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Nonlinear phase estimation via nonlinear-linear hybrid interferometers

Jian-Dong Zhang, Chuang Li, and Shuai Wang

Open Access Open Access

Abstract

Estimating nonlinear phase shifts is useful to analyze many dynamical processes. In this work, with respect to the second-order nonlinear phase shifts, we propose an estimation scheme with a coherent state as the input and balanced homodyne detection as the readout strategy. The measurement setup is a nonlinear-linear hybrid interferometer composed of an optical parametric amplifier and a beam splitter. We analyze the precision and determine the optimal parameters of our scheme. The precision is compared with the quantum Cramér–Rao bound as well as the precision of a conventional nonlinear interferometer. In view of the fact that photon loss is ubiquitous, the effects of transmission loss and detection loss on the precision are addressed. At low gain, our scheme is superior to the scheme using a conventional nonlinear interferometer, and the precision can saturate the quantum Cramér–Rao bound.

© 2022 Optica Publishing Group

1. INTRODUCTION

Quantum interferometry is a science that makes use of quantum resources to achieve high-precision measurements. In comparison with classical interferometry, there are some approaches in quantum interferometry to improve the precision. One of the most commonly used approaches is to replace the classical coherent states with exotic quantum states as inputs. In this regard, the estimation schemes using two-mode squeezed vacuum states, entangled coherent states, and N00N states have been shown to achieve the Heisenberg-limited precision [1,2]. On the other hand, precision surpassing the shot-noise limit is attainable by performing photon-number-resolving detection instead of traditional intensity-based detection at the output. Related to this, parity detection [3–5], on–off detection [6,7], and projective detection [8] have been studied and applied in a lot of estimation schemes. In addition to the above two approaches, reforming the interferometric setups is also an effective way. To be specific, one can deploy novel interferometers to improve the precision.

Nonlinear interferometers introduced by Yurke et al. are a kind of novel interferometer, which replaces two beam splitters in conventional interferometers with two nonlinear devices such as optical parametric amplifiers (OPAs) [9]. In general, a nonlinear interferometer needs to meet the conditions of phase matching ${\theta _2} - {\theta _1} = \pi$ and gain matching ${g_1} = {g_2}$, where ${\theta _1}$ (${\theta _2}$) and ${g_1}$ (${g_2}$) are the phase and gain parameter of the first (second) OPA, respectively. The improvement in precision originates from the squeezing effect induced by the first OPA. Over the past decade, many efforts have been paid to the estimation schemes for linear phase shifts using nonlinear interferometers. With coherent states as the inputs, Plick et al. discussed the precision using intensity detection [10], and Xin et al. analyzed the precision using parity detection [11]. For a nonlinear interferometer fed by coherent and squeezed vacuum states, the precisions based on homodyne detection [12,13] and parity detection [14] were studied. The precisions with thermal states [15] or displaced squeezed states [16] as the inputs were also investigated. We addressed the precision using arbitrary states combined with Fock states and parity detection [17].

With the development of nonlinear optics, there has been increasing interest in second-order nonlinear phase estimation. Recently, many studies have been devoted to this subject using nonlinear interferometers. Seth et al. investigated the precision of a nonlinear interferometer with intensity detection [18]. Jiao et al. proposed a linear-nonlinear nested interferometer and studied the precision using coherent states and homodyne detection [19]. More recently, Chang et al. analyzed the precision of a nonlinear interferometer based on coherent states and homodyne detection [20].

The aforementioned schemes show the precision surpassing the shot-noise scaling. However, phase matching and gain matching of the nonlinear interferometer increase the difficulty of implementation. In addition, the second OPA can absorb the photons stimulated by the first OPA, which is deleterious to precision. To circumvent these problems, some modified nonlinear interferometers have been proposed, such as pumped-up nonlinear interferometers [21], truncated nonlinear interferometers [22,23], and generalized truncated nonlinear interferometers [24–26]. In the context of linear phase estimation, these interferometers have showed superior precision with coherent states as the input. In this paper, we show the nonlinear phase estimation via a nonlinear-linear hybrid interferometer. The precision of our scheme is compared with that of a conventional nonlinear interferometer as well as the quantum Cramér–Rao bound. Our scheme is simple in structure, and the precision is significantly improved at low gain when compared with the scheme using a nonlinear interferometer.

