-iw_dt}+\epsilon^*b^\dagger e^{iw_dt})(a(0) e^{-iwt}+a^\dagger(0) e^{iwt}) \\= \epsilon b a(0) e^{-i(w_d+w)t}+\epsilon b a^\dagger(0) e^{-i(w_d-w)t}+\epsilon^*b^\dagger a(0) e^{i(w_d-w)t}+\epsilon^*b^\dagger a^\dagger(0) e^{i(w_d+w)t}$$

In the Rotating Wave Approximation (RWA), it is assumed that $|w_d-w|<<w_d+w$ is true, which is reasonable for $w_d\approx w$. So, the higher frequency terms $w_d+w$ oscillate very quickly and average out to zero at the time-scales at which the system is probed. This is a fully quantum mechanical treatment of the problem.

When the drive is introduced in the semi-classical approximation, it is a complex number. The semi-classical approximation is the case in which the light/atom in the cavity is treated quantum mechanically (hence, they are represented as operators) and the drive is treated classically (hence, it is a complex number). Then, the coherent drive is:

$$E\propto \epsilon e^{-iw_dt}+\epsilon^* e^{iw_dt} $$

Here, the coherent drive is explicitly time dependent because that's how one describes the electromagnetic wave classically, as a traveling wave. The interaction term would be represented as:

$$H_d \propto (\epsilon e^{-iw_dt}+\epsilon^*e^{iw_dt})(a+a^\dagger)$$

Under RWA, it gives the $H_d$ in the question.