Identical atoms in the Dicke model

[lfqm]

Hey guys, I've recently read about the Tavis-Cummings and Dicke models and I got a little bit confused about them. They are suppoused to model N identical atoms interacting with a one-mode EM field, however the atomic operators are defined in the basis (for the case of two atoms):

$$\left\{{|e_{1},e_{2}>, |e_{1},g_{2}>, |g_{1},e_{2}>, |g_{1},g_{2}>}\right\}$$

which obviously makes a distinction between the atoms.
Then it gets even more confusing, as they start working in a spin basis $$\left\{{|j,m>}\right\}$$ which makes the atoms identical for the case j=N/2... I don't even undestand why they fix j=N/2

Concretely, my question is: What is the basis of the hilbert space the Dicke hamiltonian is acting on (the atomic part)?

The $$2^N$$ elements basis (distinguishable atoms), the $$\displaystyle\sum_{j=0}^{N/2}(2j+1)$$ elements basis (considering all spin values) or the $$N+1$$ elements basis (fixing j=N/2).

And what is the form of the atomic operators in this basis?

Thanks

 

[f95toli]

I don't think I can answer your question (I am not even sure I understand it).

However, the Dicke model deals with pseudo-spins, i.e. collective excitation of an ensemble of spin 1/2 systems. The N/2 factor comes from re-writing the sums over N spins as collective pseudo-spins (the collective angular momentum). You basically re-write it so that you end up with a Hamiltonian with no sums in it, even though you are dealing with an ensemble; i.e. the basis would certainly be of the size N/2 since that is the size of representations of the pseuo-spin operators.

Note that for all systems I can think of the atoms are identical from an "EM" point of view in that they have the same energy splitting and (ideally) the same coupling to the field mode of interest; But this does not imply that they are indistinguishable, they would certainly be separated spatially. There has also been quite a bit work done looking at generalized models, i.e. what happens if the coupling is non-uniform etc. which means you can waves and so on.

A typical experimental realization of the Dicke model would be an ensemble of e.g. ions in a high-Q cavity. The ions are tuned (using for example a Zeeman shift) so that they behave like spin 1/2 systems (i.e. simple two-level systems) and have energies resonant with the cavity frequency.