Open Access
Add to BibSonomy
Abstract
By performing two local displacement operations (LDOs) inside an SU(1,1) interferometer, called as the displacement-assisted SU(1,1) [DSU(1,1)], both the phase sensitivity based on homodyne detection and quantum Fisher information (QFI) with and without photon losses are investigated in this paper. In this DSU(1,1) interferometer, we focus our attention on the extent to which the introduced LDO affects the phase sensitivity and the QFI, even in the realistic scenario. Our analyses show that the estimation performance of DSU(1,1) interferometer is always better than that of SU(1,1) interferometer without the LDO, especially for the phase precision of the former in the ideal scenario closer to the Heisenberg limit via the increase of the LDO strength. Different from the latter, the robustness of the former can be also enhanced markedly by regulating and controlling the LDO. Our findings would open an useful view for quantum-improved phase estimation of optical interferometers.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Quantum metrology is an excellent candidate of parameter estimation theory to serve as the high-precision requirement of various quantum information tasks [1–4], such as quantum sensor [1] and quantum imaging [2]. Thus, how to find a high-precision method in quantum metrology has become a general consensus among scientists. To this end, the optical interferometers, e.g., Mach-Zehnder interferometer (MZI) [5–10] and SU(1,1) interferometer [11–17], are often used for understanding the subtle phase variations thoroughly. Generally, in optical-interferometer systems, it is possible to obtain the higher precision of phase estimation using three probe strategies: generation, modification and readout [18]. In the probe generation stage, nonclassical quantum resources as the inputs of the MZI have been proven to more effectively enhance the precise measurement than its classical counterpart [19–23]. In particular, when using the NOON states [21,24], the two-mode squeezed vacuum states (TMSVS) [22] and the twin Fock states [23], the standard quantum limit (SQL) [10] can be easily beaten, even infinitely reaching at the famed Heisenberg limit (HL) [15,22]. These states, however, are extremely sensitive to noisy environments [25]. Thus, non-Gaussian resources [9,26–29] as an alternative that can be produced by taking advantage of non-Gaussian operations on an arbitrary initial state play an important role in improving the estimation performance of the MZI, even in the presence of noisy scenarios [9,29]. Apart from the generation stage, many efforts have also devoted to conceiving the probe modification by replacing the conventional beam splitters in the conventional MZI with the optical parametric amplifiers (OPAs) [11–15,17,18,30], which is also called as SU(1,1) interferometer proposed first by Yurke [13]. In this SU(1,1) interferometer with two OPAs, the first OPA (denoted as OPA$_{1}$) is used not only to obtain the entangled resources but also to eliminate amplified noise; while the usage of the second OPA (denoted as OPA$_{2}$) can result in the signal enhancement [17,18], which paves a feasible way to achieve the higher precision of phase estimation. Taking advantage of these features, an SU(1,1) interferometer scheme with the phase shift induced by a kerr medium was suggested by Chang [18]. They pointed out that the significant improvement of both the phase sensitivity and quantum Fisher information can be achieved even in the presence of photon losses. In addition, the noiseless quantum amplification of parameter-dependent processes was used to SU(1,1) interferometer, indicating how this process results in the HL [31]. Further, by using the non-Gaussian operations inside the SU(1,1) interferometer, both the phase sensitivity and the robustness of this interferometer system against the photon losses can be enhanced [32]. From these works [9,19,29,32], we also notice that the usage of non-Gaussian operations indeedy improves the estimation performance of the optical interferometers, but at the expense of the high cost of implementing these operations.
To solve the above problem, the local operations containing the local squeezing operation (LSO) [33–35] and the local displacement operation (LDO) [36] are one of the most promising choices. In particular, J. Sahota and D.F.V. James suggested a quantum-enhanced phase estimation scheme by applying the LSO into the MZI [35]. However, it should be mentioned that the LSO plays a key role in quantum metrology [35], quantum key distribution [34] and entanglement distillation [33], but the degree of the LSO is not infinite, e.g., its maximum attainable degree for the TMSVS about $1.19$ ($10.7$ dB) [37]. For this reason, here we suggest a quantum-improved phase estimation of the SU(1,1) interferometer based on the LDO, which can be called as the displacement-assisted SU(1,1) [DSU(1,1)] interferometer. Under the framework of this DSU(1,1) interferometer, we not only derive its explicit forms of both the quantum Fisher information (QFI) and the phase sensitivity based on homodyne detection, but also consider the effects of photon losses on its estimation performance. Our analyses manifest that the increase of the LDO strength is conducive to the improvement of both the QFI and the phase sensitivity, even in the presence of photon losses. In particular, this increasing LDO can narrow the gap for the phase sensitivity between with and without photon losses. This implies that the usage of the sufficiently large LDO can make the SU(1,1) interferometer systems more robust against photon losses.
