Phase sensitivity of an $\operatorname{SU}(1,1)$ interferometer via product detection

Abstract

We theoretically analyze the phase sensitivity of an $\operatorname{SU}(1,1)$ interferometer with various input states by product detection in this paper. This interferometer consists of two parametric amplifiers that play the role of beam splitters in a traditional Mach–Zehnder interferometer. The product of the amplitude quadrature of one output mode and the momentum quadrature of the other output mode is measured via balanced homodyne detection. We show that product detection has the same phase sensitivity as parity detection for most cases, and it is even better in the case with two coherent states at the input ports. The phase sensitivity is also compared with the Heisenberg limit and the quantum Cramér–Rao bound of the $\operatorname{SU}(1,1)$ interferometer. This detection scheme can be easily implemented with current homodyne technology, which makes it highly feasible. It can be widely applied in the field of quantum metrology.

1 Introduction

Quantum parameter estimation has recently drawn considerable attention for its fundamental and technological applications [1–12]. One of the most common and powerful tools in the field of precision metrology is optical interferometry, because of its extreme phase sensitivity. However, the phase sensitivity Δϕ of an interferometer with solely classical resources is bounded by $1/\sqrt{\bar{n}}$, which is well known as the shot noise limit (SNL), where n̄ is the average photon number. During the past few decades, it has been shown that this limit can be surpassed by taking advantage of quantum resources. By using particle entanglement as a resource for sensitivity enhancement [13, 14], there has been much progress toward reaching the more fundamental Heisenberg limit (HL), which is characterized by a phase sensitivity scale of $1/\bar{n}$.

One possibility for beating the SNL is to inject non-classical light into a traditional interferometer, such as a Mach–Zehnder interferometer. A number of theoretical proposals have shown that the phase sensitivity can go below the SNL using non-classical light, such as squeezed states [15, 16], two-mode squeezed states [17], Fock states [18] and NOON states [19–21].

Another possibility for beating the SNL is to use a new class of interferometers, which is described by the $\operatorname{SU}(1,1)$ group, unlike the Mach–Zehnder interferometer by the $\operatorname{SU}(2)$ group. This type of interferometer, which was first proposed by Yurke et al. [22], is shown in Fig. 1. It is configured as a Mach–Zehnder interferometer, except that the beam splitters are replaced by parametric amplifiers (PAs), such as optical parametric amplifiers or four-wave mixers. It has been pointed out that a phase sensitivity with vacuum inputs scaling as $1/[\bar{n}(\bar{n}+2)]^{1/2}$ can be achieved with intensity detection, where $\bar{n}=2 \sinh ^{2} g$ is the average photon number inside the interferometer and g is the PA interaction strength. Although the phase sensitivity reaches the HL, n̄ is small due to the limited interaction strength g.

Figure 1

(Color online) Schematic diagram of $\operatorname{SU}(1,1)$ interferometer. Two parametric amplifiers (PAs) take the place of two beam splitters in the traditional Mach–Zehnder interferometer. $g_{1}$ ($g_{2}$) and $\theta _{1}$ ($\theta _{2}$) describe the strength and phase shift introduced by the parametric processes, respectively. $\hat{a}_{i}$ and $\hat{b}_{i}$ ($i=0,1,2$) denote two modes in the different processes. The pump field between the two PAs has a π phase difference

Recently, a new theoretical scheme utilizing coherent states as inputs showed that the phase sensitivity can be largely enhanced in the interferometer as a result of the large input photon number [23]. Marino et al. [24] also studied the effect of loss on the phase sensitivity of the $\operatorname{SU}(1,1)$ interferometer with intensity detection. Sparaciari et al. [25, 26] analyzed the phase sensitivity in $\operatorname{SU}(1,1)$ interferometers initially fed by Gaussian states. It is found that incident coherent states could lead to Heisenberg scaling. Hu et al. [27] investigated the effect of phase diffusion on the performance of an $\operatorname{SU}(1,1)$ interferometer. You et al. [28] revisited the phase sensitivity in an $\operatorname{SU}(1,1)$ interferometer considering the different phase configurations. More recently, Zhang et al. [29] addressed the quantum stochastic phase estimation with $\operatorname{SU}(1,1)$ interferometer.

Experimental realizations of the $\operatorname{SU}(1,1)$ interferometer have been reported in optical systems [30, 31], spinor Bose–Einstein condensates (BECs) [32–36], a hybrid atom-light interferometer [37, 38], and a circuit quantum electrodynamics system [39]. Most recently, an alternative “pumped-up” approach was proposed [40] where all the particles can be involved in the quantum parameter estimation. It can be implemented in spinor BECs and hybrid atom-light systems. In this manuscript, we consider the $\operatorname{SU}(1,1)$ interferometer which is realized in optical system.

