  Wiener-Khinchin theorem, the fact that the autocorrelation function of a time
process has a spectral decomposition given by its power spectrum intensity, can
be used in many disciplines. However, the applications based on a quantum
counterpart of Wiener-Khinchin theorem that provides a translation between
time-energy degrees of freedom of biphoton wavefunction still remains
relatively unexplored. Here, we use a quantum Wiener-Khinchin theorem (QWKT) to
state that two-photon joint spectral intensity and the cross-correlation of
two-photon temporal signal can be connected by making a Fourier transform. The
mathematically-defined QWKT is experimentally demonstrated in
frequency-entangled two-photon Hong-Ou-Mandel (HOM) interference with the
assistance of spectrally-resolved detection. We apply this method to
spectral-domain quantum optical coherence tomography that detects
thickness-induced optical delays in a transparent sample, and show that our
method suffices to achieve great advantages in measurement precision within a
wide dynamic range and capturing time over the conventional HOM interferometric
schemes. These results may significantly facilitate the use of QWKT for quantum
information processing and quantum interferometric spectroscopy.
