A Quantum Theory of Angle and Relative Phase Measurement

Abstract

The complementarity between time and energy, as well as between an angle and a component of angular momentum, is described at three different layers of understanding. The phenomena of super-resolution are readily apparent in the quantum phase representation which also reveals that entanglement is not required. We modify Schwinger's harmonic oscillator model of angular momentum to include the case of photons. Therein the quantum angle measurement is shown to be equivalent to the measurement of the relative phase between the two oscillators. Two reasonable ways of dealing with degeneracy are shown to correspond to: a conditional measurement which takes a snapshot in absolute time (corresponding to adding probability amplitudes); and a marginal measurement which takes an average in absolute time (corresponding to adding probabilities). The sense in which distinguishability is a "matter of how long we look" is discussed and the meaning of the general theory is illustrated by taking the two oscillators to be circularly polarized photons. It is shown that an odd number of x-polarized photons will never have an angle in correspondence with the y-axis; but an even number of x-polarized photons always can! The behavior of an x-polarized coherent state is examined and the snapshot angular distributions are seen to evolve into two counter-rotating peaks resulting in considerable correspondence with the y-axis at the time for which a classical linear polarization vector would shrink to zero length. We also demonstrate how the probability distribution of absolute time (herein a measurable quantity, rather than just a parameter) has an influence on how these snapshot angular distributions evolve into a quantum version of the polarization ellipse.