Phase-insensitive amplifier gain estimation at Cramér-Rao bound for two-mode squeezed state of light

Hailong Wang; Hailong Wang; Hailong Wang; Zehua Chen; Zhongxing Fu; Yunpeng Shi; Xiong Zhang; Chunliu Zhao; Shangzhong Jin; Jietai Jing; Jietai Jing; Jietai Jing; Jietai Jing

doi:10.1364/OE.483770

1. Introduction

Squeezed states of light are a class of nonclassical light states with noise below the standard quantum limit in one quadrature component, either the amplitude or phase quadrature [1]. Due to this unique quantum feature, they have enormous applications in quantum precision measurement [2,3] and quantum sensing [4,5]. For instance, single-mode squeezed states (SMSSs) of light are employed in laser interferometer gravitational–wave observation [6–8] and in atomic magnetometers [9–11] to enhance their detection sensitivities. However, due to the use of only one sensing light in an SMSS, it is hard to eliminate all the technical noise in practice. Consequently, intensity difference measurement from a bright two-mode squeezed state (TMSS) [12,13] is an effective technique for canceling all the technical noise in the sensing process. As would be expected, the bright TMSSs have also been combined with surface plasmon resonance technology to detect small refractive index changes [14–19]. In general, all the above plasmonic mechanisms can be attributed to transmission (${T}$) sensing models based on the bright TMSSs. Transmission estimation based on bright TMSSs has also been investigated thoroughly using the quantum Cramér-Rao bound (QCRB) [20], which is a quantitative parameter for comparing the precision limits of transmission when using either pure or mixed states [21].

With reference to transmission sensing models, phase-insensitive amplifiers (PIAs) have been used to study the effects of amplification on multiple quantum correlation and multipartite quantum entanglement. For example, R. C. Pooser ${et al.}$ demonstrated that the amplification of one half of a TMSS using one PIA is possible while preserving entanglement [22]. In addition, the effects of a single gain parameter from one PIA on triple quantum correlation and tripartite quantum entanglement were analyzed [23–26]. The dependence of quadruple quantum correlation and quadripartite quantum entanglement on two gain parameters from two PIAs was also discussed [27–29]. In performing these studies, the gain value of a PIA can always be calculated as the ratio of the amplified output light power to the input light power; however, its estimation precision has not been investigated yet.

Therefore, this work serves as an extensive theoretical study of PIA gain estimation precision based on the bright TMSSs. It is organized as follows: In Sec. 2, we outline the basic theoretical framework for calculating the QCRB [30,31] of the PIA gain based on the TMSSs; then, we obtain the estimation function from the vacuum TMSS, the estimation function of the bright TMSS, and the estimation function of the coherent state. In Sec. 3, we consider another PIA with gain ${M}$ external to the system and study its noise effect on the estimation function of the bright TMSS. The results show that a scheme in which another PIA is placed on the auxiliary light beam path is more robust than two other schemes due to the weaker dependence on the squeezing parameter s and gain M. In Sec. 4, a fictitious beam splitter with transmission T is also used to simulate the noise effects of propagation loss and imperfect detection on the estimation function of the bright TMSS, and the results show that a scheme in which a fictitious beam splitter is placed before the original PIA in the probe light beam path is most robust. In Sec. 5, optimal intensity difference measurement is identified as an experimental technique that can saturate the QCRB of the bright TMSS. In Sec. 6, we briefly present the conclusions drawn from these results.

2. Estimation functions from vacuum TMSS, bright TMSS, and coherent state

The theoretical model for calculating the estimation function of the PIA gain is shown in Fig. 1. The coherent state $\hat {a}_{c}$ as the seed beam, vacuum state $\hat {a}_{0}$, and pump light (not shown in Fig. 1 for clarity) are all injected into a parametric amplifier (PA), in which the bright TMSS consisting of the amplified probe light $\hat {a}_{p0}$ and new auxiliary light $\hat {a}_{a0}$ is generated. The input-output relation of the bright TMSS in the PA obeys the following relation:

(1)$$\begin{aligned}\hat{a}_{p0}&= \cosh(s)\hat{a}_{c}+\sinh(s)\hat{a}_{0}^{\dagger},\\ \hat{a}_{a0}&= \cosh(s)\hat{a}_{0}+\sinh(s)\hat{a}_{c}^{\dagger}, \end{aligned}$$

where s is the squeezing parameter of the bright TMSS and a larger s value corresponds to a higher photon pair generation rate, and $\hat {a}_{0}^{\dagger }$ and $\hat {a}_{c}^{\dagger }$ are the creation operators of the corresponding light beams. Then, the light beam $\hat {a}_{p0}$ is also coupled into the following PIA to estimate the gain G using a similar input-output relation, i.e., $\hat {a}_{p}=\sqrt {G}\hat {a}_{p0}+\sqrt {G-1}\hat {v}^{\dagger }$, and the final twin beams can be expressed as

(2)$$\begin{aligned}\hat{a}_{p}&= \sqrt{G}\cosh(s)\hat{a}_{c}+\sqrt{G}\sinh(s)\hat{a}_{0}^{\dagger}+\sqrt{G-1}\hat{v}^{\dagger},\\ \hat{a}_{a}&= \cosh(s)\hat{a}_{0}+\sinh(s)\hat{a}_{c}^{\dagger}, \end{aligned}$$

where G and $\hat {v}$ are the gain and vacuum input, respectively, for the PIA process.

