1. Introduction

Optical interferometer, which lies at the heart of metrology, is one of the most commonly used devices measuring tiny changes of physical parameters, via phase sensing. For the traditional interferometer with classical resources being used only, its precision of phase measurement is bounded by the classical limit that scales as $1/\sqrt {n_{ps}}$, where $n_{ps}$ is the photon number sensing the phase [1,2]. This limit is also known as the standard quantum limit (SQL) which follows from the Heisenberg uncertainty principle [3].

Quantum metrology aims to utilize unique properties of quantum system to enhance the measurement precision beyond what can be achieved with only classical resources [4]. Quantum resources provide the possibility of beating the SQL. Take Mach-Zehnder interferometer (MZI) as an example, which is one of the most studied interferometric topologies. In recent decades, various schemes have been proposed to enhance the phase sensitivity of the traditional MZI [5,6]. For instances, it has been demonstrated that nonclassical states, such as N00N state [7–11] and squeezed state [12–14], is able to make the traditional MZI go below the SQL. The phase sensitivity of the traditional MZI can also be enhanced by employing the external resources [15–17]. In addition, in order to better estimate the phase information, different detection strategies based on statistical approaches, such as Bayesian analysis [18,19] and parity detection [20–22], have been proposed to beat the SQL.

Another possibility for achieving quantum-enhanced phase sensitivity is SU(1,1) interferometer (SUI) [23,24], which splits and mixes beams using nonlinear transformations. The SUI can be realized by replacing the linear beam splitters (BSs) in the MZI with parametric amplifiers, which was first proposed by Yurke et al. in 1986 [25]. The first parametric amplifier inside the SUI serves as entanglement resources, making the two modes inside the SUI entangled. The second one disentangles the modes in the two arms of the interferometer. Due to the entangled state generated from the first parametric amplifier, noiseless amplification occurs at quantum destructive interference of the SUI [26], leading to the precision enhancement in the phase estimation. The SUI is able to beat the SQL, and even approach the Heisenberg Limit (HL) with only vacuum noise entering the input ports of the SUI. Here, the HL, characterized by $1/n_{ps}$, is the fundamental quantum limit [27] in the phase estimation in the case that $n_{ps}$ is large, although it can be broken at small $n_{ps}$ [22]. In recent years, considerable attentions have been attracted in studying the SUI [28]. The SUI was first experimentally realized by Jing et al. using four-wave mixing in hot atomic assembles [29]. Signal-to-noise ratio [30] and phase sensitivity [31] enhancements of the SUI have been demonstrated in the experiment. In addition to the standard form, SUI with different configurations have also been proposed [32–40]. Although the SUI exhibits its unique advantages in precision phase measurement, its performance yields to the loss of the system. Especially, the phase sensitivity of the SUI is extremely sensitive to the internal loss of the interferometer [41]. It is this limiting factor that makes the SUI impossible to reach the HL, and even perform below the SQL [42]. Therefore, the internal loss of the SUI hinders its practical application. How to enhance the robustness of the SUI against the internal loss is still an open question.

On the other hand, entanglement distillation [43,44] is one of the key techniques in quantum communication and quantum computation, which was first proposed by Bennett et al. in 1996 [45]. In general, entanglement distillation engineers the quantum state by subtracting or adding photons from/to a Gaussian state [46]. Such photon level operations (PLOs), also known as non-Gaussian operations [47–49], allow one to probabilistically distill a smaller sub-ensemble of a weakly entangled state that owns more strength of quantum entanglement [50]. Entanglement distillation is an useful tool for overcoming the inevitable decoherence of entanglement system, and therefore makes it possible to establish strong entanglement between two distant parties. Besides PLOs [51–54], entanglement distillation protocol can also be implemented by noiseless linear amplification [55–57].

