Open Access
Add to BibSonomy
Abstract
The SU (1,1)-type atom-light hybrid interferometer (SALHI) is a kind of interferometer that is sensitive to both the optical phase and atomic phase. However, the loss has been an unavoidable problem in practical applications and greatly limits the use of interferometers. Visibility is an important parameter to evaluate the performance of interferometers. Here, we experimentally demonstrate the mitigating effect of the loss on visibility of the SALHI via asymmetric gain optimization, where the maximum threshold of loss to visibility close to 100% is increased. Furthermore, we theoretically find that the optimal condition for the largest visibility is the same as that for the enhancement of signal-to-noise ratio (SNR) to the best value with the existence of the losses using the intensity detection, indicating that visibility can act as an experimental operational criterion for SNR improvement in practical applications. Improvement of the interference visibility means achievement of SNR enhancement. Our results provide a significant foundation for practical application of the SALHI in radar and ranging measurements.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Interferometers are widely used as sensors in precision measurement [1–4]. There have been many kinds of interferometers, such as optical interferometers [5–7], atom interferometers [8–13] and atom-light hybrid interferometer (ALHI) [14,15]. Optical interferometers can measure the optical phase sensitive quantitation and are the core of optical gyroscopes [16], laser radar [17], ranging systems [18], etc. Atom interferometers can measure the atomic phase sensitive parameters, and have been demonstrated to measure the rotation rate [19], acceleration of gravity [20–22] and magnetic field [23,24]. The ALHI is sensitive to both the optical and atomic phases, which has the potential to combine the advantages of optical wave and atomic systems in precision measurement, and has been utilized to measure angular velocity, electric field and magnetic field [14,15,25,26].
In interferometry, the visibility represents the degree of interference cancellation of two beams, which can evaluate the performance of interferometers [12,21,27–29] . Low visibility has a negative effect on the measurement and causes a reduction in the SNR [28,30,31]. The SU(1, 1)-type interferometer realizes beam splitting and recombination through two parametric amplification processes [27,32–34], whose two interference arms are quantum correlated. Comparing with conventional Mach-Zehnder interferometer (MZI), SNR of SU(1, 1)-type interferometer can break standard quantum limit due to quantum correlation. There are mainly two kinds of losses in the interferometer: internal and external losses. Previous literatures have shown that the SU(1, 1)-type interferometer has the tolerance to the detection loss (that is, external loss) by increasing the gain of the wave-recombination process [34–38] . However the SU(1, 1)-type interferometer, the internal loss has a greater effect on the perfect noise cancellation between two beams [39,40]. Noise cancellation is the advantage of the SU(1, 1)-type interferometer [41]. The internal loss limit the practical application of the SU(1, 1)-type interferometer in radar and ranging measurements. Recently, in the presence of internal losses, augmenting the visibility through asymmetry is shown in the all optical SU(1, 1) interferometer [42], and here we extend to the case of SALHI.
In this paper, we experimentally and theoretically investigate the visibility optimization of SU(1, 1)-type ALHI (SALHI) with the existence of the internal loss. The conventional MZI is also given as a comparison. The visibility of SALHI and MZI decreases with the loss. However, we theoretically give an optimization condition, under which the visibility of SALHI ($\mathrm {{V_{SU}} }$) can be restored to ~100 $\%$ by optimizing the gain factor of wave recombination process to satisfy the optimization condition over a wide range of internal loss. In experiment $\mathrm {{V_{SU}} }$ is restored to $\sim$90$\%$. Furthermore, we theoretically analyze the SNR of the SALHI ($\mathrm {{SNR_{SU}} }$) and find that the optimization condition for $\mathrm {{SNR_{SU}} }$ enhancement is the same as that for the visibility restoration, which implies that as long as the $\mathrm {{V_{SU}} }$ is improved, the $\mathrm {{SNR_{SU}} }$ can be enhanced. In experiment, it is difficult to judge whether the experimental conditions are suitable to achieve the best $\mathrm {{SNR_{SU}} }$. However, the visibility is a physical quantity that is convenient to obtain and observe. Thus, the $\mathrm {{V_{SU}} }$ can be used as an experimental operational criterion for $\mathrm {{SNR_{SU}} }$ improvement. Therefore, the optimization condition and the $\mathrm {{V_{SU}} }$ restoration, have guiding significance for practical application of atom-light hybrid interferometers in the future.
