Adding a drive term on the Hamiltonian

Question

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?

Answer 1

This cavity drive Hamiltonian is based on the so called Gardiner-Collett Hamiltonian first introduced in the seminal paper C. W. Gardiner and M. J. Collett Phys. Rev. A 31, 3761 (1985) (paywalled I'm afraid). It is a model Hamiltonial to describe coupling of an external field into the cavity.

So there is not much to "derive" here. The Hamiltonian is a model that is well motivated and is found to work very well. Its main assumption is the bilinear coupling between the cavity modes and an external field. For the case of cavities specifically, a first principles derivation has been given recently in this paper.

In its most basic form, the cavity modes couple to an external continuum. The precise Hamiltonian then depends on how we drive this system (i.e. what state we put in this external continuum) and which interaction picture we work in. The case in the question seems to be assuming a monochromatic (single frequency) laser field and ignoring all other modes while working in the Schrödinger picture and performing the rotating wave approximation (as mentioned in the comments by knzhou). This constitutes a standard case that should be covered in any standard quantum optics book with a chapter on cavities.