Fig. 1. Theoretical model for calculating the estimation function of the PIA gain G. $\hat {a}_{c}$, $\hat {a}_{0}$, and $\hat {v}$ are the input light beams; $\hat {a}_{p0}$ and $\hat {a}_{a0}$ are the generated probe and auxiliary light beams, respectively, from the PA; and $\hat {a}_{p}$ and $\hat {a}_{a}$ are the two output light beams. s and G are the squeezing parameter of the PA and gain of the PIA, respectively.

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Following the technique outlined in Refs. [21,30], the bright TMSS can be completely described by the following covariance matrix $\sigma$ and mean displacement vector $\mathbf {d}$. First, the covariance matrix can be expressed as

(3)$$\sigma=2Cov\left[ \begin{matrix} (\hat{a}_{p}, \hat{a}_{p}^{\dagger}) & (\hat{a}_{p}, \hat{a}_{a}^{\dagger}) & (\hat{a}_{p}, \hat{a}_{p}) & (\hat{a}_{p}, \hat{a}_{a}) \\ (\hat{a}_{a}, \hat{a}_{p}^{\dagger}) & (\hat{a}_{a}, \hat{a}_{a}^{\dagger}) & (\hat{a}_{a}, \hat{a}_{p}) & (\hat{a}_{a}, \hat{a}_{a}) \\ (\hat{a}_{p}^{\dagger}, \hat{a}_{p}^{\dagger}) & (\hat{a}_{p}^{\dagger}, \hat{a}_{a}^{\dagger}) & (\hat{a}_{p}^{\dagger}, \hat{a}_{p}) & (\hat{a}_{p}^{\dagger}, \hat{a}_{a}) \\ (\hat{a}_{a}^{\dagger}, \hat{a}_{p}^{\dagger}) & (\hat{a}_{a}^{\dagger}, \hat{a}_{a}^{\dagger}) & (\hat{a}_{a}^{\dagger}, \hat{a}_{p}) & (\hat{a}_{a}^{\dagger}, \hat{a}_{a}) \end{matrix} \right],$$

where the diagonal terms are $Cov(\hat {a}_{p}, \hat {a}_{p}^{\dagger })=\langle \hat {a}_{p}\hat {a}_{p}^{\dagger }+\hat {a}_{p}^{\dagger }\hat {a}_{p}\rangle /2-\langle \hat {a}_{p}\rangle \langle \hat {a}_{p}^{\dagger }\rangle$ and $Cov(\hat {a}_{p}^{\dagger }, \hat {a}_{p})=\langle \hat {a}_{p}^{\dagger }\hat {a}_{p}+\hat {a}_{p}\hat {a}_{p}^{\dagger }\rangle /2-\langle \hat {a}_{p}^{\dagger }\rangle \langle \hat {a}_{p}\rangle$, and the nondiagonal terms are $Cov(\hat {a}_{p}, \hat {a}_{a})=\langle \hat {a}_{p}\hat {a}_{a}\rangle -\langle \hat {a}_{p}\rangle \langle \hat {a}_{a}\rangle$ for the upper-right half and $Cov(\hat {a}_{a}^{\dagger }, \hat {a}_{p}^{\dagger })=\langle \hat {a}_{a}^{\dagger }\hat {a}_{p}^{\dagger }\rangle -\langle \hat {a}_{a}^{\dagger }\rangle \langle \hat {a}_{p}^{\dagger }\rangle$ for the lower-left half. Using Eq. (2) and the commutation relation $[\hat {a}_{i}, \hat {a}_{j}^{\dagger }]=\delta _{ij}$ (Dirac function), we calculate each element in the covariance matrix as follows: $2Cov(\hat {a}_{p}, \hat {a}_{p}^{\dagger })=2Cov(\hat {a}_{p}^{\dagger }, \hat {a}_{p})=G-1+G\cosh (2s)$, $2Cov(\hat {a}_{a}, \hat {a}_{a}^{\dagger })=2Cov(\hat {a}_{a}^{\dagger }, \hat {a}_{a})=\cosh (2s)$, $2Cov(\hat {a}_{p}, \hat {a}_{a})=2Cov(\hat {a}_{a}, \hat {a}_{p})=2Cov(\hat {a}_{p}^{\dagger }, \hat {a}_{a}^{\dagger })=2Cov(\hat {a}_{a}^{\dagger }, \hat {a}_{p}^{\dagger })=\sqrt {G}\sinh (2s)$, and the remaining elements are all equal to zero. After arranging the above results, the covariance matrix $\sigma$ in Eq. (3) can be simply expressed as

