= Vibrational photon echo
A <vibrational photon echo> is a coherently emitted <infrared> signal in which a pulse sequence allows contributions from different vibrational frequencies to rephase. It probes <vibrational coherence> and its loss. Rephasing of optical phase is different from physically reversing a molecular reaction or restoring a lost population.
An original ideal model makes the phase cancellation explicit. Suppose a group of oscillators has angular-frequency offsets $\Delta$ distributed with density $g(\Delta)$. A rephasing pathway accumulates phase $-\Delta\tau$ during an initial interval and $+\Delta t_3$ during detection. Its static-offset contribution is
$$
E(\tau,t_3)\propto\int g(\Delta)e^{i\Delta(t_3-\tau)}\,d\Delta.
$$
For a zero-mean <normal distribution> of offsets with <standard deviation> $\sigma$,
$$
E(\tau,t_3)\propto e^{-\sigma^2(t_3-\tau)^2/2}.
$$
At $t_3=\tau$, the offset phases cancel. With stipulated $\tau=1\,\mathrm{ps}$ and $\sigma=2\,\mathrm{rad\,ps^{-1}}$, the normalized static-offset factor is $e^{-2}\approx0.135$ at $t_3=0$ and1 at rephasing. This ideal cancellation does not remove other sources of coherence decay or guarantee the same signal shape in a real experiment.
A primary peptide experiment used phase-controlled femtosecond pulses and phase matching to isolate vibrational echoes. <Heterodyne detection> measured their phase and amplitude, enabling <two-dimensional infrared spectroscopy> as pulse delays were varied.<Two-dimensional infrared spectroscopy of peptides by phase-controlled femtosecond vibrational photon echoes by M. C. Asplund and others>[Abstract, phase matching, heterodyning and multiple pulse delays.]{ref}
The chosen pathway and pulse geometry matter: not every nonlinear infrared signal is an echo. Comparing delay-dependent amplitudes and phases can separate recoverable frequency dispersion from other dynamical effects, but interpretation needs a response model, pulse characterization and an account of the sample's changing environment.
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