The remainder of this paper is organized as follows. In Section 2, we show the estimation scheme and analyze the precision. In Section 3, we compare the precision of our scheme with that of a conventional nonlinear interferometer and the quantum Cramér–Rao bound. Section 4 discusses the effects of transmission loss and detection loss on the precision. Finally, we summarize our results in Section 5.

2. ESTIMATION SCHEME AND MEASUREMENT PRECISION

In Fig. 1, we show the schematic of our estimation scheme. The whole setup can be regarded as a nonlinear-linear hybrid interferometer consisting of an OPA and a beam splitter with arbitrary transmissivity. For any non-zero transmissivity, our scheme is equal to a truncated nonlinear interferometer with balanced homodyne detection at two outputs [26]. In particular, our scheme reduces to the truncated nonlinear interferometer when the transmissivity sits at zero. A coherent state $| \alpha \rangle$ is injected into the OPA with a strong pump beam. Subsequently, the input is split into two states, one of which passes through a second-order nonlinear phase shift of $\varphi$, the parameter we would like to estimate. Finally, these two states are recombined by the beam splitter, and the amplitude quadrature of one of the outputs is obtained by balanced homodyne detection.

 figure: Fig. 1.

Fig. 1. Schematic of estimation scheme for nonlinear phase shifts based on a nonlinear-linear hybrid interferometer. P, pump; OPA, optical parametric amplifier; BS, beam splitter; BHD, balanced homodyne detection.

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For our scheme, the phase operator can be written as ${\hat U_{{\rm{NP}}}} = \exp [i{({\hat a^\dagger}\hat a)^2}\varphi]$, and the measurement operator is given by $\hat X = {\hat a^\dagger} + \hat a$, where ${\hat a^\dagger}$ and $\hat a$ are the creation and annihilation operators, respectively. In what follows, we assume that the phases of the coherent state and the OPA are $\theta$ and 0, respectively. Related to this, the operator relationship between the output and input modes is given by

$${\hat a_{{\rm{out}}}} = {\hat a_{{\rm{in}}}}\cosh g + \hat b_{{\rm{in}}}^\dagger \sinh g,$$
$${\hat b_{{\rm{out}}}} = {\hat b_{{\rm{in}}}}\cosh g + \hat a_{{\rm{in}}}^\dagger \sinh g,$$
with $g$ being the gain parameter of the OPA.

Based on this relationship and the input, the mean photon number of our scheme is found to be

$$N = {| \alpha |^2}\cosh (2g) + 2\mathop {\sinh}\nolimits^2g,$$
where ${| \alpha |^2}$ is the mean photon number of the input.

In order to calculate the precision, we give the following lemma (see Appendix A for details):

$$\left\langle \alpha \right|\hat af(\hat{a}^\dagger\hat a)| \alpha \rangle = \left\langle \alpha \right|f({\hat a^\dagger}\hat a + 1)\hat a| \alpha \rangle ,$$
and the formula
$$\left\langle {\alpha ,0} \right|\exp (c\hat{a}^\dagger\hat{a})| {\alpha ,0} \rangle = \exp [({e^c} - 1 )| \alpha |^2 ].$$

Using Eqs. (4) and (5), the expectation value of the measurement operator can be expressed as

$$\langle \hat X\rangle = | \alpha |[{e^{i({\varphi + \theta} )}}\sqrt \eta {K_2}\cosh g + i{e^{- i\theta}}\sqrt {1 - \eta} \sinh g] + {\rm{H.c.}},$$
where H.c. stands for the Hermitian conjugate and $\eta$ is the transmissivity of the beam splitter. The specific form of ${K_m}$ is given by
$$\begin{split}{K_m} &= \langle {\alpha ,0} |\hat U_{{\rm{OPA}}}^\dagger \exp (im\varphi {\hat a^\dagger}\hat a){\hat U_{{\rm{OPA}}}}| {\alpha ,0} \rangle \\ &= \frac{1}{{{{\cosh}^2}g - {e^{\textit{im}\varphi}}{{\sinh}^2}g}}\exp \left[{\frac{{({e^{\textit{im}\varphi}} - 1){{\cosh}^2}g}}{{{{\cosh}^2}g - {e^{\textit{im}\varphi}}{{\sinh}^2}g}}{{| \alpha |}^2}} \right].\end{split}$$

