Coherent feedback enhanced quantum-dense metrology in a lossy environment

Xinyun Liang; Jie Zhao; Yuhan Yan; Wenfeng Huang; Wenfeng Huang; Chun-Hua Yuan; Chun-Hua Yuan; L. Q. Chen; L. Q. Chen; L. Q. Chen

doi:10.1364/OE.519044

1. Introduction

Quantum dense metrology (QDM) plays a significant role in precise measurements for quantum sensing and detecting gravitational waves [1–13]. It realizes multiple non-commuting parameter measurements beyond the standard quantum limit (SQL) simultaneously [4,5]. Compared with a single observable detection, it can further improve the measurement result. The key to achieving QDM is using bipartite entangled states from an optical parametric amplifier (PA) [1,14–19]. One entangled mode as the meter beam couples the multiple non-commuting observations. The other mode is an external reference beam that combines with the meter beam by a splitter. The information of the observables is detected finally using two balanced homodyne detectors (HDs). Due to the noise of the two-mode entangled states being lower than the vacuum noise, the sensitivity of the measurement goes beyond the SQL. Better sensitivities can be achieved by higher entanglement degrees of the entangled state [10]. In realistic situations, the photon losses of the meter beam before or after feeling the observations are inevitable [20–24]. Photon loss causes partial signal disappearance and introduces additional uncorrelated modes, leading to a rapid decrease in entanglement and thus reducing the measurement sensitivity. Higher entanglement has worse resistance to loss. To resist the loss, one can adjust the gain of the PA to be smaller. However, the absolute sensitivity will be worse.

In this work, we employ a coherent feedback scheme to enhance the sensitivity in the loss environment. A beam splitter (BS1) as a controller that coherently feeds the reference beam back into its input port and affects the PA’s input-output relationship. A phase shifter is placed into the feedback loop to adjust the phase of the controlled field. The results demonstrate that in a lossy environment, the sensitivities of the two non-commuting measurements - phase and amplitude - are further improved in the feedback scheme. The sensitivity of the feedback scheme is beyond the SQL when suitable feedback strength is applied. The optimal feedback strength depends on the photon loss of the meter beam. Compared with the model without feedback, the model with feedback has better measurement advantages brought by entangled states within some loss ranges.

In addition to the sensitivity, long-term stabilization is also a core parameter in practical measurements. Commonly, the light field of the devices is always affected by classical noises in operation, leading to random drifts in the phase and amplitude, which reduces the stability of the sensor. For fiber systems, the classical noises can be from polarized cross-coupled lights, forward and backward scattered fields, the Shupe effect, stress deformation, etc [25–27]. To reduce the impact of the classical noises on the system, one can use self-adaption optimization algorithms or post-processing algorithms to control the phase or amplitude of light to stabilize the system, such as the Kalman filter algorithm on Allan variance and PID control, etc [28–30]. However, random drifts in phase and amplitude are mixed and cannot be separated from the error signal by intensity detection. Hence, it is impossible to confirm the exact perturbations of the phase and amplitude, failing to control precisely. Employing two HDs can separate the random drifts in phase and amplitude. One HD obtains the phase information, the other gets the amplitude information by changing the phase of the local light [31]. This approach effectively solves the problem of indistinguishable phase and amplitude drifts. The obtained error signals from two HDs enable separate control of the random drift of phase and amplitude to stabilize the system. The control accuracy depends on the measurement sensitivity of the devices. High measurement sensitivity corresponds to better control efficiency. QDM employs two-mode entangled states allowing the phase and amplitude of the system to be observed separately with sensitivities beyond SQL. When the device is stabilized over time using a two-mode entangled state, both phase and amplitude will be controlled with better accuracy than in the classical case. Under the coherent feedback-controlled QDM, the measurement sensitivities of the phase and amplitude are significantly improved, which could help further enhance the control accuracy of long-term stabilization.