In this paper, we propose to employ PLOs to enhance the robustness of the SUI against the internal loss. We consider two types of PLOs, including photon subtraction scheme (PSS) and photon addition scheme (PAS). We consider a coherent state $|\alpha \rangle$ as the input of the SUI, which is the closest analog to a classical light field and exhibits a Poisson photon number distribution with an average photon number $|\alpha |^{2}$. We find that the phase sensitivity of the SUI can be enhanced by implementing either PSS or PAS on the two arms inside the interferometer. Such enhancement becomes especially obvious when the amplitude $|\alpha |$ of the input coherent state is small. In particular for the vacuum seeded case, i.e., for $|\alpha |=0$, the HL can be even beaten by the SUI with PLOs. Furthermore, we show that the SUI with PLOs owns better robustness against the internal loss of the interferometer.

2. SUI with PLOs

Before introducing PLOs, we first take a brief review of the standard SUI. The diagrammatic sketch of an SUI is depicted in Fig. 1, excluding the dotted box. The SUI has a MZI-like topology. It replaces linear BSs of the MZI with parametric processes that can generate quantum entangled photon pairs. The evolution of the state performed by the SUI can be described as follows. For two radiation beams prepared in a factorized state $|\Psi _{\textrm {IN}}\rangle =|\psi _{a}\rangle \otimes |\psi _{b}\rangle$, we couple them via a two-mode squeezing operation defined as $\hat {U}_{S1}=\textrm {exp}[{r}_{s}({\hat {a}}\, {\hat {b}}-{\hat {a}}^{\dagger }\,{\hat {b}}^{\dagger })/2]$, where $\hat {a}$ ($\hat {a}^{\dagger }$) and $\hat {b}$ ($\hat {b}^{\dagger }$) are the annihilation (creation) operators of the corresponding modes, and $r_{s}$ quantifies the two-mode squeezing [58]. In the experiment, the two-mode squeezing operation can be realized by a nondegenerate optical parametric amplifier [59–62]. Following the first parametric process, one of the two arms (mode $a$ in Fig. 1 for example) inside the SUI undergoes a phase shift $\hat {U}_{\textrm {ph}}=\hat {U}_{\phi }\otimes \mathbb {I}$, where $\hat {U}_{\phi }=\textrm {exp}\,(-i\phi {\hat {a}}^{\dagger }{\hat {a}})$ and $\mathbb {I}$ is an identity matrix. Finally, both the two beams are coupled onto the second parametric process $\hat {U}_{S2}$, which fulfills the wave superposition in both the two output ports of the SUI and generates the corresponding interference fringes. The output state of the standard SUI can be expressed as $|\Psi ^{s}_\textrm{OUT}\rangle =\hat {U}_{S2}\hat {U}_{\textrm {ph}}\hat {U}_{S1}|\Psi _{\textrm {IN}}\rangle$.

 

Fig. 1. Diagrammatic sketch of the SUI with PLOs. The two dotted boxes in the bottom show the two types of PLOs: PSS (left) and PAS (right), respectively.

