Photon Echo, Stimulated Photon Echo, Transient Grating, Reverse Photon Echo, and Reverse Transient Grating Spectroscopies

Photon echo was one the first examples of an optical analogue of NMR. Figure 4a,b show the two photon echo WMEL diagrams. An initial pulse creates a $ga$ coherence after the first interaction. After a delay time, $\tau$, a second pulse induces the second and third interactions to create an ag coherence that reemits. Phase matching is important. For photon echo, the phase matching condition is $\vec{k}_{PE} = -\vec{k}_1 + 2 \vec{k}_2$, where the subscripts indicate the time ordering of the two pulses. The usefulness of photon echo rests on having spectral transitions that are inhomogeneously broadened so molecules that have different environments or conformations have a distribution of different transition frequencies. The temporal dependence of the excited $ga$ coherences is $e^{i\omega_{ag}t}$ which becomes $e^{i\omega_{ag}\tau}$ after the $\tau$ delay.  The temporal dependence of the final $ag$ coherence is $e^{-i\omega_{ag}t}$, conjugate to the $ga$ coherence. After a delay of $\tau$, the net phase will be zero, regardless of the ωag frequency. At this point in time, all of the coherences will be again in phase and the re-emission becomes completely coherent (it scales as N2). The large output intensity that results at time τ is called the echo. By measuring the echo intensity as function of delay time, one can measure the dephasing rate (Γag) of the ga coherence.

            Stimulated photon echo experiments are three pulse experiments so the three excitation interactions in figure 4 occur at different times. The phase matching condition is $\vec{k}_{SPE} = -\vec{k}_1 + \vec{k}_2 + \vec{k}_3$. Figure 5a-c shows there are three pathways- a) ground state bleaching, b) stimulated emission, c) excited state absorption. The sign of the polarization is (-1)n where n is the number of bra-side interactions so excited state absorption has the opposite sign from the other pathways, that is it creates intensity decreases rather than increases. After the second interaction in figure 4a,b, one has either a $gg$ ground state population (figure 4a) or an $aa$ excited state population (figure 4b). After a delay time T, the third pulse arrives to create the $ag$ output coherence. Measuring the stimulated photon echo intensity as a function of the T delay time measures the population changes of the ground and excited state populations. 

            Figure 4e,f shows the WMEL diagrams for transient grating experiments. Transient grating methods use two pulses that occur at the same time but are angled relative to each other. The phase matching condition is $\vec{k}_{TG} = \vec{k}_1 - \vec{k}_{k'} + \vec{k}_2$ where the subscripts again indicate the time ordering. These pulses are represented by the first two interactions in fig. 4e,f. They create gg (figure 4e) and aa (figure 4f) populations that are spatially modulated ($e^{i(\vec{k}_1 - \vec{k}_{1'})\cdot \vec{z}}$) to form a grating. The third pulse acts as a probe to create the ag output coherence. Its direction corresponds to its reflection off the grating created by the first two pulses. By measuring the transient grating intensity as a function of the delay between the two excitation pulses, one measures the population decay of the ground and excited state populations. A second delay can be introduced by separating the first two pulses in time. The transient grating experiment now looks very similar to the stimulated photon echo experiment. The most important difference lies in how they respond to inhomogeneous broadening. Note that the coherence formed after the first pulse is an ag coherence and so is the coherence formed after the last pulse. The phase difference created during the first delay no longer cancels the phase change formed by the last coherence; they now add, and rephasing never occurs. This form of transient grating experiment is called a non-rephasing pathway.

            Methods called reverse photon echo and reverse transient grating also exist. These methods are based on reversing the time orderings of the two pulses. In photon echo, the last two interactions occurred simultaneously. In reverse photon echo, the two simultaneous interactions now occur first. The phase matching is $\vec{k}_{RPE} = 2\vec{k}_1 - \vec{k}_2$. Similarly, the first two interactions were simultaneous in transient grating methods. In reverse transient grating experiments, the two simultaneous interactions now occur last. The phase matching is $\vec{k}_{RTG} = \vec{k}_1 - \vec{k}_2 + \vec{k}_{2'}$.  Again, the subscripts indicate the time ordering. Typical WMEL for these methods are shown in figure 4d and h. These methods have not been used appreciably except in the developing field of coherent multidimensional spectroscopy.