Yunning Lu, Zeyang Liao, Fu-Li Li, Xue-Hua Wang, "Integrable high-efficiency generation of three-photon entangled states by a single incident photon," Photonics Res. 10, 389 (2022)

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- Photonics Research
- Vol. 10, Issue 2, 389 (2022)

Fig. 1. Schematic of the system. (a) A Sagnac interferometer is a waveguide loop coupled to two external linear waveguides via a 50/50 beam splitter BS. An emitter is coupled to the waveguide loop. An optical circulator OC is used to distinguish the input and output photons. (b) Energy levels of the quantum emitter. Photons A, B, C, D are coupled to four transition paths of the emitter.

Fig. 2. (a) Frequency-space probability densities of photon A before and after scattering, | A k ( i ) ( t ) | 2 and | A k ( f ) ( t ) | 2 . The inset figure is the enlarged detail of the area around | A k ( i ) ( t ) | 2 = 0 and | A k ( f ) ( t ) | 2 = 0 . (b) Real-space probability densities of photon A before and after scattering, | A u ( i ) ( t ) | 2 and | A u ( f ) ( t ) | 2 . The parameters are Γ A = Γ B = Γ C = Γ D , γ 4 = 0.02 Γ A , δ A = 0 , ϵ A = 0.05 Γ A .

Fig. 3. (a) Real-space joint probability density | D x y z ( f ) ( t ) | 2 of B, C, D three-photon state. The parameter is ( x − c t ) Γ A / c = − 2 . (b) Real-space probability density of photons B, C, and D by integrating | D x y z ( f ) ( t ) | 2 over the position of photons C, D (B, D, or B, C). In both figures, other parameters are Γ B = Γ A , Γ C = 1.2 Γ A , Γ D = 0.5 Γ A , γ 2 = γ 3 = γ 4 = 0.02 Γ A , δ A = 0 , ϵ A = 0.05 Γ A .

Fig. 4. Frequency-space joint probability density of the three-photon state | D p q r ( f ) ( t ) | 2 . The parameters are (a) ϵ A = 0.05 Γ A ; (b) ϵ A = 0.1 Γ A . Other parameters are Γ A = Γ B = Γ C = Γ D , γ 2 = γ 3 = γ 4 = 0.02 Γ A , δ A = 0 .

Fig. 5. (a) Entanglement entropy S 1 and (b) Schmidt number K 1 of photon B and two-photon part C, D, (c) entanglement entropy S 2 and (d) Schmidt number K 2 of two-photon part B, C and photon D, as functions of Γ C / Γ A with different values of ϵ A . In all figures, red curves with triangles: ϵ A = 0.05 Γ A ; blue curves with squares: ϵ A = 0.1 Γ A . Insets in (b) and (d) show the first 25 joint eigenvalues {λ n ( 1 ) } and {λ n ( 2 ) } of the reduced density matrix of photon B and photon D with Γ C / Γ A = 1.5 , respectively. Red circles: ϵ A = 0.05 Γ A ; blue asterisks: ϵ A = 0.1 Γ A . Other parameters are: Γ B = Γ D = Γ A , γ 2 = γ 3 = γ 4 = 0.02 Γ A .

Fig. 6. Probability of (a) three-photon state P BCD , (b) photon A P A , and (c) dissipation P Dis after scattering as functions of spectrum width ϵ A of incident photon A and coupling strength Γ B . The white dashed lines in (a) and (b) show the maximum values of P BCD and the minimum values of P A along the Γ B / Γ A axis, respectively. (d) P BCD , P A , and P Dis as functions of Γ B in the monochromatic light limit ϵ A / Γ A = 10 − 4 . Other parameters are Γ C = Γ D = Γ A , γ 2 = γ 3 = γ 4 = 0.02 Γ A , δ A = 0 .

Fig. 7. (a) Entanglement entropy S 2 and (b) Schmidt number K 2 of two-photon part B, C and part D as functions of Γ D / Γ A with different values of ϵ A . In both figures, red curves with triangles: ϵ A = 0.05 Γ A ; blue curves with squares: ϵ A = 0.1 Γ A . Insets in (b) shows the first 25 joint eigenvalues {λ n ( 2 ) } of the reduced density matrix of photon D with Γ D / Γ A = 1.5 . Red circles: ϵ A = 0.05 Γ A ; blue asterisks: ϵ A = 0.1 Γ A . Other parameters are Γ B = Γ C = Γ A , γ 2 = γ 3 = γ 4 = 0.02 Γ A .

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