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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?

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    $\begingroup$ Do you know about the rotating wave approximation? $\endgroup$
    – knzhou
    Jan 29, 2021 at 4:57
  • $\begingroup$ @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? $\endgroup$ Jan 29, 2021 at 13:47

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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.

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  • $\begingroup$ Now I get it! Thank you! Indeed it's in every textbook I found but never quite explained, they simply put it there. $\endgroup$ Jan 31, 2021 at 22:16
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    $\begingroup$ @PedroPinho Glad that helped! :) Let me know if you have any follow-up questions! $\endgroup$ Jan 31, 2021 at 23:59

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