2. SU(1, 1)-type atom-light hybrid interferometer
The scheme of the SALHI is shown in Fig. 1(a), where two stimulated Raman scattering processes, SRS1 and SRS2, are used to realize the wave splitting and recombination of optical field and atomic spin wave. SRS1 generates $\hat {a}_{s_{1}}$ and $\hat {S}_{a_{1}}$ acting as two interference arms of SALHI, and then SRS2 is used to recombine $\hat {a}^{{}^{\prime }}_{s_{1}}$ and $\hat {S}^{{}^{\prime }}_{a_{1}}$. The final interference outputs are optical signal $\hat {a}_{s_{2}}$ and atomic spin wave $\hat {S}_{a_{2}}$, respectively.
Fig. 1. (a) Scheme of the SALHI, energy level of the $^{87}Rb$ atom, and frequencies of the laser. $\hat {a}_{s_{0}}$ and $\hat {S}_{a_{0}}$ are the initially input optical field and atomic spin wave, respectively. $\hat {a} _{s_{1}}$ and $\hat {S}_{a_{1}}$ are the interference arms generated from SRS1, which then evolve to $\hat {a}^{{}^{\prime }}_{s_1}$ and $\hat {S} ^{{}^{\prime }}_{a_1}$ after loss and phase shift $\varphi$, finally becoming interference outputs $\hat {a}_{s_2}$ and $\hat {S}_{a_2}$ after SRS2. SRS: stimulated Raman scattering. $|g,m\rangle$: $|5^{2}S_{1/2},F=1,F=2\rangle$; $|e_{1},e_{2}\rangle$: $|5^{2}P_{1/2},F=2 \rangle$, $|5^{2}P_{3/2}\rangle$; $W_1$ and $W_2$: Raman pump fields; OP: optical pumping field; $\Delta$: single photon detuning. (b) Experimental setup of the SALHI. PZT: piezoelectric transducer; PBS: polarization beam splitter; HWP: half-wave plate; F-R: Faraday rotator; GL: Glan prism; SMF: single-mode fiber; PD: photon detector. The diameters of $W_1$, $W_2$ and $\hat { a}_{s_{0}}$ are all 0.5mm.
When SRS is operated in single mode, which can be realized by using the seed $\hat {a}_{s_0}$ and the $W$ beam in spatial single-mode from single-mode fiber in experiment. The Hamiltonian of SRS can be written as [43]
3. Experimental setup
The experiment is performed in a cylindrical paraffin-coated $^{87}\!Rb$ vapor cell (diameter 0.5 cm, length 5 cm). As shown in Fig. 1(b), which was mounted inside a five-layer magnetic shield to reduce the stray magnetic field and heated to $75^{\circ }$C. Before the SRS1, almost all atoms are prepared in the ground state $|g\rangle$ by an optical pumping field (OP) resonant at the $|m\rangle$ $\rightarrow$ $|e_{2}\rangle$ transition. The OP pulse is 45 $\mu$s long, and its intensity is 110 mW. The $W$ field is divided into $W_{1}$ and $W_{2}$. $W_{2}$ is coupled into a 100 m-long single-mode fiber (SMF2). $W_{1}$ and initial input Stokes seed $\hat {a} _{s_{0}}$ are spatially overlapped by PBS3 and interact the atoms via SRS1. The detuning frequency $\Delta$ of $W$ is 1.2 GHz. The $\hat {a}_{s_{0}}$ beam is red tuned 6.8 GHz from the $W$ laser by an electro-optic modulator (EOM, Newport model No. 4851). After SRS1, $\hat {S}_{a_{1}}$ stays in the cell. $W_{1}$ and $\hat {a}_{s_{1}}$ exit the cell and are separated by PBS4. $\hat {a}_{s_{1}}$ is coupled into 100 m-long SMF1 and then returned back into the atomic cell with the $W_{2}$ pulse to interfere with $\hat {S} _{a_{1}}$ via SRS2. $\hat {a}_{s_{1}}$ and $W_{2}$ are temporally and spatially overlapped. The phase shift $\varphi$ of $\hat {a}_{s_{1}}$ is controlled by the PZT. The optical interference output $\hat {a}_{s_{2}}$ is detected by a photodetector after a Glan prism to filter $W_{2}$. In experiment, the internal loss $l$ of the optical interference beam is approximately 0.6, including the coupling efficiency of fiber, the transmittance of vapor cell and optical devices. The decay of the atomic spin wave is 0.4 due to atoms collisions and flying out of the interacting region during the evolution time between two SRS processes.
