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Abstract
We theoretically study the phase estimation based on a Mach-Zehnder interferometer (MZI) with a two-mode squeezed coherent state. By maximizing the quantum Fisher information, we find that the quantum Cramér-Rao bounds (QCRB) can reach sub-Heisenberg limit under the phase-matched condition. The optimal phase sensitivity can reach the sub-shot noise limit (SNL) and approach the QCRB by employing the intensity difference detection. Meanwhile, compared with the MZI fed with a coherent plus a single-mode squeezed vacuum state, this scheme can have better performance by adjusting the squeezing parameter and the mean photon number. With the same parameter, our scheme shows more sensitive phase measurement than the SU(1,1) interferometer with a coherent plus a vacuum state. We also show that the phase sensitivity of our proposal can still reach the SNL when the loss of the photon is 36%. This scheme can provide potential applications in optical sensors.
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1. Introduction
As quantum technology and quantum information developed quickly, quantum metrology has drawn much attention and has been applied in many practical fields [1–10]. A famous one is the laser interferometer gravitational wave observatory (LIGO) where a quantum beam enters the unused input port [11,12]. The objective of quantum metrology is to improve phase measurement precision based on the theory of quantum mechanics. In quantum metrology, the phase shift sensitivity can be boosted by using a single-mode squeezed vacuum beam [13]. The Mach-Zehnder interferometer (MZI) and its variants have been employed to estimate the phase shift [14–28]. Caves et al. suggested that the phase sensitivity can reach sub-shot noise limit (SNL) and even Heisenberg limit (HL) when the vacuum beam is replaced by a single-mode squeezed vacuum beam [13], which was soon demonstrated by Xiao et al. and Grangier et al. [15,16]. Later, a series of detection methods were proposed, such as balanced homodyne detection, intensity difference detection and parity detection [18,24]. To achieve the optimal phase sensitivity, countless measurement methods need to be taken into account in experiments which is an impossible task.
Nevertheless, the quantum Fisher information (QFI) and its associated quantum Cramér-Rao bounds (QCRB) can solve this problem elegantly. The QFI can characterize the maximum amount of information about an unknown parameter [17,18,24,25], while the QCRB can give the optimal theoretical value without considering the detection process. Numerous efforts have been made to obtain larger QFI for different inputs. Jarzyna et al. studied the QFI of phase shifts in the two-arm case with the MZI [28]. Lang and Caves [14] proved that the squeezed vacuum is the optimal state to inject into one input port when a coherent light is injected into the other input port of the MZI. In the phase estimation process, even with the same input, single-parameter QFI and two-parameter QFI are different [26]. In the single-phase estimation, a large number of groups impose a phase shift $\phi _0$ on one arm where $\phi _0$ is the phase to be estimated. On the other hand, some authors also assume that there is a phase shift $\phi _0/2$ in one arm and a phase shift $-\phi _0/2$ in the other arm. In this case, the phase shift $\phi _0$ is the difference between the two arms [29]. Lang and Caves proposed a two-parameter estimation scenario where $\phi _{m}$ was imposed on one arm and $\phi _{n}$ was imposed on the other arm, and they were interested in the phase difference $\phi _m-\phi _n$ [14]. Soon later, You et al. proved that two-parameter QFI can avoid the risk of using fictitious beams (unrealistic resources) in the detection process [30].
Meanwhile, the two-parameter QFI are not only employed in the MZI but also in the nonlinear interferometer, which was firstly put forward by Yurke et al. [31]. Compared with the MZI, which is also named as an SU(2) interferometer, this kind of nonlinear interferometers is named as an SU(1,1) interferometer because the transformation can be characterized by the SU(1,1) group. The employment of an SU(1,1) interferometer can surpass the SNL through the amplification process. For the SU(1,1) interferometer, the phase sensitivity can reach the HL if the inputs are two vacuum beams. The two vacuum beams become two-mode squeezed vacuum beams through an optical parameter amplification (OPA), e.g., four-wave mixing (FWM). If the two-mode squeezed vacuum beams are the inputs of the MZI, the phase sensitivity can even beat the HL and approach the Hofmann limit with the parity detection [18]. The SNL and the HL are defined as $\frac {1}{\sqrt {N_0}}$ and $\frac {1}{N_0}$, respectively. Here, for the MZI, $N_0$ is the photon number of the two input ports; while it is the photon number after the first FWM for the SU(1,1) interferometer. Under these circumstances, for the MZI with two-mode squeezed vacuum beams and the SU(1,1) interferometer fed with two vacuum beams, the total input photon numbers are the same.
