Abstract

We study the phase sensitivity of an SU(1,1) interferometer from two aspects, i.e., the phase estimation determined by the error propagation formula and that by the quantum Cramér-Rao bound (QCRB). The results show that the phase sensitivity by using the intensity detection reaches the sub-shot-noise limit with a coherent state and an m-photon-added squeezed vacuum state (m-PA-SVS) as inputs. The phase sensitivity gradually approaches the Heisenberg limit for increasing m, and the ultimate phase precision improves with the increase of m. In addition, the QCRB can be saturated by the intensity detection with inputting the m-PA-SVS.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Quantum phase estimation has been extensively studied in recent years [1–4], due to its wide applications, for example, in gravitational wave detection [5], Bose-Einstein condensate [6], quantum information processing [7], quantum imaging [8], and quantum precision measurement [9].

Optical interferometers used to estimate small phase changes are essential tools for obtaining high precision in quantum metrology. The Mach-Zehnder interferometer (MZI) characterized by two passive optical beam splitters is one of the commonly used tools in quantum metrology. When only the coherent light enters such a passive device, the phase sensitivity is limited by the shot-noise limit (SNL) [1, 10, 11], which can not be broken by classical measurement. In recent years, much efforts have been devoted to improve the phase sensitivity of the MZI beyond the SNL by using the squeezed states [12–14] and other nonclassical states [8, 15–17]. Surprisingly, another benchmark for phase sensitivity, the Heisenberg limit (HL) [18], can also be achieved by using the nonclassical states of light [19–21]. Another commonly used device is the nonlinear SU(1, 1) interferometer [22], which usually consists of two active optical elements [23], such as optical parametric amplifiers (OPA) or four-wave mixers. The nonlinear optical element OPA driven by the pump light can transfer energy from the pump light to the signal light, and this can make the signal light much stronger. Due to the enhancement of the signal light, Hudelist et al. found that the signal-to-noise ratio is 4.1 dB higher than that of the MZI [24]. William et al. found that the phase sensitivity of this nonlinear interferometer is better than the SNL when the two inputs are strong coherent light [25]. Even considering the loss of photons, the phase sensitivity is still better than the SNL [25–27]. Obviously, the SU(1,1) interferometer promises to get a better phase sensitivity than the MZI. Furthermore, The experimental realization of the SU(1,1) interferometer has aroused great interest from many groups. Gross et al. constructed a nonlinear atom interferometer whose phase precision can beyond the SNL [28]. Kong et al. proposed a scheme to amplify signal with reduced noise [29]. Chen et al. reported a hybrid atom-light interferometer which are sensitive to different types of phase shift [30]. Many studies have shown that the phase sensitivity of the SU(1,1) interferometer can be significantly enhanced by using the squeezed vacuum state (SVS). For example, Li et al. found that the phase sensitivity of a balanced-configured SU(1,1) interferometer can reach the HL with a coherent state and a squeezed vacuum state as inputs by using the parity detection [31] and the homodyne detection [32]. However, the optimal phase sensitivities can not saturate the ultimate limit set by the quantum Cramér-Rao bound (QCRB) [33, 34].

