Ultra-sensitive phase measurement based on an SU(1,1) interferometer employing external resources and substract intensity detection

Jun Liu; Yuanxiang Wang; Mingming Zhang; Jinwen Wang; Dong Wei; Hong Gao

doi:10.1364/OE.413179

1. Introduction

An optical Mach-Zehnder interferometer (MZI) has become a very useful tool for the precision measurement which has attracted a lot of attention [1–4]. Usually, a MZI, which typically contains two beam splitters (BS), is employed for the measurement of phase sensitivity. The first BS splits the beam and the second BS combines the two beams together. Inside the interferometer, the two beams experience a phase shift before the second BS. When the inputs of a MZI are a coherent beam and an vacuum beam, the phase sensitivity is limited to $\frac {1}{\sqrt {N}}$, named as shot noise limit (SNL) where $N$ is the total photon number in the interferometer.

This limit is due to the classical nature of the input coherent state and the vacuum state [5]. However, this limit can be surpassed if the vacuum state is replaced by non-classical states of light, such as squeezed states [6,7]. Furthermore, by employing N00N states and two mode squeezed vacuum states to replace the coherent and vacuum states, the phase sensitivity can approach or even beat Heisenberg limit (HL) [8,9].

In addition to the method above, another way to beat SNL is to employ an optical parametric amplifier (OPA) to replace the BS [10–14]. This interferometer is named as the SU(1,1) interferometer due to that it is described by the group SU(1,1) [15]. For the SU(1,1) interferometer, Ou et al. pointed out that the phase sensitivity with vacuum inputs can reach $\frac {1}{\sqrt {N(N+2)}}$, where N is the total photon number inside the interferometer and equal to 2$\textrm{sinh}^2r$, with r being the OPA parametric strength [16]. However, the photon number in this scheme is merely related to OPA strength r, so it is very small. On the other hand, the measurement method is balance homodyne detection where the strong local beam is only employed for detection without entering into the interferometer.

In order to solve the small photon number problem, Plick et al. proposed that the strong coherent light can ‘boost’ the photon number in an SU(1,1) interferometer with intensity detection [17]. Jing et al. had shown that the intensity of the beam after this kind interferometer can be much larger due to the amplification [18]. Recently, You et al. gave the conclusive precision bounds for SU(1,1) interferometers [19], in which they claimed that the parity detection is not the optimal detection method and the external resources need to be taken into consideration in the phase estimation process. This view was verified by Liu et al. who showed that the phase sensitivity can beat SNL in an SU(2) interferometer by employing intensity detection and external resources even with an vacuum input [20]. However, the scheme in [20] is based on the SU(2) interferometer, and a detailed scheme based on the SU(1,1) interferometer which employ the external resources to enhance the phase sensitivity is missing. So we propose to use the external resources and substract intensity detection to beat the SNL based on an SU(1,1) interferometer in this paper. It also proves that the phase sensitivity can beat SNL by employing external resources while one of the inputs is an vacuum beam.

This paper is organized as follows. In the next part, the scheme employing external resources with direct intensity detection is introduced. The phase sensitivity of this interferometer is compared with HL and SNL when the inputs are bright entangled twin beams, two mode squeezing vacuum beams and two coherent beams. The factors which can improve sensitivity are discussed. Meanwhile, the detection efficiency of the photon detectors which play an important role on the phase sensitivity are also shown. Finally, we make a conclusion for this paper.

