# Rotating wave approximation

What is meant by counter rotating terms seen in the derivation of Jaynes Cummings model and what influence does it make if they are not neglected?

$$\hat{H}_\text{int}(t) = \frac{\hbar \Omega}{2} \left(\hat{a}\hat{\sigma}_{-} e^{-i(\omega_c+\omega_a)t} +\hat{a}^{\dagger}\hat{\sigma}_{+}e^{i(\omega_c+\omega_a)t} +\hat{a}\hat{\sigma}_{+}e^{i (-\omega_c+\omega_a) t} +\hat{a}^{\dagger}\hat{\sigma}_{-}e^{-i (-\omega_c+\omega_a) t}\right).$$
Where $$a/a^{\dagger}$$ are the ladder operators for the mode of the optical cavity and $$\sigma_{+/-}$$ are the raising/lowering operators for the 2-level atom. $$\omega_a$$ the the frequency corresponding to the energy difference between the two levels of the atom and $$\omega_c$$ is the frequency of the cavity mode.
The 'counter-rotating terms' in this Hamiltonian are the terms where the sign of $$\omega_c$$ is the same as the sign of $$\omega_a$$ (in the exponential). So in this case, the counter-rotating terms are the first two terms: $$\hat{a}\hat{\sigma}_{-} e^{-i(\omega_c+\omega_a)t}$$ and $$\hat{a}^{\dagger}\hat{\sigma}_{+}e^{i(\omega_c+\omega_a)t}$$
If they are not neglected, then it makes calculations harder, which is why we tend to neglect them. If you choose to include them, then they will contribute high-frequency oscillations in the evolution of the system, this is because $$\omega_a + \omega_c$$ is a higher frequency than $$\omega_a - \omega_c$$. We can often get away with neglecting them, if we are interested in things happening at the timescale of the period of $$\omega_a - \omega_c$$, as high frequency oscillations tend to cancel out when we are dealing with a longer timescale. The greater the difference between $$\omega_a + \omega_c$$ and $$\omega_a - \omega_c$$, the less difference neglecting the counter-rotating terms will make.