Improved phase sensitivity of an SU(1,1) interferometer based on the internal single-path local squeezing operation

Abstract

Compared with passive interferometers, SU(1,1) interferometers demonstrate superior phase sensitivity due to the incorporation of nonlinear elements that enhance their ability to detect phase shifts. Nevertheless, the measurement precision of these interferometers is considerably impacted by photon losses, particularly internal losses, thereby restricting the overall accuracy of measurements. Addressing these issues is essential to fully realize the advantages of SU(1,1) interferometers in practical applications. Among the available resources in quantum metrology, squeezing stands out as one of the most practical and efficient approaches. We propose a theoretical scheme to improve the precision of phase measurement using homodyne detection by implementing the single-path local squeezing operation (LSO) inside the SU(1,1) interferometer, with the coherent state and the vacuum state as the input states. We not only analyze the effects of the single-path LSO scheme on the phase sensitivity and the quantum Fisher information (QFI) under both ideal and photon-loss cases but also compare the impact of different squeezing parameters r on the system performance. Our findings reveal that the internal single-path LSO scheme can significantly enhance the phase sensitivity and QFI by strengthening intramode correlations while weakening intermode correlations, thereby effectively improving the robustness of the SU(1,1) interferometer against photon losses.

Graphical abstract

Keywords

phase sensitivity / SU(1,1) interferometer / internal single-path local squeezing

1 Introduction

Quantum precision measurement technology is a method that utilizes quantum resources and effects to achieve measurement accuracy beyond traditional methods. It integrates multidisciplinary knowledge such as atomic physics, physical optics, electronic technology, and control technology, which leverages principles of quantum mechanics, particularly the superposition and entanglement properties of quantum states, to accomplish highly accurate measurements. This technology has a wide range of applications [1–6], including high-precision optical frequency standards and time-frequency transfer, quantum gyroscopes, atomic gravimeters, and other quantum navigation technologies, as well as quantum radar, trace atom tracking, and weak magnetic field detection in quantum sensitive detection technologies. These technologies have significant application potential in fields such as inertial navigation, next-generation time referencing, stealth target identification, global terrain mapping, medical testing [7], and fundamental physics research [8].

Phase estimation is a crucial method for precision measurement, offering a way to estimate many physical quantities that cannot be directly measured through conventional methods. Consequently, extensive research and significant advancements have been made in the field of optical interference measurement. To satisfy the demand for high precision, various optical interferometers have been proposed and developed. One of the most practical interferometers is the Mach−Zehnder interferometer (MZI), whose phase sensitivity is limited by the standard quantum-noise limit (SQL) \mathrm{\Delta }{\varphi }_{SQL}=1/\sqrt{N} (N is the average number of photons within the interferometer), with solely classical resources as the input of the MZI [9]. In recent decades, various schemes have been proposed to enhance the phase sensitivity of the traditional MZI [10]. It has been shown that using quantum states as input can enable the traditional MZI to surpass the SQL. For instance, NOON states [11], twin Fock states [12], squeezed states [13, 14], and photon-catalyzed squeezed vacuum states [15] can achieve or even exceed the Heisenberg limit (HL) \mathrm{\Delta }{\varphi }_{HL} =1/N [16].

In 1986, Yurke et al. [17] introduced the SU(1,1) interferometer, which replaced traditional beam splitters (BSs) with optical parametric amplifiers (OPAs). In the SU(1,1) interferometer comprised of two OPAs, the first OPA serves the dual purpose of generating entangled resources and suppressing amplified noise. The subsequent use of the second OPA enhances the signal, thereby providing a viable pathway to achieving higher precision in phase estimation [18]. By utilizing entangled photon states, the SU(1,1) interferometer can surpass the SQL, enabling greater precision. This technique has revolutionized phase estimation, and has become a vital tool in quantum precision measurements. In 2011, Jing et al. [19] successfully implemented this interferometer experimentally. For instance, Hudelist et al. [20] demonstrated that the gain effect of OPAs enables the SU(1,1) interferometer to exhibit higher sensitivity compared to traditional linear interferometers. This has led to a growing interest in the exploration of the SU(1,1) interferometer [21, 22]. Apart from the standard form, various configurations of SU(1,1) interferometer have also been proposed [23–28].

As previously mentioned, although SU(1,1) interferometer is highly valuable for precision measurement [29, 30], its precision is still affected by dissipation, particularly photon losses inside the interferometer [31, 32]. Consequently, to further enhance precision, non-Gaussian operations can effectively mitigate internal dissipation. Most theoretical [33–36] and experimental [37–39] studies have indicated that non-Gaussian operations can effectively enhance the nonclassicality and entanglement degrees of quantum states, thereby enhancing their potential in quantum information processing [40].