Similarly, the expectation value of the square of measurement operator is given by

$$\begin{split}\langle {\hat X^2}\rangle& = \big[{e^{i2({\varphi + \theta} )}}\eta {{| \alpha |}^2}{K_4}{{\cosh}^2}g + i{e^{i2\varphi}}\sqrt {\eta ({1 - \eta} )} ({{| \alpha |}^2} + 1)\\&\quad\times{K_2}\sinh ({2g} ) - {e^{- i2\theta}}({1 - \eta} ){{| \alpha |}^2}{{\sinh}^2}g\\&\quad - i{e^{i({\varphi + 2\theta} )}}\sqrt {\eta ({1 - \eta} )} {{| \alpha |}^2}{K_2}\sinh ({2g} ) \big] +{\rm H.c.}\\&\quad+ 2{| \alpha |^2}({\sinh ^2}g + \eta) + \cosh ({2g} ).\end{split}$$

In Fig. 2, we show the dependence of the expectation value of the measurement operator on the estimated phase. Figures 2(a) and 2(b) plot the expectation value with different gain parameters and mean photon numbers of the input, respectively. It can be seen from figures that the slope of expectation values becomes sharp and even multi-fold oscillation appears with an increase of the gain parameter or mean photon number; meanwhile, the expectation values deviating from the extrema approach zero. This result reveals that the sensitivity of the expectation value to the estimated phase increases with the increase of the gain parameter or mean photon number.

 figure: Fig. 2.

Fig. 2. Expectation value of the measurement operator against the estimated phase. (a) $|\alpha {|^2} = 1$ and $g = 0.5$, 0.7, and 0.9; (b) $g = 0.7$ and $|\alpha {|^2} = 1$, 3, and 5.

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Based on Eqs. (7) and (8), one can calculate the precision in terms of the error propagation,

$$\Delta \varphi = \frac{{\sqrt {\langle {{\hat X}^2}\rangle - {{\langle \hat X\rangle}^2}}}}{{| {\partial \langle \hat X\rangle /\partial \varphi} |}}.$$

To obtain the optimal precision, one needs to optimize the phase of the input and the transmissivity of the beam splitter. Mathematically, the simultaneous optimization of two parameters requires the use of the derivative method. While this method is strict, it is not direct or simple since the expression of the precision is complex. Here we determine the optimal parameters by observing the curve of the precision. We plot the optimal precision as a function of the transmissivity and the phase, as shown in Fig. 3. The results indicate that $\eta = 1$ and $\theta = \pi /2$ are the optimal parameters for our scheme. One can find that the optimal parameters remain the same with respect to other mean photon numbers of the input and the gain parameters. In addition, $\eta = 1$ is equivalent to the removal of the beam splitter. In other words, the truncated nonlinear interferometer is the optimal configuration for our scheme. This means that our scheme merely needs a single OPA and is much simpler than a nonlinear scheme in structure.

 figure: Fig. 3.

Fig. 3. Optimal precision with $|\alpha {|^2} = 1$ and $g = 0.7$ against transmissivity $\eta$ of the beam splitter and phase $\theta$ of the coherent state. $\eta$ ranges from 0.01 to 1 and $\theta$ ranges from 0 to $\pi$.

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 figure: Fig. 4.

Fig. 4. Shot-noise scaling (red dashed line), Heisenberg scaling (blue solid line), and precision of our scheme (green dotted line) against gain parameter $g$ with (a) $|\alpha {|^2} = 10$ and (b) $|\alpha {|^2} = 100$.

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Using the optimal parameters and Eq. (9), the optimal precision of our scheme turns out to be

$$\Delta {\varphi _{{\min}}} = \frac{{\sqrt {\cosh ({2g} )}}}{{2| \alpha |\cosh g\left[{2{{| \alpha |}^2}{{\cosh}^2}g + \cosh ({2g} )} \right]}}.$$

For our scheme, the optimal working point is $\varphi = 0$, and, at this time, we have ${K_m} = 0$. This conclusion is also reflected in Fig. 2 since the slope of the expectation value at $\varphi = 0$ is the steepest, meaning that the derivative $| {\partial \langle \hat X\rangle /\partial \varphi} |$ reaches its maximum.

