⇦Chinese Physics Letters, 2020, Vol. 37, No. 7, Article code 070301 Enhancing Phase Sensitivity in Mach–Zehnder Interferometers for Arbitrary Input States Hongbin Liang (梁宏宾)1, Jiancheng Pei (裴健成)1, and Xiaoguang Wang (王晓光)1,2* Affiliations 1Zhejiang Institute of Modern Physics and Department of Physics, Zhejiang University, Hangzhou 310027, China 2Graduate School of China Academy of Engineering Physics, Beijing 100193, China Received 25 March 2020; accepted 12 May 2020; published online 21 June 2020 Supported by the National Natural Science Foundation of China (Grant Nos. 11875231 and 11935012), the National Key Research and Development Program of China (Grant Nos. 2017YFA0304202 and 2017YFA0205700), and the Fundamental Research Funds for the Central Universities (Grant No. 2018FZA3005).
*Corresponding author. Email: xgwang1208@zju.edu.cn Citation Text: Liang H B, Pei J C and Wang X G 2020 Chin. Phys. Lett. 37 070301    Abstract To enhance the phase sensitivity of Mach–Zehnder interferometers, we use a tunable phase shift before the light beams are injected into the interferometer. The analytical result of the optimal phase shift is obtained, which only depends on the initial input states. For a non-zero optimal phase shift, the phase sensitivity of the interferometers in the output ports is always enhanced. We can achieve this enhancement for most states, including entangled and mixed states. The optimal phase shift is exhibited in three examples. Compared to previous methods, this scheme provides a general way to improve phase sensitivity and could find wide applications in optical phase estimations. DOI:10.1088/0256-307X/37/7/070301 PACS:03.67.-a, 06.90.+v, 42.50.Dv, 42.50.St © 2020 Chinese Physics Society Article Text In recent years, quantum metrology has shown widespread applications in the development of cutting-edge technology.[1–27] In quantum metrology, the question of how to obtain high precision is always a central problem, and its development has already promoted applied and theoretical physics. Quantum Fisher information (QFI) is a core concept to quantify the precision limit, according to the quantum Cramér–Rao bound (QCRB)[21–24] $$ \delta \theta \geq \frac{1}{\sqrt{m F_\theta}},~~ \tag {1} $$ where $\delta \theta$ is the standard deviation of the parameter $\theta$ (the parameter to be estimated), $F_\theta $ is the QFI for $\theta$, and $m$ is the repetition of the experiments. In this case the number of unknown parameters is more than one, and the quantum Fisher information matrix, instead of QFI, should be used to discuss the ultimate precision limit. The recent progress on the QFI matrix and multiparameter estimation can be found in Ref. [11]. The Mach–Zehnder interferometer (MZI) is a widely used model in quantum phase estimation. A variety of initial states are studied, including the coherent state, squeezed state, vacuum state, and so on.[13–16] With this model, the phase matching condition for the enhancement of phase sensitivity for one port being at even or odd states is provided.[3] Subsequently, the squeezed thermal state[12] and unbalanced photon loss[28] are also discussed. In this letter, we propose a tunable MZI through a combination of a traditional MZI and a tunable phase shift, as shown in Fig. 1. This setup may be more realistic in practice for optical platforms than the general control.[29,30] Arbitrary initial states, including both mixed states and entangled states, are considered. According to Ref. [1], the MZI can be described by $\mathfrak{su}(2) $ algebra. The phase shift can be expressed as $$ P_z=e^{{i}\varphi J_z},~~ \tag {2} $$ and the operation of the entire MZI reads $$ U_{\rm MZI} =e^{-{i}\frac{\pi}{2} J_x} e^{{i}\theta J_z} e^{{i}\frac{\pi}{2} J_x}.~~ \tag {3} $$
cpl-37-7-070301-fig1.png
Fig. 1. Scheme of Mach–Zehnder interferometer with an optimized operation. The evolution of the beam splitters can be expressed by $\exp (\pm {i} \pi J_x /2) $. The two blue rounded rectangles represent the phase shifts. The first phase shift is an optimized operation acting on the initial states, which can be expressed as $\exp ({i} \varphi J_z) $. The second phase shift can be described by $\exp ({i} \theta J_z) $, where $\theta $ is the estimated parameter. Arbitrary input states are valid in our scheme.
