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Phase sensitivity of an SU(1,1) interferometer in photon-loss via photon operations

Youke Xu, Teng Zhao, Qingqian Kang, Cunjin Liu, Liyun Hu, and Sanqiu Liu

Open Access Open Access

Abstract

We study the phase sensitivity of an SU(1,1) interferometer with photon loss by using three different photon operations schemes, i.e., performing photon-addition operation on the input port of the SU(1,1) interferometer (Scheme A), the interior of SU(1,1) interferometer (Scheme B), and both of them (Scheme C). We compare the performance of the three schemes in phase estimation by performing the same times of photon-addition operation to the mode b. The results show that Scheme B improves the phase sensitivity best in ideal case, and Scheme C performs well against internal loss, especially in the case of strong loss. All the three schemes can beat the standard quantum limit in the presence of photon loss, but Scheme B and Scheme C can break through the standard quantum limit in a larger loss range.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Quantum precision measurement is committed to improve the measurement accuracy [1]. In the process of quantum precision measurement, optical interferometers play a significant role in evaluating small phase changes, among which Mach-Zehnder interferometer (MZI) and SU(1,1) interferometer are most widely used in phase estimation [2–6]. On the other hand, how to beat the standard quantum limit (SQL) is one of the core purposes of quantum precision measurement [7]. In past decades, various schemes have been proposed to surpass the SQL and improve the phase sensitivity [8–10]. For example, Liu et al. proposed an ideal scheme to beat the SQL by using squeezed vacuum state as the input of SU(1,1) interferometer [11].

Since SU(1,1) interferometer was proposed by Yurke et al., it has more advantages in quantum precision measurement due to using of the optical parametric amplifiers (OPAs) [12]. For instance, Hudelist et al. demonstrated that due to the gain effect of OPA, SU(1,1) interferometer has better sensitivity than traditional linear interferometer [13]. In order to further improve the phase sensitivity of SU(1,1) interferometer, Li et al. proposed a scheme to approach the Heisenberg limit (HL) by employing a coherent state and a squeezed vacuum state as the inputs of SU(1,1) interferometer [14]. Recently, another nonlinear device, the Kerr nonlinear phase shift, was introduced into an SU(1,1) interferometer, which is beneficial to improve the phase sensitivity [15,16]. Moreover, in experiment, Jing et al. realized SU(1,1) interferometer for the first time by using four-wave mixing in hot atomic assembles [17]. Subsequently, various experimental schemes have been proposed to improve the phase sensitivity of SU (1,1) interferometer [13,18–22].

However, the negative effect of photon loss on the phase sensitivity in the process of phase estimation cannot be avoided [23]. Therefore, the problem to be solved in quantum precision measurement is how to reduce the negative effect of photon loss on the phase sensitivity [24–26]. Take SU (1,1) interferometer as an example, in order to improve the phase sensitivity and enhance the robustness of SU (1,1) interferometer in the presence of photon loss, Gong et al. proposed a scheme to implement phase estimation by employing an $m$-photon-subtracted squeezed vacuum state ($m$-PSSVS) and a coherent state as the inputs of SU (1,1) interferometer [27]. In addition, the results of another improved scheme showed that a non-Gaussian quantum state, generated by applying a number-conserving generalized superposition of products (GSP) operation on a two-mode squeezed vacuum state (TMSVS), as the input of MZI, can improve the phase sensitivity and resolution with photon loss [28]. Thus, in recent years, due to the remarkable nonclassical properties of the non-Gaussian states, the non-Gaussian operations used to prepare the non-Gaussian states, such as photon-subtraction, photon-addition, photon-catalysis, and coherent-superposition, are widely employed in quantum state engineering, including quantum precision measurement [29–35]. Especially in experiment, the superpositions of distinct quantum operations can be conditionally generated by using single-photon interference, which provides a realistic condition for the preparation of the non-Gaussian states [30].

More recently, an actively correlated MZI is proposed by Jiao et al., which is beneficial to improve the phase sensitivity in the presence of photon loss and contributes to beating the SQL within a certain loss range by adjusting the gain parameter of the second OPA [36]. To further weaken the effects of photon loss on the phase sensitivity, Zhang et al. proposed an modified SU(1,1) interferometer constructed of a nonlinear element and a BS to replace the second nonlinear element for eliminating the amplification effect of the second nonlinear element on the internal photon loss [37]. These schemes show that improving the structure of the interferometer can significantly reduce the negative effect of photon loss.

Moreover, another interesting method is to perform photon-subtraction or photon-addition inside SU (1,1) interferometer to enhance the robustness of SU (1,1) interferometer against the internal photon loss [38]. The advantages of this scheme are that there is no need to change the structure of interferometer, and only non-Gaussian operations are needed to perform internally to reduce the effect of internal loss, and these non-Gaussian operations, such as photon-subtraction or photon-addition, are feasible experimentally [39,40]. However, it is unclear whether the non-Gaussian operation acted on the interior of SU (1,1) interferometer has more advantages than that acted on the input port of SU (1,1) interferometer, and whether the non-Gaussian operation acted on the interior and the input port of SU (1,1) interferometer simultaneously has better performance in the presence of actual photon loss. Thus, in this paper, we shall study the phase sensitivity of SU (1,1) interferometer in the presence of the internal photon loss by performing photon-addition on the interior and the input port of SU (1,1) interferometer. In addition, we shall compare the improvement effects of the phase sensitivities of SU (1,1) interferometer in noisy environments under three different conditions, including photon-addition acted on the input port, the interior, as well as both of them.

This paper is organized as follows: In Sec. II, we briefly introduce our model. In Sec. III, we analyze the phase sensitivity in the presence of the internal photon loss. In Sec. IV, we consider the effects of photon losses on the quantum Fisher information (QFI) and compare the phase sensitivity of our scheme with some theoretical limits. Finally, we make a conclusion in the last section.

2. Model

We first give a brief description of our scheme as shown in Fig. 1. A standard SU(1,1) interferometer consists of two OPAs and a linear phase shifter, where the inputs $\left \vert \Psi \right \rangle _{in}=$ $\left \vert \alpha \right \rangle _{a}\otimes \left \vert r\right \rangle _{b}$ are a coherent state $\left \vert \alpha \right \rangle _{a}$ $(\alpha =\left \vert \alpha \right \vert e^{i\theta _{\alpha }})$ and a squeezed vacuum state (SVS) $\left \vert r\right \rangle _{b}=S_{b}(r)\left \vert 0\right \rangle,$ here $S_{b}(r)=\exp [\frac {r}{2}(b^{\dagger 2}-b^{2})]$ is the single-mode squeezing operator with the squeezing parameter $r.$ For mode $a$, it successively experiences the first OPA, the fictitious BS used to describe the photon loss, the phase shifter and the second OPA. While for mode $b$, it does not experience the phase shift process, but the photon-addition operation is successively performed on the input port and the interior, respectively. Thus, the transform relation between the output state $\left \vert \Psi \right \rangle _{out}$ and the input state $\left \vert \Psi \right \rangle _{in}$ is given by

$$\left \vert \Psi \right \rangle _{out}=N_{m,n}S_{OPA2}U_{\phi }b^{{\dagger} n}U_{Lw}S_{OPA1}b^{{\dagger} m}\left \vert \Psi \right \rangle _{in},$$
where the two-mode squeezing operator $S_{OPAj}(\xi _{j})=\exp (\xi _{j}^{\ast }ab-\xi _{j}a^{\dagger }b^{\dagger })$ ($\xi _{j}=g_{j}e^{i\theta _{g_{j}}}$ with a gain factor $g_{j}$ and a phase shift $\theta _{g_{j}}$ is the squeezing parameter, and $j=1,2$), the unitary operators $U_{Lw}=\exp [(w^{\dagger }w_{v}-ww_{v}^{\dagger })\arccos \sqrt {T_{w}}]$ ($T_{w}$ is the transmissivity of the fictitious BS, and $w=a,b$), and $U_{\phi }=\exp [i\phi (a^{\dagger }a)]$ are the mathematical version of the OPAs, the fictitious BSs and the phase shifter, respectively. $N_{m,n}$ is the normalization constant, and $a$ and $b$ are the annihilation operators for the two modes, respectively.

 figure: Fig. 1.

Fig. 1. Schematic diagram of an SU(1,1) interferometer. The two input ports of this interferometer are a coherent state $\left \vert \alpha \right \rangle _{a}$ and a squeezed vacuum states (SVS) $\left \vert r\right \rangle _{b},$ respectively. OPA is the optical parametric amplifier, the symbol of $\phi$ is corresponding to the phase shifter, and $D_{a}$ is the intensity detector. The two fictitious BSs inside of SU (1,1) interferometer are used to describe the photon loss on the mode $a$ and mode $b,$ respectively.