Comparing to optical MZI, whose two interference arms are both optical waves and the output is only sensitive to the optical phase, the SALHI goes one step futher relying atom-photon correlation. Thus the interference fringes depends on both atomic and optical phases.
4. Visibility results
In the experiment, we first measured the $\mathrm {{V_{SU}} }$ and the visibility of MZI ($\mathrm {{V_{MZ}} }$) with the same phase-sensitive particle number and internal loss as a comparison. Figure 2 (a) shows the interference fringes of the Mach-Zehnder interferometer (MZI) and SALHI at $l$=0.96, $G_{1}=3$, $G_{2}=5$ and atomic decay rate $\eta =0.4$ of $\hat {S} _{a_{1}}$. The values of the $\mathrm {{V_{MZ}} }$ and $\mathrm {{V_{SU}} }$ are 45 $\%$ and 53 $\%$, respectively. Figure 2 (b) shows the visibility value as a function of the loss rate $l$. In general, as the optical loss $l$ of $\hat {a}_{s_{1}}$ increases from 0.6 to 0.96 by variable attenuation plate, $\mathrm {V_{SU}}$ drops from 92.1$\%$ to 53$\%$, and $\mathrm {V_{MZ}}$ is always smaller than $\mathrm {V_{SU}}$ under the same loss condition.
Fig. 2. (a) The blue squares are the interference fringes of the SALHI with fixed $G_{1}$ =3, $G_{2}$ =5, $l$ =0.96 and $\eta$ =0.4, and the black triangles are the interference fringes of the MZI under the same operating conditions. (b) The visibility value as a function of the loss rate $l$. The blue squares represent $\mathrm {{V_{SU}} }$ with fixed $G_1$ = $3$, $G_2$ = $5$ and $\eta$ = $0.4$, the blue solid curve is the result of theoretical fitting according to Eq. (4), and the black triangles are the values of $\mathrm {V_{MZ}}$ measured experimentally. The pink dashed line corresponds to a visibility of 90$\%$.
In theory, according to Eq. (3), the visibility of the optical interference output $\hat {a}_{s_{2}}$ can be calculated and simplified as
5. Optimization condition
Furthermore, the largest interference visibility in Eq. (4) appears at
We call this the optimization condition. According to Eqs. (2, 3), the interference output $\hat {a}_{s_{2}}$ consists of two parts. One is $G_{2} \hat {a}_{s_{1}}^{ {\acute {}} }$ amplified from the optical arm $\hat {a}_{s_{1}}^{ {{}^{\prime }} }=\sqrt {1-l }\hat {a}_{s_{1}}$, and the other is $g_{2}\hat {S}_{a_{1}}^{ {\acute {}} }$ amplified from the atomic arm $\hat {S}_{a_{1}}^{ {{}^{\prime }} }=\sqrt { 1-\eta }\hat {S}_{a_{1}}$. When the gain factor of the wave-recombination process ($G_2$) is adjusted to satisfy the $G_{1}G_{2}\sqrt {1-l}=g_{1}g_{2} \sqrt { 1-\eta }$, the amplitudes of the two parts are equal, then the visibility of output $\hat {a}_{s_{2}}$ can reach $\sim$100$\%$.The SNR is also an important parameter to characterize the performance of an interferometer and can be calculated by SNR=$\dfrac {[(\partial \langle \hat {O} \rangle /\partial \varphi )\delta ] ^{2}}{\langle (\Delta \hat {O})^{2}\rangle }$ [44,45], where $\hat {O}$ is the measurable operator, $\delta$ is the added modulation small phase and $\langle (\Delta \hat {O})^{2}\rangle =\langle \hat {O}^{2}\rangle -\langle \hat {O}\rangle ^{2}$. Under ID, the $\mathrm {{SNR_{SU}} }$ for the output optical field is
Obviously, this condition for the $\mathrm {{SNR_{SU}} }$ under ID is same as Eq. (5), indicating that the improvement of $\mathrm {{V_{SU}}}$ corresponds to enhancement of $\mathrm {{SNR_{SU}}}$. The noises in the two arms of the interferometer are correlated, and the condition of Eq. (5) will make the correlated noise from the two arms in the output light field almost completely canceled, then $\mathrm {{SNR_{SU}}}$ of the output light field will be optimal. The optimization condition is the key point to improve $\mathrm {{V_{SU}}}$ and enhance $\mathrm {{SNR_{SU}}}$ even at large internal loss. It should be noted that the interference visibility can be restored to $\sim 100\%$ and $\mathrm {{SNR_{SU}}}$ can be enhanced to the best value in the presence of losses when the experimental conditions satisfy the optimization condition. In the interferometer, phase shift can be measured using ID and balance homodyne detection (BHD). We also give the optimization condition for BHD in appendix part, which is different to Eq. (5).
