Time-resolved measurement of the quantum states of photons using two-photon interference with short-time reference pulses

To fully utilize the energy-time degree of freedom of photons for optical quantum-information processes, it is necessary to control and characterize the temporal quantum states of the photons at extremely short time scales. For measurements of the temporal coherence of the quantum states beyond the time resolution of available detectors, two-photon interference with a photon in a short-time reference pulse may be a viable alternative. In this paper, we derive the temporal measurement operators for the bunching statistics of a single-photon input state with a photon from a weak coherent reference pulse. It is shown that the effects of the pulse shape of the reference pulse can be expressed in terms of a spectral filter selecting the bandwidth within which the measurement can be treated as an ideal projection on eigenstates of time. For full quantum tomography, temporal coherence can be determined by using superpositions of reference pulses at two different times. Moreover, energy-time entanglement can be evaluated based on the two-by-two entanglement observed in the coherences between pairs of detection times