It is evident that the use of non-Gaussian operations indeed improves the estimation performance of optical interferometers, but a high implementation cost is associated with this approach. To address the aforementioned problem, Gaussian operations such as the LSO [41–43] and the local displacement operation (LDO) [44, 45] have been regarded as promising strategies. In Ref. [45], Ye et al. investigated phase sensitivity and QFI based on homodyne detection in the presence and absence of photon losses through displacement-assisted SU(1,1) [DSU(1,1)], which involves two LDOs within the SU(1,1) interferometer. In this DSU(1,1) interferometer, the introduced LDOs have improved phase sensitivity and QFI even under realistic conditions. It is important to emphasize that the LSO has been proven crucial not only in quantum metrology [41] but also in quantum key distribution [42] and entanglement distillation [43]. These applications have demonstrated the versatility and significance of the LSO in advancing various quantum technologies, highlighting its role in enhancing precision measurements and ensuring secure communication protocols. In Ref. [41], Sahota and James proposed an innovative quantum-enhanced phase estimation scheme that applies the LSO to both paths of the MZI. Their work illustrated the potential for the LSO scheme to enhance the performance of interferometric devices, paving the way for advancements in quantum metrology and measurement techniques.

Recent advancements in the experimental realization of squeezing operations have significantly contributed to the field of quantum metrology, particularly in enhancing the sensitivity of interferometers beyond the SQL. For instance, Purdy et al. [46] experimentally demonstrated strong and continuous optomechanical squeezing of 1.7\pm 0.2 dB below the shot-noise level by exploiting the quantum interaction between laser light and a membrane mechanical resonator in an optical cavity. Meanwhile, Ono et al. [47] explored optical spin-squeezing, revealing a quantum advantage of 1.58 over the shot-noise limit for five-photon events, suggesting enhanced performance for quantum metrology applications, despite not achieving sub-shot-noise precision due to experimental imperfections. Furthermore, Zuo et al. [48] proposed and experimentally demonstrated a compact quantum interferometer that achieves a sensitivity improvement of 4.86\pm 0.24 dB beyond the SQL by using squeezed states generated within the MZI for phase sensing, resulting in a minimum detectable phase smaller than that of all present interferometers under the same phase-sensing intensity. In 2023, Kalinin et al. [49] reported the first experimental demonstration of phase sensitivity enhancement in an interferometer using Kerr squeezing, addressing the challenge of the cumbersome tilting of the squeezed ellipse in phase space. The experimental realization of squeezing operations has significantly advanced, enabling the development of highly sensitive and robust interferometric schemes that surpass classical limits. These advancements hold promise for a wide range of applications, including gravitational wave detection, quantum communication, and precision sensing. Thus, can the single-path LSO scheme implemented inside the SU(1,1) interferometer similarly enhance the QFI and phase sensitivity? Can this approach enhance the robustness of the interferometer against internal and external photon losses? Compared to Ref. [45], when using simple input states, if the single-path LSO scheme can achieve improvements, then from the perspective of quantum resource, this scheme is more advantageous.

In this paper, we propose a theoretical scheme to improve the precision of phase measurement using homodyne detection by implementing the single-path LSO scheme inside the SU(1,1) interferometer, with the coherent state and vacuum state as the input states. We also perform a comparative analysis about the effects of internal and external photon losses on interferometer performance. This paper is arranged as follows. Section 2 outlines the theoretical model of the SU(1,1) interferometer based on the internal single-path LSO scheme. Section 3 delves into phase sensitivity, encompassing both ideal and photon-loss cases. Section 4 centers on the QFI. Section 5 mainly discusses the intramode correlation, intermode correlation and quantum Fisher information matrix. Finally, Section 6 provides a summary.

