1. Introduction

With the development of quantum metrology, its theoretical and experimental progress has attracted increasing attention by scientists recently. Optical interferometer, a common tool for high-precision measurement, possesses a wide array of potential applications, including gravitational wave detection [1,2], optical lithography [3], optical gyroscope [4–6]. As depicted in Fig. 1(a), a typical Mach-Zehnder interferometer (MZI) consisting of two beam splitters works as follows: (i) an incident coherent-state beam is split into two modes by the first beam splitter; (ii) one mode experiences a relative phase shift $\phi$ while the other one retains as a reference; (iii) these two modes are then recombined at the second beam splitter. Finally, one extracts the expected information about the parameter $\phi$ by monitoring the output modes. Mathematically, the measurement quantity is regarded as an observable $A$. Accordingly, the phase sensitivity can be inferred from the variance of the observable $A$ via the linear error propagation formula $\Delta \phi = \langle \Delta A \rangle / |\partial \langle A \rangle / \partial \phi |$.

 

Fig. 1. Schemes for phase estimation: (a) the traditional MZI with the output beam $b$ being discarded, (b) the modified scheme with the output mode $b$ being re-injected into the input mode $b$. We have assumed that the output beam $b$ experiences a phase shift $\theta _0$ and photon loss $L$ before the re-injection. We set the phase shift $\theta _{\rm {LO}}=\pi /2$. HD: homodyne detection, LO: local osillator.

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As a matter of fact, the phase sensitivity in the traditional MZI is restricted by the shot-noise limit (SNL) [7], $1/\sqrt {n}$, due to the shot noise (the quantum nature of light) where $n$ is the total incident mean photon number. To improve the phase sensitivity (even beyond the SNL), there usually exist two pathways: (i) decreasing $\langle \Delta A \rangle$ (i.e. reducing the quantum noise), which could be realized by injecting a beam with a sub-shot noise, for instance, squeezed states [8–12]; (ii) increasing $|\partial \langle A \rangle / \partial \phi |$ (i.e., enhancing the effective detection signal), which could be achieved by various methods, such as utilizing nonlinear beam splitter [13–23] or using nonlinear phase shift [24–28].

In experiment, one can choose various measurement strategies, such as intensity detection [29–32], parity detection [33–36], and homodyne detection [37–40]. Particularly, we mainly focus on the homodyne detection (a measurement technique being used to monitor the quadrature of light). It is worth noting that homodyne detection is usually performed only on one output mode, while the other one is ignored in the traditional scheme [39–41] as shown in Fig. 1(a). Nevertheless, the discarded mode $b$ also contains the information about the parameter $\phi$. Naturally, we wonder whether it is able to enhance the phase sensitivity by reusing the ignored beam. Following this route, we propose an alternative scheme where the output mode $b$ is re-injected into the input port $b$ (so-called photon recycling) as shown in Fig. 1(b).

In fact, the technique of photon recycling has been therotically proposed and experimentally realized in the Michelson interferometer (MI) [30,42–46]. It has been proved that the photon recycling is an efficient technique for further signal increase, capitalizing both on an increase in circulating mean photon number and an increase in phase shift [30]. Nevertheless, the photon recycling is rarely applied in a MZI. From this point of view, it is also desired to explore the perfomance of the photon-recycled Mach-Zehnder interferometer.

In this work, we investigate the performance of this photon-recycled scheme from three related but distinct aspects: (i) the phase sensitivity via homodyne detection; (ii) the quantum Cramér-Rao bound (QCRB); (iii) the total mean photon number inside the interferometer (the photons experiencing and sensing the phase shift). First of all, it is demonstrated that this modified interferometer can realize an enhanced phase sensitivity compared with the traditional MZI. Second, we illustrate that this modified scheme can achieve a QCRB beyond the traditional one. In addition, unlike the traditional scheme where the QCRB is independent of the phase shift $\phi$, the QCRB of our scheme depends on $\phi$. Finally, we discuss the underlying mechanisms behind the enhanced phase sensitivity by analyzing the total mean photon number inside the interferometer.

This manuscript is organized as follows. In Sec. II, we briefly introduce the theoretical model of propagation of photons through this modified optical circuit. In Sec. III, we analyze the phase sensitivity via homodyne detection and QCRB in our scheme, and compare the performance between our scheme and the traditional one. In Sec. IV, we present the underlying mechanisms behind the enhanced phase sensitivity. Final remarks are given in Sec. V.

