Quantifying the rotating-wave approximation of the Dicke model

We analytically find quantitative, non-perturbative bounds to the validity of the rotating-wave approximation (RWA) for the multi-atom generalization of the quantum Rabi model: the Dicke model. Precisely, we bound the norm of the difference between the evolutions of states generated by the Dicke model and its rotating-wave approximated counterpart, that is, the Tavis-Cummings model. The intricate role of the parameters of the model in determining the bounds is discussed and compared with numerical results. Our bounds are intrinsically state-dependent and, in particular, capture a nontrivial dependence on the total angular momentum of the initial state; this behaviour also seems to be confirmed by accompanying numerical results.

💡 Research Summary

The paper presents a rigorous, non‑perturbative analysis of the rotating‑wave approximation (RWA) for the Dicke model, which describes N identical two‑level atoms (spins) collectively coupled to a single bosonic mode. The authors compare the exact Dicke Hamiltonian H_D with its RWA counterpart, the Tavis‑Cummings Hamiltonian H_TC, and derive explicit bounds on the norm of the difference between the two time‑evolution operators.

First, the authors define the Hilbert space as the tensor product of the N‑spin space (C²)^{⊗N} and the bosonic Fock space L²(R). They restrict attention to the dense subspace F_fin consisting of finite linear combinations of Hermite functions, i.e., states with a finite number of photons. This choice allows them to avoid technical difficulties associated with unbounded operators while still covering all physically relevant states.

Both Hamiltonians are written in the resonant case (ω₀ = ω) as
H_D = ω S_z ⊗ 1 + ω 1 ⊗ n̂ + λ (S_+ + S_-) ⊗ (a + a†),
H_TC = ω S_z ⊗ 1 + ω 1 ⊗ n̂ + λ (S_+ ⊗ a + S_- ⊗ a†).
Here S_{x,y,z} are collective spin operators, S_± = S_x ± i S_y, and a, a† are the usual bosonic annihilation and creation operators.

Moving to the interaction picture with respect to the free Hamiltonian H₀ = ω S_z ⊗ 1 + ω 1 ⊗ n̂, the time‑dependent interaction terms become
H₁(t) = λ