Fig. 1. (a) Schematic of QND measurement of photon number. The probe system consists of an SU(1,1)-SU(2)-concatenated atom–light hybrid interferometer. In the SU(2) interferometer in the middle box, the LRP is utilized to realize the splitting and combination of the atomic spin wave and the optical wave. a^W(0) is a coherent state, and S^a(0), S^a(1), and S^a(3) are atomic collective excitations prepared by NRP1, LRP1, and LRP2, respectively. The atomic spin wave S^a(1) experiences a phase modulation ϕAC via the AC-Stark effect by signal light b^(in) and evolves to S^a(2). The generated atomic spin wave S^a(3) of the SU(2) interferometer and the optical wave a^S(1), which is correlated with S^a(0), are combined to realize active correlation output readout via NRP2. LRP, linear Raman process; NRP, nonlinear Raman process. (b) Energy levels of the atom. The lower two energy states |g⟩ and |m⟩ are the hyperfine split ground states. The higher-energy state |e⟩ is the excited state. The strong pump field Ap1(Ap2) and strong read field Ep1(Ep2) couple the transitions |g⟩→|e⟩ and |m⟩→|e⟩, respectively. b^(in) is far off resonance with the transition |m⟩→|e⟩ by a large detuning.

Fig. 2. Lossy interferometer model with (a) internal loss and (b) external loss.

Fig. 3. Correlation coefficient C as a function of η1 and η2, where e−Γ1τ1=e−Γ2τ2=0.9, κ =10−10, g1=g2=3, Nα=1012, and Nβ=108. The correlation coefficient in the area of upper right corner and within the C0 lines can be kept above 0.6.

Fig. 4. Contour line of optimized g2/g1 as a function of η1 and η2, with e−Γ1τ1=e−Γ2τ2=0.9, where κ=10−10, g1=3, Nα=1012, and Nβ=108. The correlation coefficient in the area of upper right corner and within the line C0 (before optimization) or C1 (after optimization) can be kept above 0.6.