2. Theoretical model

In contrast to the traditional scheme where the output beam $b$ is discarded (see Fig. 1(a)), we propose an alternative one where the output mode $b$ is reused via photon recycling (the output beam $b$ is re-injected into the input port $b$ as illustrated in Fig. 1(b)). In practical experiment, the photon loss is inevitable. Therefore, we consider the effect of photon loss induced by the light propagation in the recycling arm. In addition, we assume that the output beam $b$ experiences a phase shift $\theta _0$ before it is re-injected into the input port $b$.

In our scheme as depicted in Fig. 1(b), the input-output relation is found to be

Generally, the photon loss could be modeled by adding a fictitious beam splitter as shown in Fig. 1(b). Assume the photon-recycling arm with a loss rate $L$. After passing through the fictitious beam splitter, the mode transform of field $\hat {b}^{\rm {out}}$ is given by $\hat {b}^{\rm {out}\prime } = \sqrt {1-L} \hat {b}^{\rm {out}} + \sqrt {L} \hat {v}_{b}$ ($\hat {v}_{b}$ corresponds to the vacuum state). In the presence of phase shift $\theta _0$ and photon loss $L$, the re-injected mode $b$ could be expressed as

Based on Eqs. (1) and (4), the output annihilation operators in Fig. 1(b) arrive at

Equation (5) is the output in our scheme in Fig. 1(b). To verify this result, we analyze our scheme via an alternative method and obtain the same expression. For the sake of clarity, the detailed analysis is shown in Appendix A. Since we monitor the final output mode $a$ in our scheme, Eq. (5) would be used to estimate the phase shift $\phi$.

3. Phase sensitivity

3.1 Homodyne detection

To extract the information of parameter $\phi$, we measure the quadrature of output mode $a$ via homodyne detection as show in Fig. 1(b). Mathematically, the output quadrature is defined as

Consider a coherent state $|\alpha \rangle$ as input as shown in Fig. 1(b) (the complex number $\alpha = |\alpha |e^{i\theta _{\alpha }}$ denotes the amplitude). Without loss of generality, we set the irrelevant phase $\theta _{\alpha } = 0$. In this situation, by combining Eqs. (5) and (9), it is easily found that

By inserting Eq. (12) into Eq. (11), the phase sensitivity yields

In fact, $\Lambda _1$ can be regarded as the enhancement factor of phase sensitivity. Figure 2 depicts the factor $\Lambda _1$ as a function of $\phi$ and $\theta _0$ with various losses $L=0.05, 0.10, 0.15,$ and $0.20$. It is shown that in the presence of moderate loss, the enhancement factor can reach above unity, $\Lambda _1>1$, in a horn-shaped region which indicates that our scheme could achieve a phase sensitivity beyond the traditional one. It is worth pointing out that the maximum $\Lambda _1$ is achieved around the tip of the “horn".

 

Fig. 2. The enhancement factor $\Lambda _{1}$ as a function of $\phi$ and $\theta _0$ in the presence of photon loss (a) $L=0.05$, (b) $L=0.10$, (c) $L=0.15$, and (d) $L=0.20$.

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In an attempt to gain insight into the effect of loss on the minimum phase sensitivity, we quantitatively analyze the maximum $\Lambda _{1,\rm {max}}(= \Delta \phi _{\rm {SNL}}^{\rm {Con}}/\Delta \phi _{\rm {min}}^{\rm {PR}})$ from Eq. (36). Figure 3 plots $\Lambda _{1,\rm {max}}$ as a function of loss. It indicates that the minimum phase sensitivity increases with the increase of loss. Although the factor $\Lambda _{1,\rm {max}}$ decreases, it is still larger than unity with a moderate photon loss. In particular, when $L=0.10$, the maximum enhancement of phase sensitivity is roughly equal to $\Lambda _{1,\rm {max}} \simeq 9.32$ with $\theta _0\simeq 0.3524$ rad and $\phi \simeq 2.5702$ rad.

 

Fig. 3. The maximum enhancement factor $\Lambda _{1,\rm {max}}$ as a function of photon loss $L$. QCRB: quantum Cramér-Rao bound, HD: homodyne detection.

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3.2 Quantum Cramér-Rao bound

Since the QCRB [47–49] gives the ultimate limit to precision, it is regarded as a reliable figure of merit when quantifying the performance of phase-estimation measurement scheme, which motivates us to investigate the QCRB in our scheme.

