Phase-sensitivity enhancement based on a four-beam nonlinear interferometer

We propose a different type of nonlinear interferometer, namely, four-beam nonlinear interferometer, which can realize phase-sensitivity enhancement. We compare the phase sensitivity of the four-beam nonlinear interferometer and the corresponding shot-noise limit. Our results demonstrate that the four-beam nonlinear interferometer can achieve a phase-sensitivity enhancement compared with the corresponding shot-noise limit. Moreover, taking the losses into consideration, the phase sensitivity of the four-beam nonlinear interferometer can also beat the corresponding shot-noise limit. Our scheme may find potential applications in multiparameter quantum metrology.

Figure 1

Schematic for FBNI with two FBPAs. The green boxes are phase shifters. The red, yellow, and blue beams denote the pump, probe, and conjugate beams, respectively. The input conjugate fields ${\widehat{a}}_{\mathrm{in}1}, {\widehat{a}}_{\mathrm{in}2}$ and probe field ${\widehat{a}}_{\mathrm{in}4}$ are vacuum fields, while the input probe field ${\widehat{a}}_{\mathrm{in}3}$ is a coherent field.

Figure 2

The phase sensitivities of FBNI with four photon number combinations versus $\phi$ in the lossless case with the same internal photon number and $G_1=G_2=G$. When (a) $G=1.5$, (b) $G=2$, (c) $G=2.5$, and (d) $G=3$, the phase sensitivities of FBNI with photon number combinations $N_1$ ($N_2$), $N_1+N_2, N_1+N_2+N_4$, and $N_1+N_2+N_3+N_4$ are denoted by yellow dashed, pink solid, dark blue dotted, and light blue dot curves, respectively. The corresponding SNLs are denoted by gray straight lines.

Figure 3

The phase-sensitivity scalings of FBNI in the ideal case without considering the losses. When (a) $G=1.5$, (b) $G=2$, (c) $G=2.5$, and (d) $G=3$, the phase sensitivities of FBNI and the corresponding SNLs versus the internal photon number $N_s$ are denoted by blue solid and gray dashed lines, respectively.

Figure 4

The phase-sensitivity scalings of FBNI in the case with considering the losses. When (a) $\eta =0$, (b) $\eta =0.05$, (c) $\eta =0.1$, and (d) $\eta =0.15$, the phase sensitivities of FBNI versus $G$ are denoted by blue solid curves. The phase sensitivities are normalized to the corresponding SNLs, which are denoted by gray dashed lines.

Figure 5

The phase sensitivities of FBNI with photon number at a single output port and the corresponding SNL versus $\phi$ in the lossless case with the same internal photon number and $G_1=G_2=G$. When (a) $G=1.5$, (b) $G=2$, (c) $G=2.5$, and (d) $G=3$, the phase sensitivities of FBNI with photon number $N_1$ ($N_2$), $N_3$, and $N_4$ are denoted by blue solid, yellow dot, and pink dashed curves, respectively. The corresponding SNLs are denoted by gray straight lines.

Figure 6

The phase sensitivities of FBNI with various combinations of photon numbers at any two output ports and the corresponding SNL versus $\phi$ in the lossless case with the same internal photon number and $G_1=G_2=G$. When (a) $G=1.5$, (b) $G=2$, (c) $G=2.5$, and (d) $G=3$, the phase sensitivities of FBNI with photon number combinations $N_1-N_3$ ($N_2-N_3$), $N_3-N_4, N_1-N_4$ ($N_2-N_4$), $N_1+N_3$ ($N_2+N_3$), $N_3+N_4, N_1+N_4$ ($N_2+N_4$), and $N_1+N_2$ are denoted by light blue diamond, purple dot, orange dash-dotted, dark blue dashed, pink triangle, yellow dotted, and dark green solid curves, respectively. The corresponding SNLs are denoted by gray straight lines.

Figure 7

The phase sensitivities of FBNI with various combinations of photon numbers at any three output ports and the corresponding SNL versus $\phi$ in the lossless case with the same internal photon number and $G_1=G_2=G$. When (a) $G=1.5$, (c) $G=2$, (e) $G=2.5$, and (g) $G=3$, the phase sensitivities of FBNI with photon number combinations $N_1-N_3+N_4$ ($N_2-N_3+N_4$), $N_1+N_2-N_3, N_1+N_3-N_4$ ($N_2+N_3-N_4$), $N_1-N_2-N_4$ ($N_1-N_2+N_4$), and $N_1+N_2+N_3$ are denoted by brown dot, light blue dotted, red dash-dotted, dark purple dashed, and pink solid curves, respectively. When (b) $G=1.5$, (d) $G=2$, (f) $G=2.5$, and (h) $G=3$, the phase sensitivities of FBNI with photon number combinations $N_1-N_3-N_4$ ($N_2-N_3-N_4$), $N_1-N_2-N_3$ ($N_1-N_2+N_3$), $N_1+N_3+N_4$ ($N_2+N_3+N_4$), $N_1+N_2-N_4$, and $N_1+N_2+N_4$ are denoted by dark blue dot, orange dash-dotted, yellow solid, dark green dashed, and light purple dotted curves, respectively. The corresponding SNLs are denoted by gray straight lines.

Figure 8

The phase sensitivities of FBNI with various combinations of photon numbers at four output ports and the corresponding SNL versus $\phi$ in the lossless case with the same internal photon number and $G_1=G_2=G$. When (a) $G=1.5$, (b) $G=2$, (c) $G=2.5$, and (d) $G=3$, the phase sensitivities of FBNI with photon number combinations $N_1-N_2-N_3+N_4$ ($N_1-N_2+N_3-N_4$), $N_1-N_2-N_3-N_4$ ($N_1-N_2+N_3+N_4$), $N_1+N_2+N_3-N_4, N_1+N_2-N_3+N_4$, and $N_1+N_2+N_3+N_4$ are denoted by orange dash-dotted, dark blue dashed, yellow dot, pink solid, and light blue dotted curves, respectively. The corresponding SNLs are denoted by gray straight lines.