(4)$$\sigma=\left[ \begin{matrix} G-1+G\cosh(2s) & 0 & 0 & \sqrt{G}\sinh(2s) \\ 0 & \cosh(2s) & \sqrt{G}\sinh(2s) & 0 \\ 0 & \sqrt{G}\sinh(2s) & G-1+G\cosh(2s) & 0 \\ \sqrt{G}\sinh(2s) & 0 & 0 & \cosh(2s) \end{matrix} \right].$$

Second, the mean displacement vector yields the following result:

(5)$$\mathbf{d}=\left[ \begin{matrix} \sqrt{G}\cosh(s)\alpha_{c} \\ \sinh(s)\alpha_{c}^{*} \\ \sqrt{G}\cosh(s)\alpha_{c}^{*} \\ \sinh(s)\alpha_{c} \end{matrix} \right],$$

where $\alpha _{c}$ is the complex amplitude of coherent state $\hat {a}_{c}$ and the vacuum state $\alpha _{0}$ has a complex amplitude of zero. It is easily observed that Eq. (5) is given by the mean values of the annihilation operators for the final twin light beams in Eq. (2).

From the results of Eqs. (4) and (5), we can calculate the estimation function using quantum Fisher information (QFI) instead of maximum likelihood estimation because the latter is a technique for estimating the parameters of a given distribution based on observed data. The QFI of a TMSS can be obtained from the following equation [30]:

(6)$$\begin{aligned}F_{Q}(G)&=\frac{1}{2(\left\vert \Sigma\right\vert -1)}\{\left\vert \Sigma\right\vert Tr[(\Sigma^{{-}1}\dot{\Sigma})^{2}] +\sqrt{\left\vert \Xi+\Sigma^{2}\right\vert }Tr[((\Xi+\Sigma^{2})^{{-}1} \dot{\Sigma})^{2}]+\\ &\qquad 4(\lambda_{1}^{2}-\lambda_{2}^{2})(\frac{\dot{\lambda_{2}}^{2}}{\lambda_{2} ^{4}-1}-\frac{\dot{\lambda_{1}}^{2}}{\lambda_{1}^{4}-1})\} +2\dot{\mathbf{d}}^{\dagger} \sigma^{{-}1} \dot{\mathbf{d}}, \end{aligned}$$

where $|\cdot |$ is the determinant of the corresponding matrix, $\Xi$ is the 4$\times$4 identity matrix, $\Sigma =k\sigma$ is the symplectic form of the covariance matrix in which $k=diag(1, 1, -1, -1)$, and $\dot {\Sigma }$ is the elementwise derivative with respect to the PIA gain G. Here, $\lambda _{1}$ and $\lambda _{2}$ are the two positive symplectic eigenvalues from $\Sigma$, which has four symplectic eigenvalues. In the present case, the two positive symplectic eigenvalues can be obtained from $\Sigma$ and expressed as

(7)$$\lambda_{1}=1,$$

and

(8)$$\lambda_{2}=G-\cosh(2s)+G\cosh(2s).$$

$\lambda _{1}$ is a constant, and its derivative is zero, while $\lambda _{2}$ depends on the gain G; thus, its derivative is $1+\cosh (2s)$. In addition, the derivative of the mean displacement vector from Eq. (5) can be written as

(9)$$\dot{\mathbf{d}}=\left[ \begin{matrix} \frac{\cosh(s)}{2}\sqrt{\frac{1}{G}}\alpha_{c} \\ 0 \\ \frac{\cosh(s)}{2}\sqrt{\frac{1}{G}}\alpha_{c}^{*} \\ 0 \end{matrix} \right],$$

in which the second and fourth terms are both equal to zero due to their independence from the gain G. Based on the results in Eqs. (4), (7), (8), and (9), the QFI in Eq. (6) can be calculated and simply expressed as