The remainder of this paper is arranged as follows. In section 2, we first describe the theoretical model of DSU(1,1) interferometer, and then give the relationship between the output and input operators for this interferometer. In sections 3, we analyze and discuss the phase sensitivity based on homodyne detection in DSU(1,1) interferometer with and without photon-loss scenarios, and then make a comparison about phase sensitivities containing the SQL, the HL and the DSU(1,1) interferometer schemes in the ideal case. Subsequently, we also consider the QFI of DSU(1,1) interferometer with and without photon-loss scenarios in section 4. Finally, our main conclusions are drawn in the last section.
2. DSU(1,1) interferometer and its relationship between the output and input operators
Now, let us begin with introducing the theoretical model of DSU(1,1) interferometer, whose structure is comprised of two OPAs, two LDOs and a linear phase shift, as depicted in Fig. 1(a). For simplicity, here we only consider both a squeezed vacuum state $\left \vert \xi \right \rangle _{a_{0}}= \hat {S}(\xi )\left \vert 0\right \rangle _{a_{0}}$ with the squeezing operator $\hat {S}(\xi )=\exp [(\xi ^{\ast }\hat {a}_{0}^{2}-\xi \hat {a} _{0}^{\dagger 2})/2]$ ($\xi =re^{i\theta _{\xi }}$) on the vacuum state $\left \vert 0\right \rangle _{a_{0}}$ and a coherent state $\left \vert \beta \right \rangle _{b_{0}}$ with $\beta =\left \vert \beta \right \vert e^{i\theta _{\beta }}$ as the inputs of DSU(1,1) interferometer in paths $a_{0}$ and $b_{0}$, respectively. After these input states pass through the OPA$_{1}$, both paths respectively experience the same LDO process, denoted as $\hat {D}_{a_{1}}(\gamma )=e^{\gamma \hat {a}_{1}^{\dagger }-\gamma ^{\ast } \hat {a}_{1}}$ and $\hat {D}_{b}(\gamma )=e^{\gamma \hat {b}_{1}^{\dagger }-\gamma ^{\ast }\hat {b}_{1}}$ with $\gamma =\left \vert \gamma \right \vert e^{i\theta _{\gamma }}$. In this case, the resulting state can be called as the probe state, denoted as $\left \vert \psi _{\gamma }\right \rangle$. Then, we also assume that path $a_{i},i\in \{0,1,2\}$ serves as the reference path; while in path $b_{1}$, a linear phase shifter is carryed out for producing a phase shift $\phi$ to be estimated. Finally, after paths $a_{1}$ and $b_{1}$ recombine in the OPA$_{2}$, we can extract the phase information about the value of $\phi$ by implementing the homodyne detection in path $a_{2}$. Indeed, the relationship between the output and input operators for DSU(1,1) interferometer can be given by
Fig. 1. Schematic diagram of the DSU(1,1) interferometer (a) with and (b) without photon losses, based on homodyne detection, in which a squeezed vacuum state $\left \vert \xi \right \rangle _{a_{0}}$ and a coherent state $\left \vert \beta \right \rangle _{b_{0}}$ are respectively used as the inputs of DSU(1,1) interferometer in paths $a_{0}$ and $b_{0}$. OPA$_{1}$ and OPA$_{2}$: the first and second optical parametric amplifier. LDO is a local displacement operation. $ \phi$ is a phase shift to be measured. Hom: an homodyne detection. $a_{0}(b_{0})$ and $a_{2}(b_{2})$: the input and output operators of DSU(1,1) interferometer, respectively. In (b), photon-loss scenarios occurs at after the linear phase shift in paths $a_{1}$ and $b_{1}$, in which $T$ represents the transmissivity of the fictitious beam splitter (FBS). $j_{v}$ ($j=a,b$) are the vacuum operator in path $j_{3}.$
3. Phase sensitivity of DSU(1,1) interferometer
So far, we have described the schematic of DSU(1,1) interferometer in detail. In this section, we shall present and analyze the phase sensitivity of DSU(1,1) interferometer based on the homodyne detection with and without photon-loss scenarios. Moreover, for the sake of discussion, in the following sections, we also assume that the DSU(1,1) interferometer is in the balanced case, i.e., $\theta _{2}-\theta _{1}=\pi$ and $g_{1}=g_{2}=g$ (set $\theta _{1}=0$ and $\theta _{2}=\pi$ for simplicity).
For this purpose, it is unavoidable to choose a specific detection scheme [38–43], such as homodyne detection [18,30,38], intensity detection [12,16,39], and parity detection [9,21,24,40] , for reading the phase information. In addition, compared with both intensity and parity detections, the homodyne detection is computationally convenient and compatible with existing experimental technology, thereby playing potential applications in quantum communication [44–46]. For this reason, the phase parameter $\phi$ can be estimated by exploiting the homodyne detection (see Fig. 1(a)), whose detected variable can be treated as the amplitude quadrature $\hat {X}$, i.e.,
Here, for simplicity, we have assumed that the implementation of homodyne detection is at the output path $a_{2}$. Using Eq. (3) and the error propagation formula, the phase sensitivity of DSU(1,1) interferometer without photon losses can be thus given by
Now, let us further examine the effects of photon losses on the phase sensitivity of DSU(1,1) interferometer. For this purpose, we assume that the same photon losses occur at between the phase shift and the OPA$_{2}$, as shown in Fig. 1(b). In general, under the photon-loss process, the lossy channel can be simulated by inserting the fictitious beam splitter (FBS) with a transmissivity $T$ [14,30,47]. It is worth mentioning that, the smaller the values of $T$, the more serious the photon losses.