For the second-order nonlinear phase estimation, shot-noise scaling and Heisenberg scaling are generally defined as ${N^{- 3/2}}$ and ${N^{- 2}}$, respectively [27,28]. Figure 4 gives the comparison between the precision and the shot-noise scaling and Heisenberg scaling. It can be seen that the precision lies between the shot-noise scaling and the Heisenberg scaling when $g \le 1$. For a small value of $|\alpha {|^2}$, the precision can even surpass the Heisenberg scaling at low gain, as shown in Fig. 4(a).

In addition, we simply show the advantage of nonlinear encoding by comparing our scheme with linear phase estimation. The precision of estimating the linear phase is given by [23], and the ratio of it to the precision of our scheme is defined as the advantage factor ${\cal A}$. Figure 5 shows the advantage factor as a function of gain parameter with different photon numbers. The results suggest that the advantage factor decreases with the increase of the gain parameter and finally tends to be a fixed value, which approximately equals the photon number.

 figure: Fig. 5.

Fig. 5. Advantage factor ${\cal A}$ against gain parameter $g$ with $|\alpha {|^2} = 10$ (blue solid line), $|\alpha {|^2} = 100$ (red dashed line), and $|\alpha {|^2} = 1000$ (green dotted line).

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3. COMPARISON WITH A NONLINEAR SCHEME AND THE QUANTUM CRAMÉR–RAO BOUND

In the previous section, we calculated the precision of our scheme. To demonstrate its advantage over a nonlinear scheme, in this section, we compare the precisions of these two schemes. In [20], the precision of a nonlinear scheme is given by

$$\Delta {\varphi _{{\rm{N}}}} = \frac{1}{{2| \alpha |{{\sinh}^2}g\left[{1 + 2{{\sinh}^2}g({{| \alpha |}^2} + 2)} \right]}}.$$

In order to compare the precisions of these two schemes, we define the ratio ${\cal E} = \Delta {\varphi _{\rm{N}}}/\Delta {\varphi _{{\min}}}$ as the enhancement factor. In Fig. 6, we show the enhancement factor as a function of the gain parameter and the mean photon number of the input. The blank region corresponds to the enhancement factor less than 1. For a given gain parameter, the enhancement factor remains approximately the same with the increase of the mean photon number. On the other hand, the enhancement factor decreases rapidly when increasing the gain parameter. In particular, when the gain parameter is greater than 1, the enhancement factor is less than 1. These results suggest that our scheme has the precision advantage at low gain and the precision can be improved by a factor of several orders of magnitude. In terms of Figs. 4 and 6, $g = 1$ is the watershed between our scheme and the conventional scheme based on a nonlinear interferometer. When the gain parameter is greater than 1, the conventional scheme has the advantage in precision.

 figure: Fig. 6.

Fig. 6. Enhancement factor ${\cal E}$ against the gain parameter $g$ and mean photon number $|\alpha {|^2}$ of the coherent state. $g$ ranges from 0 to 1.2, and $|\alpha {|^2}$ ranges from 1 to 100. The values of the color bar are logarithmic scale $\lg {\cal E}$.

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Next we move on to the comparison between the precision and the quantum Cramér–Rao bound, which is the fundamental limit to the precision. It is known to us that the quantum Cramér–Rao bound is the inverse square root of the quantum Fisher information. For a scheme with pure states as the input, the quantum Fisher information can be calculated in terms of

$${{\cal F}_{\rm{Q}}} = 4\left({\left\langle {\psi ^\prime \left| {\psi ^\prime} \right.} \right\rangle - \left\langle {\psi ^\prime \left| \psi \right.} \right\rangle \left\langle {\psi \left| {\psi ^\prime} \right.} \right\rangle} \right).$$
As for our scheme, the state is given by $| \psi \rangle = {\hat U_{ {{\rm{PS}}}}}{\hat U_{ {{\rm{OPA}}}}}| {\alpha ,0}\rangle$, and $| {\psi ^\prime} \rangle$ is the derivative of the state with respect to the estimated phase. Further, one can find that the quantum Fisher information can be rewritten as
$${{\cal F}_{\rm{Q}}} = 4\left({\langle {{({{\hat a}^\dagger}\hat a)}^4}\rangle - {{\langle {{({{\hat a}^\dagger}\hat a)}^2}\rangle}^2}} \right),$$
where the expectation values are taken with respect to the state ${\hat U_{ {{\rm{OPA}}}}}| {\alpha ,0} \rangle$.