Even though high quality QCRB can be achieved with two-mode squeezed vacuum beams, it is still limited to the low photon number without the ’boost’ of a coherent beam. Compared with the two-mode squeezed vacuum state, the two-mode squeezed coherent beams can be much brighter. In addition, the two-mode squeezed coherent state can be realized by the FWM process with the input of a coherent state plus a vacuum state. The coherent beam is a very cheap resource and the FWM technology is mature in both theory and experiment. Meanwhile, the intensity difference detection is a very useful detection method which has been employed in the intensity squeezing of the two-mode squeezed coherent state. And it has been widely used in the phase sensitivity based on the MZI. Under this condition, we propose to use the FWM to produce a two-mode squeezed coherent state to attain higher quality QCRB and achieve better phase sensitivity via intensity difference detection. We focus on the balanced BS in this paper. Moreover, in order to avoid the complex phase-matched condition which will make the experimental process much difficult, only the condition that one coherent beam enters into the FWM is displayed.
This article is organized as following. In the second section, we theoretically derive the QCRB and QFI with the input of two-mode squeezed coherent state based on the MZI. We also present an error propagation formula of the intensity difference measurement strategy. Then the phase sensitivity and the QCRB are analyzed. Effects of the depletion of photon detectors are also considered which show that our scheme presents robustness against the depletion of photon detectors. In the third part, we compare our scheme with a coherent beam plus a single-mode squeezed vacuum beam based on the MZI, and a coherent beam plus a vacuum beam based on an SU(1,1) interferometer. The conclusion is given in the last part.
2. Methods
2.1 QFI and QCRB in two-parameter estimation
As Fig. 1 displays, a coherent beam and a vacuum beam undergo the FWM process. Next, the output beams pass through the first beam splitter (BS). Then the beams after the first BS undergo the phase shifters $\varphi _1$ and $\varphi _2$, respectively. The intensity of the beam is taken by the photon detectors after the second BS. For the MZI, the input is a two-mode squeezed coherent state, which can be expressed as
where $|0_1,0_2\rangle$ represents the two-mode vacuum state; $\hat {D}_{\alpha }=e^{\alpha \hat {a}^{\dagger}-\alpha ^*\hat {a}}$ is the displacement operator with $\alpha =|\alpha |e^{i\theta _\alpha }$ ($|\alpha |$ and $\theta _\alpha$ are the amplitude and phase of the coherent state, respectively); $\hat {S}(\xi )$ is the squeezing operator which is defined as $\hat {S}(\xi )=e^{\xi ^*\hat {a}\hat {b}-\xi \hat {a}^{\dagger}\hat {b}^{\dagger}}$, where $\hat {a}$ $(\hat {a}^{\dagger})$ and $\hat {b}$ $(\hat {b}^{\dagger})$ are annihilation and creation operators of mode $a$ ($b$), respectively; $\xi =r_1e^{i\phi _1}$, $r_1$ and $\phi _1$ are the squeezing parameter strength and squeezing phase of the FWM process, the gain of the FWM is given by $G_1=\rm {cosh}^2 r_1$, and $\phi _1=2\phi _{\rm {p}}$ with $\phi _{\rm {p}}$ being the phase of the pump beam [32,33]. Without the ’displacement’ process, the inputs will become two-mode squeezed vacuum beams which have been proposed by many authors [18]. Note that different from other squeezed coherent states, the vacuum state is firstly displaced and then squeezed. In this case, it can be generated in an easy way in which one coherent beam and one vacuum beam enter an FWM process, which has been done in experiments. Moreover, only one vacuum beam undergoes the displacement process rather than two beams.