Theoretically, the output mode $a$ is always a pure Gaussian state in our analysis model. This is due to the facts: (i) in our proposal, the incident beam is a coherent state which is a pure Gaussian state; (ii) since the beam splitter acts as a Gaussian operation, the output is still a pure Gaussian state when a pure Gaussian state propagates through beam splitters (including the fictious BS corresponding to the photon loss).

According to Ref. [50], the QCRB for a pure Gaussian state can be written as

In this modified scheme, $\overline {X}$ and $\Gamma$ are found to be

By substituting Eq. (17) into Eq. (15), the QCRB is cast into

 

Fig. 4. The enhancement factor $\Lambda _2$ as a function of $\phi$ and $\theta _0$ with photon loss (a) $L=0.05$, (b) $L=0.10$, (c) $L=0.15$, and (d) $L=0.20$.

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According to Eq. (37), one can numerically compute the maximum $\Lambda _{\rm {2,\rm {max}}}$. As shown in Fig. 4, with the increase of loss, the factor $\Lambda _{\rm {2,\rm {max}}}$ decreases, which yields that as the loss increases, the minimum QCRB becomes worse. Although $\Lambda _{\rm {2,\rm {max}}}$ decreases, it is still larger than one. That is to say, with a moderate loss, our scheme can still achieve a QCRB beyond the tradition scheme. It is worth noting that in contrast to the traditional MZI where the QCRB is independent of $\phi$, the QCRB in our scheme depends on $\phi$.

4. Discussion

We would like to discuss the underlying mechanisms behind the enhanced phase sensitivity. Let us consider the total mean photon number, $\hat {n}_{\rm {T}}$, inside the interferometer where $\hat {n}_{\rm {T}}$ is defined as

Figure 5 illustrates $\Lambda _3$ as a function of $\phi$ and $\theta _0$ with various losses $L=0.05$, $0.10$, $0.15$ and $0.20$. Intriguingly, $\Lambda _3$ has a similar pattern to $\Lambda _1$ and $\Lambda _2$ where its maximum value appears around the tip of the “horn". It is easy to check that $\Lambda _3 \ge 1$ is always valid, which indicates that our scheme possesses more photons inside the interferometer than the conventional one. In fact, photons inside the interferometer are usually regarded as the essential resource to experience and sense the relative phase shift. Intuitively, the more resource we use, the better phase sensitivity it can achieve [51]. Therefore, the phase sensitivity is improved in this modified scheme. It is worth noting that the mean photon number inside the interferometer $\langle \hat {n}_{\rm {T}}\rangle$ depends on $\phi$, leading to the phenomenon of QCRB being dependent of $\phi$ in our scheme. For further understanding of the structure of this optical circuit, we investigate our scheme in the Michelson-interferometer form in Appendix E. It is demonstrated that our scheme is equivalent to a light-recycled Michelson interferometer.

 

Fig. 5. The factor $\Lambda _3$ as a function of $\phi$ and $\theta _0$ with loss (a) $L=0.05$, (b) $L=0.10$, (c) $L=0.15$, and (d) $L=0.20$.

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5. Conclusion

In summary we propose an alternative scheme for phase estimation in a MZI with photon recycling. We investigate the performances of our scheme, including the phase sensitivity via homodyne detection and quantum Cramér-Rao bound. It is demonstrated that this modified scheme is able to achieve an enhanced performance in contrast to the traditional MZI. Moreover, we present the physical mechanisms behind the improved phase sensitivity. We believe that our scheme may offer an another route to implement a high-precision phase estimation in a MZI exploiting photon recycling.

Appendix A: an alternative method to analyze the optical circuit in our scheme

To verify the result of Eq. (5), we provide an another method to analyze our scheme. In fact, the optical circuit in Fig. 1(b) is equivalent to a series of conventional MZIs with an iterative structure (see Fig. 6), where each output beam $b$ is injected into the following input port $b$. This iterative-structure optical system has an input-output relation

(21)$$\begin{pmatrix} \hat{a}_1^{\rm{out}} \\ \hat{b}_1^{\rm{out}} \end{pmatrix} = S_{\rm{MZI}} \begin{pmatrix} \hat{a}_1^{\rm{in}} \\ \hat{b}_1^{\rm{in}} \end{pmatrix},\quad \begin{pmatrix} \hat{a}_2^{\rm{out}} \\ \hat{b}_2^{\rm{out}} \end{pmatrix} = S_{\rm{MZI}} \begin{pmatrix} \hat{a}_1^{\rm{in}} \\ \hat{b}_2^{\rm{in}} \end{pmatrix},\quad \cdots\quad \begin{pmatrix} \hat{a}_m^{\rm{out}} \\ \hat{b}_m^{\rm{out}} \end{pmatrix} = S_{\rm{MZI}} \begin{pmatrix} \hat{a}_1^{\rm{in}} \\ \hat{b}_m^{\rm{in}} \end{pmatrix},\quad \cdots$$

