# Adding a drive term on the Hamiltonian

The Hamiltonian of a optomechanical cavity driven by a laser of frequency $$\omega_L$$ is of the following:

$$H = \hbar\omega_{cav}\hat{a}^{\dagger}\hat{a} + \hbar\Omega_{m}\hat{b}^{\dagger}\hat{b} - \hbar g_o\hat{a}^{\dagger}\hat{a}(\hat{b}+\hat{b}^{\dagger}) + H_{drive}$$

Where $$H_{drive}$$ is:

$$H_{drive} = i\hbar E(\hat{a}e^{i\omega t}-\hat{a}^{\dagger}e^{-i\omega t})$$

where E is the amplitude of the Electric Field.

All the other terms I have derives easily, my problem is with the drive term. I have already read EVERY answer in StackExchange (including this Coherent drive Hamiltonian) but I cannot understand how to derive this drive Hamiltonian. Where did the term $$(\hat{a}e^{i\omega t}-\hat{a}^{\dagger}e^{-i\omega t})$$ came from?

• Do you know about the rotating wave approximation? Commented Jan 29, 2021 at 4:57
• @knzhou I do, but this is not enough to explain why the drive term has the form it has, the purpose of the RWA is to remove the time dependence. In other words, my problem is to derive the initial form of the $H_{drive}$: why is it a combination of the creation and annihilation operators? Commented Jan 29, 2021 at 13:47