As we all know, the photon-addition and photon-subtraction are two available and effective methods for generating non-Gaussian states with high nonclassical features. Gerry et al. [35] studied the nonclassical properties of the photon-subtracted two-mode squeezed vacuum states (PS-TMSVS). Yang et al. [36] showed that the photon-added two-mode squeezed vacuum states (PA-TMSVS) offers improved phase sensitivity over both the two-mode squeezed vacuum states and the PS-TMSVS. The superiority of the PA-TMSVS is mainly due to its sub-Poissonian statistics which plays an important role in ultra-sensitive phase shift measurements [37]. For the single-mode case, Richard et al. [38] pointed out that one can obtain improved phase sensitivity by using the photon-subtracted SVS. Recently, Gong et al. [39] studied the quantum Fisher information (QFI) of the SU(1,1) interferometer and showed that the phase sensitivity can be improved when m photons are subtracted from the SVS. This amazing result has an important practical significance for the phase estimation. As far as we know, the precision of phase estimation is not only determined by the input state but also closely associated with the detection method. However, in [39] only the QFI was calculated numerically, without taking the effect of measurement into account. In our present paper, we use the intensity detection to study the phase sensitivity of the SU(1,1) interferometer with a coherent state and an m-photon-added squeezed vacuum state (m-PASVS) as inputs, and expect to obtain a better result, since the photon statistics of the PA-SVS shows sub-Poissonian statistics [40]. The phase sensitivity with the intensity detection on an SU(1,1) interferometer can be improved by using the PA-SVS. We also use the QFI to analyze the phase sensitivity and find that the ultimate phase precision can be enhanced with the help of the m-PA-SVS, and the enhancement becomes significant as m increases. Compared with the homodyne detection and the parity detection, the intensity detection offers a better phase sensitivity due to the photon-additions in the SVS.

This paper is organized as follows: In Sec. 2 we describe the model of an SU(1,1) interferometer and make a brief review of the m-PA-SVS. In Sec. 3 we study the phase sensitivity with the intensity detection. In Sec. 4 the quantum Fisher information and the quantum Cramér-Rao bound are discussed. In Sec. 5 we compare the phase sensitivity of our system with some theoretical limits and other schemes. Sec. 6 presents a brief summary.

2. Model

Figure 1 shows a typical SU(1,1) interferometer, in which the two beam splitters in a traditional MZI are replaced by two OPAs. The action of the OPA on a two-mode state is described by the unitary operator UOPA(ξ) = exp[−ξa+b + + ξab] [23], here a and b are the annihilation operators for the two modes and the coupling constant ξ=geiθg is related to the gain coefficient of the parametric amplifier. After the first OPA, mode a undergoes a phase shift ϕ, describing with the unitary operator Uϕ=eiϕa+a, then the two beams recombine in the second OPA. The relation between the output operators and the input operators is given by [25, 26]

(a2b2+)=S(a0b0+),
where
S=SOPA2SϕSOPA1,
with
SOPA1=(coshg1eiθ1sinhg1eiθ1sinhg1coshg1),
Sϕ=(eiϕ001),
SOPA2=(coshg2eiθ2sinhg2eiθ2sinhg2coshg2),
in which g1 (g2) and θ1 (θ2) are the gain factor and phase shift in the first (second) parametric amplification process, respectively. Here we consider a balanced configuration where the two OPAs have a fixed phase difference of π (θ2θ1 = π) and the same gain factor (g1 = g2 = g).

 figure: Fig. 1

Fig. 1 Schematic diagram of an SU(1,1) interferometer. The two input beams are in a coherent state and a photon-added squeezed vacuum state, respectively. OPA : optical parametric amplifier; Da and Db: detectors.

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Combining Eqs. (1)(5), we can obtain the transformations between the output operators and input operators as follows:

a2=(eiϕcosh2gsinh2g)a0[eiθ1(eiϕ1)sinhgcoshg]b0+,
b2+=[eiθ1(eiϕ1)sinhgcoshg]a0+(cosh2geiϕsinh2g)b0+.

We consider an SU(1,1) interferometer with a coherent state |αa(α=|α|eiθα) and an m-photon-added squeezed vacuum state |r, mb as inputs. The m-PA-SVS is defined as [41, 42]

|r,mb=Nmb+m|r,0b,
where |r, 0〉b = Sb(r ) |0〉b is the squeezed vacuum state, Sb(r)=exp[r2(b+2b2)] is the single-mode squeezing operator with r the squeezing parameter, and the normalization constant Nm is given by [40, 41]
Nm2=m!coshmrPm(coshr),
where Pm is the Legendre polynomial of order m [43].