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SU (1,1) interferometer can make great achievements in the field of quantum metrology mainly because the introduction of OPAs has a gain effect on phase sensitivity [22]. The OPAs process can be equivalent to the process of two-mode squeezing, and a balanced situation of the two OPAs is $\theta _{g_{2}}-\theta _{g_{1}}=\pi$ and $g_{1}=g_{2}=g$ [14]. For simplicity, we consider the balanced situation and take the phase of coherent state $\theta _{\alpha }=0,$ the phase shift in the first OPA $\theta _{g_{1}}=0,$ and the transmissivities of the two fictitious BSs $T_{a}=T_{b}=T.$ With a coherent state and an SVS as inputs, the normalization constant $N_{m,n}$ can be calculated as

$$N_{m,n}=\frac{1}{\sqrt{D_{m,n,0,0,0,0}\exp (w_{4})}},$$
with
$$\begin{aligned} w_{1} &=\sqrt{T}\alpha \lbrack (t+t_{1})\cosh g-(\tau _{1}+s_{1}+\tau +s)\sinh g]\\ &\quad+R(\tau _{1}+s_{1})(\tau +s), \end{aligned}$$
$$w_{2} =\sqrt{T}[(\tau _{1}+s_{1})\cosh g-t\sinh g]+\lambda _{1},$$
$$w_{3} =\sqrt{T}[(\tau +s)\cosh g-t_{1}\sinh g]+\lambda ,$$
$$w_{4} =w_{1}+w_{2}w_{3}\cosh ^{2}r+\frac{\sinh 2r}{4}(w_{2}^{2}+w_{3}^{2}),$$
$$\begin{aligned} D_{m,n,k,l,x,y} &=\frac{\partial ^{2m+2n+k+l+x+y}}{\partial \lambda ^{m}\partial \lambda _{1}^{m}\partial \tau ^{n}\partial \tau _{1}^{n}\partial t^{k}\partial t_{1}^{l}\partial s^{x}\partial s_{1}^{y}} \times\\ &\quad\{{\bullet} \}|_{\lambda =\lambda _{1}=\tau =\tau _{1}=t=t_{1}=s=s_{1}=0}, \end{aligned}$$
where $R=1-T.$ Here $k, l$ and $x, y$ are positive integers, and $s, s_{1}, t, t_{1}$ and $\tau, \tau _{1}, \lambda, \lambda _{1}$ are the differential variables. After the differentiation, all these differential variables take zero.

At the end of this section, we briefly introduce the photon-addition operation of our scheme. In this paper, the photon-addition operation performed on mode $b$ can be divided into three schemes as follows: (i) Scheme A is to add $m$ photons to mode $b$ of input states, i.e., $b^{\dagger m}\left \vert \Psi \right \rangle _{in};$ (ii) Scheme B is to add $n$ photons to the mode $b$ in the interior of SU (1,1) interferometer after the fictitious BSs, i.e., $b^{\dagger n}U_{Lw}S_{OPA1}\left \vert \Psi \right \rangle _{in};$ (iii) Scheme C is the successive implementation of Scheme A and Scheme B, i.e., $b^{\dagger n}U_{Lw}S_{OPA1}b^{\dagger m}\left \vert \Psi \right \rangle _{in}.$ On the other hand, the photon-addition operation can be realized by employing a non-degenerate parametric down-conversion (NDPA) with small interaction strength, which provides favorable conditions for the experimental feasibility of our scheme [39]. Moreover, Dakna et al. proposed a scheme to produce photon-added states, such as photon-added thermal state, photon-added coherent state, and photon-added squeezed state, by employing a BS [41]. Recently, due to the advantages of the non-Gaussian states in the aspect of nonclassical properties, some experimental schemes of the photon-subtraction operation, the photon-catalysis operation, and the coherent-superposition operation, are widely investigated in quantum state engineering [42–47].

3. Phase estimation based on intensity detection

Selecting an appropriate detection method to extract the phase information is the last step in the process of phase estimation. The commonly used detection methods include intensity detection, homodyne detection, and parity detection [14–16,18,28,32]. For simplicity, here we only consider intensity detection at the output ports $D_{a},$ where the photon-number operator is $N=a^{\dagger }a$ for mode $a.$ In order to obtain the phase sensitivity of SU(1,1) interferometer, we can appeal the error propagation formula [12], i.e.,

$$\Delta ^{2}\phi =\frac{\left\langle N^{2}\right\rangle -\left\langle N\right\rangle ^{2}}{\left\vert \partial _{\phi }\left\langle N\right\rangle \right\vert ^{2}},$$
where $\left \langle N\right \rangle =\left. _{out}\left \langle \Psi \right \vert a^{\dagger }a\left \vert \Psi \right \rangle _{out}\right. .$ Based on Eq. (8), the phase sensitivity can be analytically calculated as (see Appendix for more details)
$$\Delta ^{2}\phi =\frac{A_{1}+A_{2}+A_{3}-N_{m,n}^{2}A_{3}^{2}}{ N_{m,n}^{2}[\partial \phi (A_{3})]^{2}},$$
where
$$\begin{aligned} A_{1} &=\cosh ^{4}gD_{m,n,2,2,0,0}\exp (w_{4})+\sinh ^{2}2g[D_{m,n,1,1,1,1}\exp (w_{4})\\ &\quad+\frac{1}{2}\cos 2\phi D_{m,n,2,0,2,0}\exp (w_{4})]+\sinh ^{4}gD_{m,n,0,0,2,2}\exp (w_{4}), \end{aligned}$$
$$\begin{aligned} A_{2} &=2\cos \phi \sinh 2g[\cosh ^{2}gD_{m,n,2,1,1,0}\exp (w_{4})\\ &\quad+\sinh ^{2}gD_{m,n,1,0,2,1}\exp (w_{4})], \end{aligned}$$
$$\begin{aligned} A_{3} &=\cosh ^{2}gD_{m,n,1,1,0,0}\exp (w_{4})+\sinh ^{2}gD_{m,n,0,0,1,1}\exp (w_{4})\\ &\quad+\cos \phi \sinh 2gD_{m,n,1,0,1,0}\exp (w_{4}). \end{aligned}$$

First, we consider the effects of some parameters on the phase sensitivity in ideal case, i.e., $T=1.$ In order to facilitate analysis and comparison, we compare the performance of the three schemes in phase estimation by adding the same number of photons to the mode $b.$ In Fig. 2, we plot the phase sensitivity as a function of $\phi$ based on the intensity detection. It is found from Fig. 2 as follows: (i) the three schemes of photon-addition operation can improve the phase sensitivity; (ii) the phase sensitivity can be improved with increasing $m$ or $n.$ Among all these non-Gaussian operations, Scheme C ($m=2, n=2$) gives the best phase sensitivity. Here we should point out that, Scheme C ($m=2, n=2$) performs four times of photon-addition operation. (iii) If we fix the total added photon-number, i.e., say fixing $m+n=2,$ which includes Scheme A ($m=2, n=0$), Scheme B ($m=0, n=2$), and Scheme C ($m=1, n=1$), then Scheme B is the best one in this case. (iv) Scheme A is far inferior to the other two schemes in improving phase sensitivity, although it has been investigated by Guo et al. with a good performance in phase estimation.

 figure: Fig. 2.

Fig. 2. The phase sensitivity based on intensity detection as a function of $\phi$ with $\alpha =1,$ $r=1,$ $g=1,$ and $T=1.$

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Next, we further consider the effects of some other parameters on the phase sensitivity without photon loss. In Figs. 3(a)–3(c), we plot the optimal phase sensitivity (minimizing $\Delta \phi )$ versus the gain factor $g,$ the squeezing parameter $r,$ and the amplitude of coherent state $\alpha,$ respectively. It can be clearly seen from Figs. 3(a)–3(c) as follows: (i) the parameters $g$ and $r$ show gain effects on the phase sensitivity; (ii) Scheme B can obtain the highest phase sensitivity, especially in the small region of $g,$ when all the three schemes perform two times of photon-addition operation, i.e., Scheme A ($m=2,$ $n=0$), Scheme B ($m=0,$ $n=2$), and Scheme C ($m=1,$ $n=1$); (iii) the squeezing parameter $r$ has the greatest impact on the phase sensitivity of Scheme A. For example, the phase sensitivity of Scheme A with $m=2,$ and $n=0,$ is even worse than that of Scheme B with $m=0,$ and $n=1$ when $r$ is taken a value in a small region; (iv) the phase sensitivities of the three schemes can not necessarily be improved by increasing the amplitude of the coherent state; (v) when $2$ < $\alpha$ < $6,$ Scheme C can obtain the highest phase sensitivity.

 figure: Fig. 3.