For the state vector prior to the OPA$_{2}$, $\left \vert \psi _{\phi }\right \rangle,$ after going through the photon-loss channel, the output state $\left \vert \psi _{\phi }\right \rangle _{out}$ in the enlarged systems $S+E$ can be expressed as $\left \vert \psi _{out}\right \rangle = \hat {U}_{BS}^{a}\hat {U}_{BS}^{b}\left \vert \psi _{\phi }\right \rangle \left \vert 0\right \rangle _{a_{3}}\left \vert 0\right \rangle _{b_{3}}$. Thus, when passing through the OPA$_{2}$, the final output state $\left \vert \psi _{f}\right \rangle$ can be given by
In order to achieve the optimal estimation performance for the phase shifts, we illustrate the phase sensitivity $\Delta \phi$ with $T=0.6$ (dashed lines) as a function of phase shifts $\phi$ for several different values $\left \vert \gamma \right \vert =0,1,2$, as shown in Fig. 2(a). As a comparison, the solid lines represent the ideal case. As we can easily see, the minimum of phase sensitivity is always found at the point $\phi =0$, whether there is photon loss or not. Significantly, with the increase of $\left \vert \gamma \right \vert =0,1,2$, the gap of the phase sensitivity $\Delta \phi$ between with and without photon losses can be further reduced around $\phi =0$. Note that $\left \vert \gamma \right \vert =0$ corresponds to the SU(1,1) interferometer system without the LDO. These phenomena result from that the increase of the LDO strength $\left \vert \gamma \right \vert$ can still increase the slope $\partial \left \langle \hat {X}\right \rangle /\partial \phi$ of the output signal $\left \langle \hat {X}\right \rangle$ with and without photon losses, which can be shown in Fig. 2(b).
In this context, therefore, the phase sensitivity with the photon losses at the phase shift point $\phi =0$ can be calculated as
Fig. 2. When $T=0.6$, (a) phase sensitivity with homodyne detection and (b) output signal changing with $ \phi$ for different $\left \vert \gamma \right \vert =0,1,2$. As a comparison, the solid lines are the ideal case, i.e., $T=1$. Other parameters are as following: $g=r=| \beta |=1,$ $ \phi = \theta _{ \xi }=0$ and $ \theta _{ \beta }= \theta _{ \gamma }= \pi /2$.
In Fig. 3(a), we show the advantage of DSU(1,1) interferometer robust against the photon losses. Obviously, with the decrease of $T$, the phase sensitivity of DSU(1,1) interferometer systems would fade away, but this decline can be further slowed by increasing $\left \vert \gamma \right \vert =0,1,2,3$. To some extent, this phenomenon reveals that the usage of the LDO can make the whole SU(1,1) interferometer systems more robust against the photon losses, when comparing to the case without the LDO. To visualize this point, we make a comparison about the phase sensitivity between with (dashed lines) and without (solid lines) photon losses, as shown in Fig. 3(b). Surprisingly, at the same accessible parameters, the gap for the phase sensitivity between with and without photon losses narrows down with the increase of $\left \vert \gamma \right \vert =0,1,2,3$, even showing that the phase sensitivity with photon losses for $\left \vert \gamma \right \vert =2$ $(3)$ at certain small range of $g<0.436$ ($0.699$) performs better than that without both the photon losses and the LDO (black solid line). In addition, the aforementioned gap can be also further reduced by increasing the value of $g.$
Fig. 3. Under the photon losses, the corresponding phase sensitivity $\Delta \phi _{L}$ as a function of (a) $T$ at fixed $\left \vert \beta \right \vert =g=r=1,$ and of (b) $g$ at fixed $T=0.6$, $\left \vert \beta \right \vert =r=1,$ when given several different $\left \vert \gamma \right \vert =0,1,2,3$. In (b), as a comparison, the solid lines are the ideal cases, i.e., $T=1$. Other parameters are as following: $ \phi = \theta _{ \xi }=0$ and $ \theta _{ \beta }= \theta _{ \gamma }= \pi /2$.
