A time-varying signal is applied to a collection of non-degenerate absorbing two-level atoms. This signal is tuned exactly to the transition frequency _a, and is amplitude- modulated at a low-frequency wm: W₁₂(t) = W₂₁(t) = W_a + W_b cos(_mt). The modulation depth is small, so that W_b << W_a, and the modulation frequency is low,

so that _m << ₂₁ and w_m << _a. This modulation frequency wm, however, can be of the same order as the inverse relaxation rate 1/T₁ = ₁₂ + ₂₁. (1) Insert this time-modulated signal into the two-level rate equation and solve for the time-varying population difference N(t), including saturation effects. (2) Find out how the phase lag between the signal modulation W₁₂(t) and the population modulation N(t) will change with the modulation frequency _m. Hints: Assume the population difference varying like N(t) = N_a + N_b(t), with N_b << N_a, and linearize the equation by neglecting cross-products of the small terms W_b and N_b.