Manipulation of a two-photon state in a ${\chi }^{\left(2\right)}$-modulated nonlinear waveguide array

We propose to engineer the quantum state in a high-dimensional Hilbert space by taking advantage of a ${\chi }^{\left(2\right)}$-modulated nonlinear waveguide array. By varying the pump condition and the waveguide array length, the momentum correlation between the signal and idler photons can be manipulated, exhibiting bunching, antibunching, and the evolution between these two states, which are characterized by the Schmidt number. We find the Schmidt number is dependent on a structure parameter, namely the ratio of the array length and the number of channels pumped. By designing the linear profile waveguide array, the degree of spatial entanglement shows a periodic relationship with the slope of linear profile, during which a high degree of position-bunching state is suggested. The two-photon self-focusing effect is disclosed when the ${\chi }^{\left(2\right)}$ modulation in the waveguide array contains a parabolic profile, which can be designed for efficient coupling between a waveguide array and fibers. These results shed light on a feasible way to achieve desirable quantum state on a single waveguide chip by a compact engineering of ${\chi }^{\left(2\right)}$ and also suggest a degree of freedom for quantum walk and other related applications.

Figure 1

Sketch of the ${\chi }^{\left(2\right)}$-modulated nonlinear waveguide array is given. The deep blue areas are proton-exchanged waveguide channels and the red shows the domain reversal area. The red curved line is the initial position profile of the domain.

Figure 2

Pump condition, phase mismatch, and the momentum correlation is demonstrated when the pump condition is $n=0,\pm 1,\pm 2,3$ under three array lengths $L=0.3L_z,0.9L_z,4L_z$.

Figure 3

Schmidt number is given as a function of ratio $L/L_z$ when the number of pumped waveguide channels $N$ varies from 16 to 21.

Figure 4

Schmidt number is given as a function of the ratio $(L/L_z)/N$ when the number of pumped waveguide channels $N$ varies from 8 to 17. The Schmidt number for odd and even $N$ is plotted separately.

Figure 5

Momentum and position correlation of a two-photon state generated from the WA with linear domain profile.

Figure 6

Schmidt number when the slope of the linear profile increases with all waveguide channels pumped and the length of the array is $L=10Lz$.