Fig. 1. A sketch of the Mach-Zehnder interferometer with two phases $\varphi _1$ and $\varphi _2$. Two-mode squeezed coherent state comes from the four wave mixing process when the inputs are one coherent beam and one vacuum beam. For the phase estimation process, the second beam splitter belongs to the detection process. Intensity difference detection is employed as the measurement strategy. FWM: four wave mixing, B: blocker, BS: beam splitter, PD: photon detector. The dashed line indicates the vacuum beam.
Considering the potential unavailable measurement risk, two-phase estimation is necessary. In this work, we assume that the MZI is lossless and we only focus on the phase difference between the two arms $\varphi _d$. $\varphi _d=(\varphi _1-\varphi _2)/2$. Then the QCRB is given by
where $M$ is the number of repeated measurements and we only consider the case $M=1$ throughout this paper. The QFI of our proposal is $F=F_{\rm {dd}}-\frac {F_{\rm {ds}}F_{\rm {sd}}}{F_{\rm {ss}}}$, where $F_{\rm {dd}}$, $F_{\rm {ds}}$, $F_{\rm {sd}}$ and $F_{\rm {ss}}$ are the difference-difference, difference-sum, sum-difference and sum-sum Fisher matrix elements of $F$, which are given by2.2 Intensity difference detection
In order to achieve the phase sensitivity in experiments, we need to apply the error propagation formula
here $\hat {H}$ is the detection method in the experiments. In our work, the intensity difference detection method is employed. Then $\hat {H}$ can be expressed as $\hat {H}=\hat {I}_{-B}=\hat {I}_a-\hat {I}_b=\hat {a}_{\rm {4}}^{\dagger}\hat {a}_{\rm {4}}-\hat {b}_{\rm {4}}^{\dagger}\hat {b}_{\rm {4}}$, where $\hat {a}_4$ $(\hat {a}_4^{\dagger})$ and $\hat {b}_4$ $(\hat {b}_4^{\dagger})$ are the annihilation (creation) operators of the two modes after the second BS.By substituting Eqs. (12) and (13) in Appendix into Eq. (4), the slope of intensity difference can be expressed as
The input photon number of the MZI is
3. Discussion
3.1 Analysis
We consider two cases $N=0$ and $G_1=1$. The two-mode squeezed coherent state will become two-mode squeezed vacuum state if $N=0$. Then we have $F=4G_1(G_1-1)$ and therefore ${|(\partial _{\varphi } \langle \hat {I}_{\rm {}\hbox{-}{\rm B}}\rangle )|}=0$, which means the intensity difference detection is phase insensitive [18,25]. On the other hand, the inputs will become the combination of one coherent and one vacuum state if $G_1=1$. In this case, $F=N$ and $\Delta \varphi =1/|(\sqrt {N}{\rm {sin}}\varphi )|$. The optimal phase sensitivity will be limited to the SNL.
