# What are the "strong", "ultrastrong" and "deep strong" coupling regimes of the Rabi model?

The Rabi model describes a two-level system interacting via a linear coupling with a quantized harmonic oscillator, and it is described by the hamiltonian $$H_{\rm{Rabi}}=\hbar\omega\, a^\dagger a +\hbar\omega_0\,\sigma_3+\hbar g\,\sigma_1(a+a^\dagger). \tag1$$ Here $[a,a^\dagger]=1$, $\sigma_i$ are the Pauli matrices on the two-level system, and $g$ represents the coupling. When deriving such a model for a physical system, one typically needs to assume that (i) there are only two atomic energy levels involved in the experiment, (ii) there is only one mode of the relevant bosonic field (e.g. the laser field inside a cavity) that's close enough to resonance to matter, (iii) the coupling is linear and that quadrupolar-like terms like $(a+a^\dagger)^2$ either don't couple or couple too weakly, and (iv) that the two-level atom has no permanent dipole moment. Under those assumptions, equation (1) follows.

Most physical systems, on the other hand, admit a further approximation, called the rotating-wave approximation (RWA). Here the coupling term is decomposed into slow- and fast- rotating terms, in the form $$\sigma_1(a+a^\dagger)=(\sigma_++\sigma_-)(a+a^\dagger)=(\sigma_+ a+\sigma_-a^\dagger)\quad+\stackrel{\rm counter-rotating\, terms}{\overbrace{(\sigma_-a+\sigma_+a^\dagger)}}.$$ In an interaction picture, the oscillations of the terms in the first bracket, at frequencies $\omega_0$ and $\omega$, cancel out (at least approximately), whereas the so-called counter-rotating terms oscillate at frequency $\omega+\omega_0$, which is very high in usual experiments, so their effect is typically very small. Neglecting such terms leads to the Jaynes-Cummings hamiltonian, $$H_{\rm{JC}}=\hbar\omega\, a^\dagger a +\hbar\omega_0\,\sigma_3+\hbar g\,(\sigma_+ a+\sigma_-a^\dagger), \tag2$$ which is much easier to handle than the full Rabi one, and which has proved to be extremely successful in describing a wide variety of different experimental systems.

The rotating-wave approximation, however, tends to break down as the coupling increases. This has led to at least three different regimes beyond the weak coupling, Jaynes-Cummings region:

• The "strong coupling" regime at $g/\omega\lesssim0.01$
• The "ultrastrong coupling" regime at $g/\omega\gtrsim0.1$
• The "deep strong coupling" regime at $g/\omega\approx1$.

(List taken from A. Moroz, Ann. Phys. 1, 252 (2014), arXiv:1305.2595.)

I've seen these terms used before but I've never quite understood what they mean. What are the different characteristics of these three regimes? What physics should one expect, or what parameters are important, in each of them? Is there a specific reason to consider that $g/\omega\sim10^{-2}$ is already strong enough to drive very new physics, or is this simply a reflection of the fact that strong couplings are (very) hard to do experimentally for most of the interesting systems that exhibit Jaynes-Cummings physics? Why are there so many different regimes in between $g/\omega\ll1$ and $g/\omega\approx1$?

The OP's characterization of these regimes is not quite universal, and there is some variation. This is partly because there are five important timescales for a Rabi-like system: the resonance and driving frequencies, $$\omega_0$$ and $$\omega$$, and the coupling strength $$g$$, but also the cavity decay rate $$\kappa$$ and the atomic decay rate $$\gamma$$. The different arrangements of these quantities are (partly) what drives the different physical regimes. Additionally, different authors use either $$\omega$$ or $$\omega_0$$ as the relevant yardstick for $$g$$, which can muddy the waters.

A good description of some of these regimes is given by Y. Wanga and J. Yan Hawb in Phys. Lett. A 379, 779 (2015).

In short,

• In the weak coupling regime, $$g\ll \gamma, \kappa, \omega$$ and $$\omega_0$$. In this regime the RWA fully holds and the hamiltonian is that of the Jaynes-Cummings model, but care must be taken with the decoherence rates, which impede coherent JC dynamics from ocurring.

• The strong coupling regime is when $$\gamma, \kappa \ll g\ll \omega, \omega_0$$, which means that the atomic system can absorb and (coherently) re-emit a given photon many times before it leaks from the cavity or is spontaneously emitted. In this regime the Jaynes-Cummings model is fully realized, and multiple Rabi oscillations between the two eigenstates of each invariant subspace are possible.

• The ultrastrong coupling regime is when $$g$$ is strong enough to compete with the resonance or driving frequencies, so $$g \lesssim \omega_0$$, but is not yet bigger than them. Here the RWA breaks down and the Jaynes-Cummings model ceases to hold; instead, one observes phenomena like photon blockades, nonclassical state generation, breakdowns of the standard master equation, superradiant phase transitions, and ultraefficient light emission.

• The deep strong coupling regime, also called 'deep ultrastrong coupling' in some references, requires $$g \gtrsim \omega$$, the driving frequency. This regime has not yet been reached by experiments (which were capped at roughly $$g/\omega_0\lesssim 0.5$$ as of 2015), but has been the subject of some theoretical explorations (e.g. [1, 2, 3]). The physics available here is beginning to be explored and it includes the decoupling of light and matter to give a reduced spontaneous emission rate, the simulation of relativistic quantum phenomena and realizations of the Dicke spin-boson model, among others.

And also, a word of warning: the coupling strength is an important measure of the 'quantumness' of the interaction, but a strong argument can be made that it is only really relevant if you are, for whatever reason, tied to the mode frequency $$\omega$$. However, once you optimize over $$\omega$$, the coupling strength $$g$$ will often take a back seat to the cavity cooperativity $$C=g^2/\kappa \gamma$$ as the relevant figure of merit. (For an example of that in action, see e.g. this paper.) So, just something to keep in mind.

Finally, here are two interesting recent reviews on the topic, pointed out by @Wolpertinger:

• To provide some further reading on this excellent answer, I would like to point out 2 recent reviews on the topic, both available on arxiv: arxiv.org/abs/1807.11636 and arxiv.org/abs/1804.09275 Mar 1, 2019 at 14:59
• The deep strong coupling regime has been reached as of 2020 it seems, see here: doi.org/10.1038/s41586-020-2508-1 Oct 18, 2021 at 3:24