By means of Eqs. (1) and (2), the quantum Fisher information for our scheme turns out to be

$$\begin{split}{{\cal F}_{\rm{Q}}}& = 4\left[\mathop {\cosh}\nolimits^8 g(4{{| \alpha |}^6} + 6{{| \alpha |}^4} + {{| \alpha |}^2}) + \mathop {\cosh}\nolimits^6 g\mathop {\sinh}\nolimits^2 g(4{{| \alpha |}^6} \right.\\&\quad+ 32{{| \alpha |}^4} + 22{{| \alpha |}^2} + 2) + \mathop {\cosh}\nolimits^4 g\mathop {\sinh}\nolimits^4 g(11{{| \alpha |}^4}\\&\quad \left. + 40{{| \alpha |}^2} + 9) + \mathop {\cosh}\nolimits^2 g\mathop {\sinh}\nolimits^6 g(3{{| \alpha |}^2} + 6) \right].\end{split}$$

To analyze the optimality of our scheme, we consider the ratio defined as ${\cal R} = \Delta {\varphi _{{\rm{QCRB}}}}/\Delta {\varphi _{{\min}}}$. Only when the ratio sits at 1 does the readout strategy become optimal. In order to intuitively observe the variation of this ratio, we plot the ratio as a function of the gain parameter and mean photon number of the input in Fig. 7. The results show that, when the input photon number is greater than 10, the increase of the mean photon number has little effect on the ratio. That is, this ratio only changes with the gain parameter for $|\alpha {|^2} \gt 10$. On the other hand, the ratio approaches one when the gain parameter decreases, indicating that the precision can be approximately saturated by the quantum Cramér–Rao bound. As can be seen from Figs. 6 and 7, our scheme is superior to the nonlinear scheme and is the nearly optimal at low gain.

 figure: Fig. 7.

Fig. 7. Ratio ${\cal R}$ against the gain parameter $g$ and mean photon number $|\alpha {|^2}$ of the coherent state. $g$ ranges from 0 to 1, and $|\alpha {|^2}$ ranges from 1 to 100.

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4. EFFECTS OF DETECTION LOSS AND TRANSMISSION LOSS ON THE PRECISION

In this section, we study the effect of photon loss on the precision. Photon loss is the most common factor in the practical measurements. According to the theory of quantum open systems, the channel with lossy ratio $L$ can be simulated by placing a beam splitter with reflectivity of $L$. For linear phase estimation using linear interferometers, the photon loss is equivalent at any stage due to the fact that all processes are linear operations. This conclusion is not applicable to our scheme, in that the OPA and nonlinear phase shift are both nonlinear elements. In accordance with the stage of these nonlinear elements, the photon loss in our scheme can be divided into three types: preparation loss, transmission loss, and detection loss. We are not concerned here with the preparation loss because the input in our scheme is a coherent state. Specifically speaking, the preparation loss does not affect the properties of the input and only reduces the photon number into the OPA. That is, all the conclusions in the previous sections are applicable after replacing $|\alpha |$ with $\sqrt {1 - L} |\alpha |$.

We first consider the effect of detection loss. In terms of the transformation of the beam splitter to the mode operator, the expectation value of measurement operator can be written as

$$\langle {\hat X_{{{\rm{DL}}}}}\rangle = \sqrt {1 - L} \langle \hat X\rangle ,$$
and that of the square of measurement operator is given by
$$\langle \hat X_{{{\rm{DL}}}}^2\rangle = ({1 - L} )\langle {\hat X^2}\rangle + L,$$

Now we turn our attention to the effect of transmission loss. After some algebra, the expectation values are found to be

$$\langle {\hat X_{{\rm{TL}}}}\rangle = i\sqrt {1 - L} | \alpha |{e^{{i\varphi}}}{K_{2({1 - L} )}}\cosh g + {{\rm{H.c.}}}$$
and
$$\begin{split}\langle \hat X_{{{\rm{TL}}}}^2\rangle &= - ({1 - L} ){e^{i2\varphi}}{| \alpha |^2}{K_{4({1 - L} )}} {\cosh}^2g + {{\rm{H.c.}}}\\&\quad + 2({1 - L} )[{{{| \alpha |}^2}({{\sinh}^2}g + 1)} ] + \cosh (2g).\end{split}$$