On the other hand, we also make a comparison about phase sensitivities including the SQL, the HL and the DSU(1,1) interferometer schemes in ideal scenario, as shown in Fig. 4. The detailed derivations can be seen in Appendix A. It is clearly seen from Fig. 4(a) that, at fixed values of $\left \vert \beta \right \vert =$ $\left \vert \gamma \right \vert =r=1$, for a large range of $g$ (i.e., $g>0.24$), the phase precision of DSU(1,1) interferometer scheme can easily break through the SQL (solid green line), even gradually approaching to HL (solid red line) with the increase of $g$. In addition, when given $g=\left \vert \beta \right \vert =r=1$, it is found in Fig. 4(b) that the phase precision of DSU(1,1) interferometer scheme is alway superior to the SQL. In addition, when increasing the value of $\left \vert \gamma \right \vert$, the phase precision of DSU(1,1) interferometer can first approach to HL, then slightly move away from this precision limit and be finally close to HL infinitely, which can be seen from the inserting blue box in Fig. 4(b). Here, we have taken the difference $\delta =\log _{10}(\Delta \phi _{\text {DSU(1,1)}}-$HL$)$. Obviously, at the small range of $\left \vert \gamma \right \vert,$ when taking $\left \vert \gamma \right \vert =2.41$, the minimum value of the gap between the DSU(1,1) interferometer and the HL is about $10^{-2.37}$; while for the sufficiently large $\left \vert \gamma \right \vert$ $($e.g., $\left \vert \gamma \right \vert =35)$, this gap can be further shortened. Nevertheless, the phase precision of DSU(1,1) interferometer scheme (i.e., $\left \vert \gamma \right \vert \neq 0$) is alway closer to the HL than the case of SU(1,1) interferometer without the LDO (i.e., $\left \vert \gamma \right \vert =0$).
Fig. 4. Comparison about precision limits involving the SQL, the HL and the SU(1,1) interferometer with the LDOs. (a) and (b) respectively corresponds to phase sensitivity $\Delta \phi$ changing with $g$ and $| \gamma |$, when fixed $| \gamma |=| \beta |=r=1$ and $| \beta |=g=r=1$. The inserting blue box in (b) represents the difference $ \delta =\log _{10}(\Delta \phi _{\text {DSU(1,1)}}-$HL$)$. Other parameters are as following: $ \phi = \theta _{ \xi }=0$ and $ \theta _{ \beta }= \theta _{ \gamma }= \pi /2$.
4. QFI of DSU(1,1) interferometer
To directly assess the estimation performance of a unknown phase parameter without any detection strategies, it is a common approach for utilizing the QFI of the probe state since the quantum Cramér-Rao bound (QCRB) $\Delta \phi _{QCRB}$ representing the ultimate precision is in inverse proportion to the QFI (denoted as $F$). In addition, the increased value of the QFI is beneficial for achieving the high-precision estimation performance. In this context, the QFI for an arbitrary pure state in the ideal scenario can be expressed as [9,18,28,48]
The corresponding detailed derivations can be found in Appendix B, not shown here. Note that the strengths of photon losses $\eta =0$ and $\eta =1$ respectively denote the complete absorption and lossless cases. That is to say, the smaller the value of $\eta$, the more serious the photon losses. In particular, when $\eta =1$ in Eq. (15), one can obtain $F_{L}=$ $F$ corresponding to the QFI with the ideal case.
In Fig. 5(a), we show the QCRB changing with $\eta$ when given different LDO strengths $\left \vert \gamma \right \vert =0,1,2,3$. With the decrease of $\eta,$ we can find that compared to the QCRB of the SU(1,1) interferometer without the LDO (i.e., $\left \vert \gamma \right \vert =0$), the DSU(1,1) interferometer scheme can still demonstrate superior estimation performance by increasing the LDO strengths. Moreover, at fixed $\eta =0.6$ and $\left \vert \gamma \right \vert =1$ , it is also possible to further enhance the QCRB via the increasing parameters of $g$ and $\left \vert \beta \right \vert$, as seen in Fig. 5(b).
Fig. 5. The estimation performance of QCRB in photon-loss scenarios, i.e., $\Delta \phi _{L-QCRB}.$ (a) The $\Delta \phi _{L-QCRB}$ as a function of $ \eta$ for several different $\left \vert \gamma \right \vert =0,1,2,3,$ when given $\left \vert \beta \right \vert$=$g$=$1$. (b) The counterplot of the QCRB $\log _{10}\Delta \phi _{L-QCRB}$ in ($\left \vert \beta \right \vert,g$) space, when given $ \eta$=$0.6$ and $\left \vert \gamma \right \vert$=$1$. Other parameters are as following: $ \theta _{ \xi }$=$0$, $r$=$1$ and $ \theta _{ \beta }$=$ \theta _{ \gamma }$=$ \pi /2$.