In $\mathfrak{su}(2) $ algebra, the Schwinger representations of the bosons of three generators $(J_x,J_y,J_z)$ are $$\begin{align} J_x ={}& \frac{1}{2}\left(a^† b+b^† a \right), \\ J_y ={}& \frac{1}{2{i}}\left(a^† b - b^† a \right), \\ J_z ={}& \frac{1}{2}\left(a^† a - b^† b \right).~~ \tag {4} \end{align} $$ Hence, taking into account the tunable phase shift, the total transformation $U_{\rm T}$ of our device is $$ U_{\rm T} = e^{-{i}\frac{\pi}{2} J_x} e^{{i}\theta J_z} e^{{i}\frac{\pi}{2} J_x} e^{{i}\varphi J_z} = e^{-{i}\theta J_y} e^{{i}\varphi J_z}.~~ \tag {5} $$ We can then obtain the phase generator $$ G= J_y \cos \varphi - J_x \sin \varphi.~~ \tag {6} $$ Compared with the phase generator $G' = J_y $ of the traditional MZI, the phase generator in Eq. (6) depends on the manually tunable parameter $\varphi$. The phase $\varphi $ is the relative phase of two modes in the tunable phase shift. The task is to find the optimal tunable phase $\varphi_{\rm opt}$. For a non-zero optimal parameter $\varphi_{\rm opt}$, the optimized phase sensitivity in the output ports can be improved. Now we can obtain the optimal condition for maximal QFI. For an initial state $\rho_0 $, the spectral decomposition is $$ \rho_0 =\sum_{j=1}^{M} p_j \left| \psi_{j} \right\rangle \left\langle \psi_{j} \right|,~~ \tag {7} $$ where $p_j $ and $\left| \psi_{j} \right\rangle $ are the $j$th eigenvalue and eigenstate of $\rho_0 $, and $M $ is the dimension of the support of $\rho_0 $. Based on the QCRB, the QFI can be used to depict the ultimate precision limit. The calculations of the QFI and QFI matrix for non-full rank density matrices have been provided in recent years, [31,32] which have then been applied into the unitary evolution.[20] For a unitary evolution, the QFI of the total device can be expressed by[20] $$ F=\sum_{j=1}^{M} 4 p_{j} \left\langle \psi_{j} \right| \mathcal{H} ^ 2 \left| \psi_{j} \right\rangle - \sum_{j,j^\prime=1}^{M} \frac{8p_{j}p_{j^\prime}}{ p_{j}+p_{j^\prime} } \left| \left\langle \psi_{j} \right| \mathcal{H} \left| \psi_{j^\prime} \right\rangle \right|^2,~~ \tag {8} $$ where the Hermitian operator $\mathcal{H} $ is defined as $$ \mathcal{H}\equiv {i} \left(\partial_\theta U_{\rm T}^† \right) U_{\rm T} = - e^{-{i}\varphi J_z} J_y e^{{i}\varphi J_z}.~~ \tag {9} $$ Substituting Eqs. (4) and (9) into Eq. (8), the expression of the QFI for our scheme can be obtained as follows: $$ F=F_0 + A e^{-2{i}\varphi}+A^{*} e^{2{i}\varphi},~~ \tag {10} $$ where $F_0 $ is a real number and $A $ is a complex number, defined by $$\begin{align} F_0 ={}& \sum_{j=1}^{M} p_{j} \left\langle \psi_{j} \right|(2 a^† a b^† b + a^† a + b^† b) \left| \psi_{j} \right\rangle\\ & - \sum_{j,j^\prime=1}^{M} \frac{4 p_{j}p_{j^\prime}}{ p_{j}+p_{j^\prime} } \left|\left\langle \psi_{j} \right| a^† b \left| \psi_{j^\prime} \right\rangle\right|^2 ,~~ \tag {11} \end{align} $$ $$\begin{align} A ={}& \sum_{j,j^\prime=1}^{M} \frac{2p_{j}p_{j^\prime}}{ p_{j}+p_{j^\prime} } \left\langle \psi_{j} \right| a^† b \left| \psi_{j^\prime} \right\rangle \left\langle \psi_{j^\prime} \right| a^† b \left| \psi_{j} \right\rangle \\ & -\sum_{j=1}^{M} p_{j} \left\langle \psi_{j} \right| a^† a^† bb \left| \psi_{j} \right\rangle.