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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?

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In the Schrodinger picture, the Jaynes-Cummings Hamiltonian is:

$\hat{H}_\text{int}(t) = \frac{\hbar \Omega}{2} \left(\hat{a}\hat{\sigma}_{-} e^{-i(\omega_c+\omega_a)t} +\hat{a}^{\dagger}\hat{\sigma}_{+}e^{i(\omega_c+\omega_a)t} +\hat{a}\hat{\sigma}_{+}e^{i (-\omega_c+\omega_a) t} +\hat{a}^{\dagger}\hat{\sigma}_{-}e^{-i (-\omega_c+\omega_a) t}\right).$

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.

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