2 Model

This section begins with the standard SU(1,1) interferometer, as shown in Fig.1(a). The SU(1,1) interferometer, which usually consists of two OPAs and a linear phase shifter, is one of the most commonly used interferometers in quantum metrology studies. The first OPA is defined by a two-mode squeezing operator {\hat{U}}_{S_1}({\xi }_1)=\exp ({\xi }_1^{\ast } \hat{a}\hat{b}-{\xi }_1^{†} {\hat{a}}^{†}{b}^{†}), where \hat{a} (\hat{b}) and {\hat{a}}^{†} ({\hat{b}}^{†}) represent the photon annihilation and creation operators, respectively. The squeezing parameter {\xi }_1 can be represented as {\xi }_1= g_1{\mathrm{e}}^{\mathrm{i}{\theta }_1}, where g_1 being the gain factor and {\theta }_1 represents the squeezing phase shift. This parameter is crucial for shaping the interference pattern and determining the system’s phase sensitivity. After the first OPA, mode a experiences a phase shift process {\hat{U}}_{\varphi }=\exp [-\mathrm{i}\varphi ({\hat{a}}^{†}\hat{a})], while mode b remains unaltered. Subsequently, the two beams are coupled in the second OPA, characterized by the operator {\hat{U}}_{S_2}({\xi }_2)=\exp ({\xi }_2^{\ast } \hat{a}\hat{b}-{\xi }_2{\hat{a}}^{†}{\hat{b}}^{†}), where {\xi }_2= g_2{\mathrm{e}}^{\mathrm{i}{\theta }_2} with {\theta }_2-{\theta }_1=\pi. In this paper, we set the parameters g_1=g_2=g, {\theta }_1= 0, and {\theta }_2=\pi. The input states are the coherent state {|\alpha ⟩}_a and the vacuum state {|0⟩}_b, with homodyne detection performed on the output mode a.

The SU(1,1) interferometer is typically vulnerable to photon losses, especially internal ones. To simulate internal and external photon losses, the use of fictitious BSs is proposed, as depicted in Fig.1(a). The operators of these fictitious BSs can be represented as {\hat{U}}_{B_1} and {\hat{U}}_{B_2}, with {\hat{U}}_{B_1}=\exp \left[{\theta }_{T_1}\left(a^{†}{\hat{a}}_{v_1}-\hat{a}{\hat{a}}_{v_1}^{†}\right)\right] and {\hat{U}}_{B_2}=\exp \left[{\theta }_{T_2}\left(a^{†}{\hat{a}}_{v_2}-\hat{a}{\hat{a}}_{v_2}^{†}\right)\right], where {\hat{a}}_{v_1} and {\hat{a}}_{v_2} represent vacuum modes. Here, T_k (k=1,2) denotes the transmissivity of the fictitious BSs, which is associated with {\theta }_{T_k} through T_k={\cos }^2 {\theta }_{T_k}\in \left[0,1\right]. The value of transmittance equal to 1 (T_k= 1) corresponds to the ideal case without photon losses [50]. In an expanded space, the output state of the standard SU(1,1) interferometer can be represented as the following pure state:

(1)|{\mathrm{\Psi }}_{out}^0⟩={\hat{U}}_{B_2}{\hat{U}}_{S_2}{\hat{U}}_{B_1}{\hat{U}}_{\varphi }{\hat{U}}_{S_1}|{\psi }_{in}⟩,

where |{\psi }_{in}⟩ = {|\alpha ⟩}_a{|0⟩}_b{|0⟩}_{a_{v_1}}{|0⟩}_{a_{v_2}}.

In our scheme, we utilize simple and easy-to-prepare input states ({|\alpha ⟩}_a\otimes {|0⟩}_b), combined with an experimentally feasible homodyne detection. To mitigate the impact of photon losses, we introduce a Gaussian operation inside the SU(1,1) interferometer, called the single-path LSO scheme, as shown in Fig.1(b). The operator of the single-path LSO inside the SU(1,1) interferometer on mode a is given by

(2){\hat{S}}_a= \exp \left[{\displaystyle \frac{1}{2}}\left(r{\mathrm{e}}^{-\mathrm{i}\pi }{\hat{a}}^2-r{\mathrm{e}}^{\mathrm{i}\pi }{\hat{a}}^{†2}\right)\right].

In this case where the single-path LSO scheme applied inside the SU(1,1) interferometer, the output state can be expressed as the following pure state:

(3)|{\mathrm{\Psi }}_{out}^1⟩={\hat{U}}_{B_2}{\hat{U}}_{S_2}{\hat{U}}_{B_1}{\hat{U}}_{\varphi }{\hat{S}}_a{\hat{U}}_{S_1}|{\psi }_{in} ⟩.

3 Phase sensitivity

Quantum metrology utilizes quantum resources to achieve precise phase measurements [51, 52]. The main goal is to achieve highly sensitive measurements of unknown phases. Phase sensitivity is a crucial parameter since enhancing it can reduce the uncertainty in phase measurements, thereby improving the accuracy of the measurements [53]. Selecting an appropriate detection method to extract the phase information is the essential final step in the phase estimation process. This step is essential for ensuring the accuracy and reliability of measurement results. Common detection methods include homodyne detection [54, 55], parity detection [15, 56], and intensity detection [57], each offering unique trade-offs among sensitivity, complexity, and practical application. It is important to note that the phase sensitivity of different detection schemes can vary depending on the input state and the design of the interferometer [58]. Homodyne detection is feasible with current experimental techniques [59], and its theoretical calculations are relatively straightforward. Therefore, we choose to use homodyne detection at output port a to estimate phase sensitivity.