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The above model of the SUI is based on the ideal case without the consideration of loss of the system. It has been demonstrated [25] that such a lossless SUI is able to beat the SQL, and even reach the HL (see the brown dashed line in Fig. 3). However, the performance of the SUI in the practical implementation can be extremely sensitive to the loss of the system, both in terms of the internal and external loss. It has been shown that the SUI has good robustness against the external loss of the system, while is not immune to the internal loss [42]. To explain this point, the principle of the quantum enhancement of the SUI should be clarified. For an SUI, the first parametric process $\hat {U}_{S1}$ serves as an entanglement resource. It can not only amplify the input beams but also establish strong quantum correlations between them. Such quantum correlated beams makes it possible to realize the noiseless amplification at the quantum destructive interference [26] as we couple them onto the second parametric process $\hat {U}_{S2}$, which leads to the phase sensitivity enhancement for the SUI. All the quantum advantage of the SUI is due to quantum correlations introduced by the first parametric process. The internal loss of the SUI undoubtedly degrades such quantum correlations, and thus makes the performance of the SUI become worse. As to the external loss such as detection loss, it has almost no effect on the phase sensitivity of the SUI [42]. In this paper, we only consider the internal loss of the SUI. We assume that both the two modes inside the SUI are attenuated by the loss channels $\hat {U}_{L}=\hat {U}_{La}\otimes \hat {U}_{Lb}$. The attenuation processes imposed onto both the two modes can be characterized by linear BS transformations: $\hat {U}_{La}=\rm {exp}[\theta _{{a}}(\hat {{a}}^{\dagger}\,\hat {\nu }_{{a}}-\hat {{a}}\,\hat {\nu }^{\dagger}_{{a}})]$ and $\hat {U}_{Lb}=\textrm{exp}[\theta _{{b}}(\hat {{b}}^{\dagger}\,\hat {\nu }_{{b}}-\hat {{b}}\,\hat {\nu }^{\dagger}_{{b}})]$, where $\hat {\nu }_{a}$ and $\hat {\nu }_{b}$ are the vacuum modes. The transmissivity $\tau ^{2}_{k}$ ($k=a, b$) of the beam splitter is determined by $\theta _{k}$: $\tau ^{2}_{k}=\cos ^{2}\theta _{k}\in [0,1]$. The lossless case corresponds to that $\tau ^{2}_{k}=1$ [58]. The output state of the standard SUI with the internal loss considered is given by

To enhance the robustness of the SUI against the internal loss of the interferometer, we can employ PLOs to increase the quantum correlations between the two modes inside the interferometer. Two types of PLOs are considered, including PSS ($\hat {U}_{PLO\rm {i}}$) and PAS ($\hat {U}_{PLO\rm {i}i}$) as shown in the dotted boxes at the bottom of Fig. 1. PSS annihilates one photon from each mode inside the SUI, i.e., $\hat {U}_{PLO\rm {i}}=\hat {U}_{PSS}=\hat {a}\otimes \hat {b}$. On the contrary, PAS adds one photon to each mode of the SUI, i.e., $\hat {U}_{PLO\rm {i}i}=\hat {U}_{PAS}=\hat {a}^{\dagger}\otimes \hat {b}^{\dagger}$. Thus, the output state of the SUI with PSS (PAS) can be expressed as

In the experiment, the PSS can be realized using linear BSs with high-transmissivity. Each photon detection in the reflection port of the BS heralds a local success of single photon subtraction attempt [51]. The probabilistic PSS is fulfilled on the condition that the single photon subtraction simultaneously occurs at both modes inside the SUI. For the PAS, it can be implemented by low-amplitude spontaneous parametric down-conversion (SPDC) [47]. For a state entering the signal port of the SPDC, creation operators acts on this state on the condition that a single photon is detected in the idler port of the SPDC. The probabilistic PSS is fulfilled if the single photon creation simultaneously occurs at both modes inside the SUI. PLOs are based on probabilistic protocols. It is the success of photon detection that triggers the operation of the SUI with PLOs. When the PLOs fail, the phase-sensing photon numbers will not be varied, and therefore the SUI with PLOs reduces to a standard SUI.

Note that all the operations performed by the standard SUI including $\hat {U}_{S1}$, $\hat {U}_{S2}$, $\hat {U}_\textrm{ph}$ and $\hat {U}_{L}$ are Gaussian unitary. On the other hand, the PLOs are non-Gaussian operations, which transforms the input Gaussian state to be non-Gaussian. Such degaussification is able to increase the non-local unfactorizable correlations of the input state. It allows one to probabilistically distill a smaller sub-ensemble of a weakly entangled state that owns more strength of quantum entanglement [50], and therefore strengthen the nonclassicality of the input state. In this paper, we employ the PLOs to increase the quantum entanglement between the two modes inside the SUI. Such entanglement increase will lead to the quantum enhancement for the SUI in precision phase measurement.