2. Model

Figure 1 displays the scheme employed for the measurement of phase sensitivity. Considering a pump beam and a probe coherent beam as inputs entering into the vapor cell, the four wave mixing (FWM)1 is a phase insensitive process as [21]. FWM is one type of OPA. The transformation of the first FWM [22] is

(1)$$\begin{aligned} \hat{a}_{1}&=\textrm{cosh} r_1 \hat{a}_{0}+e^{i\theta_1}\textrm{sinh} r_1 \hat{b}^{\dagger}_{0},\\ \hat{b}_{1}&=\textrm{sinh} r_1 \hat{a}^{\dagger}_{0}e^{i\theta_1}+\textrm{cosh} r_1 \hat{b}_{0}, \end{aligned}$$

where $r_{1}$ is the parametric strength and $\theta _1$ is the total phase shift of the first FWM. ${\hat {a}}_0$, $\hat {b}_0$, $\hat {a}_0^{\dagger }$ and $\hat {b}_0^{\dagger }$ are annihilation and creation operators for the input probe modes and vacuum state, respectively. For the coherent input beam, it has $\langle {\hat {a}_0}\rangle = \langle {\hat {a}_0}^{\dagger }\rangle =\sqrt {N_0}$, where $N_0$ is the photon number. After FWM1, the intensity of the amplified probe beam is $\textrm{cosh}^2r_1N_0+\textrm{sinh}^2r_1$ and the intensity of the generated conjugate beam is $\textrm{sinh}^2r_1N_0+\textrm{sinh}^2r_1$. Then, the amplified probe beam enters into an SU(1,1) interferometer and the generated conjugate beam is employed for measurement directly.

Fig. 1. The scheme for phase sensitivity measurement. It can be divided into three parts: 1, the state generation; 2, an SU(1,1) interferometer; 3, the detection process. M is mirror, FWM1, FWM2 and FWM3 are four wave mixing processes and B is beam block. The dashed line means that this is an vacuum beam. The measurement method is intensity-difference detection. $\phi$ is the phase shift. Here, only one beam enters into the SU(1,1) interferometer and the conjugate beam is employed for detection. The function of the conjugate beam is similar to the local beam in balance homodyne detection. The local beam offers an external phase reference and the conjugate beam offers an intensity reference.

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The transformation of the second FWM is

(2)$$\begin{aligned} \hat{a}_{2}&=\textrm{cosh}r_{2}\hat{a}_{1}+e^{i\theta_2}\textrm{sinh}r_{2}\hat{c}^{\dagger}_{0},\\ \hat{c}_{1}&=\textrm{sinh}r_{2}\hat{a}^{\dagger}_{1}e^{i\theta_2}+\textrm{cosh}r_{2}\hat{c}_{0}. \end{aligned}$$

For the FWM2, the inputs are a quantum beam and an vacuum beam which are different from the FWM1. The photon numbers of the two arms in the interferometer are $N_1=G_2G_1N_0+G_1G_2-1$ and $N_2=G_1(G_2-1)(N_0+1)$ with $G_1=\textrm{cosh}^{2}r_1$, $G_2=\textrm{cosh}^{2}r_2$. $r_{2}$ is the parametric strength of FWM2. So the FWM2 is also a phase-insensitive process. For simplification, in the following, $\theta _1$, $\theta _2$ and $\theta _3$ are assumed to be the same which are labelled as $\theta$. $\theta _2$ is the phase shift of FWM2. and $\theta _3$ is the phase shift of FWM3. The FWM3 process is always phase-sensitive due to the inputs of three beams [23].

The total relation of this scheme is

(3)$$\hat{a}_{\textrm{out}}=M_1\hat{a}_0+M_2\hat{b}_0^{\dagger}+M_3\hat{c}_0^{\dagger},$$