To show the improvement of the SALHI compared with the conventional MZI under the same operating conditions, we calculated $\mathrm {{SNR_{MZ}}}$=$\dfrac {(1-l)(1-\eta )N_{0}sin^{2}\varphi \delta ^{2} }{[(2-l-\eta )-2\sqrt {1-l}\sqrt { 1-\eta }cos\varphi ]}$ [46,47], where $N_{0}=(2G^{2}_1-1)N_{\hat {a}_{s_0}}$ is the phase-sensitive particle number of the MZI. Figures 3 (a-c) shows the visibility and Figs. 3 (d-f) shows the SNR as a function of the optical loss $l$. Firstly, before optimization, as the loss $l$ increases, the $\mathrm {{SNR_{SU}}}$ first increase to a maximum value at $l=l_{B}$ and then decrease. In fact, $l_{B}$ is the point satisfying the optimization condition $G_{1}G_{2}\sqrt { 1-l}=g_{1}g_{2}\sqrt { 1-\eta }$, and compared with MZI, the quantum interferometers has better visibility. However, Figs. 3 (d-f) shows that $\mathrm {{SNR_{SU}}}$ is larger than $\mathrm {{SNR_{MZ}}}$ only within a small $l$ range near $l_{B}$ under a certain $G_{1}$, $G_2$ and $\eta$, and as $G_{1}$ and $\eta$ increase, this range is gradually diminished. The reason is that the increased $G_{1}$ or internal loss will bring more uncorrelated excess noise and quickly reduce the noise cancellation advantage of the SU(1, 1)-type interferometer. Therefore, when $G_{1}$, $g_{1}$ and $\eta$ are fixed, finding a suitable $G_{2}$ satisfying the optimization condition at each $l$ is an effective way to enhance $\mathrm {{ SNR_{SU}}>{SNR_{MZ}}}$ over a wider range of internal loss.
Fig. 3. (a-c) The $\mathrm {{V_{SU}}}$ before and after optimization of the output field $\hat {a}_{s_2}$ and $\mathrm {{V_{MZ}}}$ as a function of $l$ (left-hand vertical axis). The blue dash-dotted curve is $\mathrm {{V_{SU}}}$ before optimization with fixed gain factors $G_{1}$, $g_{1}$, $G_{2}$, and $g_{2}$. The red dashed curve is largest $\mathrm {{V_{SU}} }$ (right-hand vertical axis) after optimizing $G_2$. The orange dotted curve is the value of $G_{2}/G_{1}$ after optimizing $G_{2}$ for the largest $\mathrm {{V_{SU}}}$, the black solid line is $\mathrm {{V_{MZ}}}$. (d-f) The black solid is $\mathrm {{SNR_{MZ}}}$ and the blue dash-dotted curves is $\mathrm {{ SNR_{SU}}}$ before optimization with fixed gain factors $G_{1}$, $g_{1}$, $G_{2}$, and $g_{2}$, respectively, the red dashed curve is $\mathrm {{SNR_{SU} }}$ after optimizing $G_{2}$ (left-hand vertical axis). The pink circles represent the $G_{2}/G_{1}$ value at the best $\mathrm {{SNR_{SU}}}$ after optimization, and the orange dotted curve is the $G_{2}/G_{1}$ value of the largest $\mathrm {{V_{SU}}}$ after optimization (right-hand vertical axis).