~~ \tag {12} \end{align} $$ For a pure initial state $\rho_0 = | \phi_{\rm in} \rangle \langle\phi_{\rm in} | $, Eqs. (11) and (12) can be simplified to $$\begin{alignat}{1} F_0 ={}& \left\langle 2 a^† a b^† b + a^† a + b^† b \right\rangle -2 \left|\left\langle a^† b \right\rangle\right|^2 ,~~ \tag {13} \end{alignat} $$ $$\begin{alignat}{1} A ={}& \left\langle a^† b \right\rangle ^2 -\left\langle a^† a^† bb \right\rangle,~~ \tag {14} \end{alignat} $$ where the notation $\left\langle O \right\rangle = {\rm Tr} \left(\rho_0 O \right) $. Since $F_0$ is independent of $\varphi$, we only need to optimize the second and third terms of Eq. (10) over $\varphi$. Rewrite $A$ as $$ A=|A| e^{{i} \, {\rm arg}(A)},~~ \tag {15} $$ where $|A| $ is the norm of $A$, and ${\rm arg} (A) $ is the argument of $A $. Then the QFI can be expressed as $$ F=F_0 + 2 |A| \cos \left({\rm arg} (A) - 2 \varphi \right) .~~ \tag {16} $$ Utilizing this expression, it can be seen that for $|A| \neq 0 $, the optimal value $\varphi_{\rm opt} $ is $$ \varphi_{\rm opt}=\frac{1}{2} {\rm arg} (A) + k \pi,~~ \tag {17} $$ where $k $ is an integer number, and the corresponding optimal QFI is $$ F_{\rm opt}=F_0 + 2 |A|.~~ \tag {18} $$ When $\varphi=0 $, the setup is just a traditional $\mathfrak{su}(2)$ MZI and the corresponding QFI reads $$ F_{\rm g}=F_0 + 2 |A| \cos ({\rm arg} (A)).~~ \tag {19} $$ The difference between $F_{\rm opt} $ and $F_{\rm g} $ can describe the enhancement of phase sensitivity induced by the tunable phase shift. We define $D_F= F_{\rm opt} -F_{\rm g} $, and then obtain $$ D_F=2 |A|\left[ 1- \cos ({\rm arg} (A))\right].~~ \tag {20} $$ From Eq. (20), one can easily see that when $|A|>0 $ and ${\rm arg} (A) \neq 0 $, the difference function $D_F $ is always greater than zero, which means the optimized operation enhances the phase sensitivity. If we assume the initial state is a product state and the state in port B is an even (odd) state, the complex number $A $ in Eq. (12) can be simplified to $$ A = - \left\langle a^{† 2} \right\rangle \left\langle b^2 \right\rangle .~~ \tag {21} $$ Then, the optimal tunable phase $\varphi_{\rm opt} $ is $$ \varphi_{\rm opt} = \frac{1}{2} \left[ {\rm arg}\left(\left\langle b^{ 2} \right\rangle \right) -{\rm arg}\left(\left\langle a^{ 2} \right\rangle \right) +\pi \right] +k \pi,~~ \tag {22} $$ where $k $ is an integer number. When the initial state satisfies $| {\rm arg}(\langle a^{ 2} \rangle) -{\rm arg}(\langle b^{ 2} \rangle) | =\pi $, the optimal tunable $\varphi_{\rm opt} $ can be equal to zero, which is consistent with the phase matching condition in Ref. [3]. Since we make no assumptions in the calculation, all arbitrary initial input states should be valid in our scheme, including entangled states and mixed states. We consider three kinds of typical initial states