Using these equations and the error propagation, we can calculate the precisions in the presence of detection loss and transmission loss, respectively. Figure 8 shows the precision as a function of lossy ratio. For the same lossy ratio, the precision degradation caused by detection loss is more serious than that caused by transmission loss. Furthermore, the degradation in precision is slight with respect to detection loss and, however, is obvious regarding transmission loss. When the lossy ratio reaches 40%, the precision in the presence of transmission loss degrades to about half that in the lossless scenario. The reason for this difference is that the precision degradation originating from the transmission loss is amplified by the nonlinear phase process.

 figure: Fig. 8.

Fig. 8. Precisions in the presence of detection loss (red dashed line) and transmission loss (blue solid line) against lossy ratio, $g = 0.6$, $|\alpha {|^2} = 10$. The green dotted line is the QCRB.

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By combining Eqs. (10), (15), and (16), we find that the difference between the precisions with and without detection loss is only a prefactor,

$$\Delta {\varphi _{{\rm{DL}}}} = \sqrt {1 + \frac{L}{{(1 - L)\cosh (2g)}}} \Delta {\varphi _{{\min}}}.$$

For a large $g$, we have $\cosh (2g) \gg 1$, and the prefactor approaches 1. This also explains the slight degradation of the precision in the presence of detection loss. That is, our scheme is robust against photon loss.

5. CONCLUSION

In summary, we focused on nonlinear phase estimation and proposed the scheme using a nonlinear-linear hybrid interferometer. We deployed a coherent state as the input and performed the balanced homodyne detection at the output. For a low gain parameter, the precision of our scheme is superior to that of a nonlinear scheme and can be approximately saturated by the quantum Cramér–Rao bound. Additionally, we analyzed the effects of transmission loss and detection loss on the precision. The results suggest that the effect of transmission loss on precision is greater than that of detection loss; meanwhile, our scheme is robust against photon loss. Given the simple structure and resources, our scheme may make a step toward low-gain practical quantum interferometry.

APPENDIX A

Here we give the proofs of Eqs. (4), (5), and (7).

A1. Proof of Eq. (4)

The operator $f({\hat a^\dagger}\hat a)$ can be decomposed as

$$f({\hat a^\dagger}\hat a) = \sum\limits_{n = 0}^\infty f(n )| n \rangle \left\langle n \right|,$$
and the corresponding expectation value with coherent state is
$$\left\langle \alpha \right|f({\hat a^\dagger}\hat a)| \alpha \rangle = \sum\limits_{n = 0}^\infty f(n ){e^{- {\alpha ^*}\alpha}}\frac{{{{\left({{\alpha ^*}\alpha} \right)}^n}}}{{n!}}.$$

Based on the lemma,

$$| \alpha \rangle \left\langle \alpha \right|\hat a = \left({\alpha + \frac{\partial}{{\partial {\alpha ^*}}}} \right)| \alpha \rangle \left\langle \alpha \right|,$$
we have
$$\left\langle \alpha \right|\hat af({\hat a^\dagger}\hat a)| \alpha \rangle = \sum\limits_{n = 1}^\infty f(n ){e^{- {\alpha ^*}\alpha}}\frac{{{\alpha ^{*n - 1}}{\alpha ^n}}}{{({n - 1} )!}}.$$

By replacing $n$ with $n + 1$, we can get Eq. (4).

A2. Proof of Eq. (7)

In the coherent state representation, we have

$$\exp (im\varphi {\hat a^\dagger}\hat a) = \int \frac{{{d^2}{z_1}}}{\pi}\int \frac{{{d^2}{z_2}}}{\pi}\left| {{e^{\textit{im}\varphi}}{z_1}} \right\rangle | {{z_2}} \rangle \langle {{z_1}} |\left\langle {{z_2}} \right|.$$

According to the normal ordered form, we can calculate the following expression:

$$\begin{split}{\hat U_{{\rm{OPA}}}}| {\alpha ,0} \rangle& = \frac{1}{{\cosh g}}\\&\quad\times\exp \left(\!{- \frac{{{{| \alpha |}^2}}}{2} + \frac{\alpha}{{\cosh g}}{{\hat a}^\dagger} + \tanh g{{\hat a}^\dagger}{{\hat b}^\dagger}}\! \right)| {0,0} \rangle .\end{split}$$

Further, by using the integral formula,

$$\int \frac{{{d^2}z}}{\pi}\exp ({\zeta {{| z |}^2} + \xi z + \eta {z^*}} ) = - \frac{1}{\zeta}\exp \left({- \frac{{\xi \eta}}{\zeta}} \right),$$
we can obtain Eq. (4).

