In this paper it says that for a two-level system excited by two fields:

$$ V_{ab} = -\mu_{ba} E_1 e^{i \omega_1 t}+ E_3 e^{-i \omega_3 t}$$

"In steady state the off-diagonal density-matrix element $\rho_{ba}$, exhibits harmonic oscillations at an infinite number of frequencies of the form $n\omega_1 \pm m \omega_2$, where n and m are integers." If the strong field $E_1$ is treated correctly to all orders while the weak probe field $E_3$ is treated to only first order, then $\rho_{ba}$ oscillates at three dominant frequencies: $\omega_1, \omega_3,$ and $2\omega_1-\omega_3$

Any ideas how I can show explicitly how this system can be expressed as an infinite number of frequencies? This paper cites a previous paper, where it works out $\rho_{ba}(\omega_1), \rho_{ba}(\omega_3),$ and $\rho_{ba}(2\omega_1 - \omega_3)$ by fourier transforming (which I discuss in this question). Within this previous paper it's not clear to me where it uses this assumption that $E_3<<E_1$ to obtain these three special frequencies.

Additionally, I'm hoping to understand what happens in the case when $E_1$ and $E_2$ are both strong.

In this paper it says that for a two-level system excited by two fields:

$$ V_{ab} = -\mu_{ba} E_1 e^{i \omega_1 t}+ E_3 e^{-i \omega_3 t}$$

"In steady state the off-diagonal density-matrix element $\rho_{ba}$, exhibits harmonic oscillations at an infinite number of frequencies of the form $n\omega_1 \pm m \omega_2$, where n and m are integers." If the strong field $E_1$ is treated correctly to all orders while the weak probe field $E_3$ is treated to only first order, then $\rho_{ba}$ oscillates at three dominant frequencies: $\omega_1, \omega_3,$ and $2\omega_1-\omega_3$

Any ideas how I can show explicitly how this system can be expressed as an infinite number of frequencies? This paper cites a previous paper, where it works out $\rho_{ba}(\omega_1), \rho_{ba}(\omega_3),$ and $\rho_{ba}(2\omega_1 - \omega_3)$ by fourier transforming (which I discuss in this question). Within this previous paper it's not clear to me where it uses this assumption that $E_3\ll E_1$ to obtain these three special frequencies.

Additionally, I'm hoping to understand what happens in the case when $E_1$ and $E_2$ are both strong.