(10)$$\begin{aligned}F_{Q}(G) &=\frac{\sinh^{2}(s)}{G(G-1)\tanh^{2}(s)}+\frac{\cosh^{2}(s)n_{c}}{G[G\;\textrm{sech}(2s)+(G-1)]}\\ &=\frac{\left\langle \hat{n}_{p}\right\rangle ^{vac}}{G(G-1)\tanh^{2}(s)}+\frac{\left\langle \hat{n}_{p}\right\rangle ^{bright}}{G[G\;\textrm{sech}(2s)+(G-1)]}, \end{aligned}$$

where the first and second terms of the right-hand side are from the first three terms and the fourth term, respectively, in Eq. (6). As expressed in Eq. (10), $\left \langle \hat {n}_{p}\right \rangle ^{vac}=\sinh ^{2}(s)$ is the vacuum term characterizing the number of spontaneously generated photons, and $\left \langle \hat {n}_{p}\right \rangle ^{bright}=\cosh ^{2}(s)n_{c}$ is the bright seed term quantifying the number of stimulated photons. $n_{c}=|\hat {\alpha }_{c}|^{2}$ is the mean number of photons of coherent state seed light $\hat {a}_{c}$, and an $n_{c}$ value far greater than 1 will generate the bright TMSS, at which the value of $\left \langle \hat {n}_{p}\right \rangle ^{bright}$ is obviously larger than that of $\left \langle \hat {n}_{p}\right \rangle ^{vac}$. In estimating the gain G, the greater the number of probe photons is, the higher (lower) the estimation precision (estimation function value) is; thus, the gain estimation function is inversely proportional to the number of probe photons. Therefore, the QCRB for the vacuum TMSS is given by the inverse of its corresponding QFI [20]:

(11)$$\langle \Delta^{2}G \rangle^{vac}=\frac{G(G-1)\tanh^{2}(s)}{\left\langle \hat{n}_{p}\right\rangle ^{vac}},$$

here, the absolute QCRB also depends on the number of times the measurement is repeated, which is set to 1 for all cases for the following discussions about the relative advantages of the TMSS compared to the coherent state. Then, the gain estimation function is defined as the product of the QCRB in Eq. (11) and the number of probe photons (i. e., $\left \langle \hat {n}_{p}\right \rangle ^{vac}$). This makes it possible to analyze the estimation function independent of the number of probe photons while still analyzing the QCRB based on the number of probe photons. Thus, in the case of the vacuum TMSS, this function can be expressed as

(12)$$\Lambda^{vac}=G(G-1)\tanh^{2}(s),$$

as shown in Eq. (12), the estimation function of the vacuum TMSS tends toward 0 in the limit of $s\rightarrow 0$, increases with the increasing of $s$, and saturates at the value of $G(G-1)$ in the limit of $s\rightarrow \infty$. It should be noted that the input state will return to the vacuum state when $s=0$, even in this situation the amplification gain of PIA can also be estimated by measuring the mean number of spontaneously generated photons at the output of the amplifier. Using the same method, we can obtain the estimation function of the bright TMSS, i. e.,

(13)$$\Lambda^{bright}=G(G-1)+G^{2}\;\textrm{sech}(2s),$$

as expressed in Eq. (13), as s increases, the estimation function of the bright TMSS tends toward $G(G-1)$, returning to the case of the vacuum TMSS in the limit of $s\rightarrow \infty$. If s is set to zero, Eq. (13) becomes the estimation function of the coherent state, i. e., $G(2G-1)$. To discuss the relative advantage among the three estimation functions in the experimentally accessible parameter region, their dependence on the gain G and squeezing parameter s is shown in Fig. 2. As shown in Fig. 2, vacuum TMSS has the lower estimation function or higher estimation precision for a given probe number, while it can only be generated with a low mean probe photon number [32]. As we all know, amplification gain estimation enhancement can be reached by two approaches, increasing the probe photon number and utilizing quantum states of light to boost the estimation precision gained per probe photon. Therefore, the bright TMSS, which has a significantly higher mean probe number by seeding the PA used to generate them and can achieve the estimation precision of vacuum TMSS in the limit of large squeezing, should be used to overcome the limitation from the vacuum TMSS. Specifically, a probe light beam with mW-level power can be generated, and the squeezing parameter ${s}=1$ is equivalent to an intensity difference squeezing degree of $-5.8$ dB, which is easily implemented [13]. From this point of view, the bright TMSS gives an overall lower QCRB compared with the vacuum TMSS. Therefore, we will focus on the discussions about the bright TMSS in the following sections.

Fig. 2. Dependence of the estimation function from vacuum TMSS (black dashed line), estimation function of the coherent state (green dashed line), and estimation function of the bright TMSS (red solid line) on the gain G (a) and the squeezing parameter s (b). ${s}=1$ and ${G}=2$ are set in Fig. 1(a) and Fig. 1(b), respectively.

