5.2: 2D Spectroscopy from Third Order Response

These examples indicate that narrow band pump-probe experiments can be used to construct 2D spectra, so in fact the third-order nonlinear response should describe 2D spectra. To describe these spectra, we can think of the excitation as a third-order process arising from a sequence of interactions with the system eigenstates. For instance, taking our initial example with three levels, one of the contributing factors is of the form R₂:

Setting \(\tau_2=0\) and neglecting damping, the response function is

\[R_2(\tau_1,\tau_3)=p_a|\mu_{ab}|^2|\mu_{ac}|^2e^{-i\omega_{ba}\tau_1-i\omega_{ca}\tau_3} \label{7.1}\]

The time domain behavior describes the evolution from one coherent state to another—driven by the light fields:

A more intuitive description is in the frequency domain, which we obtained by Fourier transforming eq. (7.1):

\[\begin{aligned} \tilde R_2(\omega_1,\omega_3) &= \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{i\omega_1\tau_1+i\omega_3\tau_3}R_2(\tau_1,\tau_3)d\tau_1d\tau_3 \\ &=p_a|\mu_{ab}|^2|\mu_{ac}|^2\left\langle\delta(\omega_3-\omega_{ca})\delta(\omega_1-\omega_{ba})\right\rangle \\ &\equiv p_a|\mu_{ab}|^2|\mu_{ac}|^2P(\omega_3,\tau_2;\omega_1) \end{aligned} \label{7.2}\]

The function P looks just like the covariance \(\langle xy \rangle\) that describes the correlation of two variables x and y. In fact P is a joint probability function that describes the probability of exciting the system at \(\omega_{ba}\) and observing the system at \(\omega_{ca}\) (after waiting a time \(\tau_2\)). In particular, this diagram describes the cross peak in the upper left of the initial example we discussed.