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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.

Yvan Castin, with Jean Dalibard and Klaus Mølmer, formulated a stochastic wave-function approach to dissipative quantum systems. Their construction samples conditional jumps and no-jump evolution so that the ensemble reproduces a density-operator master equation.^[ref]

Using state vectors can reduce the cost of numerical simulations compared with directly storing a full density matrix, although many trajectories may be needed for precise averages. The 2008 posting reproduces a 1992 manuscript and explicitly acknowledges earlier related quantum-filtering work.

Zero-quantum coherence is a component of NMR coherence with total coherence order $p=0$, invariant under a common $z$ rotation. In a multispin system it can connect distinct basis states having the same total spin projection. This makes it different from merely specifying diagonal populations.^[ref]

For two spin one-half nuclei, consider $A=I_{+}{S}_{-}$. With $F_z=I_z+S_z$,It transfers one projection upward and the other downward, keeping their total fixed. The operator is proportional to $|\uparrow \downarrow ⟩⟨\downarrow \uparrow |$, an off-diagonal matrix element in the product basis.

An original state comparison usesThe pure-state density matrix has a zero-quantum off-diagonal pair. An equal mixture of the two basis states has the same populations and no such pair. Both have total projection zero, but their coherent properties differ.

Invariance under a common rotation does not imply that the component never evolves. For $H/\hslash ={\Omega }_I{I}_z+{\Omega }_S{S}_z$, its evolution isFor invented offsets ${\Omega }_I/(2\pi )=100\mathrm{Hz}$ and ${\Omega }_S/(2\pi )=70\mathrm{Hz}$, its coefficient completes a phase cycle in $1/30\mathrm{s}$. Adding the same offset to both nuclei leaves that difference unchanged. Unequal perturbations can still change it; the example is not a universal protection against noise.^[ref]

The zero order helps distinguish contributions in NMR phase cycling and gradient-based coherence selection. Longitudinal magnetization also has order zero, so an order label alone does not identify a unique physical component or guarantee direct detection.^[ref]