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3. Effect of noise from another PIA on the estimation function of the bright TMSS

We are motivated by Ref. [33], in which quantum noise properties from optical amplifiers and attenuators are discussed; Ref. [34], in which the effects of attenuation and amplification on the exact solution to the master equation of a nonlinear oscillator are presented and it is shown that amplification destroys quantum coherence more rapidly than does attenuation; Ref. [35], in which the behavior of the non-Gaussian state of light under the actions of probabilistic noiseless amplification and attenuation is examined and it is found that the mean-field amplitude may decrease (increase) in the process of noiseless amplification (attenuation); and Ref. [36], in which the authors analyze the general laws of continuous-variable entanglement dynamics during the deterministic attenuation and amplification of the physical signal carrying the entanglement. Therefore, the effects of noise from another PIA and a fictitious beam splitter on the estimation function of the bright TMSS will be discussed in this section and the next section, respectively. In this section, it is emphasized that the second and original PIAs have the same configuration as the PA for generating the bright TMSS, while they act as a noise source and gain medium, respectively, for one beam from the bright TMSS. As shown in Fig. 3(a), another PIA with gain M is first placed after the original PIA in the probe light beam path. Based on a similar calculation method, we obtain the estimation function for this case:

(14)$$\Lambda^{G-M}=G(2G-1)-G\{G[1-\textrm{sech}(2s)]-1+\frac{1}{M}\},$$

the second term on the right-hand side is modulated by the gain M, while Eq. (14) will be reduced to Eq. (13) when this extra PIA is absent, i.e., M is set to 1. First, the dependence of Eq. (14) on the gain G is represented by a green solid line in Fig. 4, and its estimation precision is less robust than that of the bright TMSS but more robust than that of the coherent state. Second, its dependence on the squeezing parameter s is represented by a green solid line in Fig. 5(a), and its value decreases with increasing s and approaches the value corresponding to the coherent state when $s=\{\operatorname {arcosh}[GM/(GM-M+1)]\}/2$. The dependence of this critical squeezing parameter s on the gain M is represented by a red solid line in Fig. 5(b). In the region above this red line, the estimation precision of Fig. 3(a) is always more robust than that of the coherent state; this is because more quantum sources can be used to enhance the estimation precision. Third, the green solid line in Fig. 6(a) shows that its value increases as M increases and approaches the value corresponding to the coherent state when $M=1/[1-G+G\textrm {sech}(2s)]$, where this critical equation is obtained by solving for the dependence of M on the squeezing parameter s and gain G from the above critical equation, which is not presented here. When M is smaller than this critical value, a smaller deterioration effect is coupled into the bright TMSS to ensure a higher estimation precision than that in the coherent state.

Fig. 3. Another PIA with gain M is placed after (a) and before (b) the original PIA in the probe light beam path and in the optical path of auxiliary light beam $\hat {a}_{a}$ (c).

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Fig. 4. Dependence of the estimation function of the vacuum TMSS (black dashed line), estimation function of the coherent state (blue dashed line), estimation function of the bright TMSS (red solid line), estimation function in Fig. 3(a) (green solid line), estimation function in Fig. 3(b) (cyan solid line), and estimation function in Fig. 3(c) (magenta solid line) on the gain G. Here, s and M are set to 1 and 2, respectively.

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Fig. 5. (a) Dependence of the estimation function of the vacuum TMSS (black dashed line), estimation function of the coherent state (blue dashed line), estimation function of the bright TMSS (red solid line), estimation function in Fig. 3(a) (green solid line), estimation function in Fig. 3(b) (cyan solid line), and estimation function in Fig. 3(c) (magenta solid line) on the squeezing parameter s. Here, G and M are both set to 2. (b) Dependence of the squeezing parameter $s=\{\operatorname {arcosh}[GM/(GM-M+1)]\}/2$ on the gain M. Here, G is set to 2. The pink region is the quantum-enhanced region.

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Fig. 6. (a) Dependence of the estimation function of the vacuum TMSS (black dashed line), estimation function of the coherent state (blue dashed line), estimation function of the bright TMSS (red solid line), estimation function in Fig. 3(a) (green solid line), estimation function in Fig. 3(b) (cyan solid line), and estimation function in Fig. 3(c) (magenta solid line) on the gain M. Here, s and G are set to 1 and 2, respectively. (b) The dependence of the gain $M=2/[1+\textrm {sech}(2s)]$ on the squeezing parameter s. The pink region is the quantum-enhanced region.

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Figure 3(b) shows that this extra PIA is placed before the original PIA, and its estimation function is

(15)$$\Lambda^{M-G}=G(2G-1)-G^{2}\{2-M[1+\textrm{sech}(2s)]\}.$$

It is easily observed that the value of Eq. (15) is M times larger than that of Eq. (14) because the probe light beam is amplified by a factor of M before injection into the original PIA. For this reason, its value is always larger than that of Eq. (14), and the cyan solid line in Fig. 4 verifies this observation. Its dependence on the squeezing factor s is represented by a cyan solid line in Fig. 5(a). Its critical squeezing parameter s is equal to $\{\operatorname {arcosh}[M/(2-M)]\}/2$. According to this result, when the squeezing parameter s is larger than this critical value in the gain M range of [1, 2), its estimation precision is more robust than that of the coherent state. Finally, the cyan solid line in Fig. 6(a) indicates that its value increases with increasing gain M, and when M exceeds the critical value of $2/[1+\textrm {sech}(2s)]$, its estimation precision is less robust than that of the coherent state. Figure 6(b) characterizes the dependence of $2/[1+\textrm {sech}(2s)]$ on the squeezing parameter s. In the region below the red solid line, the estimation precision is more robust than that of the coherent state due to the smaller deterioration effect from this extra PIA.