In the aspect of experiment, Dakna et al. proposed a scheme to generate the photon-added state by conditionally measuring a beam splitter [44], while Zavatta et al. employed a non-degenerate parametric down-conversion process with low pumping strength to generate the photon-added coherent state and thermal state [45–47]. In recent years, due to the potential application of photon-added squeezed vacuum state in quantum information processing, its nonclassical properties such as the sub-Poissonian statistics, antibunching effect, negativity of the Wigner function and quadrature squeezing effects, etc. have been extensively studied [40, 41, 48–50].

3. Phase sensitivity based on intensity detection

Detection is an indispensable means for extracting phase information from an interferometer, and a variety of detection methods have been proposed to improve the precision of phase estimation, such as the parity detection [31, 36, 37, 51], the homodyne detection [32, 51, 52], and the intensity detection [25,26,53,54]. In our model, the intensity detection is considered for simplicity and accuracy. We introduce the total photon-number operator N at the output ports of the SU(1,1) interferometer in the form of

N=AaNa+AbNb,
where Na=a2+a2 and Nb=b2+b2 are the photon-number operators at the output ports a and b, respectively. Aa (Ab) has a value of 1 or 0 depending on whether the detector Da (Db) is on or not. Using the variable N, the phase sensitivity Δϕ based on the intensity detection can be obtained from the error propagation formula [22]
Δ2ϕ=N2N2|ϕN|2.

In this paper we consider two kinds of intensity detection: I. Aa = 1, Ab = 0, i.e. only detector Da is on; II. Aa = 1, Ab = 1, i.e. both detectors Da and Db are on. In following calculations, for convenience, we take the phase θα = 0 and θ1 = 0. For an ideal SU(1,1) interferometer with inputting a coherent light |αa and an m-photon-added squeezed vacuum light |r, mb, we obtain

(Δ2ϕ)I=18(1+n¯m+α2)2{8[1+Bm+2n¯mn¯m2+2(1+Am+n¯m)α2+4Amα2csch22g]+csch42g[8α2csc2ϕ2+[3Bm1+2n¯m3n¯m22(n¯m+Am1)α2+4(α2Bmn¯m+n¯m2)cosh4g+(1+Bm+2n¯mn¯m2+2(1+Am+n¯m)α2)cosh8g]sec2ϕ2]},
for detection method I, and
(Δ2ϕ)II=18sinh42g(e2iϕ1)2(1+n¯m+α2)2{2e2iϕ[9Bm7+2n¯m9n¯m2+2(1+Am7n¯m)α2+(3+3Bm+6n¯m3n¯m2+(610Am+6n¯m)α2)cos2ϕ]4(e2iϕ1)2[1+Bm+2n¯mn¯m2+2(n¯m+1Am)α2]cosh4g+[1+Bm+2n¯mn¯m2+2(1+n¯m+Am)α2][8e2iϕcosϕ+(eiϕ1)4cosh8g]},
for detection method II, in which [40, 41]
n¯m=r,m|b+b|r,mb=Nm2Nm+121,
Am=r,m|b2|r,mb=Nm2cothr[Nm+12(m+1)Nm2],
Bm=r,m|b+2b2|r,mb=Nm2(Nm+224Nm+12)+2.

For m = 0, the phase sensitivity by intensity detection with coherent state and squeezed vacuum state as inputs has been studied by Li et al. [31], and their result is consistent with our (Δ2ϕ)II. Whichever detection method is chosen, the phase sensitivity of the intensity detection with inputting vacuum state is (Δ2ϕ)I = (Δ2ϕ)II = csch2 2g in our scheme, which is the same as that of Yurke [22].

Now we examine the effects of some parameters on the phase sensitivity. In Fig. 2, we plot the phase sensitivity Δϕ as a function of the phase shift ϕ with a coherent state and an m-photon-added squeezed vacuum state as inputs. We know that the smaller the value of Δϕ, the higher the phase sensitivity. We can see from Fig. 2 the following: (1) For both detection methods I and II, we notice that the optimal phase points are close to zero, but not at zero; (2) At the same phase point, the phase sensitivity improves with increasing m ; (3) The detection method I is better than the detection method II.

 figure: Fig. 2

Fig. 2 Phase sensitivity based on the intensity detection as a function of ϕ with g=1, r =1, and α=1. The solid lines correspond to the detection method I, while the dashed lines are for the detection method II.