where $\hat {O}^{\rm {in}}_k$ and $\hat {O}^{\rm {out}}_k$ ($O=a,b$; $k=1,2,\ldots$) are the annihilation operators of input and output modes in the $k$-th MZI, respectively. We have used $\hat {a}_k^{\rm {in}}=\hat {a}_1^{\rm {in}}$ ($k=2,3\cdots m$).

 

Fig. 6. The iterative-structure of series of MZIs where $m$-th output mode $b$ is injected into the $(m+1)$-th input port $b$. This model is equivalent to the scheme in Fig. 1(b).

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Before re-injection, the mode $b$ experiences the phase shift $\theta _0$ and photon loss $L$. Therefore, the input mode $b$ in $(k+1)$-th MZI is found to be

(22)$$\hat{b}_{k+1}^{\rm{in}} = \sqrt{1-L} \hat{b}_{k}^{\rm{out}} e^{{-}i \theta_0} + \sqrt{L} \hat{v}_{b},$$

where $\hat {b}_{k}^{\rm {out}}$ ($k=1,2,\ldots$) is the annihilation operator of the output mode $b$ in the $k$-th MZI, and $\hat {v}_{b}$ denotes the vacuum.

Based on Eqs. (21) and (22), the output annihilation operators of the $m$-th MZI in Fig. 6 arrive at

(23)$$\hat{a}_m^{\rm{out}} =s_{11} \hat{a}_{1}^{\rm{in}} + s_{12} \gamma^{m-1} \hat{b}_1^{\rm{in}} +s_{12} \frac{1-\gamma^{m-1}}{1-\gamma} \kappa,$$

(24)$$\hat{b}_m^{\rm{out}} =s_{21} \hat{a}_{1}^{\rm{in}} + s_{22} \gamma^{m-1} \hat{b}_1^{\rm{in}} + s_{22} \frac{1-\gamma^{m-1}}{1-\gamma} \kappa,$$

where

(25)$$\begin{aligned}\gamma = &\sqrt{1-L} e^{{-}i\theta_0}s_{22},\\ \kappa = &\sqrt{1-L} e^{{-}i\theta_0}s_{21} \hat{a}_1^{\rm{in}} + \sqrt{L} \hat{v}_b. \end{aligned}$$

Letting $m$ tend to infinity ($m \to \infty$) in Eqs. (23) and (24), one can obtain the output annihilation operators of modes $a$ and $b$ as shown in Fig. 6,

(26)$$\hat{a}^{\rm{out}} = \hat{a}^{\rm{out}}_{m}|_{m \to \infty} = (s_{11} + \frac{s_{12} s_{21} \sqrt{1-L} }{e^{i\theta_0} - s_{22}\sqrt{1-L} }) \hat{a}_1^{\rm{in}} + (1+\frac{s_{22}\sqrt{1-L} }{e^{i\theta_0} - s_{22} \sqrt{1-L} })s_{12}\sqrt{L}\hat{v}_b,$$

(27)$$\hat{b}^{\rm{out}} = \hat{b}^{\rm{out}}_{m}|_{m \to \infty} = \frac{s_{21}}{1 - s_{22} \sqrt{1-L} e^{{-}i\theta_0} } \hat{a}_1^{\rm{in}} +\frac{s_{22} \sqrt{L}}{1 - s_{22} \sqrt{1-L} e^{{-}i\theta_0} } \hat{v}_b.$$

It is easily found that Eq. (26) is the same as Eq. (5) where we arrive at the same result via two different methods.

Appendix B: derivation of the expectation values $\langle \hat {x}_a \rangle$ and $\langle \Delta \hat {x}_a \rangle$

The expectation value of output quadrature is given by

(28)$$\langle \hat{x}_a \rangle = \langle \hat{a}^{\rm{out}\dagger}+ \hat{a}^{\rm{out}} \rangle,$$

which indicates that in order to obtain $\langle \hat {x}_a \rangle$, it requires to calculate $\langle \hat {a}^{\rm {out}\dagger }\rangle$ and $\langle \hat {a}^{\rm {out}} \rangle$, respectively.