$M_1=(\sqrt {G_2G_3}e^{i\phi }+\sqrt {(G_2-1)(G_3-1)})\sqrt {G_1}$, $M_2=(\sqrt {G_2G_3}e^{i\phi }+\sqrt {(G_2-1)(G_3-1)})\sqrt {G_1-1}e^{i\theta }$ and $M_3=(\sqrt {(G_2-1)G_3}e^{i\phi }+\sqrt {G_2(G_3-1)})e^{i\theta }$, After the SU(1,1) interferometer, the photon number of the probe beam is $\langle \hat {I}_{\textrm{a}} \rangle =M_4M_1N_0+M_5M_2+M_6M_3$, $M_4=M_1^{\ast}$,$M_5=M_2^{\ast}$, $M_6=M_3^*$, which is related to the phase shift $\phi$, $G_1$, $G_2$ and $G_3$. $G_3=\textrm{cosh}^{2}r_3$ and $r_{3}$ is the parametric strength of FWM3. The minimum intensity of the probe beam after the SU(1,1) interferometer is existing at $\phi =\pi$ and the maximum intensity is at the phase point $\phi =0$. The detection signal is $\langle \hat {I}_- \rangle =\langle \hat {a}^{\dagger }_{\textrm{out}}\hat {a}_{\textrm{out}}-\hat {b}^{\dagger }_{\textrm{out}}\hat {b}_{\textrm{out}} \rangle =(M_4M_1-G_1+1)N_0+M_5M_2+M_6M_3-G_1+1$, and the slope can be displayed as

(4)$$|\frac{\delta \langle \hat{I}_-\rangle}{\delta \phi}|_{\textrm{B}}=|2\textrm{sin} \phi G_1\sqrt{G_2(G_2-1)G_3(G_3-1)}(N_0+1)|,$$

while the variance of intensity difference $\Delta ^2 \hat {I}_-$ is

(5)$$\begin{aligned} \Delta^2 \hat{I}_{\textrm{-B}}&=|\langle \hat{I}_-^2 \rangle- \langle \hat{I}_- \rangle^2|\\ &=|(M_4M_1-(G_1-1))^2N_0+(M_5M_1-\sqrt{G_1(G_1-1)}e^{-i\theta})(M_4M_2-\sqrt{G_1(G_1-1)}e^{i\theta})\\ &\;\;\;(N_0+1) +M_1M_3M_4M_6(N_0+1)|. \end{aligned}$$

The total photon number inside the interferometer is

(6)$$\begin{aligned} N_{\textrm{B}}&=\langle \hat{a}^{\dagger}_2\hat{a}_2+\hat{c}_1^{\dagger}\hat{c}_1 \rangle\\ &=(2G_1G_2-G_1)N_0+2G_1G_2-G_1-1. \end{aligned}$$

At this time, the SNL and HL are $\frac {1}{\sqrt {N_{\textrm{B}}}}$ and $\frac {1}{N_{\textrm{B}}}$. According to Eq. (6), the gain $G_3$ of the FWM3 have no effects on HL and SNL.

The phase sensitivity $\Delta \phi$ as the uncertainty in estimating a phase shift $\phi$ is

(7)$$\Delta \phi =\frac{\sqrt{\Delta^2 \hat{I}_{\textrm{-B}}}}{|(\partial_{\phi} \langle \hat{I}_{\textrm{-B}}\rangle)|}.$$

Figure 2(a) shows the phase sensitivity versus phase shift and squeezing parameter of the three FWM processes with $N_0=1000$, $r_1=1$ and $r_2=r_3=3$. The lower phase value means the better phase sensitivity. In this model, with one of the twin beams generated by FWM1 entering into the SU(1,1) interferometer and the other one being employed for detection, the phase sensitivity can beat SNL and approach HL when the phase shift $\phi$ is approaching $\pi$. In the previous SU(1,1) interferometer like [24], there is a phase shift $\frac {\pi }{2}$ between the pump beam of two FWMs. Then the second FWM will ’undo’ what the first FWM did in [24]. In this scheme, due to the lack of the phase shift $\frac {\pi }{2}$ between the pump beam, the third FWM will ’undo’ what the second FWM did when the balanced situation is $\phi =\pi$.

Fig. 2. (a) The phase sensitivity of quantum beams versus phase shift. (b) the optimal phase sensitivity versus squeezing parameter $r_1$ and (c) squeezing parameter $r_2$ and (d) squeezing parameter $r_3$. Others’ parameters are $r_1=1, r_2=r_3=3$ and $N_0=1000$.