Fig. 3. The optimal phase sensitivity based on the intensity detection as a function of (a) $g$ with $\alpha =1,$ $r=1,$ and $T=1;$ (b) $r$ with $\alpha =1,$ $g=1,$ and $T=1;$ (c) $\alpha$ with $r=1,$ $g=1,$ and $T=1.$ $\phi$ is optimized. (d) $\sigma (N)= \sqrt {\left \langle N^{2}\right \rangle -\left \langle N\right \rangle ^{2}}$ and $\left \langle N\right \rangle _{ \phi }=\left \vert \partial _{ \phi }\left \langle N\right \rangle \right \vert$ as a function of $\alpha$ with $r=1,$ $g=1,$ and $T=1.$

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In general, the phase sensitivity is monotonically decreased depending on optical power. However, we show the phase sensitivity as a function of $\alpha$ when given other parameters, which is different from most cases discussed before, as shown in Fig. 3(c). In order to clearly understand this point about the phase sensitivity has a maximum value when increasing $\alpha,$ we further examine the standard deviation $\sigma (N)=\sqrt {\left \langle N^{2}\right \rangle -\left \langle N\right \rangle ^{2}}$ of the mean photon number at the outport $D_{a}$ and the slope $\left \langle N\right \rangle _{\phi }=\left \vert \partial _{\phi }\left \langle N\right \rangle \right \vert$ as a function of $\alpha,$ respectively, by taking the original scheme ($m=0, n=0$) as an example, as shown in Fig. 3(d). From Fig. 3(d), it is clear that the standard deviation $\sigma (N)$ increases with increasing $\alpha.$ However, the slope $\left \langle N\right \rangle _{\phi }$ hardly increases in the small amplitude region compared to the standard deviation $\sigma (N).$ On the contrary, the slope $\left \langle N\right \rangle _{\phi }$ increases much faster than the standard deviation $\sigma (N)$ outside of this small range. Thus, the phase sensitivity $\Delta \phi =\frac {\sigma (N)}{\left \langle N\right \rangle _{\phi }}$ firstly increases and then decreases with $\alpha,$ which is the reason that there are maximum values when increase $\alpha$ in our scheme with intensity detection.

Through the above discussion and analysis, it is not difficult to find that, in most ideal cases, Scheme B can obtain higher phase sensitivity than the other two schemes when considering adding the same number of photons to mode $b.$ However, the negative effect of photon loss on phase sensitivity cannot be ignored in phase estimation [5]. Therefore, we next discuss the performance of the three schemes under photon loss.

Figures 4 depict the change of the phase sensitivity in the presence of the internal photon loss when $r=1,$ and $r=0.2,$ respectively. Here, $T$ is the transmissivity of the two fictitious BSs, and $T=1$ corresponds to the ideal case. From Fig. 4(a), the performance of the three schemes under photon loss is as follows: (i) all the three schemes can improve the robustness of SU (1,1) interferometer in noisy environment; (ii) under the case of $r=1,$ Scheme C ($m=1,$ $n=1$) is more suitable for phase estimation than Scheme B ( $m=0,$ $n=2$) in the presence of the severe internal photon loss ($T$ < $0.4$); (iii) Scheme A ($m=2,$ $n=0$) is more robust than Scheme B ($m=0,$ $n=2$) when there is more than 90% internal photon loss ($T$ < $0.1$); (iv) Scheme B is the best scheme in improving the phase sensitivity under general internal photon loss ($T$ > $0.4$) when all the three schemes perform two times of photon-addition operation. On the contrary, when the squeezing parameter $r$ is small ($r=0.2$), the performance of Scheme A decreases significantly in the aspect of reducing the internal loss. In addition, Scheme B ($m=0,$ $n=2$) is more robust than the scheme C ($m=1,$ $n=1$) in general case of internal photon loss ($T$ > $0.3$). Thus, the squeezing parameter $r$ plays an important role in adjusting the robustness of the three schemes in the internal noise environment.

 figure: Fig. 4.

Fig. 4. The optimal phase sensitivity as a function of $T$ with $\alpha =1,$ $g=1,$ and (a) $r=1;$ (b) $r=0.2.$

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The performance of the three schemes under photon loss can be explained by the physical model. Firstly, the performance of Scheme A is not only amplified by the first OPA, but also weakened by the internal photon loss, and this weakening effect is further amplified by the second OPA. On the contrary, for Scheme B, although the photon operation does not undergo the first OPA, it also does not undergo the process of the internal photon loss, which means that the performance improvement brought by Scheme B is only amplified by the second OPA without photon loss. Therefore, Scheme B is significantly better than Scheme A in noisy environment. In addition, Scheme C combines the advantages of Schemes A and B, making its performance is better in the whole photon loss range.

4. QFI and some theoretical limits

4.1 QFI with photon losses

In the field of quantum precision measurement, the theoretical maximum information of unknown phase shift $\phi$ can be evaluated by using QFI associated with the quantum Carmér-Rao bound (QCRB) that determines the ultimate phase precision [48]. In our scheme, the small phase change occurs in mode $a$ inside SU(1,1) interferometer, thus, for simplicity, we only consider the photon loss in mode $a$ as shown in Fig. 5. In the presence of photon loss, the method of calculating the QFI is proposed by Escher et al. [6]. The method can be briefly summarized in the following: for an arbitrary initial pure state $\left \vert \psi _{S}\right \rangle$ in the probe system $S,$ the QFI with photon loss can be calculated as [6]

$$F_{L}=\underset{\left \{ \Pi _{l}\left( \phi \right) \right \} }{\min } C_{Q}[\left \vert \psi _{S}\right \rangle ,\hat{\Pi}_{l}\left( \phi \right) ],$$
where $\hat {\Pi }_{l}\left ( \phi \right )$ is the Kraus operator which acts on the system $S$ and describes the photon losses, and $C_{Q}[\left \vert \psi _{S}\right \rangle,\hat {\Pi }_{l}\left ( \phi \right ) ]$ can be further calculated as
$$C_{Q}[\left \vert \psi _{S}\right \rangle ,\hat{\Pi}_{l}\left( \phi \right) ]=4\left[ \left \langle \hat{H}_{1}\right \rangle _{S}-\left \langle \hat{H} _{2}\right \rangle _{S}^{2}\right] ,$$
where
$$\hat{H}_{1} =\sum_{l}\frac{d\hat{\Pi}_{l}^{{\dagger} }\left( \phi \right) }{ d\phi }\frac{d\hat{\Pi}_{l}\left( \phi \right) }{d\phi },$$
$$\hat{H}_{2} =i\sum_{l}\frac{d\hat{\Pi}_{l}^{{\dagger} }\left( \phi \right) }{ d\phi }\hat{\Pi}_{l}\left( \phi \right) .$$

 figure: Fig. 5.

Fig. 5. Schematic diagram of the photon losses on the mode $a;$ the losses occurs inside this interferometer. The first fictitious BS is between the first OPA and the phase shifter, and the second fictitious BS is between the phase shifter and the second OPA. $\eta$ is the transmissivity of the two fictitious BSs, and $a_{v}$ is the vacuum noise operator.

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The Kraus operator $\hat {\Pi }_{l}\left ( \phi \right )$ is associated with the photon losses, thus, when the photon losses occur inside SU(1,1) interferometer as shown in Fig. 5, the corresponding Kraus operator is given by [6]

$$\hat{\Pi}_{l}\left( \phi \right) =\sqrt{\frac{(1-\eta )^{l}}{l!}}e^{i\phi ( \hat{n}_{a}+\lambda l)}\eta ^{\hat{n}_{a}/2}a^{l},$$
where $\hat {n}_{a}=a^{\dagger }a$ is the photon-number operator of mode $a,$ and $\eta$ is the transmissivity of the two fictitious BSs and describes the photon losses, and $\eta =0,1$ corresponding to complete absorption and lossless cases, respectively. $\lambda =0,1$ represent the photon loss after or before the phase shifts, respectively [16]. Based on Eqs. (13 ), (14), and (17), the QFI of mode $a$ in the presence of photon losses can be reformed as [6]
$$F_{L}=\frac{4\eta \left\langle n\right\rangle \left\langle \Delta n^{2}\right\rangle }{\left( 1-\eta \right) \left\langle \Delta n^{2}\right\rangle +\eta \left\langle n\right\rangle },$$
where $\left \langle n\right \rangle,$ and $\left \langle \Delta n^{2}\right \rangle$ are corresponding to the total mean photon number and the variance of total mean photon number of mode $a$ inside SU(1,1) interferometer, respectively. Thus, $\left \langle n\right \rangle,$ and $\left \langle \Delta n^{2}\right \rangle$ in our scheme can be analytically calculated as
$$\begin{aligned} \left\langle n\right\rangle &=N_{f}^{2}\left. _{f}\left\langle \Psi \right\vert n_{a}\left\vert \Psi \right\rangle _{f}\right.\\ &=N_{f}^{2}D_{m,n,1,1,0,0}\exp (f_{4}), \end{aligned}$$
$$\begin{aligned} \left\langle \Delta n^{2}\right\rangle &=N_{f}^{2}\left. _{f}\left\langle \Psi \right\vert n_{a}^{2}\left\vert \Psi \right\rangle _{f}\right. -\left\langle n\right\rangle ^{2}\\ &=N_{f}^{2}\{D_{m,n,2,2,0,0}\exp (f_{4})+D_{m,n,1,1,0,0}\exp (f_{4})\\ &\quad-N_{f}^{2}[D_{m,n,1,1,0,0}\exp (f_{4})]^{2}\}, \end{aligned}$$
with
$$\left\vert \Psi \right\rangle _{f} =b^{{\dagger} n}S_{OPA1}b^{{\dagger} m}\left\vert \Psi \right\rangle _{in},$$
$$f_{1} =\alpha \lbrack (t+t_{1})\cosh g-(\tau _{1}+s_{1}+\tau +s)\sinh g],$$
$$f_{2} =(\tau _{1}+s_{1})\cosh g-t\sinh g+\lambda _{1},$$
$$f_{3} =(\tau +s)\cosh g-t_{1}\sinh g+\lambda ,$$
$$f_{4} =f_{1}+f_{2}f_{3}\cosh ^{2}r+\frac{\sinh 2r}{4}(f_{2}^{2}+f_{3}^{2}),$$
$$N_{f}^{{-}2} =D_{m,n,0,0,0,0}\exp (f_{4}).$$

