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, |Ak(i)(t)|2 and |Ak(f)(t)|2. The inset figure is the enlarged detail of the area around |Ak(i)(t)|2=0 and |Ak(f)(t)|2=0. (b) Real-space probability densities of photon A before and after scattering, |Au(i)(t)|2 and |Au(f)(t)|2. The parameters are ΓA=ΓB=ΓC=ΓD, γ4=0.02ΓA, δA=0, ϵA=0.05ΓA.

Fig. 3. (a) Real-space joint probability density |Dxyz(f)(t)|2 of B, C, D three-photon state. The parameter is (x−ct)ΓA/c=−2. (b) Real-space probability density of photons B, C, and D by integrating |Dxyz(f)(t)|2 over the position of photons C, D (B, D, or B, C). In both figures, other parameters are ΓB=ΓA, ΓC=1.2ΓA, ΓD=0.5ΓA, γ2=γ3=γ4=0.02ΓA, δA=0, ϵA=0.05ΓA.

Fig. 4. Frequency-space joint probability density of the three-photon state |Dpqr(f)(t)|2. The parameters are (a) ϵA=0.05ΓA; (b) ϵA=0.1ΓA. Other parameters are ΓA=ΓB=ΓC=ΓD, γ2=γ3=γ4=0.02ΓA, δA=0.

Fig. 5. (a) Entanglement entropy S1 and (b) Schmidt number K1 of photon B and two-photon part C, D, (c) entanglement entropy S2 and (d) Schmidt number K2 of two-photon part B, C and photon D, as functions of ΓC/ΓA with different values of ϵA. In all figures, red curves with triangles: ϵA=0.05ΓA; blue curves with squares: ϵA=0.1ΓA. Insets in (b) and (d) show the first 25 joint eigenvalues {λn(1)} and {λn(2)} of the reduced density matrix of photon B and photon D with ΓC/ΓA=1.5, respectively. Red circles: ϵA=0.05ΓA; blue asterisks: ϵA=0.1ΓA. Other parameters are: ΓB=ΓD=ΓA, γ2=γ3=γ4=0.02ΓA.

Fig. 6. Probability of (a) three-photon state PBCD, (b) photon A PA, and (c) dissipation PDis after scattering as functions of spectrum width ϵA of incident photon A and coupling strength ΓB. The white dashed lines in (a) and (b) show the maximum values of PBCD and the minimum values of PA along the ΓB/ΓA axis, respectively. (d) PBCD, PA, and PDis as functions of ΓB in the monochromatic light limit ϵA/ΓA=10−4. Other parameters are ΓC=ΓD=ΓA, γ2=γ3=γ4=0.02ΓA, δA=0.

Fig. 7. (a) Entanglement entropy S2 and (b) Schmidt number K2 of two-photon part B, C and part D as functions of ΓD/ΓA with different values of ϵA. In both figures, red curves with triangles: ϵA=0.05ΓA; blue curves with squares: ϵA=0.1ΓA. Insets in (b) shows the first 25 joint eigenvalues {λn(2)} of the reduced density matrix of photon D with ΓD/ΓA=1.5. Red circles: ϵA=0.05ΓA; blue asterisks: ϵA=0.1ΓA. Other parameters are ΓB=ΓC=ΓA, γ2=γ3=γ4=0.02ΓA.