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In order to further confirm the enhancement of phase sensitivity by adding photons in the squeezed vacuum state, we plot the optimal phase sensitivity Δϕ versus the gain factor g as shown in Fig. 3(a), where we fix the input parameters α = 1 and r = 1. Figure 3(a) tells us the follows: (1) Δϕ decreases with increasing g for both detection methods I and II; (2) For a same value of g, the phase sensitivity improves with increasing m ; (3) The detection method I is better than the detection method II. Besides, we also find that the phase sensitivity Δϕ improves with increasing r and m, as shown in Fig. 3(b).

 figure: Fig. 3

Fig. 3 Phase sensitivity based on the intensity detection as a function of (a) g with α=1 and r =1, (b) r with g=1 and α=1. The solid lines correspond to the detection method I, while the dashed lines are for the detection method II.

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The physical meaning of Figs. 2 and 3 can be understood as follows. It is known that the phase sensitivity of an interferometer is mainly determined by the total mean photon number NT inside the interferometer. The larger NT is, the smaller Δϕ will be (the higher the phase sensitivity). In our scheme, NT=a1+a1+b1+b1 [22]. For different m, we have

NT,m=cosh2g(1+α2+n¯m)1.
For the case of m = 0, the total mean photon number is NT,0 = cosh 2g(α2 + cosh2 r) − 1, which is the same as that of [31].

In Figs. 4(a)4(b) we plot the total mean photon number NT as functions of the gain factor g and the squeezing parameter r, respectively, with m as a parameter. It can be clearly seen that NT increases with increasing g, r, and m. This is very reasonable, since the total mean photon number of the input fields increases with increasing r and m, while g is the gain factor of the OPA. Therefore, the increase of r, m, and g leads to the increase of the total mean photon number NT inside the interferometer, and in turn, leads to the decrease of Δϕ.

 figure: Fig. 4

Fig. 4 The total average photon number NT as a function of (a) g with α=1 and r =1. (b) r with α=1 and g=1.

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4. Phase sensitivity based on QFI and QCRB

It is well known that a representative method for estimating phase ϕ is to evaluate the QFI, which characterizes the maximum amount of information about the unknown phase shift ϕ that can be extracted from the interferometer by using the best detection method. Under lossless conditions, for a pure state, the expression of the QFI is given by [55, 56]

F=4[ψϕ|ψϕ|ψϕ|ψϕ|2],
where |ψϕ〉 = UϕUOPA |ψin〉 is the state vector just before the second OPA, and |ψϕ=ϕψϕ. Then the QFI can be written as [57]
F=4Δ2na,
where Δ2na=ψin|(a1+a1)2|ψinψin|(a1+a1)2|ψin2. In the present work, the input state |ψin〉 = |αa ⊗ |r, mb then for different m, we can obtain the QFI Fm as
Fm=4[α2cosh4g+(Bm+n¯mn¯m2)sinh4g]+[1+n¯m+(2Am+1+2n¯m)α2]sinh22g.

The quantum Cramér-Rao bound that determines the ultimate phase precision of an interferometer regardless of the measurement method can be written as [33, 34, 58, 59]

ΔϕF1vFm,
where v is the number of trials and we take v = 1. From Eq. (21), we can see that the larger Fm is, the smaller the lower bound of ΔϕF is, that is, the higher the phase sensitivity. In Fig. 5(a), we show that the QFI increases with increasing g and m. Using Eq. (21), we obtain the minimum phase uncertainty ΔϕF, which decreases with the increase of g, α, and r, as shown in Figs. 5(b)5(d). For given parameters g, α, and r, the PA-SVS offers a better phase sensitivity that improves with the increase of m. Therefore, adding photons in the SVS is beneficial to improve the ultimate phase precision.

 figure: Fig. 5

Fig. 5 (a) Quantum Fisher information Fm versus gain factor g for α=1 and r =1. Phase sensitivity ΔϕF as a function of (b) g with α=1 and r =1.(c) r with α=1 and g=1.(d) α with r =1 and g=1.

