Improved phase sensitivity in a quantum optical interferometer based on multiphoton catalytic two-mode squeezed vacuum states

The usage of non-Gaussian states to improve the phase sensitivity of interferometers has been studied before. In this paper, we introduce the multiphoton catalysis two-mode squeezed vacuum (MC-TMSV) state as an input of the Mach-Zehnder interferometer (MZI) and study its phase sensitivity with photon-number parity measurement and the influence of photon losses. We also study the statistical properties of the MC-TMSV state in terms of the average photon number, the Wigner function, and the Mandel-$Q$ parameter. Our results show that the MC-TMSV state can exhibit stronger nonclassical characteristics, thereby making the phase sensitivity more precise with either the increase of the photon-catalyzed number or the decrease of the transmissivity. Furthermore, we also consider the effects of photon losses involving external- and internal-loss processes of the MZI, and we find that in both cases the sensitivity with the MC-TMSV state can be significantly better than that with the TMSV state under the same parameters especially in the serious photon losses and small initial squeezing regimes. We also find that the multiphoton catalysis is more sensitive to the external photon losses compared with the internal ones. Our results here can find important applications in quantum metrology.

Figure 1

(a) Schematic diagram of the Mach-Zehnder interferometer for the detection of the phase shift when the multiphoton catalysis two-mode squeezed vacuums are injected into the first beam splitter. (b) Schematic setup for preparing the MC-TMSV state.

Figure 2

Average photon number as a function of the squeezing parameter $\lambda$ for (a) $\eta =0.2$ and $m=1$, 2, and 3 and for (b) $m=3$ and $\eta =0.1$, 0.2, and 0.3. Solid lines correspond to the TMSV state.

Figure 3

The Wigner function $W$ of the MC-TMSV state with fixed squeezed parameter $\lambda =0.7$. (a) $m=1$ and $\eta =0.2$, (b) $m=1$ and $\eta =0.4$, and (c) $m=2$ and $\eta =0.4$.

Figure 4

Plot of the Mandel $Q$-parameter of the MC-TMSV state. (a) For a given transmissivity $\eta =0.2$, Mandel $Q$-parameter as a function of the squeezed parameter $\lambda$ for different values of $m=1$, 2, and 3. (b) Mandel-$Q$ parameter as function of transmissivity $\eta$ with a fixed $\lambda =0.6$ for different values of $m=1$, 2, and 3. Solid lines correspond to the TMSV state.

Figure 5

The expectation value $〈{\mathrm{\Pi }}_b〉$ of the parity operator versus the phase shift $\varphi$ for a given squeezing parameter $\lambda =0.7$ input state of the MZI. (a) $\eta =0.3$ and $m=1$, 2, and 3. (b) $m=3$ and $\eta =0.1$, 0.2, and 0.3. Solid lines correspond to the TMSV state.

Figure 6

The quantum Fisher information $F_Q$ as the function of the squeezing parameter $\lambda$ and the transmissivity $\eta$, respectively, for some different catalytic photons $m$. (a) $\eta =0.2$. (b) $\lambda =0.5$. Solid lines correspond to the TMSV state.

Figure 7

The QFI as the function of (a) the negative volume of the WF and (b) the Mandel-$Q$ parameter for a given $\eta =0.2$ and $m=1$ (green dot-dashed line), $m=2$ (blue dotted line), and $m=3$ (red dashed line).

Figure 8

The phase uncertainty $\mathrm{\Delta }\varphi$ versus the phase shift $\varphi$ for a given squeezing parameter $\lambda =0.7$. (a) $\eta =0.2$ and $m=1$ and 2. (b) $m=3$ and $\eta =0.2$, 0.4, and 0.6. Solid lines correspond to the TMSV state.

Figure 9

The phase uncertainty $\mathrm{\Delta }\varphi$ as a function of the squeezing parameter $\lambda$ for a given $\varphi =0.5$. (a) $\eta =0.2$ and $m=1$, 2, and 3. (b) $m=3$ and $\eta =0.1$, 0.2, and 0.3. Solid lines correspond to the TMSV state.

Figure 10

Schematic diagram of the photon losses is introduced (a) in front of the parity detection and (b) between the phase shifter and the second BS.

Figure 11

In the presence of photon losses, the phase uncertainty as a function of the initial squeezed parameters $\lambda$ at $\varphi =0.05$. (a) Comparison of the ideal case and the loss case for the TMSV and the single-photon catalysis operation. (b) Comparison of the phase uncertainty of the MC-TMSV state with $m=1$, 2, and 3, and the normal TMSV (black solid line) for $10\%$ photon loss $({\kappa }_2=0.9)$ and $\eta =0.2$.

Figure 12

In the presence of photon losses, the phase uncertainty as a function of the transmissivity of the fictitious beam splitter ${\kappa }_1$ at $\varphi =0.05$ (a) for $\eta =0.2, \lambda =0.7$, and $m=1$, 2, and 3 and (b) for $m=1, \lambda =0.7$, and $\eta =0.1$, 0.2, and 0.3. Solid lines correspond to the TMSV state.

Figure 13

For photon losses in the interior of the MZI, the phase uncertainty as a function of the initial squeezing parameters $\lambda$ at $\varphi =0.05$. (a) Comparison of the ideal case and the loss case for the TMSV and the single-photon catalysis operation. (b) Comparison of the phase uncertainty of the MC-TMSV state with $m=1$, 2, and 3 and the normal TMSV (black solid line) for $10\%$ photon loss $({\kappa }_2=0.9)$ and $\eta =0.2$.

Figure 14

For photon losses in the interior of the MZI, the phase uncertainty as a function of transmissivity of fictitious beam splitter ${\kappa }_2$ at $\varphi =0.05$ (a) for $\eta =0.2,\lambda =0.7$ and $m=1$, 2, and 3, (b) for $m=1,\lambda =0.7$ and $\eta =0.1$, 0.2, and 0.3. The black solid lines correspond to the TMSV state.

Figure 15

Comparing the influence of two dissipation ways on the phase sensitivity for $m=1$ (green), $m=2$ (blue), $m=3$ (red) and photon losses in front of the parity detection (solid), in the interior of the MZI (dot-dash).