(a) Phase space dynamics of a molecular vibrational harmonic oscillator. Time evolution is equivalent to a rotation of the phase space. The Wigner function representing the vibrational quantum state is shown at two different times differing by a quarter of the vibrational period. Measurement of the position of the vibrational wave packet is equivalent to projection of the Wigner function onto the coordinate axis. The two marginal distributions shown (lines) indicate the origin of the Fourier transform, since they are proportional to the initial probability distributions of the position and momentum probability of the wave packet. (b) The diatomic molecule is excited from the ground to the first excited electronic state by a shaped optical program pulse (P) that prepares the initial quantum state of the vibrational mode—effectively programming the distribution to be Fourier transformed. A quarter vibrational period later, a reference pulse (R), generates a second vibrational wave packet that interferes with the first, converting the accumulated phase into populations of vibrational levels in the excited state. A measurement laser (M) reads out the populations by inducing transitions to a higher-lying electronic state, from which fluorescence (f) is detected. The fluorescence intensity is proportional to the population in the vibrational level to which the measurement laser is tuned.