One goal of precision metrology is to achieve the HL. For this purpose, several different detection schemes have also been thoroughly investigated, including intensity detection [23], homodyne detection [41] and parity detection [42]. It has been pointed out that an $\operatorname{SU}(1,1)$ interferometer with coherent and squeezed vacuum states as inputs can approach the HL with both homodyne and parity detection. Another kind of scheme, product detection, which was first proposed in Ref. [17], can achieve the HL scaling with two-mode squeeze states as inputs in a traditional Mach–Zehnder interferometer. Mathematically, product detection is described by $\hat{P}=(\hat{a}+\hat{a}^{\dagger })(\hat{b}-\hat{b}^{\dagger })/2i$, where â ($\hat{a}^{\dagger }$) and b̂ ($\hat{b}^{\dagger }$) are the annihilation (creation) operators for the two output modes. We adjust the relative phases of the local oscillators in homodyne detection such that the photocurrents of detectors a and b are proportional to the expectation values of the amplitude quadrature $(\hat{a}+\hat{a}^{\dagger })/\sqrt{2}$ and the momentum quadrature $(\hat{b}-\hat{b}^{\dagger })/\sqrt{2}i$. In this way, product detection can be easily implemented by homodyne technique. Compared with parity detection, product detection does not require a photon-number resolving detector, which makes it much more feasible.

In this paper, we consider a single-mode squeezed state $\vert \psi\rangle =\hat{D}(\alpha )\hat{S}(r)\vert 0\rangle$ as input, where $\hat{S}(r) = e^{(r^{*}\hat{a}^{2}-r\hat{a}^{\dagger 2})/2}$ is the squeezing operator, $\hat{D}(\alpha )=e^{(\alpha \hat{a}^{\dagger }-\alpha ^{*}\hat{a})}$ is the displacement operator [43], $r= \vert r \vert e^{i\phi _{r}}$ is the squeezing parameter, $\alpha = \vert \alpha \vert e^{i\theta _{a}}$ is the displacement amplitude, and $r^{\ast }$ and $\alpha ^{\ast }$ are the conjugate of r and α, respectively. Specifically, when $r =0$, it reduces to a coherent state, and when $\alpha =0$, it is a squeezed vacuum state. Using this method, we can easily go back to some specific inputs, such as a coherent state mixed with squeezed vacuum state [41, 42]. We show that with the coherent state and the squeezed vacuum state as inputs, the phase sensitivity with product detection can approach the HL (as in Ref. [42]) when the average photon number of the coherent and squeezed vacuum states are comparable. Furthermore, we discuss the condition for achieving the optimal phase sensitivity with different input states. Finally, we also compare the phase sensitivity with another quantum limit, the quantum Cramér–Rao bound (QCRB) [44], which quantifies the ultimate limit on the performance of the phase-estimation strategies for a given quantum state.

This paper is organized as follows: We first present the propagation of the input fields through the $\operatorname{SU}(1,1)$ interferometer in Sect. 2. We then discuss the HL in an $\operatorname{SU}(1,1)$ interferometer and compare the phase sensitivity with the HL and the QCRB in Sect. 3. After comparing different detection schemes in Sect. 4, we conclude with a summary.

2 Product detection on $\operatorname{SU}(1,1)$ interferometer

2.1 Model

Figure 1 shows the model of an $\operatorname{SU}(1,1)$ interferometer, in which two PAs play the role of beam splitters in a traditional Mach–Zehnder interferometer. We consider single-mode squeezed states as inputs. â ($\hat{a}^{\dagger }$) and b̂ ($\hat{b}^{\dagger }$) are the annihilation (creation) operators corresponding to the modes a and b, respectively. After the first PA, one output beam accumulates a phase shift ϕ. The output beams are measured via homodyne detection. The phase information can be inferred from the measured signal.

This interferometer can be modeled by the transformation between input and output modes by $\hat{V}_{2}=T_{2}T_{\phi }T_{1} \hat{V}_{0}$, where $\hat{V}_{2}=(\hat{a}_{2}, \hat{a}_{2}^{\dagger }, \hat{b}_{2}, \hat{b}_{2}^{ \dagger })^{T}$ and $\hat{V}_{0}=(\hat{a}_{0}, \hat{a}_{0}^{\dagger }, \hat{b}_{0}, \hat{b}_{0}^{ \dagger })^{T}$. $T_{1}$, $T_{\phi }$, and $T_{2}$ describe the first PA, the phase shift and the second PA, respectively:

$$\begin{aligned}& \begin{aligned} T_{1}= \begin{pmatrix} \mu _{1}&0&0&\nu _{1} \\ 0&\mu _{1}&\nu _{1}^{*}&0 \\ 0&\nu _{1}&\mu _{1}&0 \\ \nu _{1}^{*}&0&0&\mu _{1} \end{pmatrix}, \end{aligned} \end{aligned}$$