According to the initial input state, it is easily found that

(29)$$\langle \hat{a}^{\rm{in}} \rangle = \alpha, \quad \langle \hat{v}_b \rangle = 0.$$

By combining Eqs. (5), (28) and (29), one can rewrite Eq. (28) as

(30)$$\langle \hat{x}_a\rangle = \Upsilon \alpha +\Upsilon^{{\ast}} \alpha^{{\ast}},$$

where $O^{\ast }$ ($O=\Upsilon,\alpha$) is the complex conjugate of $O$ and

(31)$$\Upsilon=\frac{e^{i\theta_0} - e^{i (\phi+\theta_0)} - 2 \sqrt{1-L}}{2 e^{i (\phi+\theta_0)} + \sqrt{1-L} - e^{i \phi} \sqrt{1-L}},$$

The fluctuation $\langle \Delta \hat {x}_a \rangle$ is defined as

(32)$$\langle \Delta \hat{x}_a \rangle \equiv \sqrt{\langle (\hat{x}_a)^2\rangle - \langle \hat{x}_a \rangle^2},$$

where

(33)$$\langle (\hat{x}_a)^2\rangle = \langle (\hat{a}^{\rm{out}\dagger}+\hat{a}^{\rm{out}})^2\rangle.$$

According to Eq. (5), Eq. (33) could be recast as

(34)$$\langle (\hat{x}_a)^2\rangle = (\Upsilon \alpha +\Upsilon^{{\ast}} \alpha^{{\ast}})^2 + 1.$$

Substituting Eqs. (30), (31), and (34) into Eq. (32), one can obtain

(35)$$\langle \Delta \hat{x}_a \rangle =1.$$

Appendix C: $\Lambda _1$ and $\Lambda _2$

The explicit expression of $\Lambda _1$ is found to be

(36)$$\Lambda_1 = 4 \left|\frac{\left(\sqrt{1-L} \cos\theta_0 -1\right)\left[\left(2-L-2 \sqrt{1-L}\cos \theta_0 \right) \sin\phi-\sqrt{1-L} (\cos \phi+1)\sin \theta_0\right]}{[{-}L-(1-L) \cos\phi-2 \sqrt{1-L} \cos \theta_0+2 \sqrt{1-L} \cos (\phi+\theta_0)+3]^2}\right|,$$

and $\Lambda _2$ is given by

(37)$$\Lambda_2 = \left|\frac{2\left({-}2 \sqrt{1-L} \cos \theta_0-L+2\right)}{2 \sqrt{1-L} \Theta-(1-L) \cos \phi -L+3}\right|,$$

where

(38)$$\Theta= \cos \left(\theta _0+\phi \right)- \cos \theta _0.$$

Appendix D: derivation of the total mean number of photon inside the interferometer $\langle \hat {n}_{\rm {T}} \rangle$

Due to the energy conservation for photons propagating through a linear beam splitter, it is easy to verify that the total mean number of output photon is equal to the mean number of photon inside the interferometer

(39)$$\langle \hat{n}_{a}^{\rm{inside}} \rangle + \langle \hat{n}_{b}^{\rm{inside}} \rangle=\langle \hat{n}_{a}^{\rm{out}} \rangle + \langle \hat{n}_{b}^{\rm{out}} \rangle,$$

where $\hat {n}_{j}^{\rm {out}} \equiv \hat {j}^{\rm {out}\dagger } \hat {j}^{\rm {out}}$ ($j = a,b$). By inserting Eq. (39) into Eq. (19), the total mean photon number can be rewritten as

(40)$$\langle \hat{n}_{\rm{T}} \rangle = \langle \hat{n}_{a}^{\rm{out}} \rangle + \langle \hat{n}_{b}^{\rm{out}} \rangle=\langle \hat{a}^{\rm{out}\dagger} \hat{a}^{\rm{out}} \rangle + \langle \hat{b}^{\rm{out}\dagger} \hat{b}^{\rm{out}} \rangle.$$

From Eq. (5), the mean photon number of output mode $a$ is given by

(41)$$\langle \hat{a}^{\rm{out}\dagger} \hat{a}^{\rm{out}} \rangle = |\Upsilon|^2 |\alpha|^2,$$

where $\Upsilon$ is present in Eq. (31).