In homodyne detection, the measured variable is one of the two orthogonal components of mode a, given by \hat{X}=(\hat{a}+ {\hat{a}}^{†})/\sqrt{2}. Based on the error-propagation equation [17], the phase sensitivity can be expressed as

(4)\mathrm{\Delta }\varphi = {\displaystyle \frac{\sqrt{⟨{\mathrm{\Delta }}^2\hat{X}⟩}}{|\mathrm{\partial }⟨\hat{X}⟩/\mathrm{\partial }\varphi |}}={\displaystyle \frac{\sqrt{⟨{\hat{X}}^2⟩-⟨\hat{X}{⟩}^2}}{|\mathrm{\partial }⟨\hat{X}⟩/\mathrm{\partial }\varphi |}}.

Based on Eqs. (3) and (4), the phase sensitivity for the LSO scheme can be theoretically determined. Detailed calculation steps for the phase sensitivity \mathrm{\Delta }\varphi of the single-path LSO scheme are provided in Appendix A.

3.1 Ideal case

Initially, we consider the ideal case, T_k= 1 (where k=1,2), representing the case without photon losses. The phase sensitivity \mathrm{\Delta }\varphi is plotted as a function of \varphi for various squeezing parameters r in Fig.2. The results show that: (i) The phase sensitivity initially improves and then decreases as the phase \varphi increases, with the optimal sensitivity deviating from \varphi = 0. (ii) Implementing the single-path LSO inside the SU(1,1) interferometer significantly enhances the phase sensitivity \mathrm{\Delta } \varphi. This enhancement becomes more pronounced with higher squeezing parameters. (iii) As the value of r increases, the optimal phase sensitivity shifts away from \varphi = 0. We have compared the improvement in the QFI and phase sensitivity by performing the LSO on different modes and before or after the phase shifter in ideal case. Results show that the best improvement is achieved when the LSO is performed before the phase shifter on mode a. Therefore, we use this scheme as our research approach.

Fig.3 illustrates that the optimal phase sensitivity (minimizing \mathrm{\Delta } \varphi) is plotted against the gain factor g for different squeezing parameters. The plot confirms that the single-path LSO scheme enhances the phase sensitivity \mathrm{\Delta }\varphi. The improvement effect remains relatively constant with increasing g. Additionally, the larger the squeezing parameter r, the better the phase sensitivity \mathrm{\Delta }\varphi.

Similarly, we analyze the optimal phase sensitivity (minimizing \mathrm{\Delta } \varphi) as a function of the coherent amplitude \alpha, as shown in Fig.4. The phase sensitivity improves with the coherent amplitude \alpha, due to the increase in the average photon number with \alpha, which enhances intramode correlations [60] and quantum entanglement between the two modes. Furthermore, the enhancement effect diminishes as the coherent amplitude \alpha increases. Once more, the larger the squeezing parameter r, the better the phase sensitivity \mathrm{\Delta } \varphi.

To more comprehensively evaluate our scheme’s performance, we plot the phase sensitivity against the total average photon number N in the interferometer, as depicted in Fig.5. When N is small, the LSO does not improve the phase sensitivity. However, as N increases, the improvement effect of the LSO on phase sensitivity increases with the squeezing parameter r.

3.2 Photon-loss case

The SU(1,1) interferometer is crucial for achieving high-precision measurements, but its measurement accuracy is highly susceptible to photon losses. In this study, we focus on internal and external photon losses, corresponding to T_k\in (0,1). The optimal phase sensitivity (minimizing \mathrm{\Delta } \varphi), depicted as a function of transmittance T_k in Fig.6 for fixed g and \alpha, improves as expected with increasing transmittance T_k. Lower transmittance corresponds to higher losses, which impair the performance of phase estimation. Both internal and external photon losses degrade phase sensitivity. Notably, across the entire range of T_k, internal losses have a more pronounced effect on the system’s phase sensitivity. This is primarily because the second OPA amplifies both the signal and the internal noise. The beneficial effects of phase sensitivity also become more evident with an increase in the squeezing parameter r. In contrast, the phase sensitivity of the single-path LSO scheme is less affected, indicating that the Gaussian operations can mitigate the impact of internal or external photon losses and enhance the interferometer’s robustness against losses.