Next, we discuss the QFI of the three schemes with and without photon loss by adding the same number of photons to the mode $b,$ respectively (shown in Figs. 6). From Fig. 6(a), it is clearly seen that: (i) the QFI increases with the increase of $m$ or $n;$ (ii) the QFI of Scheme A is the highest among the three schemes; (iii) Scheme B has the least QFI, although it performs well based on the intensity detection; (iv) with the increase of $g$, the gap between the QFI of the three schemes is growing. In fact, the non-Gaussian operations can add information to the system state, while Scheme A can add photons to the input state and undergoes an OPA process. Therefore, based on the parametric amplification function of OPA, Scheme A has higher QFI than the other two schemes. On the other hand, the higher QFI means the higher QCRB accuracy. Since Scheme B performs photon-addition operation after the first OPA, the amount of information increased by Scheme B is not amplified by the first OPA, resulting in the lowest QCRB accuracy obtained by Scheme B among the three schemes.

 figure: Fig. 6.

Fig. 6. The quantum Fisher information F as a function of (a) $g$ with $r=1,$ and $\alpha =1,$ $\eta =1,$ (b) $\eta$ with $g=1,$ $r=1,$ and $\alpha =1.$

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In Fig. 6(b), we plot the QFI as a function of $\eta.$ It is shown that (i) the QFI of the three schemes with photon loss is still higher than that of the original scheme (the black solid line); (ii) the QFI of Scheme A has the fastest attenuation in the noise environment; (iii) Scheme B ($m=0,$ $n=2$) is more conducive to saving the QFI than scheme A ($m=2,$ $n=0$) when photon loss is greater than 10% ($\eta <$ $0.9$); (iv) Scheme C ($m=1,$ $n=1$) has stronger anti-noise ability in weak noise environment ($0.8$ < $\eta <$ $0.9$), and its anti-noise ability can be close to Scheme B ($m=0,$ $n=2$) in strong noise environment ($\eta <$ $0.8$).

Combining Figs. 6(a) and 6(b), it is not difficult to see that Scheme A is obviously weaker than the other two schemes in the terms of noise resistance, although Scheme A has higher fisher information in ideal case. In addition, with the increase of photon loss, Scheme B which has the minimum QFI in ideal case becomes more and more advantageous. The best scheme is Scheme C, which has good performance in both of weak and strong loss environments. Scheme C not only have the higher QFI but also have better anti-noise ability, when it adds photons to mode $b$ before the first OPA (greatly improving the QFI) and after the first OPA (greatly enhancing the anti-noise ability), respectively.

4.2 Comparison phase sensitivities and theoretical limits

In this subsection, in order to further highlight the advantages of our scheme in the presence of the internal photon loss, we compare the phase sensitivity with some theoretical limits, including the QCRB, the SQL, and the HL. The QCRB is often used to determine the ultimate phase precision of an interferometer and can be derived by the QFI [49]. Thus, in our scheme, the QCRB with photon loss is given by

$$\Delta \phi _{QCRB}=\frac{1}{\sqrt{vF_{L}}},$$
where $v$ is the number of trials and here we take $v=1$ [50]. In addition, the SQL ($\frac {1}{\sqrt {N_{T}}}$) and the HL ($\frac {1}{N_{T}}$) can be derived by the total average photon number $N_{T}$ inside the interferometer, i.e.,
$$\begin{aligned} N_{T} &=N_{f}^{2}\left. _{f}\left\langle \Psi \right\vert \left( a^{\dagger }a+b^{\dagger }b\right) \left\vert \Psi \right\rangle _{f}\right.\\ &=N_{f}^{2}[D_{m,n,1,1,0,0}\exp (f_{4})+D_{m,n,0,0,1,1}\exp (f_{4})]-1. \end{aligned}$$

For the convenience of analysis and comparison, we take $T=\eta$ and plot the optimal phase sensitivity and theoretical limits as a function of $T$ ($\eta$) as shown in Figs. 7. From Figs. 7, it is clearly seen that (i) the original scheme ($m=0, n=0$) can be saturated with the SQL under the case of $g=1, \alpha =1, r=1,$ and $T=1,$ and beat the SQL with increasing $g,$ which has been investigated by Guo $et al.$ [32]; (ii) the phase sensitivity can beat the SQL in the presence of photon loss by increasing $m$ or $n;$ (iii) Scheme B is more likely to break through the SQL in noisy environment than Scheme A, i.e., compared with increasing $m$, increasing $n$ can beat the SQL under a greater degree of photon loss (as shown in Figs. 7(b)–7(e)); (iv) both of increasing $m$ and $n$ can gradually approach the HL, but increasing $n$ can be closer to the HL; (v) Schemes C and B have similar performance in the presence of photon loss, but when the photon loss is severe, the phase sensitivity of Scheme C can further approach the QCRB (as shown in Figs. 7(e) and 7(f)). Therefore, Scheme C has a better performance in improving the robustness of the SU(1,1) interferometer and can adapt to various degrees of photon loss.

 figure: Fig. 7.

Fig. 7. The optimal phase sensitivity and theoretical limits as a function of $T$ ($\eta$) with $g=1,\ \alpha =1,$ and $r=1.$ The original scheme, Scheme A, Scheme B, and Scheme C are corresponding to (a), (b,d), (c,e) and (f), respectively.

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5. Conclusion

In summary, we compared the performances of the three photon operations schemes on the phase sensitivity of an SU(1,1) interferometer with a coherent state and an SVS as inputs in the presence of internal photon loss. By employing the three different photon operations schemes including performing photon-addition operation on the input port (Scheme A), the interior (Scheme B), and both of them (Scheme C), the phase sensitivities can be significantly improved. When the squeezing parameter $r$ is small, Scheme A performs even worse in phase estimation. Ideally, Schemes B and C can further improve the phase sensitivity compared to Scheme A, but the QFI obtained by Scheme B is the smallest among the three schemes. In contrast, due to the amplification function of the first OPA, Scheme A can obtain the highest QFI when all the three schemes perform two times of photon-addition operation to the mode $b.$

We compare the performance of the three schemes in phase estimation by performing the same times of photon-addition operation to the mode $b$, i.e., Scheme A ($m=2, n=0$), Scheme B ($m=0, n=2$), and Scheme C ($m=1, n=1$). In the case of internal loss, all the three schemes can improve the robustness of an SU(1,1) interferometer, but the performance of Scheme A is not as good as the other two schemes. Both Schemes B and C have good performance in terms of loss resistance, and Scheme B has better performance in the weak loss environment. Within a certain loss rang, the phase sensitivities of the three photon operations schemes can beat the SQL. Especially in the strong loss range, the phase sensitivity of Scheme C is closer to the QCRB. Therefore, in the weak loss environment, Scheme B can be used to better improve the phase sensitivity of an SU(1,1) interferometer, while in the strong loss environment, Scheme C is obviously better.

