  Time-frequency duality, which enables control of optical waveforms by
manipulating amplitudes and phases of electromagnetic fields, plays a pivotal
role in a wide range of modern optics. The conventional one-dimensional (1D)
time-frequency duality has been successfully applied to characterize the
behavior of classical light, such as ultrafast optical pulses from a laser.
However, the 1D treatment is not enough to characterize quantum mechanical
correlations in the time-frequency behavior of multiple photons, such as the
biphotons from parametric down conversion. The two-dimensional treatment is
essentially required, but has not been fully demonstrated yet due to the
technical problem. Here, we study the two-dimensional (2D) time-frequency
duality duality of biphotons, by measuring two-photon distributions in both
frequency and time domains. It was found that generated biphotons satisfy the
Fourier limited condition quantum mechanically, but not classically, by
analyzing the time-bandwidth products in the 2D Fourier transform. Our study
provides an essential and deeper understanding of light beyond classical wave
optics, and opens up new possibilities for optical synthesis in a
high-dimensional frequency space in a quantum manner.