3.3 Comparison with SQL and HL

Additionally, we compare the phase sensitivity with the SQL and HL in this section. The SQL and HL are defined as \mathrm{\Delta } {\varphi }_{SQL}=1/ \sqrt{N} and \mathrm{\Delta } {\varphi }_{HL}=1/N, respectively. Here N represents the total average photon number inside the interferometer before the second OPA for the ideal case [9, 61]. N can be calculated as

(5)\begin{array}{rl}N=& ⟨{\psi }_{in}|{\hat{U}}_{S_1}^{†}{\hat{S}}_a^{†}\left({\hat{a}}^{†}\hat{a}+{\hat{b}}^{†}\hat{b}\right){\hat{S}}_a{\hat{U}}_{S_1}|{\psi }_{in}⟩\\ =& Q_{1,1,0,0}+Q_{0,0,1,1},\end{array}

where the expression of Q_{x_1,y_1,x_2,y_2} is given in Appendix A.

For fixed \alpha and g, we plot the phase sensitivity \mathrm{\Delta } \varphi as a function of \varphi for a comparison with the SQL and HL of the standard SU(1,1) interferometer in Fig.7. Our findings demonstrate that: (i) In the ideal case, the phase sensitivity of the standard interferometer (without LSO) cannot surpass the SQL and HL. However, with the single-path LSO scheme, it can surpass the SQL even when the squeezing parameter r is relatively small (r=0.3). Moreover, as r increases, the sensitivity of the LSO scheme can surpass the HL. (ii) In the presence of significant photon losses, the single-path LSO scheme still demonstrates a capability to surpass the SQL when the squeezing parameter r is large. As r increases, the single-path LSO scheme approaches the HL. This indicates that the single-path LSO scheme exhibits strong robustness against photon losses, effectively addressing the challenges posed by photon losses in practical applications.

4 Quantum Fisher information

In our previous discussions, we investigated how the single-path LSO scheme affects phase sensitivity and the relationship between various relevant parameters and phase sensitivity through the use of homodyne detection. It is important to note that the phase sensitivity is influenced by the specific measurement method used. This leads us to the critical question: how can we attain the highest phase sensitivity in an interferometer that is not influenced by the choice of measurement method? In this section, we will turn our attention to the QFI, which indicates the maximum amount of information that can be extracted from the interferometer system, regardless of the measurement technique used. We will analyze the QFI under both ideal and realistic conditions.

4.1 Ideal case

For a pure state system, the QFI can be derived by [62]

(6)F=4\left[⟨{\mathrm{\Psi }}^{\mathrm{\prime }}|{\mathrm{\Psi }}^{\mathrm{\prime }}⟩-{\left|⟨{\mathrm{\Psi }}^{\mathrm{\prime }}|\mathrm{\Psi }⟩\right|}^2\right],

where |\mathrm{\Psi }⟩ is the quantum state after the phase shift and before the second OPA, and |{\mathrm{\Psi }}^{\mathrm{\prime }}⟩=\mathrm{\partial }|\mathrm{\Psi }⟩/\mathrm{\partial }\varphi . Then the QFI can be reformulated as [62]

(7)F=4⟨{\mathrm{\Delta }}^2{\hat{n}}_a⟩,

where {\hat{n}}_a={\hat{a}}^{†}\hat{a}, ⟨{\mathrm{\Delta }}^2{\hat{n}}_a⟩=⟨\mathrm{\Psi }|({\hat{a}}^{†}\hat{a}{)}^2|\mathrm{\Psi }⟩-{\left(⟨\mathrm{\Psi }|{\hat{a}}^{†}\hat{a}|\mathrm{\Psi }⟩\right)}^2.

In the ideal case, the quantum state is expressed as |\mathrm{\Psi }⟩= {\hat{U}}_{\varphi }{\hat{S}}_a{\hat{U}}_{S_1}{|\alpha ⟩}_a{|0⟩}_b. Thus, the QFI can be expressed as

(8)F=4\left(Q_{2,2,0,0}+Q_{1,1,0,0}\right)- 4{\left(Q_{1,1,0,0}\right)}^2\text{,}

where the expression for Q_{x_1,y_1,x_2,y_2} is given in Appendix A. It is possible to explore the connection between the QFI and the related parameters through the use of Eq. (8).

Fig.8 illustrates the QFI as a function of g (\alpha) for a specific \alpha (g). It is evident that higher g (\alpha) corresponds to a greater QFI. Additionally, the QFI increases with the squeezing parameter r. Furthermore, the improvement in QFI resulting from Gaussian operations becomes more significant as g (\alpha) increases.

The quantum Cramér-Rao bound (QCRB), as defined by Ref. [63], represents the minimum phase sensitivity achievable across all measurement schemes:

(9)\mathrm{\Delta }{\varphi }_{QCRB}={\displaystyle \frac{1}{\sqrt{vF}}}.