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5. Comparison and discussion

5.1. Comparison with theoretical limits

In this section, in order to evaluate the superiority of our scheme, we compare the phase sensitivity based on the two kinds of intensity detection with the Heisenberg limit ΔϕHL and the ultimate phase precision ΔϕF. The corresponding HL is an important indicator to evaluate the performance of an interferometer, which is inversely proportional to the total average photon number NT inside the interferometer, i.e. ΔϕHL=1NT. Similarly, we can obtain the shot-noise limit ΔϕSNL=1NT.

In Fig. 6 we plot the optimal phase sensitivities Δϕ with the two kinds of intensity detection as a function of the gain factor g for m = 0, 1, 2, 3. For m = 0, i.e., the case of the squeezed vacuum state, Li et al. have shown that the phase sensitivity with the homodyne detection [32] and the parity detection [31] can reach the HL in most range of the gain factor g. We can see from Fig. 6 the following: (1) Detection method I is better than the detection method II. (2) The phase sensitivity based on the detection method I can break the SNL and gradually approach the HL with the increase of g. (3) The phase sensitivity with the detection method I is more closer to the HL as m increases.

 figure: Fig. 6

Fig. 6 Pase sensitivity based on the intensity detection against g for different m, (a) m=0, (b) m=1, (c) m=2, (d) m=3. The subscript 1(2) of m corresponds to the detection method I(II). The orange dashed line is for the shot-noise limit, the purple dashed line is for the QCRB, the cyan dashed line is for the Heisenberg limit, where α=1 and r =1.

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In Fig. 7 we repeat these graphs when only the m-PA-SVS is entered into the interferometer (i.e., the other input is a vacuum state, not a coherent state). As in the case described above, the phase sensitivity of the intensity detection satisfies ΔϕHL < Δϕ < ΔϕSNL but it can saturate the QCRB with the increase of m in some range of g. Therefore, with the m-PA-SVS as input, the QCRB can be saturated by using the detection method I.

 figure: Fig. 7

Fig. 7 The same as Fig. 6 but α = 0.

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5.2. Comparison between different detection methods

Next, we compare the optimal phase sensitivities based on our intensity detection with that based on the homodyne detection [32] and the parity detection [31] of other schemes, with coherent and squeezed states as inputs of an SU(1,1) interferometer. Figures 8(a)8(d) plot the optimal phase sensitivities versus g for three different measurements, from which we notice that the optimal phase sensitivities improve with the increase of g. Figures 8(a) and 8(c) show that the optimal phase sensitivity with the homodyne detection is the worst among them, and the phase sensitivity of our intensity detection is better than that of the parity detection with the increase of m. The phase sensitivity of our intensity detection is the best among them with the m-PA-SVS as input, as shown in Fig. 8(b). In Fig. 8(d), when the coherent light is stronger, the phase sensitivity with the parity detection is better than the intensity detection, but we expect that the intensity detection will approach a better phase sensitivity by adding more photons in the squeezed vacuum state.

 figure: Fig. 8

Fig. 8 The phase sensitivity Δϕ as a function of g with (a) α = (tanh 2g)er/2 and r =0.2, (b) α=0 and r =1, (c) α = (tanh 2g)er/2 and r =0, (d) α=1 and r =0. The subscript 1(2) of m corresponds to the intensity detection method I (II).

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6. Conclusion

In summary, we have investigated the intensity detection on an SU(1,1) interferometer with a coherent state and an m-PA-SVS as inputs. We have shown that the phase sensitivity of the intensity detection can reach the sub-shot-noise limit and approach the Heisenberg limit by adding photons in the SVS. In addition, the ultimate precision of phase estimation can be improved by using the PA-SVS. Compared with the homodyne detection and the parity detection, the intensity detection has a slightly better optimal phase sensitivity with the coherent and the photon-added squeezed vacuum states as inputs. Surprisingly, with the PA-SVS as input, the QCRB can be saturated by the intensity detection.