In the third scheme, which is shown in Fig. 3(c), this extra PIA is placed in the optical path of the auxiliary light beam $\hat {a}_{a}$. After similar calculations, its estimation function can be expressed as

(16)$$\Lambda^{G/M}=G(2G-1)-G^{2}\frac{\cosh(2s)-1}{M\cosh(2s)+M-1},$$

in which the first term is the estimation function of the coherent state, while the second term is always greater than or equal to zero. Thus, Eq. (16) is less than or equal to $G(2G-1)$, i.e., its estimation precision is always more robust than that of the coherent state. When s and M are set to 1 and 2, respectively, its dependence on the gain G, which is represented as a magenta solid line in Fig. 4, is always smaller than that of the coherent state. The magenta solid line in Fig. 5(a) represents its dependence on the squeezing parameter s. Interestingly, according to Eq. (16), only when $\cosh (2s)=1$, i.e., ${s}=0$, is its value equal to that of the coherent state. Otherwise, its value is always smaller than that of the coherent state. This represents a major advantage over the two schemes presented above. For ${s}$ equal to 1, its dependence on the gain M is represented by a magenta solid line in Fig. 6(a), and its value also increases with the increase of the gain M. According to Eq. (16), only when the gain M tends toward infinity is its value equal to $G(2G-1)$, which is the estimation precision of the coherent state. In other words, due to the nonphysical meaning of an infinite value of M, any M value can ensure that its precision more robust than that of the coherent state. This is because the auxiliary light beam $\hat {a}_{a}$ from the amplification process with gain M can still provide quantum enhancement of the estimation precision under this scheme.

The scheme in Fig. 3(c) is more robust than the two other schemes because its dependence on the gain M and squeezing parameter s is not very strong. A smaller s value and any gain M value can enable its estimation precision to exceed that of the coherent state.

4. Effects of noise due to propagation loss and imperfect detection on the estimation function of the bright TMSS

Compared to the noise effect from another PIA, some losses from the propagation channel and imperfect detection are inevitable under practical application conditions. Therefore, it is important to discuss the deterioration effects of these losses due to information leakage. Here, a fictitious beam splitter is used to simulate propagation loss and imperfect detection, i.e., it is placed after (Fig. 7(a)) and before (Fig. 7(b)) the original PIA in the probe light beam path and in the optical path of the auxiliary light beam $\hat {a}_{a}$ (Fig. 7(c)). First, the estimation function in Fig. 7(a) can be written as

(17)$$\Lambda^{G-T}=G(2G-1)-G\{G[1-\textrm{sech}(2s)]+1-\frac{1}{T}\},$$

similar to Eq. (14), its second term on the right-hand side is also modulated by the transmission T, and it will be reduced to the case of the bright TMSS when the transmission T is set to 1. Its dependence on T is represented by a green solid line in Fig. 8, which shows that its value decreases with increasing T due to information recovery. Notably, its value tends toward infinity when T approaches zero because the probe light beam after the original PIA has been totally attenuated and disappeared, and no valuable information can be extracted. When its value is equal to that of the coherent state, the value of T is set equal to $1/[1+G-G\textrm {sech}(2s)]$. The dependence of this critical transmission T on the squeezing parameter s is represented by a red solid line in Fig. 9. In the region above this red line, the estimation precision is more robust than that of the coherent state.

Fig. 7. Three loss-based schemes. A fictitious beam splitter with transmission T is placed after (a) and before (b) the original PIA in the probe light beam path and in the optical path of the auxiliary light beam $\hat {a}_{a}$ (c).

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Fig. 8. Dependence of the estimation function of the vacuum TMSS (black dashed line), estimation function of the coherent state (blue dashed line), estimation function of the bright TMSS (red solid line), estimation function in Fig. 7(a) (green solid line), estimation function in Fig. 7(b) (cyan solid line), and estimation function in Fig. 7(c) (magenta solid line) on the transmission T. Here, s and G are set to 1 and 2, respectively.

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Fig. 9. Dependence of the transmission $T=1/[1+G-G\textrm {sech}(2s)]$ on the squeezing parameter s. Here, the gain G is set to 2. The pink region is the quantum-enhanced region.