Here, v denotes the number of measurements. For simplicity, we set v=1. The QCRB (denoted as \mathrm{\Delta } {\varphi }_{QCRB}) [63, 64] establishes the ultimate limit for probabilities derived from quantum system measurements. It serves as an estimator implemented asymptotically via maximum likelihood estimation and provides a measurement-independent phase sensitivity. To evaluate the optimality of the phase sensitivity achieved by the SU(1,1) interferometer with the single-path LSO scheme, we analyze the sensitivity \mathrm{\Delta } {\varphi }_{QCRB} derived from the QFI in Fig.9. It illustrates the variation of \mathrm{\Delta } {\varphi }_{QCRB} as a function of g (\alpha) for a specific \alpha (g). It is shown that \mathrm{\Delta }{\varphi }_{QCRB} improves with increasing g and \alpha. Similarly, the larger the squeezing parameter r, the better the \mathrm{\Delta } {\varphi }_{QCRB}. Furthermore, the improvement in \mathrm{\Delta } {\varphi }_{QCRB} is more obvious for small gain factor g [refer to Fig.9(a)] and small coherent amplitude \alpha [refer to Fig.9(b)].

4.2 Photon losses case

In this subsection, we extend our analysis to include the QFI in the presence of photon losses. Here, we should emphasize that the Fisher information is obtained using the state preceding the second OPA, i.e., the second OPA is not essential, as shown in Fig.10. For realistic quantum systems, Escher et al. [62] proposed a method for calculating the QFI. This method can be briefly summarized as follows.

The QFI with photon losses is calculated as detailed in Ref. [62]. After the first OPA {\hat{U}}_{S_1}, Gaussian operation {\hat{S}}_a, phase shift {\hat{U}}_{\varphi }, photon losses {\hat{U}}_B, and before detection, the output state in an expanded space is given by

(10)|{\mathrm{\Psi }}_S⟩={\hat{U}}_B{\hat{U}}_{\varphi }{\hat{S}}_a{|0⟩}_{a_v}|\psi ⟩ ,

which is a form of pure state, where |\psi ⟩={\hat{U}}_{S_1}{|\alpha ⟩}_a{|0⟩}_b.

For the case of photon losses, we can treat the system as a pure state in an extended space, similar to Eq. (3). Then following Eq. (6), we can obtain the QFI under the pure state, denoted as C_Q, which is greater than or equal to the QFI for mixed state, denoted as F_L, i.e., F_L\le C_Q. C_Q is the QFI before optimizing over all possible measurements, i.e.,

(11)C_Q=4\left[⟨\psi |{\hat{H}}_1|\psi ⟩-{\left|⟨\psi |{\hat{H}}_2|\psi ⟩\right|}^2\right],

where {\hat{H}}_1 and {\hat{H}}_2 are defined as

(12){\hat{H}}_1= \stackrel{\mathrm{\infty }}{{\displaystyle \sum }_{l=0}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}\varphi }}{\hat{\mathrm{\Pi }}}_l^{†}\left(\eta ,\varphi ,\lambda \right){\displaystyle \frac{\mathrm{d}}{\mathrm{d}\varphi }}{\hat{\mathrm{\Pi }}}_l\left(\eta ,\varphi ,\lambda \right) ,

(13){\hat{H}}_2= \mathrm{i}\stackrel{\mathrm{\infty }}{{\displaystyle \sum }_{l=0}}\left[{\displaystyle \frac{\mathrm{d}}{\mathrm{d}\varphi }}{\hat{\mathrm{\Pi }}}_l^{†}\left(\eta ,\varphi ,\lambda \right)\right]{\hat{\mathrm{\Pi }}}_l\left(\eta ,\varphi ,\lambda \right) .

Here, {\hat{\mathrm{\Pi }}}_l\left(\eta ,\varphi ,\lambda \right) =\sqrt{\frac{{\left(1-\eta \right)}^l}{l!}}{\mathrm{e}}^{\mathrm{i}\varphi \left({\hat{n}}_a-\lambda l\right)}{\eta }^{\frac{n}{2}}{\hat{a}}^l is the phase-dependent Krause operator, satisfying \sum {\hat{\mathrm{\Pi }}}_l^{†}\left(\eta ,\varphi ,\lambda \right){\hat{\mathrm{\Pi }}}_l(\eta ,\varphi ,\lambda )=1, with \lambda =0 and \lambda = -1 representing the photon losses before the phase shifter and after the phase shifter, respectively. {\hat{n}}_a= {\hat{a}}^{†}\hat{a} is the number operator, and \eta is related to the dissipation factor with \eta =1 and \eta = 0 corresponding to the cases of complete lossless and absorption, respectively. Following the spirit of Ref. [62], we can further obtain the minimum value of C_Q by optimizing over \lambda, corresponding to F_L, i.e., F_L=min{C}_Q\le C_Q. Simplifying the calculation process enables us to derive the QFI under photon losses:

(14)F_L={\displaystyle \frac{4F\eta ⟨{\hat{n}}_a⟩}{\left(1-\eta \right)F+4\eta ⟨{\hat{n}}_a⟩}},

where F is the QFI in the ideal case [24].