Appendix: derivation of Eq. (9)

In this Appendix, we give the derivation of the phase sensitivity in Eq. (9). In our scheme, the transform relation between the output state $\left \vert \Psi \right \rangle _{out}$ and the input state $\left \vert \Psi \right \rangle _{in}$ is given by

$$\left\vert \Psi \right\rangle _{out}=N_{m,n}S_{OPA2}U_{\phi }b^{{\dagger} n}U_{Lw}S_{OPA1}b^{{\dagger} m}\left\vert \Psi \right\rangle _{in}.$$

Before deriving the phase sensitivity (Eq. (9)), we introduce a formula, i.e.,

$$\begin{aligned} &\quad b^{m}S_{OPA1}^{{\dagger} }U_{Lw}^{{\dagger} }b^{n}(a^{{\dagger} k}a^{l}b^{y}b^{{\dagger} x})b^{{\dagger} n}U_{Lw}S_{OPA1}b^{{\dagger} m}\\ &=\frac{\partial ^{2m+2n+k+l+x+y}}{\partial \lambda ^{m}\partial \lambda _{1}^{m}\partial \tau ^{n}\partial \tau _{1}^{n}\partial t^{k}\partial t_{1}^{l}\partial s^{x}\partial s_{1}^{y}}\exp (\lambda _{1}b)S_{OPA1}^{{\dagger} }U_{L}^{{\dagger} }\exp [(\tau _{1}+s_{1})b]\exp (ta^{{\dagger} })\times\\ &\quad\exp (t_{1}a)\exp [(\tau +s)b^{{\dagger} }]U_{L}S_{OPA1}\exp (\lambda b^{{\dagger} })|_{\lambda =\lambda _{1}=\tau =\tau _{1}=t=t_{1}=s=s_{1}=0}. \end{aligned}$$

In order to derive Eq. (9), we use the transformation relations, i.e.,

$$S_{OPA1}^{{\dagger} }U_{La}^{{\dagger} }aU_{La}S_{OPA1} =\sqrt{T}\left( a\cosh g-b^{{\dagger} }\sinh g\right) +\sqrt{R}a_{\nu },$$
$$S_{OPA1}^{{\dagger} }U_{Lb}^{{\dagger} }bU_{Lb}S_{OPA1} =\sqrt{T}\left( b\cosh g-a^{{\dagger} }\sinh g\right) +\sqrt{R}b_{\nu },$$
$$U_{\phi }^{{\dagger} }S_{OPA2}^{{\dagger} }aS_{OPA2}U_{\phi } =e^{i\phi }a\cosh g+b^{{\dagger} }\sinh g.$$

Here $k,$ $l$ and $x,$ $y$ are positive integers, and $s,$ $s_{1},$ $t,$ $t_{1}$ and $\tau,$ $\tau _{1},$ $\lambda,$ $\lambda _{1}$ are the differential variables. After the differentiation, all these differential variables take zero.

Then, based on Eqs. (29)–(33) and Eq. (8), we can derive Eq. (9). In our scheme, the phase sensitivity can be calculated as

$$\Delta \phi =\frac{\sqrt{\left\langle N^{2}\right\rangle -\left\langle N\right\rangle ^{2}}}{\left\vert \partial _{\phi }\left\langle N\right\rangle \right\vert },$$
where
$$\left\langle N\right\rangle =N_{m,n}^{2}\left. _{N}\left\langle \Psi \right\vert U_{\phi }^{{\dagger} }S_{OPA2}^{{\dagger} }a^{{\dagger} }aS_{OPA2}U_{\phi }\left\vert \Psi \right\rangle _{N}\right. ,$$
$$\begin{aligned} \left\langle N^{2}\right\rangle &=N_{m,n}^{2}\left. _{N}\left\langle \Psi \right\vert U_{\phi }^{{\dagger} }S_{OPA2}^{{\dagger} }(a^{{\dagger} }a)^{2}S_{OPA2}U_{\phi }\left\vert \Psi \right\rangle _{N}\right.\\ &=N_{m,n}^{2}\left. _{N}\left\langle \Psi \right\vert U_{\phi }^{{\dagger} }S_{OPA2}^{{\dagger} }a^{{\dagger} 2}a^{2}S_{OPA2}U_{\phi }\left\vert \Psi \right\rangle _{N}\right. +\left\langle N\right\rangle , \end{aligned}$$
$$\left\vert \Psi \right\rangle _{N} =b^{{\dagger} n}U_{Lw}S_{OPA1}b^{{\dagger} m}\left\vert \Psi \right\rangle _{in}.$$

In order to derive the mean photon number $\left \langle N\right \rangle$ and the variance $\left \langle N^{2}\right \rangle,$ we first use Eq. (33) to calculate $U_{\phi }^{\dagger }S_{OPA2}^{\dagger }a^{\dagger }aS_{OPA2}U_{\phi }$ and $U_{\phi }^{\dagger }S_{OPA2}^{\dagger }a^{\dagger 2}a^{2}S_{OPA2}U_{\phi },$ i.e.,

$$\begin{aligned} U_{\phi }^{{\dagger} }S_{OPA2}^{{\dagger} }a^{{\dagger} }aS_{OPA2}U_{\phi } &=(e^{{-}i\phi }a^{{\dagger} }\cosh g+b\sinh g)(e^{i\phi }a\cosh g+b^{{\dagger} }\sinh g)\\ &=a^{{\dagger} }a\cosh ^{2}g+bb^{{\dagger} }\sinh ^{2}g\\ &\quad+\frac{1}{2}(e^{{-}i\phi }a^{{\dagger} }b^{{\dagger} }+e^{i\phi }ab)\sinh 2g, \end{aligned}$$
$$\begin{aligned} U_{\phi }^{{\dagger} }S_{OPA2}^{{\dagger} }a^{{\dagger} 2}a^{2}S_{OPA2}U_{\phi } &=a^{{\dagger} 2}a^{2}\cosh ^{4}g+4a^{{\dagger} }abb^{{\dagger} }\sinh ^{2}g\cosh ^{2}g+b^{2}b^{{\dagger} 2}\sinh ^{4}g\\ &\quad+(2e^{{-}i\phi }a^{{\dagger} 2}ab^{{\dagger} }+2e^{i\phi }a^{{\dagger} }a^{2}b)\sinh g\cosh ^{3}g\\ &\quad+\frac{1}{4}(e^{{-}2i\phi }a^{{\dagger} 2}b^{{\dagger} 2}+e^{2i\phi }a^{2}b^{2})\sinh ^{2}2g\\ &\quad+(2e^{{-}i\phi }a^{{\dagger} }bb^{{\dagger} 2}+2e^{i\phi }ab^{2}b^{{\dagger} })\sinh ^{3}g\cosh g. \end{aligned}$$

Next, we use Eqs. (30)–(32) to calculate a formula, i.e.,

$$\begin{aligned} W &=N_{m,n}^{2}\left. _{in}\left\langle \Psi \right\vert b^{m}S_{OPA1}^{{\dagger} }U_{Lw}^{{\dagger} }b^{n}(a^{{\dagger} k}a^{l}b^{y}b^{{\dagger} x})b^{{\dagger} n}U_{Lw}S_{OPA1}b^{{\dagger} m}\left\vert \Psi \right\rangle _{in}\right.\\ &=N_{m,n}^{2}\frac{\partial ^{2m+2n+k+l+x+y}}{\partial \lambda ^{m}\partial \lambda _{1}^{m}\partial \tau ^{n}\partial \tau _{1}^{n}\partial t^{k}\partial t_{1}^{l}\partial s^{x}\partial s_{1}^{y}}\left. _{in}\left\langle \Psi \right\vert \right. \exp (\lambda _{1}b)S_{OPA1}^{{\dagger} }U_{L}^{{\dagger} }\exp [(\tau _{1}+s_{1})b]\times\\ &\quad\exp (ta^{{\dagger} })\exp (t_{1}a)\exp [(\tau +s)b^{{\dagger} }]U_{L}S_{OPA1}\exp (\lambda b^{{\dagger} })\left\vert \Psi \right\rangle _{in}|_{\lambda =\lambda _{1}=\tau =\tau _{1}=t=t_{1}=s=s_{1}=0}\\ &=N_{m,n}^{2}\frac{\partial ^{2m+2n+k+l+x+y}}{\partial \lambda ^{m}\partial \lambda _{1}^{m}\partial \tau ^{n}\partial \tau _{1}^{n}\partial t^{k}\partial t_{1}^{l}\partial s^{x}\partial s_{1}^{y}}\exp (w_{4})|_{\lambda =\lambda _{1}=\tau =\tau _{1}=t=t_{1}=s=s_{1}=0}\\ &=N_{m,n}^{2}D_{m,n,k,l,x,y}\exp (w_{4}). \end{aligned}$$