(1)

$$\begin{aligned}& \begin{aligned} T_{\phi }= \begin{pmatrix} 1&0&0&0 \\ 0&1&0&0 \\ 0&0&e^{i\phi }&0 \\ 0&0&0&e^{-i\phi } \end{pmatrix} , \end{aligned} \end{aligned}$$

(2)

$$\begin{aligned}& \begin{aligned} T_{2}= \begin{pmatrix} \mu _{2}&0&0&\nu _{2} \\ 0&\mu _{2}&\nu _{2}^{*}&0 \\ 0&\nu _{2}&\mu _{2}&0 \\ \nu _{2}^{*}&0&0&\mu _{2} \end{pmatrix} , \end{aligned} \end{aligned}$$

(3)

where $\mu _{j}=\cosh g_{j}$, $\nu _{j}=-e^{i\theta _{j}}\sinh g_{j}$, $g_{j}$ and $\theta _{j}$ are the strength and phase shift introduced by the PA process j ($j=1,2$) [45, 46]. The phase sensitivity of this device by product detection can be obtained through an error propagation analysis,

$$ \Delta \phi = \frac{\langle \Delta \hat{P}\rangle }{ \vert \partial \langle \hat{P}\rangle /\partial \phi \vert }, $$

(4)

where the brackets denote the quantum mechanical expectation values, the product operator is defined as $\hat{P} \equiv (\hat{a}_{2} + \hat{a}_{2}^{\dagger }) (\hat{b}_{2} - \hat{b}_{2}^{\dagger })/2i$, and $\langle \Delta \hat{P}\rangle \equiv \sqrt{\langle \hat{P}^{2} \rangle - \langle \hat{P} \rangle ^{2}}$ is the standard deviation of the product operator. Usually, the $\operatorname{SU}(1,1)$ interferometer is operated in a balanced configuration in which the second PA will “undo” what the first one does [24]. In such a case, we have $\theta _{1}=0$, $\theta _{2}=\pi$, and $g_{1}=g_{2}=g$. Thus if $\phi =0$, then $\hat{a}_{2}=\hat{a}_{0}$ and $\hat{b}_{2}=\hat{b}_{0}$. We will focus on a balanced configuration throughout this paper. The input states can be generally expressed by $\hat{D}(\alpha )\hat{S}(r_{a})\vert 0\rangle _{a}\otimes \hat{D}(\beta ) \hat{S}(r_{b})\vert 0\rangle _{b}$, where $\alpha = \vert \alpha \vert e^{i\theta _{a}}$, $\beta = \vert \beta \vert e^{i\theta _{b}}$, $r_{a}=r_{1}e^{i\phi _{1}}$, and $r_{b}=r_{2}e^{i\phi _{2}}$. Various input states can be obtained from this generality.

2.2 Phase sensitivity

Product detection was firstly proposed in Ref. [17]. It can approach the HL scaling. Here we will study the phase sensitivity with various input states and compare them with the corresponding HL and QCRB.

The corresponding HL [24] is related to the total average photon number inside the interferometer $N_{\mathrm{total}}=\langle \hat{a}_{1}^{\dagger }\hat{a}_{1}+\hat{b}_{1}^{ \dagger }\hat{b}_{1}\rangle$, which is given by

$$ \begin{aligned} N_{\mathrm{total}}={}&(N_{\mathrm{PA}}+1) (N_{a}+N_{b}+N_{\mathrm{sa}}+N_{ \mathrm{sb}})+N_{\mathrm{PA}} \\ &{}-2\sqrt{N_{\mathrm{PA}}(N_{\mathrm{PA}}+2)}\sqrt{N_{a}N_{b}} \cos ( \theta _{a}+\theta _{b}), \end{aligned} $$

(5)

where $N_{\mathrm{PA}}=2\sinh ^{2}g$, $N_{a}= \vert \alpha \vert ^{2}$, $N_{b}= \vert \beta \vert ^{2}$, and $N_{\mathrm{s(a,b)}}=\sinh ^{2}r_{(1,2)}$. Thus, $N_{\mathrm{total}}$ not only depends on the strength of the process g, but also on the phase of the input states $\theta _{a}$ and $\theta _{b}$ [46]. Thus, the HL is as follows:

$$ \Delta \phi _{\mathrm{HL}}=\frac{1}{N_{\mathrm{total}}}. $$

(6)

