# Eigenstructure of the Dicke Model

I am beginning a study of the Dicke model and found a very interesting publication: "The Dicke model in quantum optics: Dicke model revisited" by Barry M Garraway in Phil. Trans. R. Soc. A (2011).

I would like to understand how he developed the eigenstates in section 2 (for the 2-atom model), what each states actually means physically, and why he can say that the triplet spin state is sufficient in developing the Dicke energy levels. The latter seems an audacious claim, to me, because I'm not sure we can say that every evolution along the triplet states will lead to superradiance.

He writes n-states in terms of the spin-states. First, what is he calling the number of excitations for the 2-atom system? He has only 2 atoms, so how can he generically use |n>, |n+/-1>? Second, how did he get the coefficients for the linear combination of spin-states? Third, how does he find the eigenenergies for this new basis? This set up is very strange to me, perhaps because the Dicke model is so new to me. The biggest question is what does this new excitation number-basis say about the 2-atom model?

• You will have to add details, and be more specific. – Norbert Schuch Sep 29 at 23:31

## 1 Answer

First, what is he calling the number of excitations for the 2-atom system?

There are two atoms coupled to a radiation mode, i.e. to a harmonic oscillator. The states $$|n\rangle$$ refer to the harmonic oscillator, the states $$|S, m_s\rangle$$ refer to the two atoms. (Each atom is assumed to have only two energy levels and thus behaves like a spin-1/2. Two spin-1/2 add up to total spin $$S=1$$ or $$S=0$$. Garraway here only considers the $$S=1$$ states.)

Second, how did he get the coefficients for the linear combination of spin-states? Third, how does he find the eigenenergies for this new basis?

He tries to find the eigenstates and eigenvalues of $$a^\dagger a + S_z$$. You can easily verify that (2.1) and (2.2) are correct by checking e.g. that $$(a^\dagger a + S_z) |n,+1\rangle = (n\omega_c + 2g\sqrt{n+1/2}) |n,+1\rangle$$.

In order to find the coefficients, he probably made the ansatz that an eigenstate of $$a^\dagger a + S_z$$ will be a linear combination $$|\psi\rangle = \alpha |m_s=1,n-1\rangle + \beta |m_s=0,n\rangle + \gamma |m_s=-1,n+1\rangle$$ and solved the eigenvalue equation $$(a^\dagger a + S_z) |\psi\rangle = \lambda |\psi\rangle$$.