Figure 3 also shows the optimal visibility and $\mathrm {{SNR_{SU}}}$ values (the left vertical axis) and corresponding $G_{2}$/$G_{1}$ value (the right vertical axis), $G_{2}$ is limited within 1$\sim$ 10 considering the experimental operability. First, $\mathrm {{V_{SU}}}$ and $\mathrm {{SNR_{SU}}}$ after optimization are larger than $\mathrm {{V_{MZ}}}$ and $\mathrm {{ SNR_{MZ}}}$ over a wide range of losses. The optimized $G_{2}$ can effectively reduce the negative impact of the internal loss on $\mathrm {{ V_{SU}}}$ and $\mathrm {{SNR_{SU}}}$. Second, the optimized $G_{2}$ value is different in the regions of $l<l_{B}$ and $l>l_{B}$. For $l>l_{B}$, the optimal $G_{2}$ value is very small and can be directly calculated according to fixed $G_{1}$, $g_{1}$, $\eta$, and $l$. For $l<l_{B}$, we can not obtain the $G_{2}$ value completely satisfying the condition, only a larger $G_{2}$ value is closer to satisfying. Therefore as $l$ increases, there is a common feature in Figs. 3 (a-c) that the optimized $G_{2}$/$G_{1}$ value first remains at the maximum value at $\leq l_{B}$ and then decrease sharply to a much smaller value at $l\geq l_{B}$. Finally, in Figs. 3 (d-f), the pink circles and the orange dotted curve completely coincide (the right vertical axis), showing that at any internal loss, the best $\mathrm {{ SNR_{SU}}}$ corresponds to the point of largest $\mathrm {{V_{SU}}}$. Optimization of $\mathrm {{V_{SU}}}$ is easy to observe in an experiment. As long as the maximum of $\mathrm {{V_{SU}}}$ is observed by optimizing $G_{2}$, we can guarantee the best performance of the SU(1, 1)-type interferometer. This is different from the approach to compensate for the impact of the external loss, where $\mathrm {{ SNR_{SU}}}$ increase monotonically with $G_2$ and is easy to judge in an experiment.
6. Visibility restoration
Next, we provide an experimental demonstration of restoration of $\mathrm {{ V_{SU}}}$ by optimizing $G_{2}$ in the SALHI. In practical applications such as radar or ranging measurement, when the parameters ($G_{1}$, $g_{1}$, $\eta$, $l$) are fixed, we can adjust only $G_{2}$ and $g_{2}$ to satisfy the optimization condition and improve the visibility. In the experiment, $W_{1}$ and $W_{2}$ are separated from the same laser as Fig. 1 (b) shown. Before the $W_{2}$ field enters the vapor cell, it passes through an attenuator and an AOM, which can be used to adjust the intensity and frequency of the $W_{2}$ field, respectively. Therefore, we can control $G_{2}$ by controlling the intensity and frequency of $W_{2}$, so that the optimal condition is satisfied to obtain the best visibility. The experimental data of the visibility after optimizing $G_{2}$ ($\mathrm {{ V_{opt}}}$) are given in Fig. 4 (a) using red dots. $\mathrm {{V_{opt}}}$ is larger than the visibility without optimization ($\mathrm {{V_{SU}}}$), and at $l=0.6$, $\mathrm {V_{opt}}$ is 95%. As $l$ increases, $\mathrm {{V_{opt}}}$ can remain at $\sim 90\%$ (see the pink dashed line), and the optimized $G_{2}$ is small at the loss $l$ of 0.6-0.96 as in the theoretical prediction, such as $\mathrm {{V_{opt}}}$=88% with $G_{2}\approx 1.3$ when $l=0.96$, and Fig. 4 (b) shows the interference fringes of the SALHI after optimization. The results show that the visibility can be restored even at large internal loss by optimizing $G_{2}$, so the negative impact of internal loss on the properties of the SALHI can be mitigated. These experimental results are well consistent with the theoretical expectations.
Fig. 4. (a) The red dots are the values of $\mathrm {{V_{opt}}}$. The pink dashed line corresponds to a visibility of $90\%$. (b) The red dots are the interference fringes of the SALHI after optimization, corresponding to $G_2 \approx$1.3 with $G_1$=3, $l$=0.96 ,and $\eta$=0.4, $\mathrm {{V_{opt}}=88\%}$.
7. Discussion and Conclusion
In conclusion, we have experimentally and theoretically researched the influence of the internal loss on the visibility of the SU(1, 1)-type ALHI. In general, the internal loss has a significant negative impact on the visibility. Moreover, we give the optimization condition $G_{1}G_{2}\sqrt {1-l}=g_{1}g_{2}\sqrt {1-\eta }$ for visibility restoration and experimentally demonstrate that the visibility can be restored to $\sim$90$\%$ over a large range of internal loss by optimizing the $G_{2}$ factor to satisfy the optimization condition. Finally, we also theoretically find that the optimization condition for $\mathrm {{SNR_{SU}}}$ enhancement is the same as that for visibility restoration. Visibility as a physical quantity that is easy to obtain and observe, which can be used as an experimental operational criterion to judge whether the $\mathrm {{SNR_{SU}}}$ is optimized. What we have found will guide significance for practical application of quantum measurement, especially for SU(1, 1)-type interferometers systems in the presence of losses.
