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

[tex] \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) [/tex]

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!