It is shown both theoretically and experimentally that the phase of a trans
ition dipole matrix element may be determined from the interference between
competing quantum mechanical paths. Specifically, we consider simultaneous
m- and n-photon excitation of a molecule to a degenerate continuum state,
leading to multiple reaction products. Variation of the relative phase of t
he electromagnetic fields produces a modulation of the product signals in t
he various channels. A relation between the phase lag between a pair of pro
duct signals and properties of the continuum is established. Information is
revealed about coupling mechanisms, the properties of weak spectroscopic t
ransitions, and the phases of quasi-bound wave functions.