The rotating wave approximation (RWA) is well justified in a regime of a small perturbation. In this limit you can neglect the so-called Bloch-Siegert and Stark shifts. You can find an explanation in this paper. But, in order to make this explanation self-contained, I will give an idea with the following model
H=Δσ3+V0sin(ωt)σ1
being, as usual σi the Pauli matrices. You can easily work out a small perturbation series for this Hamiltonian working in the interaction picture with
HI=e−iℏσ3tV0sin(ωt)σ1eiℏσ3t
producing, with a Dyson series, the following next-to-leading order correction
Texp[−iℏ∫t0HI(t′)dt′]=I−iℏ∫t0dt′V0sin(ωt′)σ1e2iℏΔσ3t′+….
Now, let us suppose that your system is in the eignstate |0⟩ of the unperturbed Hamiltonian. You will get
|ψ(t)⟩=|0⟩−iℏ∫t0dt′V0sin(ωt′)e−2iℏΔt′σ1|0⟩+…
=|0⟩−12ℏ∫t0dt′V0(eiωt′−2iℏΔt′−e−iωt′−2iℏΔt′)σ1|0⟩
Now, very near the resonance ω≈2Δ, one term is overwhelming large with respect to the other and one can write down
|ψ⟩≈|0⟩−V02ℏtσ1|0⟩+….
but in the original Hamiltonian this boils down to
HI=V0σ1sin(ωt)(cos(2Δt)+iσ3sin(2Δt))
=V02σ1(sin((ω−2Δ)t)+sin((ω+2Δ)t))
+V02σ2(cos((ω−2Δ)t)−cos((ω+2Δ)t))
≈V02σ2
with all the counter-rotating terms properly neglected with the condition ω≈2Δ applied. It is essential to emphasize that, as the applied field increases, this approximation becomes even less reliable and it is just the leading order of a perturbation series in a near-resonance regime.
This post imported from StackExchange Physics at 2014-08-05 16:17 (UCT), posted by SE-user Jon