A3. Proof of Eq. (5)

It can be found that Eq. (5) is a special case with respect to Eq. (7). By taking $g = 0$ and $im\varphi = c$, the result given by Eq. (7) reduces to Eq. (5).

Funding

Natural Science Research of Jiangsu Higher Education Institutions of China (21KJB140007); National Natural Science Foundation of China (12104193).

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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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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Figures (8)
Fig. 1.
Fig. 1. Schematic of estimation scheme for nonlinear phase shifts based on a nonlinear-linear hybrid interferometer. P, pump; OPA, optical parametric amplifier; BS, beam splitter; BHD, balanced homodyne detection.

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Fig. 2.
Fig. 2. Expectation value of the measurement operator against the estimated phase. (a) $|\alpha {|^2} = 1$ and $g = 0.5$, 0.7, and 0.9; (b) $g = 0.7$ and $|\alpha {|^2} = 1$, 3, and 5.

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Fig. 3.
Fig. 3. Optimal precision with $|\alpha {|^2} = 1$ and $g = 0.7$ against transmissivity $\eta$ of the beam splitter and phase $\theta$ of the coherent state. $\eta$ ranges from 0.01 to 1 and $\theta$ ranges from 0 to $\pi$.

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Fig. 4.
Fig. 4. Shot-noise scaling (red dashed line), Heisenberg scaling (blue solid line), and precision of our scheme (green dotted line) against gain parameter $g$ with (a) $|\alpha {|^2} = 10$ and (b) $|\alpha {|^2} = 100$.

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Fig. 5.
Fig. 5. Advantage factor ${\cal A}$ against gain parameter $g$ with $|\alpha {|^2} = 10$ (blue solid line), $|\alpha {|^2} = 100$ (red dashed line), and $|\alpha {|^2} = 1000$ (green dotted line).

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Fig. 6.
Fig. 6. Enhancement factor ${\cal E}$ against the gain parameter $g$ and mean photon number $|\alpha {|^2}$ of the coherent state. $g$ ranges from 0 to 1.2, and $|\alpha {|^2}$ ranges from 1 to 100. The values of the color bar are logarithmic scale $\lg {\cal E}$.

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Fig. 7.
Fig. 7. Ratio ${\cal R}$ against the gain parameter $g$ and mean photon number $|\alpha {|^2}$ of the coherent state. $g$ ranges from 0 to 1, and $|\alpha {|^2}$ ranges from 1 to 100.

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Fig. 8.
Fig. 8. Precisions in the presence of detection loss (red dashed line) and transmission loss (blue solid line) against lossy ratio, $g = 0.6$, $|\alpha {|^2} = 10$. The green dotted line is the QCRB.

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Equations (26)