Next, we further analyze the effects of each parameter on the QFI of the single-path LSO scheme under photon losses through numerical calculation. Fig.11 plots the QFI and QCRB as functions of transmittance \eta, from which it is observed that the QFI and the QCRB improve with the rising transmittance \eta, and the single-path LSO can enhance the QFI and the QCRB. The improved QFI increases with the transmittance \eta, while the improved \mathrm{\Delta }{\varphi }_{QCR{B}_L} decreases with the transmittance \eta.

Similar to the ideal case, Fig.12 illustrates the F_L as a function of g (\alpha) for a given \alpha (g), under photon-loss case with \eta =0.5. Similar to Fig.8, The improvement in QFI increases with increasing values of the squeezing parameter r and g (\alpha).

To gain deeper insights into the enhancement effect of the single-path LSO scheme on phase sensitivity, we compare the phase sensitivity \mathrm{\Delta } \varphi obtained via homodyne detection with the sensitivity \mathrm{\Delta }{\varphi }_{QCRB} derived from the QFI. Fig.13 shows how \mathrm{\Delta }\varphi varies with the local squeezing parameter r when g=1 and \alpha =1. The LSO scheme enables the phase sensitivity to progressively approach the QCRB with increasing r.

5 Discussion

5.1 Intramode correlation and intermode correlation

To further investigate the intrinsic mechanisms by which local squeezing operations enhance phase measurement precision, we employ second-order coherence functions [65]: the intramode correlation function g_a^{(2)}=⟨{\hat{a}}^{†2}{\hat{a}}^2⟩/{⟨{\hat{a}}^{†}\hat{a}⟩}^2 and g_b^{(2)}= ⟨{\hat{b}}^{†2}{\hat{b}}^2⟩/⟨{\hat{b}}^{†}\hat{b}{⟩}^2, as well as the intermode correlation g_{ab}^{(2)}=⟨{\hat{a}}^{†}\hat{a}{\hat{b}}^{†}\hat{b}⟩/(⟨{\hat{a}}^{†}\hat{a}⟩⟨{\hat{b}}^{†}\hat{b}⟩), to analyze the dependence of the QCRB on these two types of correlations. Notably, within the photonic coherence theory, the intramode correlation functions g_a^{(2)} and g_b^{(2)} are proportional to the probability of detecting two photons simultaneously at detectors A and B, respectively, while the intermode correlation function g_{ab}^{(2)} the likelihood of simultaneous detection of one photon at detector A and another at detector B [65], as illustrated in Fig.14.

As shown in Fig.9, it is evident that the greater the local squeezing parameter r, the lower the QCRB value, leading to a significant improvement in phase measurement precision. This can be attributed to an increased local squeezing parameter r on a single path, which strengthens the intramode correlations of the probe state while reducing the intermode correlations, as demonstrated in Fig.15 and Fig.16. Consequently, the enhancement of phase sensitivity primarily depends on intramode correlations rather than intermode correlations, consistent with the findings in references [60, 65]. According to Ref. [60], mode entanglement is not necessary for achieving quantum-enhanced interferometric measurements. Separable states offer greater flexibility and simpler preparation processes compared to entangled states, thus allowing for higher QFI through strong intramode correlations.

5.2 Matrix-based quantum Fisher information

Current research commonly employs single-parameter QFI to analyze single-phase estimation in interferometric systems. However, in many practical scenarios where different phase shifts may occur in each arm, both the phase sum {\varphi }_s= {\varphi }_1+ {\varphi }_2 and phase difference {\varphi }_d= {\varphi }_1-{\varphi }_2 become unknown parameters requiring simultaneous estimation. The two-parameter problem requires the Quantum Fisher Information Matrix (QFIM) for comprehensive phase estimation analysis [66, 67]. Notably, differences exist between single-parameter and multi-parameter estimation methods, as thoroughly discussed in Ref. [66]. To strengthen analytical rigor, this work adopts the QFIM approach in Ref. [66] for estimation of both phase parameters ({\varphi }_s and {\varphi }_d) in our system.