2. Model and theory

In any measurement device, not only the phase of the light field could be modulated, but the amplitude may also be modulated. To high-measurement sensitivity, it is necessary to measure both amplitude and phase when doing detection. QDM provides simultaneous measurements of the phase and amplitude with both sensitivities beyond the SQL, where the theoretical model is in Fig. 1(a). An optical PA with gains $G$ and $g$ is used to create two entangled beams $a_{1}$ and $b_{1}$. After the meter beam $a_{1}$ coupling a amplitude $\epsilon$ and a phase $\delta$, a BS with reflectivity $R$ and transmittance $T$ will split $a_{1}$ and $b_{1}$ for achieving the phase and amplitude measurements, respectively. In a lossless situation, the input-output relations of system are given by

(1)$$\hat{a}_{out} =\sqrt{T}e^{i\delta }e^{-\epsilon }e^{i\varphi_{1} }\left( G \hat{a}_{0}+g\hat{b}_{0}^{{\dagger} }\right) -\sqrt{R}e^{i\varphi_{2}}\left( g\hat{a} _{0}^{{\dagger} }+G\hat{b}_{0}\right),$$

(2)$$\hat{b}_{out} =\sqrt{R}e^{i\delta }e^{-\epsilon }e^{i\varphi_{1}}\left( G \hat{a}_{0}+g\hat{b}_{0}^{{\dagger} }\right) +\sqrt{T}e^{i\varphi_{2}}\left( g\hat{a} _{0}^{{\dagger} }+G\hat{b}_{0}\right),$$

where $\hat {a}_{i}$ ($\hat {a}^{\dagger }_{i}$) is the annihilation operator (the creation operator) of the light field $a_{i}$, $\hat {b}_{i}$ ($\hat {b}^{\dagger }_{i}$) is the annihilation operator (the creation operator) of the light field $b_{i}$. $\varphi _1$ ($\varphi _2$) is the phase difference between local light and field $a_1$ ($b_1$). In order to achieve the high precision of the measurement for both phase $\delta$ and amplitude $\epsilon$ simultaneously, $T$ and $R$ are locked at $1/2$. Considering the modulations $(\delta,\epsilon )\ll 1$, i.e., $e^{i\delta }e^{-\epsilon }\simeq 1+i\delta -\epsilon$, $a_{0}$ is coherent state with mean-photon number $\left \vert \alpha \right \vert ^{2}$, $b_{0}$ is vacuum state, then the phase sensitivity $\Delta \delta _{0}=\sqrt {\langle \Delta ^2\hat {Y}_{b_{out}} \rangle }/|\partial \langle \hat {Y}_{b_{out}}\rangle /{\partial \delta } |$ from HD1 and amplitude sensitivity $\Delta \epsilon _{0}=\sqrt {\langle \Delta ^2\hat {X}_{a_{out}} \rangle }/|\partial \langle \hat {X}_{a_{out}}\rangle /{\partial \epsilon } |$ from HD2 are given by [10]

(3)$$\Delta \delta_{0} =\Delta \epsilon _{0}=\frac{G-g}{\sqrt{{2\left\vert \alpha \right\vert ^{2}G^{2}}}},$$

where $\hat {Y}_{b_{out}}=\hat {b}_{out}+\hat {b}_{out}^\dagger$ with $\varphi _1=\varphi _2=\pi /2$, $\hat {X}_{a_{out}}=\hat {a}_{out}+\hat {a}_{out}^\dagger$ with $\varphi _1=\varphi _2=0$, $G^2-g^2=1$. The Eq. (3) shows the sensitivities beyond the SQL due to the noise term $(G-g)$ brought by entangled state being lower than the vacuum noise, where SQL of the joint measurement here is defined as $1/\sqrt {2 \langle \hat {a}^\dagger _1\hat {a}_1 \rangle }$.

Fig. 1. (a) Theoretical model for QDM with a single loss $l_{a}$ in the meter beam. (b) Coherent feedback scheme for enhanced sensitivities in the lossy environment. Using an additional linear beam splitter (BS1) as “controller” that coherent feeds the one output port of the PA into its input port. The feedback strength is determined by the reflectivity $k$ of BS1. $k=0$ represents the scheme without feedback. $\phi$ is the phase shift in the feedback loop. $v$ is vacuum state that brings a loss rate $l$ in the feedback loop.