Next, we compare the QCRB with the SNL and the HL. Lower phase value implies better sensitivity. It is not difficult to find that $F$ is larger than $N_B$, therefore $\frac {1}{\sqrt {F}}$ is smaller than $\frac {1}{\sqrt {N_B}}$, which means that the QCRB will always be better than the SNL. Moreover, the QCRB can beat the HL if $F$ is larger than $N_B^2$. Let $N_C=F-N_B^2$ and we have
As shown in Fig. 2(a), with low photon number $N$ of the coherent beam, the QCRB can beat the HL. The black line means that the QCRB reaches the HL when $N_C=0$. The QCRB can beat the HL when $N_C$ is larger than zero. With $r_1=0.5$, the QCRB can beat the HL when $N$ is less than 1.24. With $N=1$, the QCRB can reach the sub-HL when $r_1$ is less than 0.88 ($G_1\approx 2$). Figure 2(b) shows the phase sensitivity versus the phase shift with $N=1$ and $r_1=0.88$. We can see that in this case, the QCRB can reach the HL. And the optimal phase sensitivity with intensity difference detection can only reach sub-SNL. In Fig. 2(c), though the QCRB can beat HL, the phase sensitivity can only reach sub-SNL. Moreover, the optimal phase sensitivity with intensity difference detection is worse than SNL while the QCRB is better than HL in Fig. 2(d). Meanwhile, in Figs. 2(b), 2(c) and 2(d), the optimal phase sensitivity can be achieved when the phase shift is close to $k \pi$. By averaging the phase shift over $[0, 2\pi ]$, we find that the averaged phase sensitivity of intensity difference detection is worse than the SNL in Figs. 2(b), 2(c) and 2(d).Fig. 2. (a) $N_C$ versus photon number and parametric strength. Phase sensitivity versus phase shift in (b) with $N=1$ and $r_1=0.88$, in (c) with $N=1$ and $r_1=0.5$ and in (d) with $N=0.1$ and $r_1=1.3$. In (a), the QCRB beats the HL when $N_C$ is higher than zero. The black line indicates $N_C=0$ where the QCRB reaches the HL. The black dots mean the parameter values used in (b), (c) and (d). ID: phase sensitivity with intensity difference detection, SNL: shot noise limit, HL: Heisenberg limit, QCRB: quantum Cramér-Rao bounds.
The phase sensitivity versus photon number and squeezing parameter strength is shown in Figs. 3(a) and 3(b). As shown in Fig. 3(a), with $G_1$=2, the QCRB can reach sub-HL when the photon number is less than 1. The lower value represents the better sensitivity. In Fig. 3(a), when N is larger than 0.3, the value of the phase sensitivity achieved by the intensity difference detection is below SNL. Then the phase sensitivity can beat SNL and reach sub-SNL. When N is below 0.3, though the QCRB can beat HL, the optimal phase sensitivity with intensity difference detection is worse than SNL. With the increase of the photon number, the QCRB and phase sensitivity become better. However, in this case, both the QCRB and the phase sensitivity can only reach sub-SNL. Meanwhile, the phase sensitivity approaches the QCRB with the increase of the photon number. Figure 3(b) shows the phase sensitivity versus the parametric strength $r_1$. Increasing the squeezing parameter $r_1$ can boost the phase sensitivity. The QCRB can beat the HL when $r_1<0.88$. And it can reach sub-SNL with the increase of $r_1$. In Fig. 3(b), the value of the phase sensitivity with intensity difference detection is always below SNL, which means that it can always reach sub-SNL. At the same time, the phase sensitivity with the intensity difference detection can never approach the QCRB with $N=1$. Figures 3(c) and 3(d) show the optimal phase point versus the photon number and parametric strength. The optimal phase point is the phase shift where optimal phase sensitivity can be achieved. In Fig. 3(c), with $G_1=2$, the optimal phase point increases quickly and approaches constant when $N$ is larger than $10^2$. In Fig. 3(d), the optimal phase point decreases from $\pi /2$ to $0$ when the parametric strength increases from 0 to 3.
Fig. 3. Phase sensitivity versus photon number (a) and parametric strength (b). Optimal phase point with intensity difference detection versus photon number (c) and parametric strength (d). $r_1=0.88$ in (a) and (c). $N=1$ in (b) and (d). In (c) and (d), $\phi _1=2\theta _{\alpha }$. Others’ are the same with Fig. 2.
3.2 Depletion of photon detectors
In this part, we consider the influence of the detection depletion of photon detectors. According to Fig. 1, detection depletion of photon detectors occurs during the measurement process, hence the QFI is unchanged. Then we only need to consider the depletion of the detector in the error propagation formula.
As shown in Fig. 4, the detection depletion can be equivalently treated as fictitious beam splitters placed before the detectors. The input-output relationships of the fictitious beam splitters are
Fig. 4. The model of detection process with depletion. FBS: fictitious beam splitter with the same transmissivity $T$.
