Intramode-correlation–enhanced simultaneous multiparameter-estimation precision

The multiparameter estimation in quantum metrology is significant for a large number of applications for quantum information tasks. Nevertheless, how mode correlations affect the multiparameter-estimation precision with and without photon-loss cases remains ambiguous. For this reason, here, we present the relationship between the quantum Cramér-Rao bound and the mode correlations in the framework of the simultaneous two-phase estimation for a three-mode Mach-Zehnder interferometer induced by the tritter which is composed of three cascaded beam splitters and phase shifters. By choosing different input resources, including a two-mode squeezed vacuum state $\otimes$ a coherent state (scheme I) and a twin squeezed vacuum state $\otimes$ a coherent state (scheme II), the results show that compared with scheme II, scheme I shows a better estimation performance even in the presence of photon losses. The reason may be that the intramode correlation of the probe state for scheme I is stronger than that for scheme II, while the intermode correlation of the probe state for the former is weaker than that for the latter. This allow us to open vital perspectives on the essence of enhanced multiparameter-estimation precision.

Figure 1

(a) Schematic diagram of the tritter, which is composed of beam splitters and phase shifters. (b) Schematic diagram of the simultaneous two-phase estimation strategy for the three-mode MZI induced by the tritter. (c) Sketch of the distinction between the intramode correlations $g_a^{\left(2\right)}$ and $g_b^{\left(2\right)}$ and the intermode correlation $g_{a,b}^{\left(2\right)}$, where the intramode correlation functions $g_a^{\left(2\right)}$ and $g_b^{\left(2\right)}$ are, respectively, proportional to the probability of detecting two photons simultaneously at detector $A$ and detector $B$, while the intermode correlation function $g_{a,b}^{\left(2\right)}$ describes the probability of observing one photon at detector $A$ and detector $B$ simultaneously.

Figure 2

The QCRB as a function of the total average input photon number $N$. The black and blue solid lines correspond to schemes I and II, respectively.

Figure 3

Both (a) the intramode correlation function $g_a^{\left(2\right)}$ and (b) the intermode correlation function $g_{a,b}^{\left(2\right)}$ of the probe state change with the total average input photon number $N.$ The black and blue solid lines correspond to schemes I and II, respectively.

Figure 4

The QCRB for schemes (a) I and (b) II under the ideal cases as a function of both the intramode correlation function $g_a^{\left(2\right)}$ and the intermode correlation function $g_{a,b}^{\left(2\right)}$ in the range of $N\in [1,200]$.

Figure 5

Schematic diagram of the photon losses that occur during the probe-modification process. ${\eta }_j (j=a,b,c)$ are the transmissivities of the fictitious beam splitter; $j_v (j=a,b,c)$ are the vacuum operators.

Figure 6

$S_{\mathrm{I}\left(\mathrm{II}\right)}$ as a function of the total average input photon number $N$. The black and blue solid lines correspond to schemes I and II, respectively.

Figure 7

For schemes I (black lines) and II (blue lines), the QCRB as a function of (a) the transmissivity $\eta$ at fixed $N=50$ and (b) the total average input photon number $N$ at fixed $\eta =0.6$. The dashed and solid lines correspond to the effects of photon losses and no photon losses, respectively.

Figure 8

Function $g_a^{\left(2\right)}$ and the intermode correlation function $g_{a,b}^{\left(2\right)}$ in the range of $N\in [1,200].$

Figure 9

The QCRB as a function of the squeezing parameter $r$ and the coherent amplitude $\left|\gamma \right|$ with $r_1=r_2=r$ and $\alpha =\beta =\gamma .$ The orange and blue surfaces respectively correspond to schemes I and II under the ideal cases, while the green and red surfaces respectively correspond to schemes I and II under the photon-loss cases.