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In practical applications, the photon loss of the meter beam $a_{1}$ is unacceptable, leading to worse sensitivities. Considering a loss $l_{a}$ in the meter beam seen in Fig. 1(a), we have the input-output relations

(4)$$\hat{a}_{out} =\sqrt{T}e^{i\delta }e^{-\epsilon }e^{i\varphi_{1} }\left[\sqrt{1-l_a}\left( G \hat{a}_{0}+g\hat{b}_{0}^{{\dagger} }\right)+\sqrt{l_a}\hat{v}_{a} \right ] -\sqrt{R}e^{i\varphi_{2} }\left( g\hat{a} _{0}^{{\dagger} }+G\hat{b}_{0}\right) ,$$

(5)$$\hat{b}_{out} =\sqrt{R}e^{i\delta }e^{-\epsilon }e^{i\varphi_{1} }\left[\sqrt{1-l_a}\left( G \hat{a}_{0}+g\hat{b}_{0}^{{\dagger} }\right)+\sqrt{l_a}\hat{v}_{a} \right ] +\sqrt{T}e^{i\varphi_{2} }\left( g\hat{a} _{0}^{{\dagger} }+G\hat{b}_{0}\right),$$

where $\hat {v}_{a}$ is the annihilation operator of the vacuum state introduced by loss $l_{a}$. In the lossy situation, the phase sensitivity $\Delta \delta _{l_{a}}$ and amplitude sensitivity $\Delta \epsilon _{l_{a}}$ are given by

(6)$$\Delta \delta _{l_{a}}=\Delta \epsilon _{l_{a}}=\sqrt{\frac{\left( 2-l_{a}\right) \left( G^{2}+g^{2}\right) -4\sqrt{1-l_{a}}gG+l_{a}}{4\left\vert \alpha \right\vert ^{2}\left( 1-l_{a}\right) G^{2}}}.$$

It shows that the sensitivities are affected by loss rate $l_{a}$. The loss $l_a$ leads to the signal decreases and brings additional noise, resulting in the measurement sensitivity degradation. Here, we employ measurement-free single-port feedback control to enhance the sensitivities in the lossy environment. The theoretical scheme is in Fig. 1(b). The additional BS1 acts as a controller, feeding the reference beam $b_{1}$ into its input port. The feedback strength is determined by the BS1’s reflectivity $k$, where

(7)$$\hat{b}_{f} =\sqrt{1-k}\hat{b}_{0}-\sqrt{k}\hat{b}_{1},$$

(8)$$\hat{b}_{t} =\sqrt{k}\hat{b}_{0}+\sqrt{1-k}\hat{b}_{1}.$$

When $k=0$, the scheme will reduce into the original case in Fig. 1(a). Considering unavoidable losses in practical applications, we add an additional loss $l$ in the feedback process. The input-output relations in the feedback scheme with losses are

(9)$$\begin{aligned}\hat{a}_{out} &=\frac{\sqrt{T}\sqrt{1-l_{a}}e^{i\delta }e^{-\epsilon }e^{i\varphi_1 }}{1+ \sqrt{k(1-l)}Ge^{{-}i\phi }}\left[ \left( G+\sqrt{k(1-l)}e^{{-}i\phi }\right) \hat{a}_{0}+g \sqrt{\left( 1-k\right) (1-l)}e^{{-}i\phi }\hat{b}_{0}^{{\dagger}} + \sqrt{l}g\hat{v}^\dagger\right]\\ &-\frac{\sqrt{R}e^{i\varphi_2}}{1+\sqrt{k(1-l)}Ge^{i\phi }}\left[ g\sqrt{\left( 1-k\right) } \hat{a}_{0}^{{\dagger} }+\left( \sqrt{1-l}Ge^{i\phi }+\sqrt{k}\right) \hat{b}_{0}+G\sqrt{l(1-k)}\hat{v}\right]\\ & +\sqrt{Tl_{a}}e^{i\delta }e^{-\epsilon }e^{i\varphi_1 }\hat{v}_{a}, \end{aligned}$$