Based on Eqs. (37)–(40), the mean photon number $\left \langle N\right \rangle$ and the variance $\left \langle N^{2}\right \rangle$ can be further calculate as

$$\begin{aligned} \left\langle N\right\rangle &=N_{m,n}^{2}[\cosh ^{2}gD_{m,n,1,1,0,0}\exp (w_{4})+\sinh ^{2}gD_{m,n,0,0,1,1}\exp (w_{4})\\ &\quad+\cos \phi \sinh 2gD_{m,n,1,0,1,0}\exp (w_{4})], \end{aligned}$$
$$\begin{aligned} \left\langle N^{2}\right\rangle &=N_{m,n}^{2}\{\cosh ^{4}gD_{m,n,2,2,0,0}\exp (w_{4})+\sinh ^{2}2g[D_{m,n,1,1,1,1}\exp (w_{4})\\ &\quad+\frac{1}{2}\cos 2\phi D_{m,n,2,0,2,0}\exp (w_{4})]+\sinh ^{4}gD_{m,n,0,0,2,2}\exp (w_{4})\\ &\quad+2\cos \phi \sinh 2g[\cosh ^{2}gD_{m,n,2,1,1,0}\exp (w_{4})\\ &\quad+\sinh ^{2}gD_{m,n,1,0,2,1}\exp (w_{4})]\}+\left\langle N\right\rangle , \end{aligned}$$
where the normalization constant (Eq. (2)) can be derived by using Eq. (40) and the normalization condition $_{out}\left \langle \Psi |\Psi \right \rangle _{out}=1$ when $k=l=x=y=0.$ In addition, we can obtain the QFI by using Eq. (18), Eq. (21), and Eq. (40) under the case of $T=1.$

Funding

Major Discipline Academic and Technical Leaders Training Program of Jiangxi Province (20204BCJL22053); National Natural Science Foundation of China (11964013).

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 may be obtained from the authors upon reasonable request.

References

1. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96(1), 010401 (2006). [CrossRef]  

2. R. Demkowicz-Dobrzański, M. Jarzyna, and J. Kolodynski, “Quantum limits in optical interferometry,” Prog. Optics. 60, 345–435 (2015). [CrossRef]  

3. C. M. Caves, “Quantum-Mechanical Radiation-Pressure Fluctuations in an Interferometer,” Phys. Rev. Lett. 45(2), 75–79 (1980). [CrossRef]  

4. V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5(4), 222–229 (2011). [CrossRef]  

5. D. Li, B. T. Gard, Y. Gao, C. H. Yuan, W. P. Zhang, H. Lee, and J. P. Dowling, “Phase sensitivity at the Heisenberg limit in an SU (1, 1) interferometer via parity detection,” Phys. Rev. A 94(6), 063840 (2016). [CrossRef]  

6. 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]  

7. X. Zuo, Z. Yan, Y. Feng, J. Ma, X. Jia, C. Xie, and K. Peng, “Quantum interferometer combining squeezing and parametric amplification,” Phys. Rev. Lett. 124(17), 173602 (2020). [CrossRef]  

8. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: Beating the standard quantum limit,” Science 306(5700), 1330–1336 (2004). [CrossRef]  

9. P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum Metrology with Two-Mode Squeezed Vacuum: Parity Detection Beats the Heisenberg Limit,” Phys. Rev. Lett. 104(10), 103602 (2010). [CrossRef]  

10. R. Birrittella, J. Mimih, and C. C. Gerry, “Multiphoton quantum interference at a beam splitter and the approach to Heisenberg-limited interferometry,” Phys. Rev. A 86(6), 063828 (2012). [CrossRef]  

11. J. Liu, W. X. Liu, S. Li, D. Wei, H. Gao, and F. Li, “Enhancement of the angular rotation measurement sensitivity based on SU(2) and SU(1,1) interferometers,” Photonics Res. 5(6), 617–622 (2017). [CrossRef]  

12. 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]  

13. F. Hudelist, J. Kong, C. J. Liu, J. T. Jing, Z. Y. Ou, and W. P. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5(1), 3049 (2014). [CrossRef]  

14. D. Li, C. H. Yuan, Z. Y. Ou, and W. P. Zhang, “The phase sensitivity of an SU(1,1) interferometer with coherent and squeezed-vacuum light,” New J. Phys. 16(7), 073020 (2014). [CrossRef]  

15. O. Seth, X. F. Li, H. N. Xiong, J. Y. Luo, and Y. X. Huang, “Improving the phase sensitivity of an SU(1,1) interferometer via a nonlinear phase encoding,” J. Phys. B: At., Mol. Opt. Phys. 53(20), 205503 (2020). [CrossRef]  

16. S. K. Chang, W. Ye, H. Zhang, L. Y. Hu, J. H. Huang, and S. Q. Liu, “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]  

17. J. Jing, C. Liu, Z. Zhou, Z. Y. Ou, and W. Zhang, “Realization of a nonlinear interferometer with parametric amplifiers,” Appl. Phys. Lett. 99(1), 011110 (2011). [CrossRef]  

18. S. S. Liu, Y. B. Lou, J. Xin, and J. T. Jing, “Quantum Enhancement of Phase Sensitivity for the Bright-Seeded SU(1,1) Interferometer with Direct Intensity Detection,” Phys. Rev. Appl. 10(6), 064046 (2018). [CrossRef]  

19. W. Du, J. Jia, J. F. Chen, Z. Y. Ou, and W. P. Zhang, “Absolute sensitivity of phase measurement in an SU(1,1) type interferometer,” Opt. Lett. 43(5), 1051 (2018). [CrossRef]  

20. M. Manceau, G. Leuchs, F. Khalili, and M. Chekhova, “Detection Loss Tolerant Supersensitive Phase Measurement with an SU(1,1) Interferometer,” Phys. Rev. Lett. 119(22), 223604 (2017). [CrossRef]  

21. B. Chen, C. Qiu, S. Chen, J. Guo, L. Q. Chen, Z. Y. Ou, and W. Zhang, “Atom-light hybrid interferometer,” Phys. Rev. Lett. 115(4), 043602 (2015). [CrossRef]  

22. B. E. Anderson, P. Gupta, B. L. Schmittberger, T. Horrom, C. Hermann-Avigliano, K. M. Jones, and P. D. Lett, “Phase sensing beyond the standard quantum limit with a variation on the SU(1, 1) interferometer,” Optica 4(7), 752 (2017). [CrossRef]  

23. A. M. Marino, N. V. Corzo Trejo, and P. D. Lett, “Effect of losses on the performance of an SU(1, 1) interferometer,” Phys. Rev. A 86(2), 023844 (2012). [CrossRef]  

24. K. B. Jiang, C. J. Brignac, Y. Weng, M. B. Kim, H. Lee, and J. P. Dowling, “Strategies for choosing path-entangled number states for optimal robust quantum-optical metrology in the presence of loss,” Phys. Rev. A 86(1), 013826 (2012). [CrossRef]  

25. X. L. Hu, D. Li, L. Q. Chen, K. Y. Zhang, W. P. Zhang, and C. H. Yuan, “Phase estimation for an SU(1,1) interferometer in the presence of phase diffusion and photon losses,” Phys. Rev. A 98(2), 023803 (2018). [CrossRef]  

26. Z. D. Chen, C. H. Yuan, H. M. Ma, D. Li, L. Q. Chen, Z. Y. Ou, and W. P. Zhang, “Effects of losses in the atom-light hybrid SU(1,1) interferometer,” Opt. Express 24(16), 17766 (2016). [CrossRef]  

27. Q. K. Gong, X. L. Hu, D. Li, C. H. Yuan, Z. Y. Ou, and W. P. Zhang, “Intramode-correlation-enhanced phase sensitivities in an SU (1, 1) interferometer,” Phys. Rev. A 96(3), 033809 (2017). [CrossRef]  

28. H. Zhang, W. Ye, C. P. Wei, C. J. Liu, Z. Y. Liao, and L. Y. Hu, “Improving phase estimation using number-conserving operations,” Phys. Rev. A 103(5), 052602 (2021). [CrossRef]  

29. 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]  

30. A. Zavatta, V. Parigi, M. S. Kim, H. Jeong, and M. Bellini, “Experimental Demonstration of the Bosonic Commutation Relation via Superpositions of Quantum Operations on Thermal Light Fields,” Phys. Rev. Lett. 103(14), 140406 (2009). [CrossRef]  

31. R. Birrittella and C. C. Gerry, “Quantum optical interferometry via the mixing of coherent and photon-subtracted squeezed vacuum states of light,” J. Opt. Soc. Am. B 31(3), 586–593 (2014). [CrossRef]  

32. 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]  

33. K. Zhang, Y. H. Lv, Y. Guo, J. T. Jing, and W. M. Liu, “Enhancing the precision of a phase measurement through phase-sensitive non-Gaussianity,” Phys. Rev. A 105(4), 042607 (2022). [CrossRef]  

34. Y. K. Xu, S. K. Chang, C. J. Liu, L. Y. Hu, and S. Q. Liu, “Phase estimation of an SU(1,1) interferometer with a coherent superposition squeezed vacuum in a realistic case,” Opt. Express 30(21), 38178 (2022). [CrossRef]  