Similarly, according to Eq. (6), one can obtain

(42)$$\langle \hat{b}^{\rm{out}\dagger} \hat{b}^{\rm{out}} \rangle = |\Xi|^2 |\alpha|^2,$$

where

(43)$$\Xi=\frac{i e^{i \theta _0} (1+e^{i \phi })}{2 e^{i (\theta _0+\phi )}-\sqrt{1-L} e^{i \phi }+\sqrt{1-L}}.$$

The total mean photon number is then found to be

(44)$$\langle \hat{n}_{\rm{T}} \rangle = \Lambda_3 |\alpha|^2,$$

where

(45)$$\Lambda_3=|\Upsilon|^2+|\Xi|^2 =\frac{2 \sqrt{1-L} \Theta-(1-L) \cos \phi -2L +4}{2 \sqrt{1-L} \Theta-(1-L) \cos \phi -L+3},$$

with $\Theta$ being defined in Eq. (38).

Appendix E: our scheme in the Michelson-interferometer form

The conventional MZI is equivalent to a MI where two beam splitters in the MZI can be reduced to one in the MI. From this point of view, our scheme (the photon-recycled MZI) can be regarded as a light-recycled MI as shown in Fig. 7. According to the input and output relationship in the light-recycled MI [52]

$$\begin{aligned} \hat{a}_{a_1} &= (\hat{a}_{a_0} + i \hat{a}_{b_0})/\sqrt{2}, \quad \quad \hat{a}_{b_1} = (i \hat{a}_{a_0} + \hat{a}_{b_0})/\sqrt{2},\\ \hat{a}'_{a_1} &= \hat{a}_{a_1} e^{{-}i (\phi + \theta_{a1})}, \quad \quad \hat{a}'_{b_1} = \hat{a}_{b_1}e^{{-}i \theta_{b1}},\\ \hat{a}'_{b_0} &= (i \hat{a}'_{a_1} + \hat{a}'_{b_1})/\sqrt{2}, \quad \quad \hat{a}_{b_0} = t_0 \hat{a}_{v_b} + r_0 \hat{a}'_{b_0} e^{{-}i \theta_{0}}, \end{aligned}$$

and

(46)$$\hat{a}^{\rm{out}} = (\hat{a}'_{a_1} + i \hat{a}'_{b_1})/\sqrt{2}, \quad \hat{a}_{b_0} = \hat{a}'_{b_0}e^{{-}i \theta_0},$$

with $r_0 = \sqrt {1-L}$ and $t_0 = \sqrt {L}$ being the reflected and transmitted coefficients of the mirror $M_0$, we can obtain the quadrature of the final output beam $a$

(47)$$\begin{aligned}\langle \hat{x}_a \rangle &= \langle \hat{a}^{\rm{out}\dagger} + \hat{a}^{\rm{out}} \rangle\\ &=\frac{e^{i \theta_0} (1 - e^{i \phi}) - 2 \sqrt{1-L}}{2 e^{i(\theta_0 + \phi)} + \sqrt{1-L} (1 - e^{i\phi})} \alpha + \frac{e^{{-}i \theta_0} (1 - e^{{-}i \phi}) - 2 \sqrt{1-L}}{2 e^{{-}i(\theta_0 + \phi)} + \sqrt{1-L} (1 - e^{{-}i\phi})} \alpha^{{\ast}}\\ &= \Upsilon \alpha +\Upsilon^{{\ast}} \alpha^{{\ast}}. \end{aligned}$$

which is consistent with the results of our scheme in Eq. (12) (we have set the irrelevant phases $\theta _{a1} = 0$ and $\theta _{b1} = 0$ and initial input state $|\alpha \rangle \otimes |0\rangle$).

 

Fig. 7. Our scheme in the Michelson-interferometer form. The two mirrors $M_{a1}$ and $M_{b1}$ of interferometer arms are highly reflective.

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The photon recycling, as a mature technique to improve the phase sensitivity, has been proposed and experimentally realized in the Michelson interferometer, for instance, the power-recycled Michelson interferometer and the signal-recycled Michelson interferometer [42]. According our analysis, it is found that our scheme could achieve a phase sensitivity in the same order of magnitude as that of the power-recycled or signal-recycled Michelson interferometer. Inspired by the enhanced phase sensitivity in the power-recycled or signal-recycled Michelson interferometer with squeezed-state input [12,32,53], we would like to investigate the case of our scheme with the squeezed light in our subsequent work.

Funding

National Natural Science Foundation of China (12104423, 12175204, 62201535, 11974111, 12234014, 11654005); Innovation Program for Quantum Science and Technology (2021ZD0303200); Shanghai Municipal Science and Technology Major Project (2019SHZDZX01); National Key Research and Development Program of China (2016YFA0302001); Shanghai Talent Program.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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