(10)$$\begin{aligned}\hat{b}_{out} &=\frac{\sqrt{R}\sqrt{1-l_{a}}e^{i\delta }e^{-\epsilon }e^{i\varphi_1 }}{1+ \sqrt{k(1-l)}Ge^{{-}i\phi }}\left[ \left( G+\sqrt{k(1-l)}e^{{-}i\phi }\right) \hat{a}_{0}+g \sqrt{\left( 1-k\right) (1-l)}e^{{-}i\phi }\hat{b}_{0}^{{\dagger}} + \sqrt{l}g\hat{v}^\dagger\right]\\ &+\frac{\sqrt{T}e^{i\varphi_2}}{1+\sqrt{k(1-l)}Ge^{i\phi }}\left[ g\sqrt{\left( 1-k\right) } \hat{a}_{0}^{{\dagger} }+\left( \sqrt{1-l}Ge^{i\phi }+\sqrt{k}\right) \hat{b}_{0}+G\sqrt{l(1-k)}\hat{v}\right]\\ & +\sqrt{Rl_{a}}e^{i\delta }e^{-\epsilon }e^{i\varphi_1}\hat{v}_{a}, \end{aligned}$$

where $\phi$ is the total phase shift of the field $b_{f}$ and $\hat {v}$ ($\hat {v}^\dagger$) is the annihilate operator (the creation operator) of vacuum state $v$ in the feedback loop. By setting $\phi =\pi$ for best squeezing [32], we obtain the sensitivities of the phase and amplitude measurements

(11)$$\Delta \delta_f =\Delta \epsilon_f =\sqrt{\frac{\left( 1-l_{a}\right) H+ F-4g\sqrt{1-k}\sqrt{\left( 1-l_{a}\right)} U +l_{a}P^{2}}{4\left\vert \alpha \right\vert ^{2}\left( 1-l_{a}\right) U^{2}}},$$

where $U =G-\sqrt {k(1-l)}$, $H=U^2+(1-l)(1-k)g^2+g^2 l$, $F=( G\sqrt {1-l}-\sqrt {k})^2+g^2(1-k)+G^2l(1-k)$, and $P= 1-G\sqrt {k(1-l)}$. The above Eq. (11) shows that the loss rate $l_{a}$, the feedback strength $k$, and the loss $l$ in the feedback process influence the sensitivities. A suitable $k$ can enhance the sensitivities and reduce the impact of loss $l_a$. However, the additional loss $l$ influences the feedback effect. When $l=0$ meaning a perfect feedback process, we have the mean-photon number of output field $b_t$ given by $I^{out}_{(0)}=(1-k)g^2(1+\left \vert \alpha \right \vert ^{2})/(1-G\sqrt {k})^2$. This expression shows the output intensity limit after the feedback process. When the loss $l$ occurs, the output intensity of field $b_t$ becomes $I^{out}_{(l)}=(1-k)g^2(1+\left \vert \alpha \right \vert ^{2})/(1-G\sqrt {k(1-l)})^2$. Comparing these two expressions, the parameters $k$, $l$, and $G$ in the lossy feedback process are required to satisfy

(12)$$Gkl\le2\sqrt{k}(1-\sqrt{1-l}).$$

This ensures that the system is able to maintain stability and performance in the presence of uncertainties and disturbances. Under a certain feedback strength $k$, the gain $G$ relates to loss $l$ in the feedback process. In the following discussion, we take $G=2\sqrt {k}(1-\sqrt {1-l})/(kl)$ with $(k,l)>0$ for simply under the feedback scheme. With the increasing $l$, the gain $G$ increases to keep steady output. Under the feedback scheme, the optical field is continuously amplified in the loop until the system reaches a steady state, showing an increase in output photon number compared with the no-feedback model. Since the phase/amplitude-sensing photon numbers of the original model and the feedback model are different, we use SQL$_0$ to represent the standard limit of the original model, and SQL$_1$ to represent the case of the feedback model, where they are given as follows:

(13)$$\text{SQL}_0 =\frac{1} {\sqrt{2(1-l_a)(G^2|\alpha |^2+g^2)} },$$

(14)$$\text{SQL}_1 = \frac{1}{\sqrt{2(1-l_a)\left [ U^2|\alpha |^2+(1-l)(1-k)g^2+lg^2 \right ] /P^2} }.$$

3. Results and discussions

In the following discussion, we mainly show the results of the phase sensitivity since the amplitude sensitivity is the same as the phase sensitivity when $T=R=1/2$. Firstly, we study the impact of loss $l_a$ on sensitivity $\Delta \delta _{l_{a}}$ as shown in Fig. 2(a). It shows that the sensitivity $\Delta \delta _{l_{a}}$ gradually changes worse as $l_a$ increases. For $G=2$ (blue solid line), the sensitivity $\Delta \delta _{l_{a}}$ is worse than SQL$_0$ (blue dashed line) when $l_a\ge 0.67$, meaning the destruction of the measurement advantage brought by the two-mode entangled state. When the gain $G$ is changed to 3 (red solid line), the sensitivity $\Delta \delta _{l_{a}}$ performs better. However, the sensitivity is worse than SQL$_0$ (red dashed line) when $l_a\ge 0.5$, indicating that the disappearance speed of measurement advantage is faster than the case of that when $G=2$. Figure 2(b) shows the sensitivity difference between $\Delta \delta _{l_{a}}$ and SQL$_0$. Enhancing $G$ can effectively improve the absolute sensitivity. However, the measurement advantage from the entangled state is less resistance to loss. In the lossy environment, when $\Delta \delta _{l_{a}}=\text {SQL}_0$, we can obtain the relationship between the gain G and loss rate $l_a$ as

(15)$$G_{\max}\simeq\frac{2-l_a}{l_a},$$

considering large number of coherent photons ($|\alpha |^2\gg 1$). Adjusting the gain $G$ smaller than $G_{\max }$ can maintain the quantum advantage in the lossy environments, whereas the absolute sensitivity will become worse. It is significant to improve the absolute sensitivity of the measurement while the sensitivity still surpasses the SQL for maintaining the measurement advantage in the lossy environment.

Fig. 2. (a) Phase sensitivity of no-feedback model in Fig. 1(a) versus loss rate $l_a$ when $G=2$ (blue solid line) and $G=3$ (red solid line). The dashed lines represent the corresponding SQL$_0$. (b) The sensitivity difference (dB) between $\Delta \delta _{l_{a}}$ and SQL$_0$ as a function of loss $l_a$ form (a). The value is larger than zero, indicating that the sensitivity $\Delta \delta _{l_{a}}$ is beyond the SQL$_0$. The other parameters are $T=R=1/2$, and $\left \vert \alpha \right \vert ^{2}=10^{15}$.

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Now, we study the impact of the feedback scheme on the phase measurement sensitivity in the lossy environment. Figure 3 shows the phase sensitivities of the feedback model and origin model (left-hand vertical axis) as a function of the feedback strength $k$ when $l_a=0.2$ in (a) and $l_a=0.7$ in (b). The value of gain $G$ is at the right-hand vertical axis. When $l_a=0.2$ corresponding to a low loss rate, the sensitivity $\Delta \delta _{l_{a}}$ (yellow solid line) goes beyond the SQL$_0$ (purple dashed line) within the range of $G$ less than 9, indicating that the gain $G$ of the original model cannot higher than 9 for keeping measurement advantage, limiting the improvement of absolute sensitivity. Naturally, the parameter gain G cannot be increased infinitely in the actual situation. It requires a significantly strong driving field to achieve large gain G, which will bring additional high-order nonlinear noise during the PA process, leading to worse sensitivity. When the feedback structure is employed, one can see that the sensitivity $\Delta \delta _{f}$ (blue solid line) is better than $\Delta \delta _{l_{a}}$ in all ranges of $k$, showing the absolute sensitivity can effectively be improved further under the feedback model. At the ranges of $k\in (0.59,0.85)$, the sensitivity $\Delta \delta _f$ is beyond the SQL$_1$, indicating an improvement in absolute accuracy while displaying the measurement advantage. By setting $\partial (\Delta \delta _f)/\partial k=0$, we can obtain the optimal value of feedback strength $k$ for the best sensitivity. It shows that the optimal feedback strength needs to be adjusted correspondingly with changes in loss $l_a$. In Fig. 2(a), the optimal feedback strength $k_{opt}$ is located at 0.7 when $l_a=0.2$, $l=0.1$, and $G\approx 1.23$.