3. Results and discussions

In general, to study how well an interferometer can estimate the phase difference, one need to measure its phase sensitivity. For measuring an observable $\hat {O}$, the phase sensitivity $\Delta \phi$ of an interferometer can be defined by error-propagation analysis:

In this paper, we consider the intensity sum detection [42,63] as the measurement scheme, that is,

For the initial input state, we assume that the SUI is seeded by a coherent state $|\alpha \rangle$ with amplitude $|\alpha |$, while the other port of interferometer is in vacuum, i.e., $|\Psi _\textrm{IN}\rangle =|\psi _{a}\rangle \otimes |\psi _{b}\rangle =|\alpha \rangle \otimes |0\rangle$. By using Eqs. (2)–(4), the phase sensitivity of the SUI with PLOs can be calculated (see the Appendix for detail).

3.1 Vacuum seeded

We first focus on the case that the SUI is vacuum-seeded, i.e., $|\alpha |^{2}=0$. In this special case, it has been demonstrated that the standard SUI with no loss involved is able to reach the HL [24,25,42]. Here, we show that better performance can be achieved by the SUI with the help of PLOs, when the interferometer is vacuum-seeded. Figure 2(a) shows the behavior of the phase sensitivity $\Delta \phi$ of the vacuum-seeded SUI with PLOs as a function of $\phi$ in the lossless case, where the two-mode squeezing degrees of the parametric amplifiers inside the SUIs are fixed to be the same for both the PSS (red solid line) and PAS (blue dash-dotted line). For comparison, we also plot the phase sensitivity of the standard SUI in the brown dashed line. As shown in Fig. 2(a), the phase sensitivity of the SUI is dependent on $\phi$. There exists an optimal working phase point $\phi _\textrm{opt}$ where the best phase sensitivity of the SUI can be achieved. It is clear that the phase sensitivity of the SUI can be further enhanced, when either PSS or PAS is performed on the two modes inside the interferometer. Such enhancement can be explained as follows. First, it should be noted that the performance of the SUI is dependent on the quantum entanglement between the two modes inside the interferometer. The more entanglement the two modes inside share, the better the quantum noise cancellation [64] occurs at the output ports of the SUI, which leads to the precision enhancement in the phase estimation. Along this line, a common method to enhance the phase sensitivity of the SUI is to increase the gains of the parametric amplifiers (i.e., the strength of the two-mode squeezers) of the SUI [65]. On the other side, entanglement distillation scheme (i.e., PSS and PAS) allows one to probabilistically distill a smaller sub-ensemble of a weakly entangled state that owns more strength of quantum entanglement [66]. Therefore, it is expectable that the performance of the SUI can be enhanced as either PSS or PAS is performed on the two arms inside the interferometer. To compare the SUIs with the SQL and HL, we fix the phase-sensing photon numbers $n_{ps}$ inside the interferometer to be the same for the standard SUI and the SUIs with PLOs. Our results are shown in Fig. 2(b), where the SQL ($1/\sqrt {n_{ps}}$) and HL ($1/n_{ps}$) are marked by the green squares and black circles, respectively. It is clear that the SUI with PSS is advantageous over that with PAS. This is because PLOs will vary $n_{ps}$ of the SUI. PSS subtracts two photons from the arms of the interferometer, while PAS adds two. To achieve equal $n_{ps}$, the two-mode squeezing degrees of the parametric amplifiers for the SUI with PSS is larger than that of the SUI with PAS, i.e., $r_\textrm{i}>r_\textrm{ii}$. This leads to the fact that the SUI with PSS is advantageous over that with PAS, for a given $n_{ps}$.

 

Fig. 2. Phase sensitivity of the vacuum-seeded SUI with PLOs versus $\phi$ in the lossless case, (a), we fix the SUIs with given two-mode squeezing degrees of the parametric amplifiers inside the interferometers. (b), we fix the SUIs with given phase-sensing photon numbers. The parameters in plotting (a) and (b) are as follows: {$|\alpha |^{2}=0, \tau ^{2}=1, \sinh ^{2}{r_{s}}=\sinh ^{2}{r_\textrm{i}}=\sinh ^{2}{r_\textrm{ii}}=2$} and {$|\alpha |^{2}=0$, $n_{ps}=4$, $\tau ^{2}=1$, $\sinh ^{2}{r_{s}}=n_{ps}/2$, $\sinh ^{2}{r_\textrm{i}}=n_{ps}/2+1$ and $\sinh ^{2}{r_\textrm{ii}}=n_{ps}/2-1$}.