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a ^ o u t = a ^ i n cosh ⁡ g + b ^ i n † sinh ⁡ g ,
b ^ o u t = b ^ i n cosh ⁡ g + a ^ i n † sinh ⁡ g ,
N = | α | 2 cosh ⁡ ( 2 g ) + 2 sinh 2 ⁡ g ,
⟨ α | a ^ f ( a ^ † a ^ ) | α ⟩ = ⟨ α | f ( a ^ † a ^ + 1 ) a ^ | α ⟩ ,
⟨ α , 0 | exp ⁡ ( c a ^ † a ^ ) | α , 0 ⟩ = exp ⁡ [ ( e c − 1 ) | α | 2 ] .
⟨ X ^ ⟩ = | α | [ e i ( φ + θ ) η K 2 cosh ⁡ g + i e − i θ 1 − η sinh ⁡ g ] + H . c . ,
K m = ⟨ α , 0 | U ^ O P A † exp ⁡ ( i m φ a ^ † a ^ ) U ^ O P A | α , 0 ⟩ = 1 cosh 2 g − e im φ sinh 2 g exp ⁡ [ ( e im φ − 1 ) cosh 2 g cosh 2 g − e im φ sinh 2 g | α | 2 ] .
⟨ X ^ 2 ⟩ = [ e i 2 ( φ + θ ) η | α | 2 K 4 cosh 2 g + i e i 2 φ η ( 1 − η ) ( | α | 2 + 1 ) × K 2 sinh ⁡ ( 2 g ) − e − i 2 θ ( 1 − η ) | α | 2 sinh 2 g − i e i ( φ + 2 θ ) η ( 1 − η ) | α | 2 K 2 sinh ⁡ ( 2 g ) ] + H . c . + 2 | α | 2 ( sinh 2 g + η ) + cosh ⁡ ( 2 g ) .
Δ φ = ⟨ X ^ 2 ⟩ − ⟨ X ^ ⟩ 2 | ∂ ⟨ X ^ ⟩ / ∂ φ | .
Δ φ min = cosh ⁡ ( 2 g ) 2 | α | cosh ⁡ g [ 2 | α | 2 cosh 2 g + cosh ⁡ ( 2 g ) ] .
Δ φ N = 1 2 | α | sinh 2 g [ 1 + 2 sinh 2 g ( | α | 2 + 2 ) ] .
F Q = 4 ( ⟨ ψ ′ | ψ ′ ⟩ − ⟨ ψ ′ | ψ ⟩ ⟨ ψ | ψ ′ ⟩ ) .
F Q = 4 ( ⟨ ( a ^ † a ^ ) 4 ⟩ − ⟨ ( a ^ † a ^ ) 2 ⟩ 2 ) ,
F Q = 4 [ cosh 8 ⁡ g ( 4 | α | 6 + 6 | α | 4 + | α | 2 ) + cosh 6 ⁡ g sinh 2 ⁡ g ( 4 | α | 6 + 32 | α | 4 + 22 | α | 2 + 2 ) + cosh 4 ⁡ g sinh 4 ⁡ g ( 11 | α | 4 + 40 | α | 2 + 9 ) + cosh 2 ⁡ g sinh 6 ⁡ g ( 3 | α | 2 + 6 ) ] .
⟨ X ^ D L ⟩ = 1 − L ⟨ X ^ ⟩ ,
⟨ X ^ D L 2 ⟩ = ( 1 − L ) ⟨ X ^ 2 ⟩ + L ,
⟨ X ^ T L ⟩ = i 1 − L | α | e i φ K 2 ( 1 − L ) cosh ⁡ g + H . c .
⟨ X ^ T L 2 ⟩ = − ( 1 − L ) e i 2 φ | α | 2 K 4 ( 1 − L ) cosh 2 g + H . c . + 2 ( 1 − L ) [ | α | 2 ( sinh 2 g + 1 ) ] + cosh ⁡ ( 2 g ) .
Δ φ D L = 1 + L ( 1 − L ) cosh ⁡ ( 2 g ) Δ φ min .
f ( a ^ † a ^ ) = ∑ n = 0 ∞ f ( n ) | n ⟩ ⟨ n | ,
⟨ α | f ( a ^ † a ^ ) | α ⟩ = ∑ n = 0 ∞ f ( n ) e − α ∗ α ( α ∗ α ) n n ! .
| α ⟩ ⟨ α | a ^ = ( α + ∂ ∂ α ∗ ) | α ⟩ ⟨ α | ,
⟨ α | a ^ f ( a ^ † a ^ ) | α ⟩ = ∑ n = 1 ∞ f ( n ) e − α ∗ α α ∗ n − 1 α n ( n − 1 ) ! .
exp ⁡ ( i m φ a ^ † a ^ ) = ∫ d 2 z 1 π ∫ d 2 z 2 π | e im φ z 1 ⟩ | z 2 ⟩ ⟨ z 1 | ⟨ z 2 | .
U ^ O P A | α , 0 ⟩ = 1 cosh ⁡ g × exp ⁡ ( − | α | 2 2 + α cosh ⁡ g a ^ † + tanh ⁡ g a ^ † b ^ † ) | 0 , 0 ⟩ .
∫ d 2 z π exp ⁡ ( ζ | z | 2 + ξ z + η z ∗ ) = − 1 ζ exp ⁡ ( − ξ η ζ ) ,
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