Quantum statistics is defined by Hilbert space products between the eigenstates associated with state preparation and measurement. The same Hilbert space products also describe the dynamics generated by a Hamiltonian when one of the states is an eigenstate of energy E and the other represents an observable B. In this paper, we investigate this relation between the observable time evolution of quantum systems and the coherence of Hilbert space products in detail. It is shown that the times of arrival for a specific value of B observed with states that have finite energy uncertainties can be used to derive the Hilbert space product between eigenstates of energy E and eigenstates of the dynamical variable B. Quantum phases and interference effects appear in the form of an action that relates energy to time in the experimentally observable dynamics of localized states. We illustrate the relation between quantum coherence and dynamics by applying our analysis to several examples from quantum optics, demonstrating the possibility of explaining non-classical statistics in terms of the energy-time relations that characterize the corresponding transformation dynamics of quantum systems.

Multi-photon interference results in modulations of output probabilities with phase shift periods that are much shorter than 2 Pi. Here, we investigate the physics behind these statistical patterns in the case of well-defined photon numbers in the input and output modes of a two-path interferometer. We show that the periodicity of the multi-photon interference is related to the weak value of the unobserved intensity difference between the two arms of the interferometer. This means that the operator relations between the photon number differences in input, path, and output can be used to determine the periodicity of the experimentally observed quantum interference, establishing an important link between the classical causality of random phase interference and quantum effects that depend on the superposition of classically distinct possibilities.Comment: 15 pages, including 6 figures; clarifications of approximation and assumptions added in section IV. Final version for publication in Phys. Rev.

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