When considering two arms with unknown phase shifts, the phase shifter operator takes the form of

(15){\hat{U}}_{{\varphi }_M}={\mathrm{e}}^{-\mathrm{i}{\varphi }_1({\hat{a}}^{†}\hat{a})}{\mathrm{e}}^{-\mathrm{i}{\varphi }_2({\hat{b}}^{†}\hat{b})}={\mathrm{e}}^{-\mathrm{i}{\hat{g}}_s{\varphi }_s}{\mathrm{e}}^{-\mathrm{i}{\hat{g}}_d{\varphi }_d},

with {\hat{g}}_s=\left({\hat{a}}^{†}\hat{a}+{\hat{b}}^{†}\hat{b}\right)/2, {\hat{g}}_d=\left({\hat{a}}^{†}\hat{a}-{\hat{b}}^{†}\hat{b}\right)/2, and {\varphi }_1, {\varphi }_2 are the unknown phases on modes a and b, respectively. These phases can also be described using the phase sum {\varphi }_s= {\varphi }_1+ {\varphi }_2 and the phase difference {\varphi }_d= {\varphi }_1-{\varphi }_2.

In multiparameter estimation, the QCRB is determined via the QFIM. For the estimation of {\varphi }_s and {\varphi }_d, the QFIM is given by a 2\times 2 matrix:

(16)F_Q=\left(\begin{array}{cc}{F}_{ss}& F_{sd}\\ F_{ds}& F_{dd}\end{array}\right),

where F_{ij}=4\left(⟨{\hat{g}}_i{\hat{g}}_j⟩-⟨{\hat{g}}_i⟩⟨{\hat{g}}_j⟩\right) and the subscripts s and d denote {\varphi }_s and {\varphi }_d, respectively. The diagonal elements of this matrix correspond to the QFI for the respective parameters, while the off-diagonal entries provide information about the parameter correlations. The QCRB for simultaneous two-phase estimation can be derived, that is, the lower bound on the estimation uncertainty [66–69]:

(17){\left|\mathrm{\Delta }\varphi \right|}^2⩾ {\left|\mathrm{\Delta }{\varphi }_{QCRB}\right|}^2=\mathrm{T}\mathrm{r}\left(F_Q^{-1}\right).

As shown in Fig.17, the results are consistent with the QCRB trend derived from single-phase estimation using QFI: (i) The estimation precision improves with increasing g and \alpha. (ii) The enhancement of \mathrm{\Delta } {\varphi }_{QCRB} becomes more pronounced with a larger squeezing parameter r.

To conclude, intramode correlations are the primary factor in boosting the QFI and reducing the QCRB, directly driving the improvement in phase sensitivity. In practical applications, using separable states with strong intramode correlations may be a more direct and efficient strategy. Therefore, the single-path LSO scheme is a highly effective and easily implementable method for enhancing measurement precision. It should be noted that although the single-path LSO scheme has significant advantages, it also faces challenges such as environmental noise and the maintenance of high-quality squeezing states. Future research might explore hybrid schemes that combine single-path squeezing with other quantum enhancement techniques to further enhance measurement accuracy and robustness.

6 Conclusion

In this paper, we have analyzed the impact of the single-path LSO scheme on the phase sensitivity and the QFI in both ideal and photon-loss cases. Additionally, we have investigated how the squeezing parameter r of single-path LSO scheme, the gain coefficient g of OPAs, the amplitude \alpha of coherent state, and the transmittance T_k of BSs affect system performance. Through analytical comparison, we have verified that the single-path LSO scheme improves the SU(1,1) interferometer’s measurement accuracy by strengthening intramode correlations and weakening intermode correlations, thereby enhancing robustness against internal and external photon losses. Across the entire T_k range, internal losses affect the system’s phase sensitivity more significantly than external losses. This suggests that reducing internal photon losses is crucial for improving overall performance when optimizing quantum interferometer design.

Gaussian operations are widely used in quantum optics and information processing due to their ease of implementation and high energy efficiency. They allow for effective quantum state manipulation with relatively simple experimental setups, making them suitable for various applications. In contrast, non-Gaussian operations, though capable of generating diverse quantum resources, present significant challenges. They are usually more complex to implement, often requiring advanced experimental techniques and precise system parameter control. They are usually more complex to implement, often requiring advanced experimental techniques and precise system parameter control. Our study highlights Gaussian operations’ potential to enhance quantum measurement and information processing system performance.

In summary, the single-path LSO scheme plays a crucial role in reducing internal and external photon losses in the SU(1,1) interferometer, significantly enhancing quantum measurement accuracy. The Gaussian operation not only enhances measurement reliability but also opens doors to new insights and methodologies in future quantum technology applications. Looking ahead, research may explore more advanced squeezing schemes and their potential applications across diverse quantum systems, potentially driving significant advancements in high-precision quantum sensing and metrology.

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