Fig. 3. Sensitivities $\Delta \delta _{l_{a}}$ and $\Delta \delta _{f}$ as a function of feedback strength $k$ when loss rate $l_{a}=0.2$ in (a) and $l_{a}=0.7$ in (b) (left-hand vertical axis). The value of gain $G$ is at the right-hand vertical axis. SQL$_0$ (purple dashed lines) represents the standard quantum limit of the model without feedback. SQL$_1$ (red dashed lines) represents the standard quantum limit of the model with feedback. Under the feedback scheme, the sensing photon number is increased compared with the original model due to the continuous amplification in the feedback loop, resulting in $\text {SQL}_0\ne \text {SQL}_1$. Black circles represent $\Delta \delta _{l_{a}}=\text {SQL}_0$. The other parameters are $l=0.1$, $\phi =\pi$, $T=R=1/2$ and $\left \vert \alpha \right \vert ^{2}=10^{15}$.

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When $l_a=0.7$ corresponds to the large loss rate, the absolute sensitivity $\Delta \delta _{l_{a}}$ changes worse, and the maximum value of $G$ of the original model is approximately equal to 1.86 for maintaining the quantum advantage. Compared with the original model, the employed feedback scheme can make the absolute sensitivity of the measurement better while surpassing the SQL$_1$ at $k\approx 0.95$. Even though the feedback scheme can effectively enhance the absolute sensitivity, the vacuum noise introduction from the dark input port of BS1 leads to an upper limit for enhancement.

Figure 4 shows the sensitivity difference as a function of loss $l_a$ when the feedback strength $k$ is set to the optimal value and the parameter $G=2\sqrt {k_{\text {opt}}}(1-\sqrt {1-l})/(k_{\text {opt}}l)$. The blue solid line indicates that the absolute sensitivity under the feedback model is better than case of the original model at all loss $l_a$ when $l=0.1$. In view of the measurement advantage, the degree of surpassing SQL under the feedback scheme (red dashed line) is higher than the original case (green dashed line) when $l_a\le 0.64$. Furthermore, when the loss $l$ in feedback loop is changed from 0.1 (blue solid line) to 0.5 (purple dashed line), the sensitivity difference decreases, i.e., the sensitivity is worse. The increment between them decreases as $l_a$ increases, indicating that the loss $l_a$ in the meter beam affects the feedback enhancement results.A perfect feedback process benefits sensitivity enhancement.

Fig. 4. Sensitivity difference on a logarithmic (20log$_{10}$) scale as a function of loss $l_a$ when the feedback strength $k$ is operated at the optimal value and $G=2\sqrt {k_{\text {opt}}}(1-\sqrt {1-l})/(k_{\text {opt}}l)$. Blue solid line (purple dashed line) represents the sensitivity compared with the case without feedback when $l=0.1$ ($l=0.5$). Red (green) dashed line represents the sensitivity of the feedback model (original model) compared with the SQL$_1$ (SQL$_0$). The other parameters are $\phi =\pi$, $T=R=1/2$, and $\left \vert \alpha \right \vert ^{2}=10^{15}$.