Additionally, the question we are concerned about is whether the phase sensitivity of DSU(1,1) interferometer scheme based on homodyne detection can saturate the QCRB? Toward this end, Fig. 6 shows the phase sensitivity of DSU(1,1) interferometer with homodyne detection (solid lines) and the QCRB (dashed lines) in the presence of photon losses $T($or $\eta )$ for several different $\left \vert \gamma \right \vert =0,1,2,3$. We can see that the phase sensitivity of DSU(1,1) interferometer scheme based on homodyne detection at the same parameters cannot saturate the QCRB. Nevertheless, compared to the case without the LDO (black lines), the gap between the phase sensitivity of DSU(1,1) interferometer scheme with homodyne detection and the QCRB can be shortened via the increase of $\left \vert \gamma \right \vert$, especially for the ideal scenario (i.e., $T($or $\eta )=1$). This also means that the application of LDO into SU(1,1) interferometer is beneficial for the corresponding phase sensitivity to saturate the QCRB.
Fig. 6. Comparisons about the QCRB (dashed lines) and the phase sensitivity based on homodyne detection (solid lines) changing with the strengths of photon losses $T($or $ \eta )$ when given $\left \vert \gamma \right \vert =0,1,2,3$. Other parameters are as following: $| \beta |=r=1$, $ \theta _{ \xi }=0$ and $ \theta _{ \beta }= \theta _{ \gamma }= \pi /2$.
From a practical point of view, finally, the implementation of LDOs in both paths do not guarantee an exact match. Hence, based on Fig. 1(a), we also consider the effects of the imbalance between the upper ($\left \vert \gamma _{2}\right \vert$) and lower $\left \vert \gamma _{1}\right \vert$ arm displacements into both the phase sensitivity and the QCRB. We find that both the phase sensitivity with homodyne detection and the QCRB are only related to the LDO in the path $b_{1}$ (i.e., the upper arm displacement), rather than the lower arm displacement. In other words, for both balanced and unbalanced LDOs, the corresponding phase sensitivity with homodyne detection and the QCRB are only affected by the LDO of path $b_{1}$, independent of the case of path $a_{1}$. The reason for this phenomenon may be that the LDO in the path is only used as the reference shown in Fig. 1, not going through the phase shift. Moreover, the imbalance between the upper and lower arm displacements would affect the HL and SQL, which results from Eq. (22). As depicted in Fig. 7, when $\left \vert \gamma _{1}\right \vert =0$ and $\left \vert \gamma _{2}\right \vert =1$, the corresponding phase precision with respect to both HL and SQL would be reduced. Even so, compared to the SU(1,1) interferometer without LDOs (thick-black solid line), the phase precision for the DSU(1,1) interferometer with $\left \vert \gamma _{2}\right \vert =1$ (blue dashed line) can be more accessible to the HL, especially in the case of $\left \vert \gamma _{1}\right \vert =0$ and $\left \vert \gamma _{2}\right \vert =1$.
Fig. 7. The effects of the imbalance between the upper ($\left \vert \gamma _{2}\right \vert$) and lower $\left \vert \gamma _{1}\right \vert$ arm displacements into the phase sensitivity. Dot-dashed lines are the precision limits (e.g., SQL and HL) with $\left \vert \gamma _{2}\right \vert =1$ and $\left \vert \gamma _{1}\right \vert =0$. Thick black line is the phase sensitivity for SU(1,1) interferometer without LDOs. Note that for the two cases of ($\left \vert \gamma _{2}\right \vert,\left \vert \gamma _{1}\right \vert$)$\in \{$(1,0), (1,1)$\},$the corresponding phase sensitivity for DSU(1,1) interferometer is consistent (blue dashed line). Other parameters are as following: $| \beta |=r=1$, $ \phi = \theta _{ \xi }=0$ and $ \theta _{ \beta }= \theta _{ \gamma }= \pi /2$.
5. Conclusions
In summary, we have presented the positive contribution of the LDO for improving the estimation performance of SU(1,1) interferometer in terms of both the QCRB and the phase sensitivity based on homodyne detection. The numerical results show that the increase of the LDO strength is conducive to the improved estimation performance of the QCRB and the phase sensitivity. In particular, for the ideal case, the phase sensitivity of DSU(1,1) interferometer scheme can easily break through the SQL, even approaching to the HL. From a realistic point of view, we also consider both the QCRB and the phase sensitivity in the presence of photon losses. Our analyses indicate that, when given the same parameters, the DSU(1,1) interferometer scheme with respect to both the QCRB and the phase sensitivity can be superior to the SU(1,1) interferometer without the LDO in the photon-loss scenarios. In addition, both the phase sensitivity with homodyne detection and the QCRB is only related to the LDO in the path $b_{1}$ (i.e., the upper arm displacement), rather than the lower arm displacement; while the imbalance between the upper and lower arm displacements would affect the HL and SQL. Finally, we also find that the sufficiently large LDO can not only strengthen the robustness of SU(1,1) interferometer systems against photon losses, but also help us to saturate the QCRB, when comparing to the case without LDO.
However, the DSU(1,1) interferometer scheme we proposed still has some limitations. For example, we only take the photon losses into account, and other practical scenarios are not considered, such as thermal noise and phase diffusion. Moreover, we only carry out ideal homodyne detection at the output end, ignoring its actual detection efficiency. Finally, does the LSO that is seen as one of local operations improve the phase precision of SU(1,1) interferometer? These limitation factors will be considered in our future research work.