Fig. 5. Phase sensitivity versus transmissivity of photon detectors. Red line: $N=1$ and $r_1=0.5$. Blue line: $N=10^3$ and $r_1=0.5$. Black line: $N=1$ and $r_1=1.5$. Others’ are the same with Fig. 2.
3.3 Other schemes with different input states
First, we compare our scheme with the one which uses the MZI with the input of a single-mode squeezed plus a coherent state, a scheme that can increase the sensitivity of the gravitational-wave detector GEO 600 and has been employed in LIGO [34,35]. Meanwhile, it has similar form of a two-mode squeezed state. The single-mode squeezed vacuum state can be shown as $|\psi \rangle =\hat {S_2}(\xi _2)|0\rangle$, where $\hat {S_2}(\xi _2)=e^{[\xi _2^*\hat {a}\hat {a}-\xi _2\hat {a}^{\dagger}\hat {a}^{\dagger}]/2}$ is the single-mode squeezing operator, $\xi _2=r_2e^{i\phi _2}$, $r_2$ and $\phi _2$ are the squeezing parameter strength and the phase of the single-mode squeezed vacuum state, respectively. The two-parameter QFI for this scenario is $F_2=N_2e^{2r_2}+{\rm {sinh}}^2r_2$ [14] and the optimal phase sensitivity can reach $\frac {1}{\sqrt {N_2e^{2r_2}}}$ with balanced homodyne detection [24,36]. Here, $N_2$ is the photon number of the coherent state. Figure 6 shows the phase sensitivity versus the photon number $N$ and the parametric strength $r_1$. In Figs. 6(a) and (b), with $N_2={\rm {sinh}^2r_2}$, both $\frac {1}{\sqrt {F_2}}$ and $\frac {1}{\sqrt {N_2e^{2r_2}}}$ can reach the HL scaling. And they always have better performance than that in Fig. 2 when $N$ is more than 1. The QCRB with two-mode squeezed coherent state can beat that of a coherent state plus a squeezed vacuum state when the photon number $N$ is less than 0.4. However, the optimal phase sensitivity with intensity difference detection only can reach sub-SNL or is even worse than the SNL (It can be found in the inset of Fig. 6(a)). In Figs. 6(c) and 6(d), with $G_2=3$ $(G_2={\rm {cosh}^2r_2})$ and $N_2=N_B-G_2+1$, $N_2$ is much larger than $\rm {sinh}^2r_2$. Then the QCRB and the optimal phase sensitivity with a coherent state plus a single-mode squeezed vacuum state can only reach sub-SNL. Meanwhile, $\frac {1}{\sqrt {N_2e^{2r_2}}}$ also approaches $\frac {1}{\sqrt {F_2}}$. With $r_1=1.5$, the QCRB and optimal phase sensitivity are always better than that of the MZI with a coherent state plus a single-mode squeezed vacuum state, as shown in Fig. 6(c). In Fig. 6(d), with $r_2$ being larger than 1.3 and $N=10^3$, the phase sensitivity with two-mode squeezed coherent state is better than that of a single-mode squeezed vacuum plus a coherent state. In this case, the optimal phase sensitivity with intensity difference detection also approaches the QCRB. Note that $N_2 \gg \rm {sinh}^2r_2$ is more reasonable in practical experiments since the current squeezed record of single-mode squeezed vacuum state is about 15 dB [37]. Then the phase sensitivity with two-mode squeezed coherent state is better than that with a coherent state plus a single-mode squeezed vacuum state. Meanwhile, for the record of the intensity difference squeezing of two-mode squeezed coherent state, it is only about 8.8 dB. However, higher intensity squeezing degree is foreseeable. For example, the squeezing degree can surpass 10 dB in the phase sensitive FWM [38].