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To show the phase sensitivity scaling, we change the phase-sensing photon number $n_{ps}$ of the SUI. For each $n_{ps}$, the best phase sensitivity of the interferometer is taken as the y-value with the optimal working phase point $\phi _\textrm{opt}$ being chosen. Figure 3 shows the phase sensitivity scaling of the vacuum-seeded SUI with PSS and PAS in the lossless case, which is plotted in the red solid and blue dash-dotted lines on a log$_{10}$-log$_{10}$ scale, respectively. The behavior of $\phi _\textrm{opt}$ as a function of log$_{10}n_{ps}$ is shown in the inset of Fig. 3. The corresponding results of the standard SUI are also plotted in Fig. 3 with brown dashed lines. The SQL and HL are marked by the green squares and black circles, respectively. As shown in Fig. 3, the phase sensitivity of the standard SUI approaches the HL with the increasing of $n_{ps}$. For the SUI with PSS, our results show that its phase sensitivity is better than that of the standard SUI, and is able to beat the HL, for any $n_{ps}>1$. In addition, we note that the performance of the SUI with PAS is worse than the standard SUI, when $n_{ps}$ is small. This is because PAS adds two extra photons to the arms of the standard SUI. When $n_{ps}$ is small, it mainly comes from the PAS, and therefore the entangled photons generated from the parametric amplifier become weaken. An extreme case is that $n_{ps}\leq 2$, which invalidates the parametric amplifiers of the SUI. The performance of the SUI with PAS becomes better with the increasing of $n_{ps}$. When $n_{ps}$ is large enough, the precision enhancements introduced by PSS and PAS tend to be the same.

 

Fig. 3. Phase sensitivity scaling of the vacuum-seeded SUI with PLOs in the lossless case. The parameters in plotting the brown, red solid and blue dash-dotted lines are as follows: {$|\alpha |^{2}=0$, $\sinh ^{2}{r_{s}}=n_{ps}/2$, $\phi =\phi _\textrm{opt}$, $\tau ^{2}=1$}, {$|\alpha |^{2}=0$, $\sinh ^{2}{r_\textrm{i}}=n_{ps}/2+1$, $\phi =\phi _\textrm{opt}$, $\tau ^{2}=1$} and {$|\alpha |^{2}=0$, $\sinh ^{2}{r_\textrm{ii}}=n_{ps}/2-1$, $\phi =\phi _\textrm{opt}$, $\tau ^{2}=1$}.

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The above results are based on the lossless case. In the following, we take the internal loss of the SUI into consideration, which can largely affect the performance of the standard SUI [42]. The corresponding results are shown in Fig. 4, where $\tau ^{2}$ is assumed to be $0.8$ (corresponding to setting the attenuation coefficients of the loss channels inside the SUI to be $0.2$). The red solid and blue dash-dotted lines are the phase sensitivity scaling of the vacuum-seeded SUI with PSS and PAS, the brown dashed line is that of the standard SUI, the green squares are the SQL, and black circles are the HL. The behavior of $\phi _\textrm{opt}$ as a function of log$_{10}n_{ps}$ is shown in the inset of Fig. 3. By comparing the brown dashed lines in Fig. 3 and Fig. 4, it is clear that the internal loss degrades the phase sensitivity of the standard SUI, which makes it impossible to reach the HL. However, the PLOs can enhance the robustness of the standard SUI against the internal loss. Especially for the PSS as shown by the red solid line in Fig. 4, it can be found that the phase sensitivity of the SUI with PSS is better than that of the standard SUI, for any $n_{ps}>1$. More interestingly, the SUI with PSS can still beat the HL in the lossy case, when $n_{ps}$ is small. For the SUI with PAS as shown by the blue dash-dotted line in Fig. 4, its phase sensitivity becomes advantageous over that of the standard SUI with the increasing of $n_{ps}$.