Appendix
Appendix A: Phase sensitivity based on homodyne detection in the ideal scenario
Combining Eqs. (1) and (3), one can derive the variance $\Delta ^{2}\hat {X}$ as
Substituting Eqs. (16) and (18) into the error propagation formula shown in Eq. (4), the explicit expression of the phase sensitivity of DSU(1,1) interferometer in the ideal scenario can be given by
In particular, when $\phi =\theta _{\xi }=0$, the variance $\Delta ^{2}\hat {X }=e^{-2r}/2.$ Moreover, by utilizing the results given in Eq. (18) at the phase-shift point $\phi =0,$ one can find the absolute value of the derivative of $\left \langle \hat {X}\right \rangle$
Finally, after achieving $\sin \theta _{\beta }=\sin \theta _{\gamma }=1$ by taking $\theta _{\beta }=\theta _{\gamma }=\pi /2$, one can obtain Eq. (9).
After presenting the phase sensitivity of DSU(1,1) interferometer based on homodyne detection, here we also derive phase sensitivities including the DSU(1,1), the SQL and the HL scheme in the ideal scenario. For this purpose, the total mean photon number inside DSU(1,1) interferometer should be introduced, which can be defined as
Appendix B: The QFI for DSU(1,1) interferometer
For DSU(1,1) interferometer with the input state $\left \vert \psi _{in}\right \rangle =\left \vert \xi \right \rangle _{a_{0}}\otimes \left \vert \beta \right \rangle _{b_{0}},$ the QFI in the ideal scenario can be given by Eq. (12) with $\Gamma _{m}=$ $\left \langle \psi _{\gamma }\right \vert \hat {b}_{1}^{\dagger m}\hat {b}_{1}^{m}\left \vert \psi _{\gamma }\right \rangle,(m=1,2)$ to be calculated. For this reason, here we need to introduce the characteristic function (CF), in which for the any probe state $\left \vert \psi _{\gamma }\right \rangle$ of DSU(1,1) interferometer, its CF can be expressed as
Therefore, substituting Eq. (27) into Eq. (12), one can obtain the explicit expression of the QFI for DSU(1,1) interferometer in the ideal case.
Due to the existence of photon losses, on the other hand, it is not suitable for deriving the QFI via the conventional method given in Eq. (11). To solve this problem, a novel variational method was proposed by Escher [51], which has been used in the photon-loss single-(or multi-)parameter estimation systems [18,52,53]. Based on the aforementioned work [51], below we would derive the explicit form of the QFI in the photon-loss DSU(1,1) interferometer.
Originally, we first denote the probe state as $\left \vert \psi _{\gamma }\right \rangle \equiv \left \vert \psi _{\gamma }\right \rangle _{S}$ where $\left \vert \psi _{\gamma }\right \rangle _{S}$ is an initial probe state of DSU(1,1) interferometer system $S$. Because of the photon losses, the encoding process of the probe state $\left \vert \psi _{\gamma }\right \rangle$ to an unknown phase $\phi$ is no longer the unitary evolution, so that the system $S$ is expanded into the enlarged one $S+E$ ($E$ represents the photon-loss environment system). Under such circumstances, the initial probe state $\left \vert \psi _{\gamma }\right \rangle _{S}$ in the enlarged systems $S+E$ experiences the unitary phase encoding process $\hat {U} _{S+E}(\phi )$, which can be described as [51]
Upon substituting Eqs. (29) into (32), so that
Thus, combining Eqs. (30) and (33), Eq. (32) can be rewritten as [51]
Funding
Scientific Research Startup Foundation at Nanchang Hangkong University (EA202204230); Jiangxi Provincial Natural Science Foundation (20232BAB211032); National Natural Science Foundation of China (11534008, 62161029, 91536115).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. T. J. Proctor, P. A. Knott, and J. A. Dunningham, “Multiparameter Estimation in Networked Quantum Sensors,” Phys. Rev. Lett. 120(8), 080501 (2018). [CrossRef]
2. X. Yu, X. Zhao, L. Y. Shen, et al., “Maximal quantum Fisher information for phase estimation without initial parity,” Opt. Express 26(13), 16292–16302 (2018). [CrossRef]
3. S. Y. Lee, Y. Jo, T. Jeong, et al., “Observable bound for Gaussian illumination,” Phys. Rev. A 105(4), 042412 (2022). [CrossRef]