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Figure 7(b) shows another scheme, in which the fictitious beam splitter is placed before the original PIA, which means that the probe light beam is attenuated by a factor of 1$-$T, and its corresponding estimation function is

(18)$$\Lambda^{T-G}=G(2G-1)-G^{2}T[1-\textrm{sech}(2s)],$$

in which the first term is the estimation function of the coherent state, and the value of $1-\textrm {sech}(2s)$ in the second term is greater than or equal to zero; thus, Eq. (18) can be regarded as the difference between the estimation function of the coherent state and a positive number. Therefore, only when the transmission T or the squeezing parameter s is set to zero will its value be the same as that of the coherent state. Under the conditions ${s}=1$ and ${G}=2$, its dependence on the transmission T is represented by a cyan solid line in Fig. 8.

In the final scheme, this fictitious beam splitter is placed in the optical path of the auxiliary light beam $\hat {a}_{a}$. Following a similar method, its estimation function can be calculated as

(19)$$\Lambda^{G/T}=G(2G-1)-G^{2}\frac{(2T-1)[\cosh(2s)-1]}{T\cosh(2s)-T+1},$$

its form is similar to that of Eq. (16); the first term is the estimation function of the coherent state, and the second term is the modulated term. Its dependence on the transmission T is represented as a magenta solid line in Fig. 8, and its value decreases with increasing T. According to Fig. (8) and Eq. (19), when the transmission of the auxiliary light beam is below 50%, the second term will become negative, and the value in Eq. (19) will be larger than that of the coherent state. At $T=50\%$ transmission of the auxiliary light beam, the second term will be equal to zero, and it will return to the case of the coherent state. If the auxiliary light beam is perfectly detected, i.e., $T=1$, it will be reduced to the case of the bright TMSS. Thus, from this point of view, low loss or at least half of the auxiliary light beam is required to obtain quantum enhancement compared to the coherent state under the present scheme.

Interestingly, a deep projection exists between Eq. (16) and Eq. (19), i.e., they yield the same value when $M=T/(2T-1)$. Specifically, $T=1/2$ and $T=1$ correspond to $M\rightarrow \infty$ and $M=1$, respectively, and the range of $1/2<T<1$ fully covers the range of the gain $M$.

Similarly, the scheme in Fig. 7(b) is more robust than the two other schemes because the requirements on the transmission T and squeezing parameter s are easy to satisfy, which can guarantee that its estimation precision is beyond the performance of the coherent state.

5. Optimal intensity difference measurement saturation of the QCRB

The estimation function of the bright TMSS was presented in Sec. II; however, determining what kind of optimal optical measurement can saturate it is not a trivial task. Therefore, in this section, the optimal intensity difference measurement will be demonstrated to be an optimal option. First, the intensity difference operator can be written as

(20)$$\hat{n}_{-}^{\beta}=\hat{n}_{p}-\beta\hat{n}_{a},$$

where $\hat {n}_{p}$ and $\hat {n}_{a}$ are the numbers of photons of the probe and auxiliary light beams, respectively, and $\beta$ is an adjustable factor that can be used to minimize the intensity difference noise. Thus, in the following, its noise or variance must be calculated as

(21)$$\begin{aligned}\left\langle \Delta^{2}\hat{n}_{-}^{\beta}\right\rangle &=\left\langle \Delta^{2}\hat{n}_{p}\right\rangle +\beta^{2}\left\langle \Delta ^{2}\hat{n}_{a}\right\rangle -2\beta Cov(\hat{n}_{p},\hat{n} _{a})\\ & =G^{2}\left\langle \Delta^{2}\hat{n}_{p}\right\rangle _{0} +G(G-1)\left\langle \hat{n}_{p}\right\rangle _{0}\\ &\quad +\beta^{2}\left\langle\Delta^{2}\hat{n}_{a}\right\rangle _{0} -2G\beta(\left\langle \hat{n}_{p}\hat{n}_{a}\right\rangle _{0}-\left\langle \hat{n}_{p}\right\rangle _{0}\left\langle \hat {n}_{a}\right\rangle_{0}), \end{aligned}$$

where $\left \langle \Delta ^{2}\hat {n}_{p}\right \rangle$ and $\left \langle \Delta ^{2}\hat {n}_{a}\right \rangle$ are the variances of the probe and auxiliary light beams, respectively; $Cov(\hat {n}_{p},\hat {n}_{a})$ is the covariance between the final twin light beams; and the subscript 0 represents the original twin beams generated from the PA. Second, by computing the partial derivative with respect to $\beta$ from Eq. (21), we obtain $\beta _{opt}$, which minimizes the intensity difference noise:

(22)$$\beta_{opt}=G\frac{\left\langle \hat{n}_{p}\hat{n}_{a}\right\rangle _{0}-\left\langle \hat{n}_{p}\right\rangle _{0}\left\langle \hat {n}_{a}\right\rangle _{0}}{\left\langle \Delta^{2}\hat{n}_{a}\right\rangle _{0}}.$$