 

Fig. 4. Phase sensitivity scaling of the vacuum-seeded SUI with PLOs in the lossy case. The parameters in plotting the brown, red solid and blue dash-dotted lines are as follows: {$|\alpha |^{2}=0$, $\sinh ^{2}{r_{s}}=n_{ps}/2$, $\phi =\phi _\textrm{opt}$, $\tau ^{2}=0.8$}, {$|\alpha |^{2}=0$, $\sinh ^{2}{r_\textrm{i}}=n_{ps}/2+1$, $\phi =\phi _\textrm{opt}$, $\tau ^{2}=0.8$} and {$|\alpha |^{2}=0$, $\sinh ^{2}{r_\textrm{ii}}=n_{ps}/2-1$, $\phi =\phi _\textrm{opt}$, $\tau ^{2}=0.8$}.

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3.2 Bright seeded

The above discussions are based on the vacuum-seeded SUI. Its phase-sensing photons are all from the SPDC (i.e., two-mode squeezing operation) occurring in the first parametric amplifier of the SUI. Such phase-sensing photons generated from quantum resources are able to enormously enhance the phase sensitivity of the SUI. However, the degree of the two-mode squeezing is limited by the imperfect factors in the experiment, such as saturation effect of the parametric amplifier [67–69] and unavoidable loss. To make the SUI achieve a high absolute phase sensitivity, bright-seeded scheme is commonly used, where $|\alpha |^{2}\neq 0$.

In the following, we study the phase sensitivity of the bright-seeded SUI with PLOs in the lossless case, which can be calculated using Eqs. (11) (15) in the Appendix. We first assume that the two-mode squeezing degrees of the parametric amplifiers inside the bright-seeded SUIs are the same for both the PSS and PAS, $r_\textrm{i}=r_\textrm{ii}$. Then for different amplitudes of the input coherent states $|\alpha |^{2}=1, 10,$ and $10^{2}$, the phase sensitivity $\Delta \phi$ of the bright-seeded SUI with PLOs versus $\phi$ is shown in Fig. 5(a), Fig. 5(b), and Fig. 5(c), respectively. The red solid and blue dash-dotted lines are the SUI with PSS and PAS, the brown dashed line is the standard SUI. By comparing the three subfigures (a)-(c) in Fig. 5, it is clear that the SUI with PLOs is advantageous over the standard SUI in the bright-seeded case. Such advantage is especially obvious at small $|\alpha |^{2}$. The enhancement effect of the PLOs on the phase sensitivity of the SUI fades, with the increasing of $|\alpha |^{2}$. The phase sensitivity of the SUI with PLOs is almost the same with that of the standard SUI, when $|\alpha |^{2}$ is large. This phenomenon can be explained as follows, qualitatively. First, the phase-sensing photons $n_{ps}$ inside the standard bright-seeded SUI depends on two factors: the amplitude $|\alpha |$ of the input coherent state and the degree of the two-mode squeezing of the first parametric amplifier of the SUI. The former factor determines the classical resources employed by the interferometer, while the latter one quantifies the quantum enhancement of the SUI. For a given degree of the two-mode squeezing, the phase-sensing photons are mainly dominated by the classical resources with the increasing of $|\alpha |^{2}$, leading to the approach of the standard bright-seeded SUI to a classical MZI, and almost no quantum enhancement can be achieved. As to the PLOs such as PSS and PAS, although they are able to increase the quantum resources of the SUI via introducing non-local unfactorizable correlations to the two modes inside the interferometer, the phase-sensing photons $n_{ps}$ enhancement caused by such quantum resources increase is negligible, when $n_{ps}$ is mainly dominated by the classical resources (i.e., when $|\alpha |^{2}$ is large). Therefore, the PLOs can hardly enhance the phase sensitivity of the standard bright-seeded SUI at large $|\alpha |^{2}$. To compare the bright-seeded SUIs with the SQL and HL, we fix the phase-sensing photon numbers $n_{ps}$ inside the interferometer to be the same for the standard SUI and the SUIs with PLOs. For different $n_{ps}$, our results are shown in Figs. 5(d)–5(f), where the SQL ($1/\sqrt {n_{ps}}$) and HL ($1/n_{ps}$) are marked by the green squares and black circles, respectively. It can be found that the bright-seeded SUI with PSS is able to beat the HL, when $n_{ps}$ is small. Different from the PSS, the bright-seeded SUI with PAS is unable to beat the HL in the case that $n_{ps}$ is small, as shown by the blue dash-dotted line in Fig. 5(d). This can be explained by the same reason as we have discussed in the vacuum-seeded case.

