Where does the 1/√N factor in the Dicke model arise from?

[lfqm]

Hello colleagues, hope you can help me.

The Dicke model describes a system of N two-level atoms cooperatively interacting with a single mode of an electromagnetic field as follows:

\hat{H}_{D}=\omega_{A}\hat{J}_{z}+\omega_{F}\hat{a}^{\dagger}\hat{a}-\frac{\gamma}{\sqrt{N}}\left(\hat{J}_{-}\hat{a}^{\dagger}+\hat{J}_{+}\hat{a}\right)-\frac{\gamma}{\sqrt{N}}\left(\hat{J}_{+}\hat{a}^{\dagger}+\hat{J}_{-}\hat{a}\right)

In the case of a single atom the factor turns into 1, and the hamiltonian of the N atoms should be the sum of N single atom hamiltonians, right? So, where does the 1/√N factor arise from?

Thanks!

 

[DrDu]

I think you fix the asypmtotics in the high N limit. Specifically, using a Holstein Primakoff transformation, you can express the spin operators in terms of harmonic oscillator operators, whence a sqrt(N) appears which gets canceled by the definition of the coupling constant:
http://rsta.royalsocietypublishing.org/content/roypta/369/1939/1137.full.pdf

 

[DrDu]

Actually, it was the first link in google searching for "Dicke model" not related to fat female models ("dicke" means fat, in German) :-)