2.2.1 Vacuum input states $\vert 0\rangle _{a}\otimes \vert 0\rangle _{b}$

With vacuum inputs, the optimal phase sensitivity is found to be $\Delta \phi _{\mathrm{opt}}=1/ \sqrt{N_{\mathrm{PA}}(N_{\mathrm{PA}}+2)}$, while the corresponding HL is $\Delta \phi _{\mathrm{HL}}=1/N_{\mathrm{PA}}$. The QCRB in this case is also $1/\sqrt{N_{\mathrm{PA}}(N_{\mathrm{PA}}+2)}$ [42]. Figure 2 compares the phase sensitivity $\Delta \phi _{\mathrm{V}}$ with the SNL, HL, and QCRB, as functions of g. We assume that the interferometer is seeded with pure Gaussian states throughout the paper. As g increases, the phase sensitivity Δϕ becomes closer to the HL and it can saturate the QCRB. As shown in Ref. [42], intensity detection, homodyne detection, and parity detection can saturate the QCRB as well. From an experimental perspective, intensity detection is much more preferred in this case.

Figure 2

(Color online) Optimal phase sensitivity via product detection as a function of g with a vacuum as inputs. The blue line represents the HL, the dashed purple line represents the shot noise limit (SNL), the dot-dashed red line represents the quantum Cramér–Rao bound (QCRB), the dotted green line represents the phase sensitivity via product detection in the ideal case, the dashed red line represents the phase sensitivity via product detection with photon loss $L=0.05$, and the dashed brown line represents the phase sensitivity via product detection with photon loss $L=0.10$

2.2.2 One coherent input state $\vert \alpha\rangle _{a}\otimes \vert 0\rangle _{b}$

With only one coherent state at the input ports, the first PA works as a phase insensitive amplifier. The phase sensitivity with product detection at $\phi =0$ is given by

$$ \Delta \phi =\sqrt{ \frac{1+2N_{a}+2N_{a}\cos 2\theta _{a}}{\kappa (1+N_{a}+N_{a}\cos 2\theta _{a})^{2}}}, $$

(7)

where $\kappa =N_{\mathrm{PA}}(N_{\mathrm{PA}}+2)$. The optimal phase sensitivity is obtained at $\theta _{a}=0$. When $N_{a} \gg 1$, the optimal phase sensitivity can be simplified to

$$ \Delta \phi _{\mathrm{opt}}\simeq \frac{1}{\sqrt{\kappa (N_{a}+1)}}. $$

(8)

This agrees with the parity detection result [42]. Both detection schemes cannot saturate the QCRB. The corresponding HL is $\Delta \phi _{\mathrm{HL}}=1/[(N_{\mathrm{PA}}+1)N_{a}+N_{ \mathrm{PA}}]$. The QCRB can be found as $1/[\kappa (2N_{a}+1)+2N_{a}(N_{\mathrm{PA}}+2)]^{1/2}$ in Ref. [42]. As shown in Fig. 3, the optimal phase sensitivity can beat the SNL, but it does not surpass the HL. With the increase in g, the phase sensitivity via product detection becomes closer to the QCRB, as shown in Fig. 3(a). As $N_{a}$ increases, the phase sensitivity can be approximated as $1/\sqrt{N_{\mathrm{PA}} N_{\mathrm{total}}}$. It is improved by a factor of $1/\sqrt{N_{\mathrm{PA}}}$ compared with the SNL as shown in Fig. 3(b). The enhancement arises from the amplification of the parametric process.

Figure 3

(Color online) Optimal phase sensitivity via product detection as a function of (a) g and (b) $N_{a}$ with one coherent input state. The blue line represents the HL, the dashed purple line represents the shot noise limit (SNL), the dot-dashed red line represents the quantum Cramér–Rao bound (QCRB), the dotted green line represents the phase sensitivity via product detection in the ideal case, the dashed red line represents the phase sensitivity via product detection with photon loss $L=0.05$, and the dashed brown line represents the phase sensitivity via product detection with photon loss $L=0.10$

2.2.3 Two coherent input states $\vert \alpha\rangle _{a}\otimes \vert \beta\rangle _{b}$

With two coherent states at the input ports, the first PA works as a phase sensitive amplifier. The amplification will depend on the relative phase between the two input beams. For simplicity, here we assume that two coherent input states have the same photon number, $N_{a}=N_{b}$. The phase sensitivity at $\phi =0$ is obtained by

$$ \begin{aligned} \Delta \phi ={}&\sqrt{1+4N_{a}+2N_{a}( \cos 2\theta _{a}-\cos 2\theta _{b})} \\ &{}\times \bigl[4N_{a}\bigl(\sin \theta _{a}\sin \theta _{b}\sinh ^{2}g-\cos \theta _{a}\cos \theta _{b}\cosh ^{2}g\bigr) \\ &{}+ N_{a}(2+\cos 2\theta _{a}-\cos 2\theta _{b})\sinh 2g+\sinh 2g\bigr]^{-1}. \end{aligned} $$

