Fig. 3: Characterization of high-dimensional frequency entanglement.
Spatial beating of discrete frequency-entangled photon pairs of a, d two-dimensional frequency entanglement when setting τ1 = 0.12 ps (36 μm), b, e four-dimensional frequency entanglement when setting τ1 = 0.27 ps (80 μm) and c, f six-dimensional frequency entanglement when setting τ1 = 0.37 ps (110 μm). a–c Theoretical prediction and experimental results of the normalized coincidence rate, d–f real and imaginary parts of the restricted density matrix. d The superposition states \(\left|{\omega }_{m}{\omega }_{n}\right\rangle\) correspond to the basis of \(\left|{\omega }_{1}{\omega }_{1}\right\rangle\), \(\left|{\omega }_{1}{\omega }_{2}\right\rangle\), \(\left|{\omega }_{2}{\omega }_{1}\right\rangle\) and \(\left|{\omega }_{2}{\omega }_{2}\right\rangle\) in the two-dimensional space; e The superposition states \(\left|{\omega }_{m}{\omega }_{n}\right\rangle\) correspond to the basis of\(\left|{\omega }_{1}{\omega }_{n}\right\rangle\), \(\left|{\omega }_{2}{\omega }_{n}\right\rangle\), \(\left|{\omega }_{3}{\omega }_{n}\right\rangle\), and \(\left|{\omega }_{4}{\omega }_{n}\right\rangle\) (n = 1, 2, 3, and 4) in the four-dimensional space; f The superposition states \(\left|{\omega }_{m}{\omega }_{n}\right\rangle\) correspond to the basis of\(\left|{\omega }_{1}{\omega }_{n}\right\rangle\), \(\left|{\omega }_{2}{\omega }_{n}\right\rangle\), \(\left|{\omega }_{3}{\omega }_{n}\right\rangle\), \(\left|{\omega }_{4}{\omega }_{n}\right\rangle\), \(\left|{\omega }_{5}{\omega }_{n}\right\rangle\), and \(\left|{\omega }_{6}{\omega }_{n}\right\rangle\) (n = 1, 2, 3, 4, 5, and 6) in the six-dimensional space. The error bars are estimated by statistical methods assuming a Poisson distribution.
