where σ σ z ^ ^ is the Pauli z spin operator for the two eigenstates | e ⟩ ⟩ and | g ⟩ ⟩ of the isolated two level system separated in energy by ℏ ℏ ω ω 0 ; σ σ ^ ^ + = | e ⟩ ⟩ ⟨ ⟨ g | and σ σ ^ ^ − − = | g ⟩ ⟩ ⟨ ⟨ e | are the raising and lowering operators of the two level system; a ^ ^ † † and a ^ ^ are the creation and annihilation operators for photons of energy ℏ ℏ ω ω in the cavity mode; and