(9)

Under the condition $\theta _{a}=0$ and $\theta _{b}=\pi$, the optimal phase sensitivity is achieved (here we assume that $N_{a}\gg 1$):

$$ \begin{aligned} \Delta \phi _{\mathrm{opt}}\simeq \bigl\{ 2 N_{a}\bigl[(N_{\mathrm{PA}}+2) (\sqrt{ \kappa }+1)+\kappa \bigr]+ \kappa \bigr\} ^{-1/2}. \end{aligned} $$

(10)

The corresponding HL is given by

$$ \Delta \phi _{\mathrm{HL}}= \frac{1}{2N_{a}(N_{\mathrm{PA}}+1+\sqrt{\kappa })+N_{\mathrm{PA}}}. $$

(11)

According to the general formalism for the QCRB of Gaussian states by Gao et al. [47], $\Delta \phi _{\mathrm{QCRB}}$ is given by

$$ \Delta \phi _{\mathrm{QCRB}}=\bigl\{ \kappa +4 N_{a}\bigl[\kappa +1+ \sqrt{\kappa }(N_{ \mathrm{PA}}+1)\bigr]\bigr\} ^{-1/2}. $$

(12)

Figure 4 shows that the optimal phase sensitivity can beat the SNL, but not the HL, and it is very close to the QRCB.

Figure 4

(Color online) Optimal phase sensitivity via product detection as a function of (a) g and (b) $N_{a}$ with two coherent input states. The blue line represents the HL, the dashed purple line represents the shot noise limit (SNL), the dot-dashed red line represents the quantum Cramér–Rao bound (QCRB), the dotted green line represents the phase sensitivity via product detection in the ideal case, the dashed red line represents the phase sensitivity via product detection with photon loss $L=0.05$, and the dashed brown line represents the phase sensitivity via product detection with photon loss $L=0.10$

2.2.4 Coherent state mixed with squeezed vacuum state $\vert \alpha\rangle _{a}\otimes \hat{S}(r_{b})\vert 0\rangle _{b}$

With the coherent state and the squeezed vacuum state as inputs, the phase sensitivity at $\phi =0$ is given by

$$ \Delta \phi = \sqrt{ \frac{4(1+4N_{a}\cos ^{2}\theta _{a})(\cosh 2r_{2}+\cos \phi _{2}\sinh 2r_{2})}{\sinh ^{2} 2g(1+4N_{a}\cos ^{2}\theta _{a}+\cosh 2r_{2}+\cos \phi _{2}\sinh 2r_{2})^{2}}}. $$

(13)

Under the condition $\phi _{2}=\pi$ and $\theta _{a}=\pi$, the optimal phase sensitivity is achieved

$$ \Delta \phi _{\mathrm{opt}}=\sqrt{ \frac{1+4N_{a}}{\kappa (\cosh r+2N_{a}\cosh r+2N_{a}\sinh r)^{2}}}. $$

(14)

We assume that $N_{a}\gg 1$ and denote $r_{2}$ by r, the optimal phase sensitivity is reduced to $1/[\kappa (N_{a}e^{2r}+\cosh r e^{r})]^{1/2}$. This result is similar to parity detection [42]. The corresponding HL is $1/[(N_{\mathrm{PA}}+1)(N_{a}+N_{s})+N_{\mathrm{PA}}]$, where $N_{s}=\sinh ^{2} r$. Meanwhile, the QCRB is given by $1/[2N_{a}(N_{\mathrm{PA}}+2)+N^{2}_{\mathrm{PA}} \sinh ^{2}(2r)/2 + \kappa (2N_{a}\cosh r e^{r}+\cosh ^{2}r)]^{1/2}$. As pointed out in Ref. [42], when $\vert \alpha \vert \simeq \tanh (2g) e^{r}/2$, the phase sensitivity can approach the HL. We plot the phase sensitivity $\Delta \phi _{\mathrm{opt}}$ as a function of the interaction strength g in Fig. 5. The phase sensitivity is always below the SNL and close to the HL.

