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J. Eur. Opt. Society-Rapid Publ. 21, 3( 2025)
the phase-matching kernel is a highly oscillation function for both frequency arguments x and w in a wide region, giving birth to many fine ripples. Even near the line x + w = x p, the function U( x, x p � x) is still highly oscillating, see Figure 1c. Asfz is the compensation of spatial frequency by the crystal, any mismatch Dk( x, w) from�fL to0canfulfill the phase-matching condition Dk( x, w)+ fz = 0atacertain position. As a result, the SPDC processes take place in the active zone – fL Dk( x, w) 0, which can also be known from the properties of Fresnel integral. With larger chirping parameter f, more frequencies are allowed to be down-converted from the pump. It allows the phase mismatching with wider frequency range to be compensated, which further expands the frequency response range of the biphoton. One then concludes that the frequencies contribute to the SPDC processes quite equally in the presence of chirp.
The BSA is a function with complicated quadratic phase mismatching factor caused by the interaction between the chirped QPM PPLN crystal and the three waves’ phase mismatching. And the chirped QPM plays a substantial roles for this special part. There would be an inevitable ultrabroad response frequency caused by the chirped poled periods for the QPM PPLN crystal.
The BSA function T( x, w) can be operated by Schmidt decomposition with the product of the orthogonal eigen basis in the form of x and w. T( x, w) can be expressed by the eigen basis of T written as follows [ 31 ]:
Tðx; wÞ ¼ X n pffiffiffiffiffi k nun x ð Þv n ðwÞ
Where k n is the eigenvalue corresponding the Schmidt modes u n and v n which defined as the eigenvectors for the signal and idler photons.
Schmidt decomposition is expected to quantize calculate the entanglement [ 17, 31, 33, 34 ] generated by SPDC. This decomposition process for SPDC involves two factors: the bandwidth of pump pulse and the structure of the nonlinear crystal. For the bandwidth of pump, it means that the frequencies of signal and idler would be independent and not be one-one matching. However, they often show strong correlation for both of them are related to the frequency of pump pulse. The correlation range of frequency for idler light is restricted by the bandwidth of pump light, and the detected signal frequency [ 34 ].
The biphoton amplitude is symmetric, T( x, w) = T( w, x), since Dk( x, w)= Dk( w, x). It enables one to make the following Schmidt decomposition
rffiffiffiffiffi T 0 Tðx; wÞ ¼ X1 k n u n ðxÞu n ðwÞ; k 1 P k 2 P P 0 2 n¼1 ð4Þ
ð5Þ p ffiffiffiffiffi
where k n, un( x) areknownastheSchmidtcoefficients and Z the Schmidt modes respectively, satisfying
u m ðxÞu nðxÞdx ¼ d mn. As a result, the biphoton state can be expressed as
jbii
¼ X1 n¼1 pffiffiffiffiffi j 2: un i k n ð6Þ
where |
jn: u k i |
are |
distinguishable |
Fock |
states |
with |
|
nP photons in the mode u k, and the coefficients satisfy
1 n¼1 k n ¼ 1.
In literature, the Schmidt number K ¼ 1 = P n k2 n is known as a good measure of entanglement in pure biphoton
|
states, and is can also be calculated in the following way |
ZZ |
2 |
|
|
jTðx; wÞj dxdw |
|
|
K ¼
ZZ
|
Z
|
2
:
Tðx; x 0
ÞT ðx 0; wÞdx
dxdw
|
ð7Þ |
When the spectral bandwidth of the pump is much smaller than the typical distance between two neighboring ripples in the phase-matching kernel, the spectral envelope a( x, w) become a narrow peak function along the line x + w = x p. Because a( x p �x, x p �w)= a( x, w), the biphoton amplitude should satisfy
Tðx; wÞ ¼’ Tðx p � x; x p � wÞ: ð8Þ One may then introduce an analogue of the parity operator ^b about the frequency center x p / 2: ^bf ðxÞ ¼ f ðx p � xÞ for
any function f. Because ^b 2 ¼ 1, the parity operator ^b has two eigenvalues ± 1. Consequently, the Schmidt coefficients should be degenerate, and the Schmidt modes can be of odd parity or even parity.
3 Results and analysis
The Schmidt decomposition of wavefunction for biphoton shows a remarkable property that the probability of the particles in Schmidt modes within corresponding condition in the range of 0? 100 % [ 33 ], it means that if the one of pairs photons can be detected in Schmidt mode u n, and another photon can be detected, within the certain probability in adjoint mode v n in the same mode index p. ffiffiffiffiffi And the biphoton probability is represented by the k n of Schmidt decomposition in the modes { u n, v n } [ 33 ]. The entangled photons can be probed in the homologous modes with certain probability as the entangled one has been detected in the same index. The entanglement can be detected between the two independent frequencies. In this paper, we only consider the pairs in the homologous modes for the two entangled biphoton, the vector value of k n represents the paired probability in the same modulus index [ 31 ]. Of course some other peculiar parameters such as azimuthal angles which shows orbital angular movement in SPDC configuration has been considered and discussed in Ref. [ 17 ], would be ignored in this paper.
Figures 2a – 2c shows the Schmidt modes distributions in various cases, one may notice that the Schmidt modes shown in these Figures are symmetrically distributed with the x / x p = 0.5 and generated biphoton with ultrabroad bandwidth.
It can be found by the simulation that with the appropriate chirp rate, the spectral response frequency Dx is relatively wide in each Schmidt mode of entangled biphoton generated by SPDC in chirped QPM crystal. The most common structures spectral distribution in Schmidt modes