as examples to illustrate our scheme. In the first case, we consider an entangled state. In the remaining two cases, we consider mixed states $\rho_0= \rho_a \otimes \left| \phi_b \right\rangle \! \left\langle \phi_b \right| $, where $\rho_a $ is a mixed state in port A and $\left| \phi_b \right\rangle $ is a pure state in port B. The first example we consider for the initial input state is the following entangled state $$ \left| \phi \right\rangle = \frac{1}{N_\phi}\left(\left| \alpha \right\rangle_a \left| \beta \right\rangle_b + e^{{i}\varphi_{\rm r}} \left| -\alpha \right\rangle_a \left| -\beta \right\rangle_b\right),~~ \tag {23} $$ where $\varphi_{\rm r} $ is a relative phase, and the normalization $$ N_\phi^2 = 2\left(1+\cos\varphi_{\rm r} \,e^{-2\left(\left|\alpha\right|^{2}+\left|\beta\right|^{2}\right)}\right).~~ \tag {24} $$ Here $\left| \pm \alpha \right\rangle_a $ and $\left| \pm \beta \right\rangle_b $ are coherent states, and we denote $\alpha = \left| \alpha \right| e^{{i} \varphi_{\alpha}} $ and $\beta = \left| \beta \right| e^{{i} \varphi_{\beta}} $. The total average photon number $\bar{n}_{\rm T} $ then reads $$\begin{align} \bar{n}_{\rm T} ={}& \left\langle \phi \right| \left(a^† a+ b^† b\right) \left| \phi \right\rangle \\ ={}& \frac{4-N_{\phi}^{2}}{N_{\phi}^{2}} \left(\left|\alpha\right|^{2}+\left|\beta\right|^{2}\right).~~ \tag {25} \end{align} $$ Since the initial state is a pure state, from Eqs. (13) and (14), one can obtain $$\begin{align} F_0 ={}& 16\varGamma\left|\alpha\right|^{2}\left|\beta\right|^{2} +\bar{n}_{\rm T},~~ \tag {26} \end{align} $$ $$\begin{align} A ={}& -8 e^{2{i}\left(\varphi_{\beta}-\varphi_{\alpha}\right) } \mathit{\varGamma}\left|\alpha\right|^{2}\left|\beta\right|^{2} ,~~ \tag {27} \end{align} $$ where $\varGamma = (N_{\phi}^{2} - 2)/N_{\phi}^{4} $. Thus, the QFI equals $$ F=\bar{n}_{\rm T}+32\mathit{\varGamma}\left|\alpha\right|^{2}\left|\beta\right|^{2} \sin^2 \left(\varphi_{\beta}-\varphi_{\alpha} - \varphi \right) .~~ \tag {28} $$ If $N_\phi =\sqrt{2} $, the optimized term $A=0 $, which means that the tunable phase shift cannot enhance the phase sensitivity under this condition. When $N_\phi \neq \sqrt{2} $, we can obtain the argument of $A $ as $$ {\rm arg} (A) = \begin{cases} 2\left(\varphi_{\beta}-\varphi_{\alpha}\right) , & 0 < N_\phi < \sqrt{2}, \\ 2\left(\varphi_{\beta}-\varphi_{\alpha}\right) \pm \pi, & \sqrt{2} < N_\phi < 2. \end{cases}~~ \tag {29} $$ Then, we obtain the analytical optimal phase $\varphi_{\rm opt} $ as $$ \varphi_{\rm opt} = \begin{cases} \varphi_{\beta}-\varphi_{\alpha}+ k \pi, & 0 < N_\phi < \sqrt{2}, \\ \varphi_{\beta}-\varphi_{\alpha}+\frac{\pi}{2}+ k \pi, & \sqrt{2} < N_\phi < 2, \end{cases}~~ \tag {30} $$ where $k $ is an integer number. The corresponding optimal QFI is $$ F_{\rm opt} = \begin{cases} \bar{n}_{\rm T}, & 0 < N_\phi < \sqrt{2}, \\ \bar{n}_{\rm T} + 32\mathit{\varGamma}\left|\alpha\right|^{2}\left|\beta\right|^{2} , & \sqrt{2} < N_\phi < 2. \end{cases}~~ \tag {31} $$ The condition $0 < N_\phi < \sqrt{2} $ is equivalent to $\cos \varphi_{\rm r} < 0 $, whereas $\sqrt{2} < N_\phi < 2 $ is equivalent to $\cos \varphi_{\rm r} >0 $. From the expression of the optimal QFI in Eq. (31), we know that the optimal phase sensitivity can reach the standard quantum limit. We can also see that when the relative phase satisfies $\cos \varphi_{\rm r} >0 $, there exists an extra term in the optimal QFI $F_{\rm opt} $, which makes the phase sensitivity better than the standard quantum limit. Figure 2 shows the QFI as a function of the parameter $\varphi $. The yellow line represents the QFI $F_{\rm g} $ with no tunable phase shift. The QFI with a certain range of parameter $\varphi $ is greater than $F_{\rm g} $, whereas the QFI with the remaning range is less than $F_{\rm g} $. We can see that the optimal $\varphi_{\rm opt} $ is the maximum point of QFI. The black bracket shows the difference function $D_F $, and with the parameters in Fig. 2, we can obtain $D_F \approx 0.076 F_{\rm g} $.
cpl-37-7-070301-fig2.png
Fig. 2. QFI versus the parameter $\varphi $. The parameters are $| \alpha | = | \beta | =1 $, $\gamma = \pi/4 $ and $\varphi_{\beta}-\varphi_{\alpha}=\pi/6 $. The blue line shows the QFI $F_\varphi $ changing with parameter $\varphi $, and the yellow line $F_{\rm g} $ marks the QFI without optimization. The red dashed line marks the optimal parameter $\varphi_{\rm opt} $, which equals $\varphi_{\rm opt} = \varphi_{\beta}-\varphi_{\alpha}+\pi/2 =2\pi/{3} $. The black bracket marks the difference function $D_F $ between the optimal $F_{\rm opt} $ and $F_{\rm g} $.
In the second example, we consider the mixed state $\rho_0= \rho_a \otimes \left| \beta \right\rangle_{b b} \! \left\langle \beta \right| $ as the initial input state. The squeezed thermal state $\rho_a $ in port A is $$ \rho_a = \sum_{n=0}^{\infty}\frac{\bar{n}_{\rm th}^n}{\left(\bar{n}_{\rm th} +1\right)^{n+1} } S_a(\xi) \left| n\left\rangle_{ a\,a } \right\langle n \right| S_a^†(\xi),~~ \tag {32} $$ where $\bar{n}_{\rm th} $ is the average thermal photon number. The squeezing operator $S_a (\xi) $ is defined as $$ S_a (\xi) = e^{\frac{1}{2} \left(-\xi a^{† 2} +\xi^{*} a^{2} \right)},~~ \tag {33} $$ where the squeezing parameter $\xi = r e^{{i} \varphi_{\xi}} $. The average photon number of port A is $\bar{n}_a = (2 \bar{n}_{\rm th} +1) \sinh^2 r + \bar{n}_{\rm th}$. The state $| \beta \rangle_{b} $ in port B is a coherent state with the parameter $\beta = | \beta | e^{{i} \varphi_{\beta}} $. The average photon number of coherent state $| \beta \rangle_{b} $ is $\bar{n}_b=| \beta |^2 $. Based on Eqs. (11), (12) and (16), we obtain $$\begin{alignat}{1} F=\bar{n}_a+\bar{n}_b\frac{\cosh (2r)+\sinh (2r) \cos \left(2\varphi_{\beta} -\varphi_{\xi}-2\varphi \right) }{2\bar{n}_{\rm th}+1}.~~ \tag {34} \end{alignat} $$ The optimal $\varphi_{\rm opt} $ then reads $$ \varphi_{\rm opt} = \varphi_{\beta} -\frac{1}{2} \varphi_{\xi}+k\pi ,~~ \tag {35} $$ where $k $ is an integer number. The optimal QFI is $$ F_{\rm opt} = \bar{n}_a +\bar{n}_b\frac{ e^{2r} }{2\bar{n}_{\rm th}+1}.