Funding

National Natural Science Foundation of China (Grant Nos. 61775062, 11574092, 61378012, 91121023, and 60978009); National Basic Research Program of China (Grant No. 2013CB921804); the Innovation Project of Graduate School of South China Normal University (2017LKXM088).

References

1. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693 (1981). [CrossRef]  

2. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006). [CrossRef]   [PubMed]  

3. M. Zwierz, C. A. Pérez-Delgado, and P. Kok, “General optimality of the Heisenberg limit for quantum metrology,” Phys. Rev. Lett. 105, 180402 (2010). [CrossRef]  

4. V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222 (2011). [CrossRef]  

5. J. Abadie, B. P. Abbott, R. Abbott, T. D. Abbott, M. Abernathy, C. Adams, R. Adhikari, C. Affeldt, B. Allen, and G. S. Allen, “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962 (2011). [CrossRef]  

6. S. M. Roy and S. L. Braunstein, “Exponentially enhanced quantum metrology,” Phys. Rev. Lett. 100, 220501 (2008). [CrossRef]   [PubMed]  

7. J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777 (2012). [CrossRef]  

8. S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled Fock states for robust quantum optical metrology, imaging, and sensing,” Phys. Rev. A 78, 063828 (2008). [CrossRef]  

9. M. V. Chekhova and Z. Y. Ou, “Nonlinear interferometers in quantum optics,” Adv. Opt. Photonics 8, 104 (2016). [CrossRef]  

10. M. Hillery and L. Mlodinow, “Interferometers and minimum-uncertainty states,” Phys. Rev. A 48, 1548 (1993). [CrossRef]   [PubMed]  

11. G. M. D’Ariano and M. G. A. Paris, “Lower bounds on phase sensitivity in ideal and feasible measurements,” Phys. Rev. A 49, 3022 (1994). [CrossRef]  

12. M. Xiao, L. A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59, 278 (1987). [CrossRef]   [PubMed]  

13. P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-light-enhanced polarization interferometer,” Phys. Rev. Lett. 59, 2153 (1987). [CrossRef]   [PubMed]  

14. H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura, D. W. Berry, and T. C. Ralph, “Quantum-enhanced optical-phase tracking,” Science 337, 1514 (2012). [CrossRef]   [PubMed]  

15. Q. S. Tan, J. Q. Liao, X. G. Wang, and F. Nori, “Enhanced interferometry using squeezed thermal states and even or odd states,” Phys. Rev. A 89, 053822 (2014). [CrossRef]  

16. A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733 (2000). [CrossRef]   [PubMed]  

17. Y. M. Zhang, X. W. Li, W. Yang, and G. R. Jin, “Quantum Fisher information of entangled coherent states in the presence of photon loss,” Phys. Rev. A 88, 043832 (2013). [CrossRef]  

18. S. Boixo, A. Datta, M. J. Davis, S. T. Flammia, A. Shaji, and C. M. Caves, “Quantum metrology: dynamics versus entanglement,” Phys. Rev. Lett. 101, 040403 (2008). [CrossRef]   [PubMed]  

19. K. P. Seshadreesan, P. M. Anisimov, H. Lee, and J. P. Dowling, “Parity detection achieves the Heisenberg limit in interferometry with coherent mixed with squeezed vacuum light,” New J. Phys. 13, 083026 (2011). [CrossRef]  

20. M. J. Holland and K. Burnett, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. 71, 1355 (1993). [CrossRef]   [PubMed]  

21. L. Pezzé and A. Smerzi, “Mach-Zehnder interferometry at the Heisenberg limit with coherent and squeezed-vacuum light,” Phys. Rev. Lett. 100, 073601 (2008). [CrossRef]   [PubMed]  

22. B. Yurke, S. L. McCall, and J. R. Klauder, “SU (2) and SU (1, 1) interferometers,” Phys. Rev. A 33, 4033 (1986). [CrossRef]  