Fig. 6. Phase sensitivity versus photon number (a), (c) and parametric strength (b), (d). $r_1=0.88$ in (a), $N=1$ in (b) and $N_2={\rm {sinh}^2r_2}=N_B/2$ in (a) and (b). $r_1=1.5$ in (c), $N=10^3$ in (d), $G_2=3$ and $N_2=N_B-G_2+1$ in (c) and (d). TMS QCRB and SMS QCRB are the quantum Cramér-Rao bounds when the inputs are two-mode squeezed coherent state and single-mode squeezed vacuum state plus a coherent state, respectively, and they are equal to $\frac {1}{\sqrt {F}}$ and $\frac {1}{\sqrt {F_2}}$. SMS BHD is the optimal phase sensitivity $\frac {1}{\sqrt {N_2e^{2r_2}}}$ with balanced homodyne detection and a single-mode squeezed vacuum state plus a coherent state. For the two scenarios, they have the same SNL and HL since $N_2+G_2-1=N_B$. Others’ are the same with Fig. 2.
Next, we compare our scheme with the SU(1,1) interferometer. As shown in Fig. 7, a coherent beam and a vacuum beam enter the SU(1,1) interferometer. For the first part, it can be treated as the generating process of a two-mode squeezed coherent beam as that in Fig. 1. The SU(1,1) interferometer with the input of a coherent beam plus a vacuum beam has been investigated by many groups [30,31,36,39,40]. In order to present a direct comparison, we assume that the two FWMs of SU(1,1) interferometers have the same parametric strength, and the phase shift of the pump beams is zero [33]. In addition, the photon number of the coherent state is equal to $N$. Then Eq. (7) can be applied in the SNL and HL in this case. So far, with the input of a coherent plus a vacuum state, the reported two-parameter QFI is $F_3=4G_1(G_1-1)(N+1)$ [30] and the optimal phase sensitivity can reach $\frac {1}{\sqrt {4G_1(G_1-1)N}}$ by employing balanced homodyne detection [39]. More comparison details can be found in Fig. 8 in Appendix B. In the MZI, we focus on the phase difference between the two arms. In contrast, the sum of the phases in the two arms of the SU(1,1) interferometer are investigated in [30]. Meanwhile, the gain of the second FWM of the SU(1,1) interferometer can boost the phase sensitivity and extend the sub-SNL phase range [36]. If $r_1=0$, the input of the MZI is one coherent state plus one vacuum state. For the SU(1,1) interferometer, it means that the FWM is absent and the phase sensitivity can not be attained. In addition, without the ’displacement’ process, $N=0$, $F=F_3$ and the phase sensitivity of the SU(1,1) interferometer can reach sub-HL [40].
Fig. 7. A sketch of the SU(1,1) interferometer with the input of a coherent beam plus a vacuum beam. It contains two four-wave mixing processes. Others’ are the same with Fig. 1.
Fig. 8. Phase sensitivity versus photon number and parametric strength. $r_1=0.88$ in (a) and $N=1$ in (b). MZ QCRB and SU QCRB are the quantum Cramér-Rao bounds of the Mach-Zehnder interferometer and SU(1,1) interferometer, respectively, and they are equal to $\frac {1}{\sqrt {F}}$ and $\frac {1}{\sqrt {F_3}}$. BHD is the optimal phase sensitivity with balanced homodyne detection based on the SU(1,1) interferometer and it is $\frac {1}{\sqrt {4G_1(G_1-1)N}}$. Others’ are the same with Fig. 2.
4. Conclusion
In conclusion, this paper presents the phase sensitivity with the input of a two-mode squeezed coherent state. Under the phase-matched condition, the QCRB can reach sub-HL and the optimal phase sensitivity with intensity difference detection can only reach sub-SNL with $N=1$. Both the QCRB and the optimal phase sensitivity with the intensity difference measurement can only reach sub-SNL under the condition that $N=10^2$. Meanwhile, even with up to $36{\% }$ depletion of the photon detectors, the phase sensitivity of this scheme can still reach the SNL, which implies that the scenario is robust against the external depletion. Note that it is an inadequacy of our work that the enhancement over HL is only evident in the low photon number regime. Nevertheless, the current FWM technique allows a relatively large squeezing parameter to be realizable. Therefore, the photon number of the output beams could be large enough to be detected. The phase sensitivity of this scheme is even better than that of the MZI with a coherent state plus a single-mode squeezed vacuum state with different squeezing parameters. Furthermore, compared with the scenario of a coherent state plus a vacuum state based on an SU(1,1) interferometer, this proposal always has better phase sensitivity. Meanwhile, other detection methods, such as the balanced homodyne detection and parity detection, can be used to achieve the phase sensitivity. As the experimental techniques develop, both the degenerate FWM and non-degenerate FWM with frequency converter can be used in this scenario. This scheme also shows the advantage of quantum squeezing in the area of phase measurement and displays its potential superiority in other areas of precision measurements.