35. D. Braun, P. Jian, O. Pinel, and N. Treps, “Precision measurements with photon-subtracted or photon-added Gaussian states,” Phys. Rev. A 90(1), 013821 (2014). [CrossRef]  

36. G. F. Jiao, Q. Wang, Z. F. Yu, L. Q. Chen, W. P. Zhang, and C. H. Yuan, “Effects of losses on the sensitivity of an actively correlated Mach-Zehnder interferometer,” Phys. Rev. A 104(1), 013725 (2021). [CrossRef]  

37. J. D. Zhang, C. L. You, C. Li, and S. Wang, “Phase sensitivity approaching the quantum Cramé-Rao bound in a modified SU(1,1) interferometer,” Phys. Rev. A 103(3), 032617 (2021). [CrossRef]  

38. J. Xin, “Phase sensitivity enhancement for the SU(1,1) interferometer using photon level operations,” Opt. Express 29(26), 43970 (2021). [CrossRef]  

39. A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306(5696), 660–662 (2004). [CrossRef]  

40. J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92(15), 153601 (2004). [CrossRef]  

41. M. Dakna, L. Knöll, and D. G. Welsch, “Photon-added state preparation via conditional measurement on a beam splitter,” Opt. Commun. 145(1-6), 309–321 (1998). [CrossRef]  

42. S. Y. Lee and H. Nha, “Quantum state engineering by a coherent superposition of photon subtraction and addition,” Phys. Rev. A 82(5), 053812 (2010). [CrossRef]  

43. Y. Kurochkin, A. S. Prasad, and A. I. Lvovsky, “Distillation of the Two-Mode Squeezed State,” Phys. Rev. Lett. 112(7), 070402 (2014). [CrossRef]  

44. T. J. Bartley, G. Donati, J. B. Spring, X. M. Jin, M. Barbieri, A. Datta, B. J. Smith, and I. A. Walmsley, “Multiphoton state engineering by heralded interference between single photons and coherent states,” Phys. Rev. A 86(4), 043820 (2012). [CrossRef]  

45. C. Kumar, Rishabh, and S. Arora, “Realistic non-Gaussian-operation scheme in parity-detection-based Mach-Zehnder quantum interferometry,” Phys. Rev. A 105(5), 052437 (2022). [CrossRef]  

46. L. Y. Hu, Z. Y. Liao, and M. S. Zubairy, “Continuous-variable entanglement via multiphoton catalysis,” Phys. Rev. A 95(1), 012310 (2017). [CrossRef]  

47. Y. S. Ra, A. Dufour, M. Walschaers, C. Jacquard, T. Michel, C. Fabre, and N. Treps, “Non-Gaussian quantum states of a multimode light field,” Nat. Phys. 16(2), 144–147 (2020). [CrossRef]  

48. X. X. Jing, J. Liu, W. Zhong, and X. G. Wang, “Quantum Fisher Information of Entangled Coherent States in a Lossy Mach-Zehnder Interferometer,” Commun. Theor. Phys. 61(1), 115–120 (2014). [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. L. Seveso, M. A. C. Rossi, and M. G. A. Paris, “Quantum metrology beyond the quantum Cramé-Rao theorem,” Phys. Rev. A 95(1), 012111 (2017). [CrossRef]  

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Figures (7)
Fig. 1.
Fig. 1. Schematic diagram of an SU(1,1) interferometer. The two input ports of this interferometer are a coherent state $\left \vert \alpha \right \rangle _{a}$ and a squeezed vacuum states (SVS) $\left \vert r\right \rangle _{b},$ respectively. OPA is the optical parametric amplifier, the symbol of $\phi$ is corresponding to the phase shifter, and $D_{a}$ is the intensity detector. The two fictitious BSs inside of SU (1,1) interferometer are used to describe the photon loss on the mode $a$ and mode $b,$ respectively.

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Fig. 2.
Fig. 2. The phase sensitivity based on intensity detection as a function of $\phi$ with $\alpha =1,$ $r=1,$ $g=1,$ and $T=1.$

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Fig. 3.
Fig. 3. The optimal phase sensitivity based on the intensity detection as a function of (a) $g$ with $\alpha =1,$ $r=1,$ and $T=1;$ (b) $r$ with $\alpha =1,$ $g=1,$ and $T=1;$ (c) $\alpha$ with $r=1,$ $g=1,$ and $T=1.$ $\phi$ is optimized. (d) $\sigma (N)= \sqrt {\left \langle N^{2}\right \rangle -\left \langle N\right \rangle ^{2}}$ and $\left \langle N\right \rangle _{ \phi }=\left \vert \partial _{ \phi }\left \langle N\right \rangle \right \vert$ as a function of $\alpha$ with $r=1,$ $g=1,$ and $T=1.$

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Fig. 4.
Fig. 4. The optimal phase sensitivity as a function of $T$ with $\alpha =1,$ $g=1,$ and (a) $r=1;$ (b) $r=0.2.$

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Fig. 5.
Fig. 5. Schematic diagram of the photon losses on the mode $a;$ the losses occurs inside this interferometer. The first fictitious BS is between the first OPA and the phase shifter, and the second fictitious BS is between the phase shifter and the second OPA. $\eta$ is the transmissivity of the two fictitious BSs, and $a_{v}$ is the vacuum noise operator.

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Fig. 6.
Fig. 6. The quantum Fisher information F as a function of (a) $g$ with $r=1,$ and $\alpha =1,$ $\eta =1,$ (b) $\eta$ with $g=1,$ $r=1,$ and $\alpha =1.$

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Fig. 7.
Fig. 7. The optimal phase sensitivity and theoretical limits as a function of $T$ ($\eta$) with $g=1,\ \alpha =1,$ and $r=1.$ The original scheme, Scheme A, Scheme B, and Scheme C are corresponding to (a), (b,d), (c,e) and (f), respectively.

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Equations (42)