23. C. Sparaciari, S. Olivares, and M. G. A. Paris, “Gaussian-state interferometry with passive and active elements,” Phys. Rev. A 93, 023810 (2016). [CrossRef]  

24. F. Hudelist, J. Kong, C. J. Liu, J. T. Jing, Z. Y. Ou, and W. P. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014). [CrossRef]   [PubMed]  

25. W. N. Plick, J. P. Dowling, and G. S. Agarwal, “Coherent-light-boosted, sub-shot noise, quantum interferometry,” New J. Phys. 12, 083014 (2010). [CrossRef]  

26. A. M. Marino, N. V. Corzo Trejo, and P. D. Lett, “Effect of losses on the performance of an SU (1, 1) interferometer,” Phys. Rev. A 86, 023844 (2012). [CrossRef]  

27. Z. Y. Ou, “Enhancement of the phase-measurement sensitivity beyond the standard quantum limit by a nonlinear interferometer,” Phys. Rev. A 85, 023815 (2012). [CrossRef]  

28. C. Gross, T. Zibold, E. Nicklas, J. Estève, and M. K. Oberthaler, “Nonlinear atom interferometer surpasses classical precision limit,” Nature 464, 1165 (2010). [CrossRef]   [PubMed]  

29. J. Kong, F. Hudelist, Z. Y. Ou, and W. P. Zhang, “Cancellation of internal quantum noise of an amplifier by quantum correlation,” Phys. Rev. Lett. 111, 033608 (2013). [CrossRef]   [PubMed]  

30. B. Chen, C. Qiu, S. Y. Chen, J. X. Guo, L. Q. Chen, Z. Y. Ou, and W. P. Zhang, “Atom-Light Hybrid Interferometer,” Phys. Rev. Lett. 115, 043602 (2015). [CrossRef]   [PubMed]  

31. D. Li, B. T. Gard, Y. Gao, C. H. Yuan, W. P. 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, 063840 (2016). [CrossRef]  

32. D. Li, C. H. Yuan, Z. Y. Ou, and W. P. Zhang, “The phase sensitivity of an SU (1, 1) interferometer with coherent and squeezed-vacuum light,” New J. Phys. 16, 073020 (2014). [CrossRef]  

33. C. W. Helstrom, “Quantum detection and estimation theory,” J. Stat. Phys 1, 231 (1969). [CrossRef]  

34. S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439 (1994). [CrossRef]   [PubMed]  

35. R. Carranza and C. C. Gerry, “Photon-subtracted two-mode squeezed vacuum states and applications to quantum optical interferometry,” J. Opt. Soc. Am. B 29, 2581 (2012). [CrossRef]  

36. Y. Ouyang, S. Wang, and L. J. Zhang, “Quantum optical interferometry via the photon-added two-mode squeezed vacuum states,” J. Opt. Soc. Am. B 33, 1373 (2016). [CrossRef]  

37. C. C. Gerry and J. Mimih, “Heisenberg-limited interferometry with pair coherent states and parity measurements,” Phys. Rev. A 82, 013831 (2010). [CrossRef]  

38. R. Birrittella and C. C. Gerry, “Quantum optical interferometry via the mixing of coherent and photon-subtracted squeezed vacuum states of light,” J. Opt. Soc. Am. B 31, 586 (2014). [CrossRef]  

39. Q. K. Gong, X. L. Hu, D. Li, C. H. Yuan, Z. Y. Ou, and W. P. Zhang, “Intramode-correlation-enhanced phase sensitivities in an SU (1, 1) interferometer,” Phys. Rev. A 96, 033809 (2017). [CrossRef]  

40. L. Y. Hu and H. Y. Fan, “Nonclassicality of photon-added squeezed vacuum and its decoherence in thermal environment,” J. Mod. Optic 57, 1344 (2010). [CrossRef]  

41. Z. Zhang and H. Fan, “Properties of states generated by excitations on a squeezed vacuum state,” Phys. Lett. A 165, 14 (1992). [CrossRef]  

42. H. Y. Fan, X. G. Meng, and J. S. Wang, “New form of Legendre polynomials obtained by virtue of excited squeezed state and IWOP technique in quantum optics,” Commun. Theor. Phys. 46, 845 (2006). [CrossRef]  

43. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integration Series and Products (Academic, 2014).

