Simulating a pump-probe experiment of a collection of Lorentz oscillators

Question

I am having some fundamental misunderstanding I think in my simulation of a pump-probe experiment. Take, for example, a medium that has a resonant frequency at 266 nm. I am pumping it with a Gaussian beam centered at that wavelength with some defined bandwidth. Assuming, for simplicity, the sample is a collection of Lorentz oscillators, and foregoing any time-dependence on the population (just a simple zero-time signal), my model for the resonant polarization is:

$$ P_\mathrm{resonant} = \frac{Ne^2}{m_0}\frac{1}{(\omega_0^2 - \omega^2 - i\gamma\omega)} \epsilon$$

There are constants (density of atoms, etc.), but what's important here is whether or not we are at the resonant frequency. Where $\epsilon$ is my incoming electric field, of which I will define as:

$$\epsilon = \mathrm{Re}(\epsilon_0(\exp(-i(\omega t + \phi)))) = \epsilon_0\cos(\omega t + \phi) $$

Thus, my signal will be: $\epsilon_\mathrm{signal} = \epsilon_\mathrm{pump/probe} + P_\mathrm{resonant}$

What I am doing is determining what $P_\mathrm{resonant}$ is at each wavelength/frequency of my Gaussian pulse. However, I'm not sure how to deal with the electric field, as I'm interested in the resulting signal; I am thinking it is equivalent to the amplitude of the interfered field (the modulus of it), but I can't seem to figure it out; I would like to pass my probe pulse (another Gaussian) through the sample and measure its interaction (whether it is transient gain or absorption), but I am not certain how to achieve that part. Can anybody help me understand this part?

Answer 1

When you write a polarization response of the form $$ P_\mathrm{resonant} = \frac{Ne^2}{m_0}\frac{1}{(\omega_0^2 - \omega^2 - i\gamma\omega)} \epsilon $$ you're basically already committed to a spectral view of the problem: this polarization describes the response to a monochromatic driver, $$ P_\mathrm{resonant}(\omega) = \frac{Ne^2}{m_0}\frac{1}{(\omega_0^2 - \omega^2 - i\gamma\omega)} \epsilon(\omega), $$ in the understanding that you've taken your real-world time-dependent driver and Fourier transformed it into a bunch of monochromatic components, $$ \epsilon(t) = \int \epsilon(\omega) e^{-i\omega t} \mathrm d\omega, $$ and that the total response of your medium is the combination of the response at each of the frequencies involved (in this case, the continuum inside the bandwidth of your driving laser).