5. Appendix
5.1 Appendix A
In the process of parameter estimation, the QCRB is given by [14,26]
$\Delta \varphi$ is the variance of the parameter $\varphi$ and $F(\varphi )$ is the QFI. $M=1$ is the number of repeated measurements. The phase estimation in this paper is treated as a two-parameter estimation process. Then Fisher information matrix is defined as According to [14,24–26], when the input is pure state,In the QFI calculation process, two kinds of relationships are necessary. For the transformation of the FWM process, it can be written as
For input and output states of the MZI, it can be displayed as $|\psi _{\rm {out}} \rangle =\hat {U}\hat {B}|\psi _{\rm {in}} \rangle$, where $\hat {B}$ is the unitary transformation operator of the BS and $\hat {U}$ is the phase shift unitary operators. Here, $\hat {B}=e^{-i\hat {J}\pi /4}$ and $\hat {J}=\hat {a}^{\dagger}_1\hat {b}_1+\hat {b}_1^{\dagger}\hat {a}_1$. $\hat {U}=e^{i(\varphi _1 \hat {a}_2^{\dagger}\hat {a}_2+\varphi _2 \hat {b}_2^{\dagger}\hat {b}_2)}=e^{i\hat {N}_s\varphi _s}e^{i\hat {N}_d\varphi _d}$, $\hat {N}_s=(\hat {a}_2^{\dagger}\hat {a}_2+\hat {b}_2^{\dagger}\hat {b}_2)$, $\hat {N}_d=(\hat {a}_2^{\dagger}\hat {a}_2-\hat {b}_2^{\dagger}\hat {b}_2)$. $|\psi _{\rm {out}} \rangle$ is the state after the phase shift and $|\psi _{\rm {in}} \rangle$ is the state before the first BS. $|\psi _{\rm {0}} \rangle$ is the combination of a coherent state and a vacuum state. So, the QFI matrix elements can be shown as
5.2. Appendix B
As shown in Fig. 8(a), all the phase sensitivities become better with increasing $N$, and $\frac {1}{\sqrt {4G_1(G_1-1)N}}$ approaches $\frac {1}{\sqrt {F_3}}$. With $r_1=0.88$, both $\frac {1}{\sqrt {F_3}}$ and $\frac {1}{\sqrt {4G_1(G_1-1)N}}$ are better than the SNL and worse than the HL. In the low photon number region $N=1$, the QCRB of the SU(1,1) interferometer is better than the optimal phase sensitivity with intensity difference detection. However, it is much worse than the optimal phase sensitivity with intensity difference detection when $N=10^2$. In Fig. 8(b), the QCRB of the MZI is always better than that of the SU(1,1) interferometer. And optimal phase sensitivity with intensity difference strategy is better than the optimal phase sensitivity with balanced homodyne detection. Meanwhile, the phase sensitivity with intensity difference detection is worse than the QCRB of the SU(1,1) interferometer. Under this condition, the phase sensitivity with two-mode squeezed coherent state is better than that of the SU(1,1) interferometer with a coherent plus a vacuum state even though they have the same SNL and HL.
Funding
National Natural Science Foundation of China (12104189, 12104190, 12204312); Natural Science Foundation of Jiangsu Province (BK20210874); General project of natural science research in colleges and universities of Jiangsu Province (20KJB140008); Jiangxi Provincial Natural Science Foundation (20224BAB211014).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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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