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One possible application of QDM in device is long-term measurements since the phase and amplitude of the light field can be measured simultaneously with precision beyond SQL. It is well known that the phase and amplitude of the light field randomly drift due to the classical noises when the device is in practical applications, which influences long-term precision measurements. Usually, to stabilize the system, one can add an electro-optic modulator (EOM) or acoustic-optic modulator (AOM) on the optical path to control the phase or amplitude of the light field. The EOM or AOM command is determined by the error signal $i_{E}$, which can be obtained by detecting the output of the light field. With the intensity detection (ID), the error signal can be given by

(16)$$i_{E}\propto \Delta I\left( \epsilon_{\text{noise}} ,\sin \left( \delta_\text{noise} \right) \right) ,$$

including both phase and amplitude pieces of modulated information caused by noises. $\Delta I$ is the intensity shift of the output light field caused by random drifts in amplitude and phase. When using the error signal $i_{E}$ to control one parameter by the EOM or the AOM, the final accuracy of control will be affected by another parameter. Using two HDs instead of ID, the random drifts in phase and amplitude can be separated from the error signal, which allows to control of phase and amplitude individually. The theoretical model is shown in Fig. 5. An error signal $i_{E1}$ that only includes phase drift information is obtained by HD1, while an error signal $i_{E2}$ that only includes amplitude drift information is obtained by HD2,

(17)$$i_{E1}\propto \alpha\left( \delta_{\text{noise}} \right), i_{E2}\propto \alpha\left( \epsilon_{\text{noise}} \right).$$

These two error signals $i_{E1}$ and $i_{E2}$ are then given to the controllers, such as PID, to achieve control of the phase and amplitude, respectively. The measurement sensitivity of the device limits the control accuracy. The above attention theoretical model for QDM in Fig. 1(a) allows the measurement sensitivities of phase and amplitude beyond the SQL under the suitable parameters conditions. Such a scheme could independently control the phase and amplitude of the light field with high accuracy, which would help address the impact of classical noises on the device. Under the feedback scheme, the measurement sensitivities are improved, which could achieve better control accuracy of long-term stabilization.

Fig. 5. Scheme for individually controlling the phase and amplitude of the light field. An error signal ($i_{E1}$) that only includes the phase drift information of the light field is obtained by HD1, while another error signal ($i_{E2}$) that only includes the amplitude drift information of the light field is obtained by HD2. These two error signals are then given to the controller 1 (C1) and controller 2 (C2) to order the electro-optic modulator (EOM) and acoustic-optic modulator (AOM), respectively.

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4. Conclusion

In conclusion, we have studied the single-port coherent feedback-controlled QDM in the lossy environment. Utilizing the beam splitter as the controller feeds the reference beam back into its input port. The sensitivities of the phase and amplitude with the feedback scheme are enhanced significantly compared with the case without feedback. The optimal feedback strength of the feedback model adjusts based on the loss to maintain the measurement advantage provided by the two-mode entangled state. The results hold significance for practical applications of feedback-based quantum sources. For the long-term stabilization of any device, the phase control of the light field is often affected by amplitude shifts, thereby compromising precise control. By employing two HDs, one can independently obtain the random drifts in phase and amplitude, which reduces the impact from each other when doing control. The accuracy of control is directly related to the sensitivity of measurement. A better measurement sensitivity leads to higher control efficiency. QDM employs a two-mode entangled state that simultaneously measures both phase and amplitude with sensitivities beyond the SQL. When using two-mode entangled states for long-term stabilization, the control accuracy of both phase and amplitude is superior to classical methods. Furthermore, the coherent feedback-controlled QDM greatly improves the measurement sensitivities of both phase and amplitude, which could enhance the control accuracy for long-term stabilization.

Funding

National Natural Science Foundation of China (U23A2075, 12274132, 11974111); Fundamental Research Funds for the Central Universities; Shanghai Municipal Education Commission (202101070008E00099); China Postdoctoral Science Foundation (2023M741187).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Coherent feedback enhanced quantum-dense metrology in a lossy environment

Abstract

1. Introduction

2. Model and theory

3. Results and discussions

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Equations (17)

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