Substituting Eq. (22) into Eqs. (20) and (21) yields

(23)$$\left\langle \hat{n}_{-}^{\beta_{opt}} \right\rangle=G\left\langle \hat{n}_{p}\right\rangle _{0}-\beta_{opt}\left\langle \hat{n}_{a}\right\rangle _{0},$$

and

(24)$$\begin{aligned}\left\langle \Delta^{2}\hat{n}_{-}^{\beta_{opt}}\right\rangle & =G^{2}\left\langle \Delta^{2}\hat{n}_{p}\right\rangle _{0}+G(G-1)\left\langle \hat{n} _{p}\right\rangle _{0}\\ &\quad -G^{2}\frac{(\left\langle \hat{n}_{p}\hat{n}_{a}\right\rangle _{0}-\left\langle \hat{n}_{p}\right\rangle _{0}\left\langle \hat {n}_{a}\right\rangle _{0})^{2}}{\left\langle \Delta^{2}\hat{n} _{a}\right\rangle _{0}}, \end{aligned}$$

respectively, and the related physical properties of the original twin beams in Eqs. (23) and (24) need to be calculated. They are obtained from Eq. (1) as follows:

(25)$$\begin{aligned}\left\langle \Delta^{2}\hat{n}_{p}\right\rangle _{0} &= \cosh^{2} (s)\cosh(2s)n_{c},\\ \left\langle \Delta^{2}\hat{n}_{a}\right\rangle _{0} &= \sinh^{2} (s)\cosh(2s)n_{c},\\ \left\langle \hat{n}_{p}\right\rangle _{0} &= \cosh^{2}(s)n_{c},\\ \left\langle \hat{n}_{a}\right\rangle _{0} &= \sinh^{2}(s)n_{c},\\ \left\langle \hat{n}_{p}\hat{n}_{a}\right\rangle _{0}-\left\langle \hat{n}_{p}\right\rangle _{0}\left\langle \hat{n}_{a}\right\rangle _{0} &= 2\sinh^{2}(s)\cosh^{2}(s)n_{c}. \end{aligned}$$

Finally, error propagation analysis [18,19] is used to determine the uncertainty in estimating the gain G by using Eqs. (23) and (24), which is given by

(26)$$\left\langle \Delta^{2}G\right\rangle = \frac{\left\langle \Delta ^{2}\hat{n}_{-}^{\beta_{opt}}\right\rangle }{\left\vert \frac{\partial G\left\langle \hat{n}_{p}\right\rangle _{0}}{\partial G}\right\vert ^{2}} =G(G-1)+G^{2}\;\textrm{sech}(2s).$$

This expression is equal to Eq. (13), which indicates that the optimal intensity difference measurement can saturate the estimation function of the bright TMSS.

6. Conclusions

In this study, first, we theoretically characterized estimation functions from the vacuum TMSS, coherent state, and the bright TMSS scenario. The absolute estimation ability of the bright TMSS is strongest because it has more probe photons than the vacuum TMSS and higher estimation precision than the coherent state. Second, we studied the effects of noise from another PIA with gain M and a fictitious beam splitter with transmission T on the estimation precision of the bright TMSS. The results show that a scheme in which another PIA is placed in the auxiliary light beam path is more robust than two other schemes and that a scheme in which the fictitious beam splitter is placed before the original PIA in the probe light beam path is the most robust. Third, optimal intensity difference measurement was confirmed to be an optimal measurement technique that can saturate the QCRB using the current technology. Although this work focused on the TMSS, the estimation function of the SMSS can also obtained by replacing the term $\textrm {sech}(2s)$ with $e^{-2s}$ in the related results, e.g., Eqs. (13), (14), (15), (17), and (18). Therefore, this research opens a new avenue for quantum metrology based on PIAs.

Funding

Natural Science Foundation of Zhejiang Province (LY22A040007); the Fundamental Research Funds for the Provincial Universities of Zhejiang (2021YW29); National Natural Science Foundation of China (11804323, 12225404, 11874155, 91436211, 11374104); Science and Technology Commission of Shanghai Municipality (2021-01-07-00-08-E00100); Program of Shanghai Academic Research Leader (22XD1400700); the Basic Research Project of the Shanghai Science and Technology Commission (20JC1416100); Natural Science Foundation of Shanghai (17ZR1442900); Minhang Leading Talents (201971); Shanghai Municipal Education Commission (2019SHZDZX01); 111 Project (B12024).

Disclosures

The author declares 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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Phase-insensitive amplifier gain estimation at Cramér-Rao bound for two-mode squeezed state of light

Abstract

Corrections

1. Introduction

2. Estimation functions from vacuum TMSS, bright TMSS, and coherent state

3. Effect of noise from another PIA on the estimation function of the bright TMSS

4. Effects of noise due to propagation loss and imperfect detection on the estimation function of the bright TMSS

5. Optimal intensity difference measurement saturation of the QCRB

6. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Equations (26)

Optics Express