Appendix: Comparison of optimization conditions of ID and BHD
1. The optimization conditions of seed light field $\hat {a}_{s_0}$ input
After considering the losses $l$ and $\eta$, the optimal condition for the best $\rm {SNR_{SU}}$ under the ID is $G_{1}G_{2}\sqrt {1-l}=g_{1}g_{2}\sqrt {1-\eta }$, Obviously, this corresponds to the condition of the largest visibility. However, under the BHD, the quadrature component of interference output at phase dark point $\varphi =0$ is $\hat {X}_{\hat a_{s_{2}}}$,
From the Eq. (3), the visibility expression is:
However, under the BHD,
In previous paper [14], we theoretically studied the ${\rm {SNR_{SU}}}$ using homodyne detection only considering optical loss $l$. In this paper, we further study the visibility and ${\rm {SNR_{SU}}}$ using ID and BHD with both losses $l$ and $\eta$ because these two losses are always exist simultaneously in practical application. We find that whether with optical input seed or initial atomic seed, the optimization condition for best ${\rm {SNR_{SU}}}$ using ID is same as that of largest visibility, but different with that using BHD.
Therefore, here we experimentally measure the signal using ID. We can intuitively judge whether the optimization conditions for best ${\rm {SNR_{SU}}}$ is achieved according to the visibility restoration. And furthermore, compared with BHD, the ID device is simpler and more suitable for practical application of the SALHI in radar and ranging measurements.
Funding
National Natural Science Foundation of China (11654005, 11874152, 11974111, 91536114); Shanghai Municipal Science and Technology Major Project (2019SHZDZX01); Innovation Program of Shanghai Municipal Education Commission (202101070008E00099); ;National Key Research and Development Program of China (2016YFA0302001); Fundamental Research Funds for the Central Universities and the Fellowship of China Postdoctoral Science Foundation (2020TQ0193).
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.
References
1. X. Nie, J. Huang, Z. Li, W. Zheng, C. Lee, X. Peng, and J. Du, “Experimental demonstration of nonlinear quantum metrology with optimal quantum state,” Sci. Bull. 63(8), 469–476 (2018). [CrossRef]
2. B. P. Abbott (LIGO Scientific Collaboration and Virgo Collaboration), et al., “Gw170817: observation of gravitational waves from a binary neutron star inspiral,” Phys. Rev. Lett. 119(16), 161101 (2017). [CrossRef]
3. B. P. Abbott (LIGO Scientific Collaboration, Virgo Collaboration, 1M2H Collaboration, Dark Energy Camera GW-EM Collaboration, DES Collaboration, DLT40 Collaboration, Las Cumbres Observatory Collaboration, VINROUGE Collaboration and MASTER Collaboration), et al., “A gravitational-wave standard siren measurement of the hubble constant,” Nature 551(7678), 85–88 (2017). [CrossRef]
4. J. Liu, W. Liu, S. Li, D. Wei, H. Gao, and F. Li, “Enhancement of the angular rotation measurement sensitivity based on su (2) and su (1, 1) interferometers,” Photonics Res. 5(6), 617–622 (2017). [CrossRef]
5. M. Xiao, L.-A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59(3), 278–281 (1987). [CrossRef]
6. R. S. Bondurant and J. H. Shapiro, “Squeezed states in phase-sensing interferometers,” Phys. Rev. D 30(12), 2548–2556 (1984). [CrossRef]
7. O. Steuernagel and S. Scheel, “Approaching the heisenberg limit with two-mode squeezed states,” J. Opt. B: Quantum Semiclassical Opt. 6(3), S66–S70 (2004). [CrossRef]
8. C. Gross, T. Zibold, E. Nicklas, J. Esteve, and M. K. Oberthaler, “Nonlinear atom interferometer surpasses classical precision limit,” Nature 464(7292), 1165–1169 (2010). [CrossRef]
9. Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. Saravanan, S. R. Segal, and S. Wu, “Atom michelson interferometer on a chip using a bose-einstein condensate,” Phys. Rev. Lett. 94(9), 090405 (2005). [CrossRef]
10. M. Weitz, B. C. Young, and S. Chu, “Atomic interferometer based on adiabatic population transfer,” Phys. Rev. Lett. 73(19), 2563–2566 (1994). [CrossRef]
11. P. Hamilton, M. Jaffe, P. Haslinger, Q. Simmons, H. Müller, and J. Khoury, “Atom-interferometry constraints on dark energy,” Science 349(6250), 849–851 (2015). [CrossRef]
12. B. Estey, C. Yu, H. Müller, P.-C. Kuan, and S.-Y. Lan, “High-resolution atom interferometers with suppressed diffraction phases,” Phys. Rev. Lett. 115(8), 083002 (2015). [CrossRef]
13. M. Kasevich and S. Chu, “Measurement of the gravitational acceleration of an atom with a light-pulse atom interferometer,” Appl. Phys. B 54(5), 321–332 (1992). [CrossRef]
14. B. Chen, C. Qiu, S. Chen, J. Guo, L. Chen, Z. Ou, and W. Zhang, “Atom-light hybrid interferometer,” Phys. Rev. Lett. 115(4), 043602 (2015). [CrossRef]
15. C. Qiu, S. Chen, L. Chen, B. Chen, J. Guo, Z. Ou, and W. Zhang, “Atom–light superposition oscillation and ramsey-like atom–light interferometer,” Optica 3(7), 775–780 (2016). [CrossRef]
16. A. Matsko, A. Savchenkov, V. Ilchenko, and L. Maleki, “Optical gyroscope with whispering gallery mode optical cavities,” Opt. Commun. 233(1-3), 107–112 (2004). [CrossRef]
17. P. A. Rosen, S. Hensley, I. R. Joughin, F. K. Li, S. N. Madsen, E. Rodriguez, and R. M. Goldstein, “Synthetic aperture radar interferometry,” Proc. IEEE 88(3), 333–382 (2000). [CrossRef]
18. T. Kubota, M. Nara, and T. Yoshino, “Interferometer for measuring displacement and distance,” Opt. Lett. 12(5), 310–312 (1987). [CrossRef]
19. T. Gustavson, P. Bouyer, and M. Kasevich, “Precision rotation measurements with an atom interferometer gyroscope,” Phys. Rev. Lett. 78(11), 2046–2049 (1997). [CrossRef]
20. A. Peters, K. Y. Chung, and S. Chu, “High-precision gravity measurements using atom interferometry,” Metrologia 38(1), 25–61 (2001). [CrossRef]
21. P. Hamilton, M. Jaffe, J. M. Brown, L. Maisenbacher, B. Estey, and H. Müller, “Atom interferometry in an optical cavity,” Phys. Rev. Lett. 114(10), 100405 (2015). [CrossRef]
22. X. Wu, Z. Pagel, B. S. Malek, T. H. Nguyen, F. Zi, D. S. Scheirer, and H. Müller, “Gravity surveys using a mobile atom interferometer,” Sci. Adv. 5(9), eaax0800 (2019). [CrossRef]
23. M.-K. Zhou, Z.-K. Hu, X.-C. Duan, B.-L. Sun, J.-B. Zhao, and J. Luo, “Precisely mapping the magnetic field gradient in vacuum with an atom interferometer,” Phys. Rev. A 82(6), 061602 (2010). [CrossRef]
24. C. F. Ockeloen, R. Schmied, M. F. Riedel, and P. Treutlein, “Quantum metrology with a scanning probe atom interferometer,” Phys. Rev. Lett. 111(14), 143001 (2013). [CrossRef]
25. Y. Wu, J. Guo, X. Feng, L. Chen, C.-H. Yuan, and W. Zhang, “Atom-light hybrid quantum gyroscope,” Phys. Rev. Appl. 14(6), 064023 (2020). [CrossRef]
26. S. Chen, L. Chen, Z. Ou, and W. Zhang, “Quantum non-demolition measurement of photon number with atom-light interferometers,” Opt. Express 25(25), 31827–31839 (2017). [CrossRef]
27. F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5(1), 3049 (2014). [CrossRef]
28. J. Kong, J. Jing, H. Wang, F. Hudelist, C. Liu, and W. Zhang, “Experimental investigation of the visibility dependence in a nonlinear interferometer using parametric amplifiers,” Appl. Phys. Lett. 102(1), 011130 (2013). [CrossRef]
