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Vibrational coherence is a definite phase relationship between components of vibrational motion. In a quantum description, it concerns off-diagonal entries of a density matrix in a specified vibrational energy basis, rather than only the populations of those levels. The basis must be stated because quantum coherence is basis-dependent.

An original two-level illustration compareswith an incoherent mixture ${\rho }_{\mathrm{mix}}=\frac{1}{2}\mathrm{diag}(1,1)$. Both have equal quantum state populations, but only the first has a nonzero off-diagonal element. For a displacement operator with real off-diagonal matrix element $q_{01}$ and zero diagonal elements, its expectation in the pure state is $q_{01}\cos \varphi$, while the mixture gives zero. Relative phase can therefore affect an observable without changing the populations.

For an ensemble of classical vibrational oscillators, the analogous distinction is between individual motion and the sum of their phases. Oscillators can retain their energies while a spread of phases reduces the net coherent signal. A weak ensemble signal need not mean every molecule has stopped vibrating. Conversely, coherence does not require every possible vibration to have one common phase.

A primary peptide experiment measured phase and amplitude of vibrational photon echoes using optical heterodyne detection. It provides an experimental setting in which phase relationships are observable rather than merely a drawing convention.^[ref]

Dephasing and vibrational relaxation describe different aspects of dynamics: losing a phase relationship and changing vibrational populations need not occur at the same rate. Extracting either from a measured trace requires the pulse sequence, detector response and a stated molecular model.

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 isFor a zero-mean normal distribution of offsets with standard deviation $\sigma$,At $t_3=\tau$, the offset phases cancel. With stipulated $\tau =1\mathrm{ps}$ and $\sigma =2\mathrm{rad}\mathrm{p}{\mathrm{s}}^{-1}$, the normalized static-offset factor is $e^{-2}\approx 0.135$ 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.^[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.