~~ \tag {36} $$ The QFI is exponentially enhanced by the squeezing. Tan et al.[12] considered a special case with $\varphi_{\xi} =\varphi_{\beta} = 0 $, and their result is consistent with Eq. (36). In the third example, we consider another mixed state. A single photon state may lose a photon in the evolution, so we consider a mixed state to represent this case. Similar to a mixed state of a two-level system, we could write the single photon state with photon loss as $$\begin{alignat}{1} \rho_{\rm loss} ={}& p \left| 0\right\rangle \left\langle 0\right| + \sqrt{p\left(1-p \right) } e^{-{i}\varphi_a-\gamma} \left| 0\right\rangle \left\langle 1\right| \\ &+\left(1-p \right) \left| 1\right\rangle \left\langle 1\right|+\sqrt{p\left(1-p \right) } e^{{i}\varphi_a-\gamma} \left| 1\right\rangle \left\langle 0\right|,~~ \tag {37} \end{alignat} $$ where $p $ denotes the probability of photon loss, $\gamma >0 $ is the dephasing parameter and $\varphi_a $ is the relative phase between the vacuum state $| 0\rangle $ and the single photon sate $| 1\rangle $. Here we consider the mixed state $\rho_0 = \rho_a \otimes | \phi_b \rangle \langle \phi_b | $, with $\rho_a = \rho_{\rm loss}$. We choose the state in port B to be $| \phi_b \rangle = \frac{1}{\sqrt{2}} (| 0 \rangle+ e^{{i} \varphi_b} | 1 \rangle) $, with $\varphi_b $ being the relative phase in port B. According to Eqs. (11), (12) and (16), we obtain $$\begin{align} F ={}& p\left(1-p \right) e^{-2\gamma}\cos^2 \left(\varphi_b - \varphi_{a} -\varphi \right) \\ & +p^2-3p +\frac{5}{2}.~~ \tag {38} \end{align} $$ When the loss probability $p=0 $ or $p=1 $, the initial state becomes the pure state, and the complex term $A $ becomes zero. When $p\left(1-p \right) \neq 0 $, there exists an optimal phase $$ \varphi_{\rm opt} = \varphi_b-\varphi_{a}+k \pi,~~ \tag {39} $$ where $k $ is an integer number. The optimal QFI is $$ F_{\rm opt} = p\left(1-p \right) e^{-2\gamma} +p^2-3p +\frac{5}{2}.~~ \tag {40} $$ The optimal $F_{\rm opt} $ decreases when the loss probability $p $ increases. Therefore, when $p \rightarrow 0 $, the QFI in Eq. (40) trends to a maximum of $F_{\rm opt}\rightarrow \frac{5}{2} $. In summary, we introduce a new scheme to enhance the phase sensitivity of MZIs, which consists of a traditional $\mathfrak{su}(2)$ interferometer and a tunable phase shift. By analyzing the QFI, the general expression of the optimal angle for the tunable phase shift is given. Our result shows that the final precision limit using most states can be enhanced via this tunable phase shift. An entangled coherent state and two typical mixed states are discussed as examples, which shows a comprehensive validity of our scheme. Due to the mature realization of $\mathfrak{su}(2)$ interferometers in linear optical platforms, our scheme introduces a direct and general method to improve phase sensitivity and may be experimentally realized in the near future. 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