29. M. I. Kolobov, E. Giese, S. Lemieux, R. Fickler, and R. W. Boyd, “Controlling induced coherence for quantum imaging,” J. Opt. 19(5), 054003 (2017). [CrossRef]
30. Z. Ou, “Enhancement of the phase-measurement sensitivity beyond the standard quantum limit by a nonlinear interferometer,” Phys. Rev. A 85(2), 023815 (2012). [CrossRef]
31. M. Kacprowicz, R. Demkowicz-Dobrzański, W. Wasilewski, K. Banaszek, and I. Walmsley, “Experimental quantum-enhanced estimation of a lossy phase shift,” Nat. Photonics 4(6), 357–360 (2010). [CrossRef]
32. B. Yurke, S. L. McCall, and J. R. Klauder, “Su (2) and su (1, 1) interferometers,” Phys. Rev. A 33(6), 4033–4054 (1986). [CrossRef]
33. K. Zheng, M. Mi, B. Wang, L. Xu, L. Hu, S. Liu, Y. Lou, J. Jing, and L. Zhang, “Quantum-enhanced stochastic phase estimation with the su (1, 1) interferometer,” Photonics Res. 8(10), 1653–1661 (2020). [CrossRef]
34. Y. Liu, J. Li, L. Cui, N. Huo, S. M. Assad, X. Li, and Z. Ou, “Loss-tolerant quantum dense metrology with su (1, 1) interferometer,” Opt. Express 26(21), 27705–27715 (2018). [CrossRef]
35. A. M. Marino, N. C. Trejo, and P. D. Lett, “Effect of losses on the performance of an su (1, 1) interferometer,” Phys. Rev. A 86(2), 023844 (2012). [CrossRef]
36. M. Manceau, G. Leuchs, F. Khalili, and M. Chekhova, “Detection loss tolerant supersensitive phase measurement with an su (1, 1) interferometer,” Phys. Rev. Lett. 119(22), 223604 (2017). [CrossRef]
37. E. Giese, S. Lemieux, M. Manceau, R. Fickler, and R. W. Boyd, “Phase sensitivity of gain-unbalanced nonlinear interferometers,” Phys. Rev. A 96(5), 053863 (2017). [CrossRef]
38. M. Manceau, F. Khalili, and M. Chekhova, “Improving the phase super-sensitivity of squeezing-assisted interferometers by squeeze factor unbalancing,” New J. Phys. 19(1), 013014 (2017). [CrossRef]
39. J. Xin, H. Wang, and J. Jing, “The effect of losses on the quantum-noise cancellation in the su (1, 1) interferometer,” Appl. Phys. Lett. 109(5), 051107 (2016). [CrossRef]
40. W. Du, J. Chen, Z. Ou, and W. Zhang, “Quantum dense metrology by an su (2)-in-su (1, 1) nested interferometer,” Appl. Phys. Lett. 117(2), 024003 (2020). [CrossRef]
41. D. Li, B. T. Gard, Y. Gao, C.-H. Yuan, W. Zhang, H. Lee, and J. P. Dowling, “Phase sensitivity at the heisenberg limit in an su (1, 1) interferometer via parity detection,” Phys. Rev. A 94(6), 063840 (2016). [CrossRef]
42. Y. Michael, I. Jonas, L. Bello, M.-E. Meller, E. Cohen, M. Rosenbluh, and A. Pe’er, “Augmenting the sensing performance of entangled photon pairs through asymmetry,” Phys. Rev. Lett. 127(17), 173603 (2021). [CrossRef]
43. K. Hammerer, A. S. Sørensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82(2), 1041–1093 (2010). [CrossRef]
44. B. E. Anderson, P. Gupta, B. L. Schmittberger, T. Horrom, C. Hermann-Avigliano, K. M. Jones, and P. D. Lett, “Phase sensing beyond the standard quantum limit with a variation on the su (1, 1) interferometer,” Optica 4(7), 752–756 (2017). [CrossRef]
45. Z.-D. Chen, C.-H. Yuan, H.-M. Ma, D. Li, L. Chen, Z. Ou, and W. Zhang, “Effects of losses in the atom-light hybrid su(1, 1) interferometer,” Opt. Express 24(16), 17766–17778 (2016). [CrossRef]
46. R. Demkowicz-Dobrzanski, U. Dorner, B. Smith, J. Lundeen, W. Wasilewski, K. Banaszek, and I. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80(1), 013825 (2009). [CrossRef]
47. U. Dorner, R. Demkowicz-Dobrzanski, B. Smith, J. Lundeen, W. Wasilewski, K. Banaszek, and I. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102(4), 040403 (2009). [CrossRef]