 

Fig. 5. Phase sensitivity of the bright-seeded SUI with PLOs versus phase in the lossless case. In each subfigure of (a)-(c), the two-mode squeezing degrees of the parametric amplifiers are assumed to be the same for the standard SUI and SUIs with PLOs, i.e., $r=r_{s}=r_\textrm{i}=r_\textrm{ii}$. In each subfigure of (d)-(f), the phase-sensing photon numbers inside the SUIs are assumed to be the same, and therefore the two-mode squeezing degrees of the parametric amplifiers inside the SUIs are $\sinh ^{2}{r_{s}}=(n_{ps}-|\alpha |^{2})/(2|\alpha |^{2}+2)$, $\sinh ^{2}{r_\textrm{i}}=(n_{ps}-|\alpha |^{2}+2)/(2|\alpha |^{2}+2)$ and $\sinh ^{2}{r_\textrm{ii}}=(n_{ps}-|\alpha |^{2}-2)/(2|\alpha |^{2}+2)$, respectively. The parameters in plotting (a), (b), (c), (d), (e) and (f) are as follows: $\{|\alpha |^{2}=1, \tau ^{2}=1, \sinh ^{2}{r}=3\}$, $\{|\alpha |^{2}=10, \tau ^{2}=1, \sinh ^{2}{r}=3\}$, $\{|\alpha |^{2}=10^{2}, \tau ^{2}=1, \sinh ^{2}{r}=3\}$, $\{|\alpha |^{2}=1, \tau ^{2}=1, n_{ps}=4\}$, $\{|\alpha |^{2}=10, \tau ^{2}=1, n_{ps}=4\}$ and $\{|\alpha |^{2}=10^{2}, \tau ^{2}=1, n_{ps}=4\}$.

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Figure 6 shows the phase sensitivity scaling of the bright-seeded SUI with PSS and PAS in the lossless case, which are plotted in the red solid and blue dash-dotted lines, respectively. The brown dashed line is the phase sensitivity scaling of standard SUI. The green squares are the SQL and black circles are the HL. The inset in Fig. 6 shows the behavior of $\phi _\textrm{opt}$ as a function of log$_{10}n_{ps}$. To make a fair comparison, we assume that the two-mode squeezing degrees of the parametric amplifiers are the same for the SUIs, and we vary the amplitudes of the input coherent states to manipulate $n_{ps}$. As shown in Fig. 6, the phase sensitivity of the bright-seeded SUI with either PSS or PAS is better than that of the standard SUI. Such phase sensitivity enhancement is especially obvious at small $n_{ps}$. With the increasing of $n_{ps}$, the phase sensitivity of the SUI with PLOs becomes worse and approaches to the SQL. This trend is reasonable and has been explained above in detail.

