  We theoretically propose a novel quantum interferometer in which the NOON
state interferometer (NOONI) is combined with the Hong-Ou-Mandel interferometer
(HOMI). This interferometer combined the advantages of both the NOONI that
depends on biphoton frequency sum, and the HOMI that depends on biphoton
frequency difference into a single interferometer. It can thus simultaneously
obtain the spectral correlation information of biphotons in both frequency sum
and difference by taking the Fourier transform from a single time-domain
quantum interferogram, which provides a method for complete spectral
characterization of an arbitrary two-photon state with exchange symmetry. A
direct application of such an interferometer can be found in quantum
Fourier-transform spectroscopy where direct spectral measurement is difficult.
Furthermore, as it can realize the measurement of time intervals on three
scales at the same time, we expect that it can provide a new method in quantum
metrology. Finally, we discuss another potential application of such an
interferometer in the generation and characterization of high-dimensional and
phase-controlled frequency entanglement.