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The optimal phase sensitivities vary with the squeezing parameter $r_1$, $r_2$ and $r_3$ as displayed in Figs. 2(b), 2(c) and 2(d). When $r_2=r_3=3$, the optimal phase sensitivity can beat SNL with $r_1<2.12$. However, the optimal phase sensitivity becomes worse than SNL with the increase of $r_1$. When $r_1=1$ and $r_3=3$, the phase sensitivity can beat SNL only with $1.145<r_2<4.809$. According to the definition of SNL and HL, the photon number is only related to the first and second FWM. So SNL and HL remain unchanged with the increase of $r_3$. However, with the increase of the squeezing parameter $r_3$, optimal phase sensitivity also tends to be lower. When $r_3$ is approaching 2.114 or larger, the phase sensitivity is approaching constant.

If the squeezing strength $r_1=0$, this scheme will reduce to be an ordinary one where only a coherent beam and an vacuum beam enter into the SU(1,1) interferometer without external resources [24]. When $r_1=0$ and $N_0=0$, the scheme in this paper will be the one as [16]. So the conditions above are not taken into consideration.

If the inputs of the FWM1 are two vacuum beams with $N_0=0$, the beams after the FWM1 will be two-mode squeezing vacuum beams. One of them enters into the SU(1,1) interferometer and the other one is employed for the measurement. Then the photon number in the SU(1,1) interferometer will be

(8)$$\begin{aligned} N_{\textrm{two}}&=\langle \hat{a}^{\dagger}_2\hat{a}_2+\hat{c}_1^{\dagger}\hat{c}_1 \rangle\\ &=2G_1G_2-G_1-1. \end{aligned}$$

The $N_{\textrm{two}}$ which represent the photon number in the interferometer is more than 1, so HL and SNL is less than 1.

The corresponding phase sensitivity can be expressed as

(9)$$\Delta \phi =\frac{\sqrt{m_1}}{|\frac{\delta \langle \hat{I}_-\rangle}{\delta \phi}|_{\textrm{two}}},$$

where the slope in Eq. (9) is

(10)$$|\frac{\delta \langle \hat{I}_-\rangle}{\delta \phi}|_{\textrm{two}}=|2\textrm{sin} \phi G_1\sqrt{G_2(G_2-1)G_3(G_3-1)}|$$

and $m_1=|(M_5M_1-\sqrt {G_1(G_1-1)}e^{-i\theta })(M_4M_2-\sqrt {G_1(G_1-1)}e^{i\theta })+M_1M_3M_4M_6|$. In [20], Liu et al. found that in the SU(2) interferometer, the phase sensitivity with bright entangled twin beams as inputs can beat SNL and approach HL. Conversely, the phase sensitivity with two mode squeezing vacuum beams as inputs is worse than SNL. However, for the SU(1,1) interferometer, when the inputs are two mode squeezing vacuum beams with only one entering into the interferometer, the phase sensitivity can beat SNL and approach HL as Fig. 3(a). Meanwhile, in Fig. 3(b), when the parametric strength $r_1$ is less than 2.322, the phase sensitivity is better than SNL. As the increase of $r_1$, the phase sensitivity is worse than SNL. In Fig. 3(c), the phase sensitivity can beat SNL with $r_2>0,7679$. As shown in Fig. 3(d), when $r_3$ is better than 1.953, the phase sensitivity can beat SNL and approach constant. Figure 3 has displayed that the phase sensitivity with the two mode squeezing vacuum beams can beat SNL. However, considering the low photon number of the two mode squeezing vacuum beams, the phase sensitivity is worse than that with bright entangled twin beams as inputs. Meanwhile, for the inputs of two mode squeezing vacuum beams, the optimal phase sensitivity is also realized near at $\phi =\pi$ as shown in Fig. 3(a).