Figure 5

(Color online) Optimal phase sensitivity via product detection as a function of g with the coherent state mixed with the squeezed vacuum state. The blue line represents the HL, the dashed purple line represents the shot noise limit (SNL), the dot-dashed red line represents the quantum Cramér–Rao bound (QCRB), the dotted green line represents the phase sensitivity via product detection in the ideal case, the dashed red line represents the phase sensitivity via product detection with photon loss $L=0.05$, and the dashed brown line represents the phase sensitivity via product detection with photon loss $L=0.10$. The parameter values are as follows: $r=2$, $\vert \alpha \vert \simeq \tanh (2g) e^{r}/2$

2.2.5 Two squeezed vacuum states $\hat{S}(r_{a})\vert 0\rangle _{a}\otimes \hat{S}(r_{b})\vert 0\rangle _{b}$

For simplicity, we consider the case in which $r_{1}=r_{2}=r$. When $\phi _{1}=\phi _{2}=0$ or π, the optimal phase sensitivity at $\phi =0$ is achieved

$$ \Delta \phi _{\mathrm{opt}}=\frac{1}{\sqrt{\kappa (2N_{s}+1)}}. $$

(15)

The corresponding HL is $1/[2(N_{\mathrm{PA}}+1)N_{s}+N_{\mathrm{PA}}]$. The QCRB is $1/[(1+N_{\mathrm{PA}})^{2}\cosh 4r -1]^{1/2}$. The optimal phase sensitivity depends not only on the strength g, but also on the squeezing parameter r, as shown in Fig. 6(a) and (b). Figure 6(a) shows that the phase sensitivity with product detection cannot approach the HL as g increases, while it can easily go below the SNL. When r decreases to zero, phase sensitivity goes back to the case with vacuum inputs, as shown in Fig. 6(b). When r goes to infinity, phase sensitivity would follow the SNL scaling (not shown in Fig. 6). The phase sensitivity is improved by a factor of $1/\sqrt{\cosh 2r}$ compared to the case of vacuum inputs, which comes from noise reduction of the squeezed vacuum states. In this case, the phase sensitivity cannot saturate the QCRB.

Figure 6

(Color online) Optimal phase sensitivity via product detection as a function of (a) g and (b) r with two squeezed vacuum states. The blue line represents the HL, the dashed purple line represents the shot noise limit (SNL), the dot-dashed red line represents the quantum Cramér–Rao bound (QCRB), the dotted green line represents the phase sensitivity via product detection in the ideal case, the dashed red line represents the phase sensitivity via product detection with photon loss $L=0.05$, and the dashed brown line represents the phase sensitivity via product detection with photon loss $L=0.10$

3 Discussion

3.1 Comparison between various detections

We have studied the behavior of the phase sensitivity of an $\operatorname{SU}(1,1)$ interferometer with various input states via product detection. Table 1 summarizes the phase sensitivity for different input states with product detection, parity detection and homodyne detection. It also shows the QCRB for different input states.

Table 1 Phase sensitivities with various detections and QCRB of an $\operatorname{SU}(1,1)$ interferometer with different input states

With vacuum states as inputs, product detection, parity detection and intensity detection all can saturate the QCRB. However, homodyne detection is not available in this case, because the expectation value of the amplitude quadrature operator is always zero, which is independent of phase shift ϕ. From the viewpoint of experimental implementation, intensity detection is greatly preferred because of its simplicity.

In the case with one coherent state as the input (see Fig. 7(a)), product detection achieved the same phase sensitivity as parity detection, and it was slightly better than homodyne detection and intensity detection. All four detection schemes cannot saturate the QCRB. Furthermore, when $N_{a}$ becomes larger than one, all four detections show the characterization of the SNL scaling, except they included an improvement factor of $1/\sqrt{\kappa }$, which comes from the amplification of the parametric process. When the input photon number is small, product detection is preferred. Otherwise, homodyne detection is the best choice since only one output light beam is measured.

Figure 7

(Color online) Optimal phase sensitivity as a function of g with different detections in the presence of various input states: (a) one-coherent state $\vert \alpha \rangle \otimes \vert 0\rangle$, (b) two-coherent state $\vert \alpha \rangle \otimes \vert \beta \rangle$, (c) coherent and squeezed-vacuum state $\vert \alpha \rangle \otimes \hat{S}(r)\vert 0\rangle$, (d) two-squeezed-vacuum state $\hat{S}(r)\vert 0\rangle \otimes \hat{S}(r)\vert 0\rangle$. The blue line represents the product detection, the dashed purple line represents the intensity detection, the dot-dashed red line represents the homodyne, and the dotted green line represents the parity detection

With two coherent states as inputs, the conditions for achieving the optimal phase sensitivity are different as shown in Fig. 7(b). For product detection, the input states is $\vert \alpha e^{i\pi}\rangle \otimes \vert \alpha\rangle$. The condition for parity detection and homodyne detection is $\vert \alpha e^{i\pi /2}\rangle \otimes \vert \alpha\rangle$, where we assume α is real. From Ref. [42], we know that parity detection is much worse than homodyne detection. On the other hand, product detection can approach the QCRB, as shown in Fig. 4, and is better than homodyne detection. Therefore, with two coherent states as inputs, product detection is the best scheme. However, all four schemes cannot approach the HL.