 

Fig. 6. Phase sensitivity scaling of the bright-seeded SUI with PLOs in the lossless case. Here, we fix the SUIs with given two-mode squeezing degrees of the parametric amplifiers inside the interferometers, i.e., $r=r_{s}=r_\textrm{i}=r_\textrm{ii}$. The parameters in plotting the brown dashed, red solid and blue dash-dotted lines are as follows: {$\sinh ^{2}{r}=1$, $|\alpha |^{2}=(n_{ps}-2\sinh ^{2}{r})/(2\sinh ^{2}{r}+1)$, $\phi =\phi _\textrm{opt}$, and $\tau ^{2}=1$}, {$\sinh ^{2}{r}=1$, $|\alpha |^{2}=(n_{ps}-2\sinh ^{2}{r}+2)/(2\sinh ^{2}{r}+1)$, $\phi =\phi _\textrm{opt}$, and $\tau ^{2}=1$} and {$\sinh ^{2}{r}=1$, $|\alpha |^{2}=(n_{ps}-2\sinh ^{2}{r}-2)/(2\sinh ^{2}{r}+1)$, $\phi =\phi _\textrm{opt}$, and $\tau ^{2}=1$}.

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In what follows, we take the internal loss into our consideration. Figure 7 shows the phase sensitivity scaling of the bright-seeded SUI with PLOs in the lossy case. The red solid and blue dash-dotted lines are the phase sensitivity scaling of the bright-seeded SUI with PSS and PAS, the brown dashed line is that of the standard SUI, the green squares are the SQL, and black circles are the HL. The behavior of $\phi _\textrm{opt}$ as a function of log$_{10}n_{ps}$ is shown in the inset of Fig. 7. By comparing the corresponding lines in Fig. 6 and Fig. 7, it can be found that the phase sensitivity of the bright-seeded SUI with PLOs is degraded by the internal loss of the interferometer. Nevertheless, both PSS and PAS are able to increase the robustness of the bright-seeded SUI against the internal loss of the interferometer, compared with that of the standard SUI.

 

Fig. 7. Phase sensitivity scaling of the bright-seeded SUI with PLOs in the lossy case. Here, we fix the SUIs with given two-mode squeezing degrees of the parametric amplifiers inside the interferometers, i.e., $r=r_{s}=r_\textrm{i}=r_\textrm{ii}$. The parameters in plotting the brown dashed, red solid and blue dash-dotted lines are as follows: {$\sinh ^{2}{r}=1$, $|\alpha |^{2}=(n_{ps}-2\sinh ^{2}{r})/(2\sinh ^{2}{r}+1)$, $\phi =\phi _\textrm{opt}$, and $\tau ^{2}=0.8$}, {$\sinh ^{2}{r}=1$, $|\alpha |^{2}=(n_{ps}-2\sinh ^{2}{r}+2)/(2\sinh ^{2}{r}+1)$, $\phi =\phi _\textrm{opt}$, and $\tau ^{2}=0.8$} and {$\sinh ^{2}{r}=1$, $|\alpha |^{2}=(n_{ps}-2\sinh ^{2}{r}-2)/(2\sinh ^{2}{r}+1)$, $\phi =\phi _\textrm{opt}$, and $\tau ^{2}=0.8$}.

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4. Conclusion

SUI is a new type of nonlinear quantum interferometer, which exhibits unique advantages in the phase estimation. However, the performance of the SUI is extremely dependent on the internal loss of the interferometer. In this paper, we propose to employ PLOs to enhance the phase sensitivity of the SUI. Such enhancement is due to the fact that PLOs increase the entanglement shared between the two modes inside the SUI. Two types of PLOs are considered including PSS and PAS. For a standard SUI seeded by a coherent light, we theoretically demonstrate that its phase sensitivity can be further enhanced by performing either PSS or PAS on both the two modes inside the interferometer. More interestingly, we find that both PSS and PAS are able to enhance the robustness of the SUI against the internal loss of the interferometer. The phase sensitivity enhancement introduced by the PLOs is especially obvious when the amplitude of the input coherent state is small. Such an SUI with PLOs may provide new ideas for the application of the SUI in quantum metrology.