Dicke State

Collective spin operators§

Define§

$$\hat S_\pm = \sum_i{\hat\sigma_{(i)}^\pm} \qquad \hat S_e=\sum_i{\hat\sigma_e^{(i)}}\qquad \hat S_g=\sum_i{\hat\sigma_g^{(i)}} \qquad \hat S_z=\frac{1}{2}\sum_i{\hat\sigma_z^{(i)}}$$

Commutation§

With linearity of commutator, easily we see that this Definition of collective spin not only make sense, but also obey single atom spin commutation relation.

$$\left[\hat S^+, \hat S^-\right]=2\hat S_z \qquad \text{etc...}$$

Effective z-value component spin§

Just as single atom spin could have a z-component, here, easily we note that from

$$S_z=\frac{1}{2}\sum_i{\hat\sigma_z^{(i)}} = \sum_{i}\hat\sigma_e^{(i)}-\frac{N}{2}$$

z-component will be in $\{N/2, N/2 - 1, \dots,-N/2\}$, call this $m$

Basis§

So, sufficiently by $S$ and $m$ ( ignoring relative dephase, or so called inhomogeneous broadening ), we define a basis

$$\ket{S,m}=\frac{1}{\sqrt{\binom{N}{n_e}}}\sum_{\text{perm}}\ket{\underbrace{eee\dots}_{n_e} \underbrace{ggg\dots}_{n_g}}$$

Where $n_e=S+m$, $n_g=S-m$