The advent of broadband femtosecond laser pulses in the 1980s brought with it the observation of oscillatory signals arising from coherent quantum-beat signals in time-resolved spectroscopy measurements of atomic, semiconductor, and molecular samples. Much of that initial work focused on the vibrational wavepacket signals in transient–absorption spectroscopy measurements of the pigment–protein complex known as rhodopsin, the pigment of human vision.
In 2011-2013, Paul Arpin, Scott McClure, and I had loads of fun in Toronto measuring quantum beats in phycobilin light-harvesting proteins extracted (by Tihana Mirkovic) from crypotphyte algae. Although the pulse compression was not perfect—and we used an averaging scheme that my group at NYU later showed to be suboptimal—the signals were super strong and both Scott and Paul wrote papers. As a control, we somewhat arbitrarily picked the laser dye cresyl violet. That led to its own saga.
Most quantum beats in molecular systems arise from oscillatory vibrational wavepackets created by the in-phase excitation of several quanta of one or more vibrational modes of the molecule by the femtosecond pump pulse. Most measurements were interpreted by envisioning and modeling a Gaussian wavepacket propagating in time on the excited-state electronic potential-energy surface. This is highly related to the famous displaced harmonic-oscillator (DHO) model. The Gaussian wavepacket model arose to such prominence likely because of its special characteristic: It stays together as it propagates in a harmonic potential. Other shapes of wavepackets dissociate as they propagate. (A notable exception is the group of Paul Champion, who in the early 2000s developed and used the effective linear response theory.)
A question worth pondering: How accurate is the Gaussian wavepacket model? Mathematically, it takes an infinite number of vibrational quanta to produce a perfect Gaussian shape. How many quanta are excited by the femtosecond pump pulse? The answer depends on how different the two equilibrium positions are from each other – the displacement (∆). This brings up the all-important Franck–Condon factors, which express the degree of overlap between each vibrational eigenstate of the ground- and excited-state potential energy surfaces. The value of each Franck–Condon factor depends on ∆.
It seemed that researchers knew this: For example in Andrei Tokmakoff’s notes there is a depiction of how a two-quanta (n=0 and n=1) wavepacket oscillates in time.
Enter Paul Arpin. He realized there was not yet a publication modeling the vibrational quantum beats in TA spectroscopy, beginning with the Franck–Condon factors. After some discussion, we worked this out for the DHO model using the doorway–window approximation for TA spectroscopy and published what is now the first in a series of papers in 2020. We focused on the main observables: A sharp amplitude node—usually at a detection frequency near the fluorescence maximum—and a phase shift of approximately π at the same detection frequency.
In the Gaussian wavepacket model, one explains the ubiquitous amplitude node and phase shift as interference between forward- and backward-propagating waves. I never really understood this explanation very well. By contrast, in our 2020 report, Paul and I showed that the amplitude node and phase shift are explained almost trivially if one considers that the wavepacket is not Gaussian but rather composed of only the n=0 and n=1 vibrational quanta.
Due to that single aspect, we felt we were on to something. Hence, after that initial report, we have continued adding models. The key difference among nearly all of these models is the analytic form of the Franck–Condon factors. Sometime in 2021 we decided the name of femtosecond coherence spectra (FCS) best described this family of models. The list (as of March 2023) now includes:
- 1D DHO fundamental oscillations
- 1D DHO overtone oscillations
- quasi-2D DHO combination-band oscillations
- purely electronic wavepacket oscillations
- 1D displaced Morse oscillator (DMO) oscillations
- 1D unequal-curvature DHO fundamental oscillations
- 2D DHO with Duschinsky rotation oscillations
These models all have analytic expressions based on a relatively small number of parameters. Hence, we have even made a first attempt at extracting microscopic parameters (such as ∆) from measured spectra.