To take advantage of the large photon number of the coherent state and the noise reduction of the squeezed vacuum state, coherent states mixed with squeezed vacuum states as inputs have been studied [41, 42]. In this case (see Fig. 7(c)), product detection and parity detection behave similarly. Both schemes can beat the SNL and perform better than homodyne detection. Under the specific condition $\vert \alpha \vert \simeq \tanh (2g) e^{r}/2$, the phase sensitivity can approach the HL. Compared to parity detection, product detection is much easier to implement with current homodyne technology. Therefore, it is better to estimate the phase via product detection in this situation.

With both squeezed vacuum states as inputs, the optimal phase sensitivity with product detection and parity detection are the same as depicted in Fig. 7(b). Compared with the case of vacuum states as inputs, it is improved by a factor $1/\sqrt{\cosh 2r}$ as a result of the noise reduction of the squeezed vacuum states.

In general, product detection behaves similarly to parity detection. With both coherent states as inputs, product detection performs better than parity detection. Moreover, it does not require a photon-number resolving detector, which makes it much more feasible than parity detection. It can be easily realized with current homodyne technology.

From Figs. 2, 3, 4, 5, and 6, it is easy to find that the HL can be either above or below the QCRB. In fact, these results are not contradictory. Because HL and QCRB are obtained from different aspects. HL is related to the total mean photon number inside the interferometer while QCRB is dependent on quantum estimation theory which sets the ultimate limit for a set of probabilities that originated from measurements on a quantum system and gives a detection-independent phase sensitivity. For any detection (e.g. intensity detection, homodyne detection, parity detection), the corresponding phase sensitivity is not able to beat the QCRB, but it may reach below the HL.

3.2 Effect of photon loss on the phase sensitivity with product detection

In optical experiment, photon loss is inevitable. Therefore, it is necessary to investigate the effect of photon loss on the performance of an $\operatorname{SU}(1,1)$ interferometer with product detection. Particularly, we concentrate mainly on the photon loss of modes $a_{1}$ and $b_{1}$ inside the interferometer. Consider modes $a_{1}$ and $b_{1}$ experience photon loss $L_{1}$ and $L_{2}$, respectively. For simplicity, we assume that $L_{1}=L_{2}=L$. Under this situation, one can obtain

$$\begin{aligned} \hat{a}_{1}^{\mathrm{loss}} = \sqrt{1-L} \hat{a}_{1} + \sqrt{L} \hat{a}_{1}^{\mathrm{V}}, \\ \hat{b}_{1}^{\mathrm{loss}} = \sqrt{1-L} \hat{b}_{1} + \sqrt{L} \hat{b}_{1}^{\mathrm{V}}, \end{aligned}$$

(16)

where $\hat{a}_{1}^{\mathrm{loss}}$ ($\hat{b}_{1}^{\mathrm{loss}}$) is the annihilation operator of the mode a (b) after experiencing photon loss, $\hat{a}_{1}^{\mathrm{V}}$ and $\hat{b}_{1}^{\mathrm{V}}$ are the annihilation operators corresponding to the vacuum states.

Combining Eqs. (1), (2), (3), (4), and (16), one can obtain the phase sensitivity via product detection with photon loss. To compare between the ideal and the loss situations, we plot the phase sensitivities via product detection with photon loss $L=0.05$ and $L=0.10$ in Figs. 2, 3, 4, 5, 6, and 7, respectively. It is easily found that photon loss degrades the phase sensitivity via product detection. Nevertheless, with a small loss, the phase sensitivity via product detection is still able to reach below the SNL in some case.

3.3 Comparison with the previous works

Although both the previous researches [25–29] and our work consider the performance of quantum interferometer, there still is significant difference between them. First of all, unlike these previous papers mainly involving the QCRB, the intensity detection, or the homodyne detection, our work concentrates on the product detection which is still missing in an $\operatorname{SU}(1,1)$ interferometer. Secondly, we compare product detection with other three detections (intensity, parity, homodyne). We present which detection strategy can lead to the better sensitivity in the presence of different state inputs. It is found that product detection is the optimal detection scheme with two-equal-coherent-state input. From this point of view, our results make sense for the optimal choice of meansurement scheme in phase estimation.

4 Conclusion

In summary, we investigated the product detection on an $\operatorname{SU}(1,1)$ interferometer with pure Gaussian states as inputs. We have shown that product detection can approach the HL when the coherent beam and squeezed vacuum beam have roughly equal photon numbers. Compared to parity detection, product detection has a slightly better optimal phase sensitivity with two coherent states as the input, and has the same or a similar phase sensitivity as those of the other four inputs. Product detection always has better phase sensitivity than homodyne detection when non-vacuum states are seeded. We also show the QCRB with various input states. It is difficult for the QCRB to be saturated with product and parity detection and with homodyne detection when non-vacuum states are seeded. The search for the optimal scheme that can saturate the QCRB continues.