Fig. 3. The phase sensitivity with two mode squeezing vacuum beams versus (a) phase shift; Optimal phase sensitivity versus (b) parametric strength $r_1$ and (c) parametric strength $r_2$ and (d) parametric strength $r_3$. Others’ parameters are $r_1=1, r_2=r_3=3$ and $N_0=0$. TMSV means two mode squeezing vacuum.

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For comparison, we also study the phase sensitivity measurement with two coherent beams as inputs. The intensities of the two coherent beams are the same as that of bright entangled twin beams generated by FWM1. The intensity of the coherent beam which enters into the SU(1,1) interferometer is $G_1N_0+G_1-1$ and the intensity of another one employed for the detection is $(G_1-1)(N_0+1)$. The input-output relationship is

(11)$$\hat{a}_{out}=u_1 \hat{a}_1+u_2 \hat{b}_0^{\dagger},$$

$u_1=\sqrt {G_2G_3}e^{i\phi }+\sqrt {(G_2-1)(G_3-1)}$, $u_2=\sqrt {(G_2-1)G_3}e^{i\phi +i\theta }+\sqrt {G_2(G_3-1)}e^{i\theta }$, $u_3=u_1^*$, $u_4=u_2^*$. The slope can be expressed as

(12)$$ |\frac{\delta \langle \hat{I}_-\rangle}{\delta \phi}|_{\textrm{C}}=|2\textrm{sin} \phi \sqrt{G_2(G_2-1)G_3(G_3-1)}(G_1N_0+G_1)|,$$

while the variance of the coherent beams are described by

(13)$$\begin{aligned} \Delta \hat{I}_{\textrm{-C}}^2&=|\langle \hat{I}_-^2 \rangle- \langle \hat{I}_- \rangle^2|\\ &=|(G_1u_1^2u_3^2+G_1u_1u_2u_3u_4 +G_1-1)N_0\\ &\;\;\;+(G_1-1)(u_1^2u_3^2+1)+G_1u_1u_2u_3u_4|. \end{aligned}$$

The photon number inside the interferometer is

(14)$$N_{\textrm{C}}=\hat{a}_2^{\dagger}\hat{a}_2+\hat{b}_2^{\dagger}\hat{b}_2=(2G_2-1)G_1N_0+2G_1G_2-G_1-1.$$

According to Eqs. (6) and (14), for the SU(1,1) interferometer, the two coherent beams and bright entangled twin beams have the same intensities and the photon numbers inside the interferometer are same. So SNL and HL are same while the inputs are quantum and classical beams. Meanwhile, the corresponding slope of the bright entangled twin beam is same to the slope of the coherent beam. In addition, in an SU(2) interferometer, the quantum and classical beams have the same slope, so the optimal phase sensitivity can be realized by the achievement of the optimal intensity difference squeezing. For the SU(1,1) interferometer, due to the same slope, it means that the phase sensitivity can be replaced by intensity squeezing. However, considering the complexity of the scheme, intensity squeezing are not displayed here. By employing external resources, the phase sensitivity with the inputs of two coherent beams can beat SNL and approach HL. In an SU(2) interferometer, while the external resources are employed, the phase sensitivity in fact is worse than SNL. For the SU(1,1) interferometer, with external resources, the phase sensitivity can beat SNL as displayed in Fig. 4(a). The optimal phase sensitivity can be achieved near at $\phi =\pi$. Figures 4(b) 4(c) and 4(d) show the optimal phase sensitivities versus parametric $r_1$, $r_2$ and $r_3$. While $r_1$ varies from 0.1 to 5, the phase sensitivity can always beat SNL. While $r_2$ is larger than 0.7547, the phase sensitivity is better than SNL. For the parametric strength $r_3$, the phase sensitivity can beat SNL with $r_3$ being larger than 1.872. Meanwhile, notice that in Fig. 4(b), the increase of the parametric strength $r_1$ only represents the increase of the photon number.