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| Ψ ⟩ o u t = N m , n S O P A 2 U ϕ b † n U L w S O P A 1 b † m | Ψ ⟩ i n ,
N m , n = 1 D m , n , 0 , 0 , 0 , 0 exp ⁡ ( w 4 ) ,
w 1 = T α [ ( t + t 1 ) cosh ⁡ g − ( τ 1 + s 1 + τ + s ) sinh ⁡ g ] + R ( τ 1 + s 1 ) ( τ + s ) ,
w 2 = T [ ( τ 1 + s 1 ) cosh ⁡ g − t sinh ⁡ g ] + λ 1 ,
w 3 = T [ ( τ + s ) cosh ⁡ g − t 1 sinh ⁡ g ] + λ ,
w 4 = w 1 + w 2 w 3 cosh 2 ⁡ r + sinh ⁡ 2 r 4 ( w 2 2 + w 3 2 ) ,
D m , n , k , l , x , y = ∂ 2 m + 2 n + k + l + x + y ∂ λ m ∂ λ 1 m ∂ τ n ∂ τ 1 n ∂ t k ∂ t 1 l ∂ s x ∂ s 1 y × { ∙ } | λ = λ 1 = τ = τ 1 = t = t 1 = s = s 1 = 0 ,
Δ 2 ϕ = ⟨ N 2 ⟩ − ⟨ N ⟩ 2 | ∂ ϕ ⟨ N ⟩ | 2 ,
Δ 2 ϕ = A 1 + A 2 + A 3 − N m , n 2 A 3 2 N m , n 2 [ ∂ ϕ ( A 3 ) ] 2 ,
A 1 = cosh 4 ⁡ g D m , n , 2 , 2 , 0 , 0 exp ⁡ ( w 4 ) + sinh 2 ⁡ 2 g [ D m , n , 1 , 1 , 1 , 1 exp ⁡ ( w 4 ) + 1 2 cos ⁡ 2 ϕ D m , n , 2 , 0 , 2 , 0 exp ⁡ ( w 4 ) ] + sinh 4 ⁡ g D m , n , 0 , 0 , 2 , 2 exp ⁡ ( w 4 ) ,
A 2 = 2 cos ⁡ ϕ sinh ⁡ 2 g [ cosh 2 ⁡ g D m , n , 2 , 1 , 1 , 0 exp ⁡ ( w 4 ) + sinh 2 ⁡ g D m , n , 1 , 0 , 2 , 1 exp ⁡ ( w 4 ) ] ,
A 3 = cosh 2 ⁡ g D m , n , 1 , 1 , 0 , 0 exp ⁡ ( w 4 ) + sinh 2 ⁡ g D m , n , 0 , 0 , 1 , 1 exp ⁡ ( w 4 ) + cos ⁡ ϕ sinh ⁡ 2 g D m , n , 1 , 0 , 1 , 0 exp ⁡ ( w 4 ) .
F L = min { Π l ( ϕ ) } C Q [ | ψ S ⟩ , Π ^ l ( ϕ ) ] ,
C Q [ | ψ S ⟩ , Π ^ l ( ϕ ) ] = 4 [ ⟨ H ^ 1 ⟩ S − ⟨ H ^ 2 ⟩ S 2 ] ,
H ^ 1 = ∑ l d Π ^ l † ( ϕ ) d ϕ d Π ^ l ( ϕ ) d ϕ ,
H ^ 2 = i ∑ l d Π ^ l † ( ϕ ) d ϕ Π ^ l ( ϕ ) .
Π ^ l ( ϕ ) = ( 1 − η ) l l ! e i ϕ ( n ^ a + λ l ) η n ^ a / 2 a l ,
F L = 4 η ⟨ n ⟩ ⟨ Δ n 2 ⟩ ( 1 − η ) ⟨ Δ n 2 ⟩ + η ⟨ n ⟩ ,
⟨ n ⟩ = N f 2 f ⟨ Ψ | n a | Ψ ⟩ f = N f 2 D m , n , 1 , 1 , 0 , 0 exp ⁡ ( f 4 ) ,
⟨ Δ n 2 ⟩ = N f 2 f ⟨ Ψ | n a 2 | Ψ ⟩ f − ⟨ n ⟩ 2 = N f 2 { D m , n , 2 , 2 , 0 , 0 exp ⁡ ( f 4 ) + D m , n , 1 , 1 , 0 , 0 exp ⁡ ( f 4 ) − N f 2 [ D m , n , 1 , 1 , 0 , 0 exp ⁡ ( f 4 ) ] 2 } ,
| Ψ ⟩ f = b † n S O P A 1 b † m | Ψ ⟩ i n ,
f 1 = α [ ( t + t 1 ) cosh ⁡ g − ( τ 1 + s 1 + τ + s ) sinh ⁡ g ] ,
f 2 = ( τ 1 + s 1 ) cosh ⁡ g − t sinh ⁡ g + λ 1 ,
f 3 = ( τ + s ) cosh ⁡ g − t 1 sinh ⁡ g + λ ,
f 4 = f 1 + f 2 f 3 cosh 2 ⁡ r + sinh ⁡ 2 r 4 ( f 2 2 + f 3 2 ) ,
N f − 2 = D m , n , 0 , 0 , 0 , 0 exp ⁡ ( f 4 ) .
Δ ϕ Q C R B = 1 v F L ,
N T = N f 2 f ⟨ Ψ | ( a † a + b † b ) | Ψ ⟩ f = N f 2 [ D m , n , 1 , 1 , 0 , 0 exp ⁡ ( f 4 ) + D m , n , 0 , 0 , 1 , 1 exp ⁡ ( f 4 ) ] − 1.
| Ψ ⟩ o u t = N m , n S O P A 2 U ϕ b † n U L w S O P A 1 b † m | Ψ ⟩ i n .
b m S O P A 1 † U L w † b n ( a † k a l b y b † x ) b † n U L w S O P A 1 b † m = ∂ 2 m + 2 n + k + l + x + y ∂ λ m ∂ λ 1 m ∂ τ n ∂ τ 1 n ∂ t k ∂ t 1 l ∂ s x ∂ s 1 y exp ⁡ ( λ 1 b ) S O P A 1 † U L † exp ⁡ [ ( τ 1 + s 1 ) b ] exp ⁡ ( t a † ) × exp ⁡ ( t 1 a ) exp ⁡ [ ( τ + s ) b † ] U L S O P A 1 exp ⁡ ( λ b † ) | λ = λ 1 = τ = τ 1 = t = t 1 = s = s 1 = 0 .
S O P A 1 † U L a † a U L a S O P A 1 = T ( a cosh ⁡ g − b † sinh ⁡ g ) + R a ν ,
S O P A 1 † U L b † b U L b S O P A 1 = T ( b cosh ⁡ g − a † sinh ⁡ g ) + R b ν ,
U ϕ † S O P A 2 † a S O P A 2 U ϕ = e i ϕ a cosh ⁡ g + b † sinh ⁡ g .
Δ ϕ = ⟨ N 2 ⟩ − ⟨ N ⟩ 2 | ∂ ϕ ⟨ N ⟩ | ,
⟨ N ⟩ = N m , n 2 N ⟨ Ψ | U ϕ † S O P A 2 † a † a S O P A 2 U ϕ | Ψ ⟩ N ,
⟨ N 2 ⟩ = N m , n 2 N ⟨ Ψ | U ϕ † S O P A 2 † ( a † a ) 2 S O P A 2 U ϕ | Ψ ⟩ N = N m , n 2 N ⟨ Ψ | U ϕ † S O P A 2 † a † 2 a 2 S O P A 2 U ϕ | Ψ ⟩ N + ⟨ N ⟩ ,
| Ψ ⟩ N = b † n U L w S O P A 1 b † m | Ψ ⟩ i n .
U ϕ † S O P A 2 † a † a S O P A 2 U ϕ = ( e − i ϕ a † cosh ⁡ g + b sinh ⁡ g ) ( e i ϕ a cosh ⁡ g + b † sinh ⁡ g ) = a † a cosh 2 ⁡ g + b b † sinh 2 ⁡ g + 1 2 ( e − i ϕ a † b † + e i ϕ a b ) sinh ⁡ 2 g ,
U ϕ † S O P A 2 † a † 2 a 2 S O P A 2 U ϕ = a † 2 a 2 cosh 4 ⁡ g + 4 a † a b b † sinh 2 ⁡ g cosh 2 ⁡ g + b 2 b † 2 sinh 4 ⁡ g + ( 2 e − i ϕ a † 2 a b † + 2 e i ϕ a † a 2 b ) sinh ⁡ g cosh 3 ⁡ g + 1 4 ( e − 2 i ϕ a † 2 b † 2 + e 2 i ϕ a 2 b 2 ) sinh 2 ⁡ 2 g + ( 2 e − i ϕ a † b b † 2 + 2 e i ϕ a b 2 b † ) sinh 3 ⁡ g cosh ⁡ g .
W = N m , n 2 i n ⟨ Ψ | b m S O P A 1 † U L w † b n ( a † k a l b y b † x ) b † n U L w S O P A 1 b † m | Ψ ⟩ i n = N m , n 2 ∂ 2 m + 2 n + k + l + x + y ∂ λ m ∂ λ 1 m ∂ τ n ∂ τ 1 n ∂ t k ∂ t 1 l ∂ s x ∂ s 1 y i n ⟨ Ψ | exp ⁡ ( λ 1 b ) S O P A 1 † U L † exp ⁡ [ ( τ 1 + s 1 ) b ] × exp ⁡ ( t a † ) exp ⁡ ( t 1 a ) exp ⁡ [ ( τ + s ) b † ] U L S O P A 1 exp ⁡ ( λ b † ) | Ψ ⟩ i n | λ = λ 1 = τ = τ 1 = t = t 1 = s = s 1 = 0 = N m , n 2 ∂ 2 m + 2 n + k + l + x + y ∂ λ m ∂ λ 1 m ∂ τ n ∂ τ 1 n ∂ t k ∂ t 1 l ∂ s x ∂ s 1 y exp ⁡ ( w 4 ) | λ = λ 1 = τ = τ 1 = t = t 1 = s = s 1 = 0 = N m , n 2 D m , n , k , l , x , y exp ⁡ ( w 4 ) .
⟨ N ⟩ = N m , n 2 [ cosh 2 ⁡ g D m , n , 1 , 1 , 0 , 0 exp ⁡ ( w 4 ) + sinh 2 ⁡ g D m , n , 0 , 0 , 1 , 1 exp ⁡ ( w 4 ) + cos ⁡ ϕ sinh ⁡ 2 g D m , n , 1 , 0 , 1 , 0 exp ⁡ ( w 4 ) ] ,
⟨ N 2 ⟩ = N m , n 2 { cosh 4 ⁡ g D m , n , 2 , 2 , 0 , 0 exp ⁡ ( w 4 ) + sinh 2 ⁡ 2 g [ D m , n , 1 , 1 , 1 , 1 exp ⁡ ( w 4 ) + 1 2 cos ⁡ 2 ϕ D m , n , 2 , 0 , 2 , 0 exp ⁡ ( w 4 ) ] + sinh 4 ⁡ g D m , n , 0 , 0 , 2 , 2 exp ⁡ ( w 4 ) + 2 cos ⁡ ϕ sinh ⁡ 2 g [ cosh 2 ⁡ g D m , n , 2 , 1 , 1 , 0 exp ⁡ ( w 4 ) + sinh 2 ⁡ g D m , n , 1 , 0 , 2 , 1 exp ⁡ ( w 4 ) ] } + ⟨ N ⟩ ,
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