4. E. Oelker, L. Barsotti, S. Dwyer, et al., “Squeezed light for advanced gravitational wave detectors and beyond,” Opt. Express 22(17), 21106–21121 (2014). [CrossRef]
5. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23(8), 1693–1708 (1981). [CrossRef]
6. W. Ge, K. Jacobs, Z. Eldredge, et al., “Distributed Quantum Metrology with Linear Networks and Separable Inputs,” Phys. Rev. Lett. 121(4), 043604 (2018). [CrossRef]
7. J. D. Zhang and S. Wang, “Unbalanced beam splitters enabling enhanced phase sensitivity of a Mach-Zehnder interferometer using coherent and squeezed vacuum states,” Phys. Rev. A 107(4), 043704 (2023). [CrossRef]
8. J. Y. Wu, N. Toda, and H. F. Hofmann, “Quantum enhancement of sensitivity achieved by photon-number-resolving detection in the dark port of a two-path interferometer operating at high intensities,” Phys. Rev. A 100(1), 013814 (2019). [CrossRef]
9. H. Zhang, W. Ye, C. P. Wei, et al., “Improving phase estimation using number conserving operations,” Phys. Rev. A 103(5), 052602 (2021). [CrossRef]
10. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: Beating the standard quantum limit,” Science 306(5700), 1330–1336 (2004). [CrossRef]
11. W. Du, J. Kong, G. Z. Bao, et al., “SU(2)-in-SU(1,1) Nested Interferometer for High Sensitivity, Loss-Tolerant Quantum Metrology,” Phys. Rev. Lett. 128(3), 033601 (2022). [CrossRef]
12. L. L. Guo, Y. F. Yu, and Z. M. Zhang, “Improving the phase sensitivity of an SU(1,1) interferometer with photon added squeezed vacuum light,” Opt. Express 26(22), 29099 (2018). [CrossRef]
13. B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1,1) interferometers,” Phys. Rev. A 33(6), 4033–4054 (1986). [CrossRef]
14. D. Li, C. H. Yuan, Z. Y. Ou, et al., “The phase sensitivity of an SU(1,1) interferometer with coherent and squeezed-vacuum light,” New J. Phys. 16(7), 073020 (2014). [CrossRef]
15. D. Li, B. T. Gard, Y. Gao, et al., “Phase sensitivity at the Heisenberg limit in an SU(1,1) interferometer via parity detection,” Phys. Rev. A 94(6), 063840 (2016). [CrossRef]
16. W. N. Plick, J. P. Dowling, and G. S. Agarwal, “Coherent-light-boosted, sub-shot noise, quantum interferometry,” New J. Phys. 12(8), 083014 (2010). [CrossRef]
17. S. K. Chang, C. P. Wei, H. Zhang, et al., “Enhanced phase sensitivity with a nonconventional interferometer and nonlinear phase shifter,” Phys. Lett. A 384(29), 126755 (2020). [CrossRef]
18. S. K. Chang, W. Ye, H. Zhang, et al., “Improvement of phase sensitivity in an SU(1,1) interferometer via a phase shift induced by a Kerr medium,” Phys. Rev. A 105(3), 033704 (2022). [CrossRef]
19. R. Carranza and C. C. Gerry, “Photon-subtracted two-mode squeezed vacuum states and applications to quantum optical interferometry,” J. Opt. Soc. Am. B 29(9), 2581 (2012). [CrossRef]
20. H. Kwon, K. C. Tan, T. Volkoff, et al., “Nonclassicality as a Quantifiable Resource for Quantum Metrology,” Phys. Rev. Lett. 122(4), 040503 (2019). [CrossRef]
21. J. P. Dowling, “Quantum optical metrology-the lowdown on high-N00N states,” Contemp. Phys. 49(2), 125–143 (2008). [CrossRef]
22. P. M. Anisimov, G. M. Raterman, A. Chiruvelli, et al., “Quantum Metrology with Two-Mode Squeezed Vacuum: Parity Detection Beats the Heisenberg Limit,” Phys. Rev. Lett. 104(10), 103602 (2010). [CrossRef]
23. R. A. Campos, C. C. Gerry, and A. Benmoussa, “Optical interferometry at the Heisenberg limit with twin Fock states and parity measurements,” Phys. Rev. A 68(2), 023810 (2003). [CrossRef]
24. I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328(5980), 879–881 (2010). [CrossRef]
25. T. Ono and H. F. Hofmann, “Effects of photon losses on phase estimation near the Heisenberg limit using coherent light and squeezed vacuum,” Phys. Rev. A 81(3), 033819 (2010). [CrossRef]
26. W. Ye, H. Zhong, Q Liao, et al., “Improvement of self-referenced continuous-variable quantum key distribution with quantum photon catalysis,” Opt. Express 27(12), 17186 (2019). [CrossRef]
27. D. Braun, P. Jian, O. Pinel, et al., “Precision measurements with photon-subtracted or photon-added Gaussian states,” Phys. Rev. A 90(1), 013821 (2014). [CrossRef]