Appendix

1.1 1.1 Phase sensitivity $\Delta \phi _{\mathrm{onecoh}}^{\mathrm{I}}$

Phase sensitivity with intensity detection with one-coherent-state input is given by

$$\begin{aligned} \Delta \phi _{\mathrm{onecoh}}^{\mathrm{I}} ={}&\bigl[\operatorname{csch}^{4}(2 g) \bigl( \sec ^{2}(\phi /2) \bigl((2 N_{a}+1) \cosh (8 g)-1 \bigr)+2 N_{a} \csc ^{2}( \phi /2)\bigr) \\ &{}-8 (2 N_{a}+1)\bigr]^{1/2}{\bigl[2\sqrt{2}(N_{a}+1) \bigr]^{-1}}. \end{aligned}$$

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The corresponding optimal phase is found to be

$$\begin{aligned} \phi _{\mathrm{onecoh,opt}}^{\mathrm{I}} ={}&{-}2 \arctan \bigl\{ \bigl[{ \operatorname{csch}^{2}(4 g) \bigl( \sqrt{2N_{a} \bigl((2 N_{a}+1) \cosh (8 g)-1\bigr)}-2 N_{a}\bigr)} \bigr]^{1/2} \\ & {} \times {(4 N_{a}+2)^{-1/2}}\bigr\} . \end{aligned}$$

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1.2 1.2 Phase sensitivity $\Delta \phi _{\mathrm{cohsqz}}^{\mathrm{I}}$

Phase sensitivity with intensity detection with coherent and squeezed-vacuum state input is given by

$$\begin{aligned} \Delta \phi _{\mathrm{cohsqz}}^{\mathrm{I}} ={}& {-}\bigl\{ \operatorname{sech}^{4} g \csc ^{2} \phi (32 \bigl(\cosh g+\cosh (3 g)\bigr)^{2} \operatorname{csch}^{2} g \cos \phi \bigl(4 e^{-2 r} N_{a}+ \cosh (4 r) \\ & {}-1 \bigr) -8 \coth ^{2} g \cos (2 \phi ) \bigl(4 \sinh ^{2}(2 g) \bigl(4 N_{a} \cosh (2 r)+\cosh (4 r)-1 \bigr) \\ & {} -8 N_{a} \bigl(\cosh (4 g)+3\bigr) \sinh (2 r)\bigr)-64 N_{a} \bigl(3 \cosh (4 g)+5\bigr) \coth ^{2} g \cosh (2 r) \\ & {} +64 N_{a} \bigl(3 \cosh (4 g)+1\bigr) \coth ^{2} g \sinh (2 r)-2 \bigl(4 \cosh (4 g)+3 \cosh (8 g)+9\bigr) \\ & {} \times \operatorname{csch}^{4} g \sinh ^{2}(2 r))\bigr\} ^{1/2}\bigl\{ 16 \bigl(2 N_{a}+ \cosh (2 r) \bigr)\bigr\} ^{-1}. \end{aligned}$$

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1.3 1.3 Phase sensitivity $\Delta \phi _{\mathrm{twosqz}}^{\mathrm{I}}$

Phase sensitivity with intensity detection with two-squeezed-vacuum state input is given by

$$\begin{aligned} \Delta \phi _{\mathrm{twosqz}}^{\mathrm{I}} ={}&\frac{1}{2\sqrt{2}} \operatorname{csch}(2 g) \operatorname{sech}(2 r) \bigl(\operatorname{csch}^{2}(2 g) \bigl(\sec ^{2} ( \phi /2 ) \bigl(\cosh (8 g) \cosh (4 r)-1\bigr) \\ & {} +2 \sinh ^{2}(2 r) \csc ^{2} (\phi /2 )\bigr)-4 \cosh (4 g) \cosh (4 r)+4\bigr)^{1/2}. \end{aligned}$$

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The corresponding optimal phase is found to be

$$\begin{aligned} \phi _{\mathrm{twosqz,opt}}^{\mathrm{I}} ={}&2 \arctan \bigl\{ \bigl[ \operatorname{csch}^{2}(4 g) \bigl(\operatorname{sech}(4 r) \bigl(\sqrt{2\sinh ^{2}(2 r) \bigl(\cosh (8 g) \cosh (4 r)-2\bigr)} \\ & {} +1\bigr)-1\bigr)\bigr]^{1/2}/{\sqrt{2}}\bigr\} . \end{aligned}$$

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