  The classical Wiener-Khinchin theorem (WKT), which can extract spectral
information by classical interferometers through Fourier transform, is a
fundamental theorem used in many disciplines. However, there is still need for
a quantum version of WKT, which could connect correlated biphoton spectral
information by quantum interferometers. Here, we extend the classical WKT to
its quantum counterpart, i.e., extended WKT (e-WKT), which is based on
two-photon quantum interferometry. According to the e-WKT, the
difference-frequency distribution of the biphoton wavefunctions can be
extracted by applying a Fourier transform on the time-domain Hong-Ou-Mandel
interference (HOMI) patterns, while the sum-frequency distribution can be
extracted by applying a Fourier transform on the time-domain NOON state
interference (NOONI) patterns. We also experimentally verified the WKT and
e-WKT in a Mach-Zehnder interference (MZI), a HOMI and a NOONI. This theorem
can be directly applied to quantum spectroscopy, where the spectral correlation
information of biphotons can be obtained from time-domain quantum interferences
by Fourier transform. This may open a new pathway for the study of light-matter
interaction at the single photon level.
