I'm doing a literature review on the history of model development in the scope of light-matter interaction. I've studied in depth the two key tools for a lot of the research in this field, two level systems and harmonic oscillators (and models coupling them and considering dissipations) and the abundance of literature from reviews and textbooks on top of the original papers was great. After the basics, I wanted to generalize two level systems coupled to many quantized field modes, and then many two level systems coupled to many quantized field modes - all that is available and I had no issues finding the topics of interest.

Beyond, that I was seeking $N$-Level systems coupled to one or more quantized field modes. I have found some papers but not what I was hoping for. I found a lot of work on three level systems with restrictions on the type of transitions between states (Xi, Lambda and V types) also with some rotating wave approximations imposed. I'm seeking an $N$-Level generalization of the Rabi model without constraints on transitions between states and without RWA.

The Rabi model I'm referring to is one where a two level system is coupled via a dipole interaction to a single quantized field mode.

$$ \hat{H}_R= -\hbar \omega_A \hat{\sigma}_z + \hbar \omega_F\hat{N} + \hbar g \hat{\sigma}_x(\hat{a} + \hat{a}^\dagger)$$

Generalizations of this to $N$-level systems with no RWA and no restrictions on state transitions is what I am looking for. I tried google scholar using some of those labels and key words but no luck. Any help with references is appreciated!

  • $\begingroup$ You might want to look at the Dicke model. It is not exactly what the OP asks for, but a rather interesting and far reaching extension. $\endgroup$
    – Roger V.
    Feb 24, 2022 at 8:16
  • $\begingroup$ @RogerVadim it should be exactly the same as the Dicke model under the constraint that all of the atoms are the same (working in the symmetric subspace will give an $N+1$-level system coupled to a quantized field as per OP's request( $\endgroup$ Feb 24, 2022 at 18:40

1 Answer 1


First a general note about deriving these things; in practice, doing away with the rotating wave approximation (RWA) is tantamount to stopping your derivation at that step and saying 'I'm done.' There are ways to systematically go beyond RWA, like the Magnus expansion, but there's probably not much sense in doing that unless you have a particular problem to solve. So if you know how to derive the equations of motion for an N-level system coupled to a harmonic oscillator with the RWA, then you also know how to derive them without the RWA as well.

It looks like section 3.3 of this thesis covers much of what you're asking for, in the sense that it covers N-level systems coupled to a continuum (and then you can simply add more copies of the resulting master equation to model coupling to multiple independent fields). The end of that section gives a brief overview of some of the non-trivial things you have to take into account in this situation. In particular, the derivation in the thesis assumes that the various transitions in the system are either exactly degenerate or well-separated in frequency. Bullet point 1 at the end points out that you will see beating in the general case, and refers you to reference 120.

If you want to focus on the case of a system coupled to a discrete number of oscillators instead of a continuum, you can consider Eqs. 3.41 and 3.42, but replacing the integral with a sum. I don't know of any resources that do a careful analysis of the physics that you expect from these systems beyond 3 levels, but this at least gives you a derivation of their equations of motion.


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