Fig. 4. The phase sensitivity with two coherent beams as inputs versus (a) phase shift and optimal phase sensitivity versus (b) parametric strength $r_1$ and (c) parametric strength $r_2$ and (d) parametric strength $r_3$. Others’ parameters are $r_1=1, r_2=r_3=3$ and $N_0=1000$.

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3. Detection efficiency

Next, we consider the SU(1,1) interferometer with bright entangled twin beams as input while the detection efficiency of the photon detectors is not 1.

In the new scheme as shown in Fig. 5, there are two attenuators in the path where the transmissivities are $T_1$ and $T_2$ which represent the detection efficiency of the photon detectors [25]. Then the slope in this case can be expressed as

(15)$$|\frac{\delta \langle \hat{I}_-\rangle}{\delta \phi}|_{\textrm{T}}=|2\textrm{sin} \phi T_2G_1\sqrt{G_2(G_2-1)G_3(G_3-1)}(N_0+1)|,$$

while the variance of intensity difference of $\Delta ^2 \hat {I}_-$ is

(16)$$\begin{aligned} \Delta^2 \hat{I}_{\textrm{-T}}&=|\langle \hat{I}_-^2 \rangle- \langle \hat{I}_- \rangle^2|\\ &=|(T_2M_4M_1-T_1(G_1-1))^2N_0+(T_2M_5M_1-T_1\sqrt{G_1(G_1-1)}e^{-i\theta})\\ &\;\;\;(T_2M_4M_2-T_1\sqrt{G_1(G_1-1)}e^{i\theta})(N_0+1)+T_2^2M_1M_3M_4M_6(N_0+1)|. \end{aligned}$$

Fig. 5. The scheme for phase sensitivity measurement with attenuators which represent the detection efficiency of the photon detector.

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Compared with the scheme in Fig. 2, while there are two attenuators in the new scheme, the photon number inside the interferometer is the same. So SNL and HL keep the same. According to Fig. 6(a), the phase sensitivity can always beat SNL with the increase of $T_1$. Meanwhile, the phase sensitivity also becomes worse with the increase of $T_1$. For $T_2$, the phase sensitivity becomes better when $T_2$ is larger. It can beat SNL and approach HL while $T_1>0.03$. Then the external resources and detection efficiency need to be taken into consideration, which will be helpful in the process of phase estimation [26–34].

Fig. 6. Optimal phase sensitivity with the increase of transmissivity (a) $T_1$ with $T_2$ being 0.8 and (b) $T_2$ with $T_1$ being 1. Others’ parameters are same as Fig. 2.

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4. Conclusion

In conclusion, this paper gives a detailed discussion about the phase sensitivity of an SU(1,1) interferometer where external resources are employed. When bright entangled twin beams, two mode squeezing vacuum beams and two coherent beams are employed as the inputs, the phase sensitivities can beat SNL and approach HL by varying the squeezing parameter of the FWM process. Meanwhile, the detection efficiency of the detectors which play an important role on the sensitivity are also shown. Compared with [20], only the SU(1,1) interferometer with one of the inputs being an vacuum beam is considered. However, if the vacuum beam is replaced by the squeezing beam and $etc$, the phase sensitivity may be better. The external resources in the construction open the possibility for new schemes in optical metrology which will be very helpful in the quantum Fisher information process [27]. Considering the maturity of experimental technology, this scheme will be very helpful in LIGO, VIRGO and many other interferometers, such as the wide-field SU(1,1) interferometer and the enhancement of plasmonic sensing [35,36].

Funding

Natural Science Foundation of Shaanxi Province (2019JM-279); National Natural Science Foundation of China (11774286).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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Ultra-sensitive phase measurement based on an SU(1,1) interferometer employing external resources and substract intensity detection

Abstract

1. Introduction

2. Model

3. Detection efficiency

4. Conclusion

Funding

Disclosures

References

Cited By

Figures (6)

Equations (16)

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