28. Y. Ouyang, S. Wang, and L. J. Zhang, “Quantum optical interferometry via the photon-added two-mode squeezed vacuum states,” J. Opt. Soc. Am. B 33(7), 1373 (2016). [CrossRef]
29. H. Zhang, W. Ye, C. P. Wei, et al., “Improved phase sensitivity in a quantum optical inter ferometer based on multiphoton catalytic two-mode squeezed vacuum states,” Phys. Rev. A 103(1), 013705 (2021). [CrossRef]
30. X. Y. Hu, C. P. Wei, Y. F. Yu, et al., “Enhanced phase sensitivity of an SU(1,1) interferometer with displaced squeezed vacuum light,” Front. Phys. 11(3), 114203 (2016). [CrossRef]
31. G. S. Agarwal and L. Davidovich, “Quantifying quantum-amplified metrology via Fisher information,” Phys. Rev. Res. 4(1), L012014 (2022). [CrossRef]
32. J. Xin, “Phase sensitivity enhancement for the SU(1,1) interferometer using photon level operations,” Opt. Express 29(26), 43970–43984 (2021). [CrossRef]
33. S. L. Zhang and P. V. Loock, “Local Gaussian operations can enhance continuous-variable entanglement distillation,” Phys. Rev. A 84(6), 062309 (2011). [CrossRef]
34. V. C. Usenko and F. Grosshans, “Unidimensional continuous-variable quantum key distribution,” Phys. Rev. A 92(6), 062337 (2015). [CrossRef]
35. J. Sahota and D. F. V. James, “Quantum-enhanced phase estimation with an amplified Bell state,” Phys. Rev. A 88(6), 063820 (2013). [CrossRef]
36. J. Fiurasek, “Improving entanglement concentration of Gaussian states by local displacements,” Phys. Rev. A 84(1), 012335 (2011). [CrossRef]
37. Y. J. Wang, W. H. Zhang, R. X. Li, et al., “Generation of -10.7 dB unbiased entangled states of light,” Appl. Phys. Lett. 118(13), 134001 (2021). [CrossRef]
38. S. Ataman, “Optimal Mach-Zehnder phase sensitivity with Gaussian states,” Phys. Rev. A 100(6), 063821 (2019). [CrossRef]
39. S. Ataman, “Phase sensitivity of a Mach-Zehnder interferometer with single-intensity and difference-intensity detection,” Phys. Rev. A 98(4), 043856 (2018). [CrossRef]
40. Y. Chen, J. Gao, J. Han, et al., “Orbital-angular-momentum-enhanced phase estimation using non-Gaussian states with photon loss,” Phys. Rev. A 108(2), 022613 (2023). [CrossRef]
41. M. Lipka and M. Parniak, “Single-Photon Hologram of a Zero-Area Pulse,” Phys. Rev. Lett. 127(16), 163601 (2021). [CrossRef]
42. V. Cimini, F. Albarelli, I. Gianani, et al., “Semiparametric estimation of the Hong-Ou-Mandel profile,” Phys. Rev. A 104(6), L061701 (2021). [CrossRef]
43. F. Kaiser, P. Vergyris, D. Aktas, et al., “Quantum enhancement of accuracy and precision in optical interferometry,” Light: Sci. Appl. 7(3), 17163 (2017). [CrossRef]
44. F. Grosshans and P. Grangier, “Continuous Variable Quantum Cryptography using Coherent States,” Phys. Rev. Lett. 88(5), 057902 (2002). [CrossRef]
45. N. Gisin, G. Ribordy, W. Tittel, et al., “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002). [CrossRef]
46. S. L. Braunstein and H. J. Kimble, “Teleportation of Continuous Quantum Variables,” Phys. Rev. Lett. 80(4), 869–872 (1998). [CrossRef]
47. J. Xin, H. L. Wang, and J. T. Jing, “The effect of losses on the quantum-noise cancellation in the SU(1,1) interferometer,” Appl. Phys. Lett. 109(5), 051107 (2016). [CrossRef]
48. Y. M. Zhang, X. W. Li, W. Yang, et al., “Quantum Fisher information of entangled coherent states in the presence of photon loss,” Phys. Rev. A 88(4), 043832 (2013). [CrossRef]
49. S. L. Braunstein and C. M. Caves, “Statistical Distance and the Geometry of Quantum States,” Phys. Rev. Lett. 72(22), 3439–3443 (1994). [CrossRef]
50. S. Luo, “Wigner-Yanase Skew Information and Uncertainty Relations,” Phys. Rev. Lett. 91(18), 180403 (2003). [CrossRef]
51. B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys. 7(5), 406–411 (2011). [CrossRef]
52. J. D. Yue, Y. R. Zhang, and H. Fan, “Quantum-enhanced metrology for multiple phase estimation with noise,” Sci. Rep. 4(1), 5933 (2014). [CrossRef]
53. F. Albarelli, M. Mazelanik, M. Lipka, et al., “Quantum Asymmetry and Noisy Multimode Interferometry,” Phys. Rev. Lett. 128(24), 240504 (2022). [CrossRef]
54. M. Zwierz, C. A. Perez-Delgado, and P. Kok, “General Optimality of the Heisenberg Limit for Quantum Metrology,” Phys. Rev. Lett. 105(18), 180402 (2010). [CrossRef]
