= Vibrational coherence
<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 compares
$$
|\psi\rangle=\frac{|0\rangle+e^{i\phi}|1\rangle}{\sqrt2},\qquad
\rho_{\rm pure}=\frac12\begin{pmatrix}1&e^{-i\phi}\\e^{i\phi}&1\end{pmatrix}
$$
with an incoherent mixture $\rho_{\rm mix}=\tfrac12\operatorname{diag}(1,1)$. 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 $q_{01}\cos\phi$, 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.<Two-dimensional infrared spectroscopy of peptides by phase-controlled femtosecond vibrational photon echoes by M. C. Asplund and others>[Abstract, phase matching, heterodyning and echo phase/amplitude.]{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.
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