44. M. Dakna, L. Knöll, and D. G. Welsch, “Photon-added state preparation via conditional measurement on a beam splitter,” Opt. Commun. 145, 309 (1998). [CrossRef]  

45. A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660 (2004). [CrossRef]   [PubMed]  

46. A. Zavatta, S. Viciani, and M. Bellini, “Single-photon excitation of a coherent state: Catching the elementary step of stimulated light emission,” Phys. Rev. A 72, 023820 (2005). [CrossRef]  

47. A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106 (2007). [CrossRef]  

48. K. Si, X. H. Ji, and H. Y. Jia, “Nonclassicality of photon-added squeezed vacuum states,” Chin. Phys. B 19, 064205 (2010). [CrossRef]  

49. S. Xu, X. X. Xu, C. J. Liu, H. L. Zhang, and L. Y. Hu, “Hermite polynomial excited squeezed vacuum state: Generation and nonclassical properties,” Optik 144, 664 (2017). [CrossRef]  

50. S. Y. Liu, Y. Z. Li, L. Y. Hu, J. H. Huang, X. X. Xu, and X. Y. Tao, “Nonclassical properties of Hermite polynomial excitation on squeezed vacuum and its decoherence in phase-sensitive reservoirs,” Laser Phys. Lett. 12, 045201 (2015). [CrossRef]  

51. C. Oh, S. Y. Lee, H. Nha, and H. Jeong, “Practical resources and measurements for lossy optical quantum metrology,” Phys. Rev. A 96, 062304 (2017). [CrossRef]  

52. 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, 752 (2017). [CrossRef]  

53. S. S. Szigeti, R. J. Lewis-Swan, and S. A. Haine, “Pumped-up SU (1, 1) interferometry,” Phys. Rev. Lett. 118, 150401 (2017). [CrossRef]  

54. B. E. Anderson, B. L. Schmittberger, P. Gupta, K. M. Jones, and P. D. Lett, “Optimal phase measurements with bright-and vacuum-seeded SU (1, 1) interferometers,” Phys. Rev. A 95, 063843 (2017). [CrossRef]  

55. X. X. Jing, J. Liu, W. Zhong, and X. G. Wang, “Quantum Fisher Information of Entangled Coherent States in a Lossy Mach-Zehnder Interferometer,” Commun. Theor. Phys. 61, 115 (2014). [CrossRef]  

56. M. Jarzyna and R. Demkowicz-Dobrzański, “Quantum interferometry with and without an external phase reference,” Phys. Rev. A 85, 011801 (2012). [CrossRef]  

57. X. Yu, X. Zhao, L. Y. Shen, Y. Y. Shao, J. Liu, and X. G. Wang, “Maximal quantum Fisher information for phase estimation without initial parity,” Opt. Express 26, 16292 (2018). [CrossRef]   [PubMed]  

58. R. Demkowicz-Dobrzański, M. Jarzyna, and J. Kołodyński, “Quantum limits in optical interferometry,” Prog. Optics (Elsevier, 2015), pp. 345-435. [CrossRef]  

59. G. Tóth and I. Apellaniz, “Quantum metrology from a quantum information science perspective,” J. Phys. A: Math. Theor. 47, 424006 (2014). [CrossRef]  

References

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  • |
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  1. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693 (1981).
    [Crossref]
  2. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
    [Crossref] [PubMed]
  3. M. Zwierz, C. A. Pérez-Delgado, and P. Kok, “General optimality of the Heisenberg limit for quantum metrology,” Phys. Rev. Lett. 105, 180402 (2010).
    [Crossref]
  4. V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222 (2011).
    [Crossref]
  5. J. Abadie, B. P. Abbott, R. Abbott, T. D. Abbott, M. Abernathy, C. Adams, R. Adhikari, C. Affeldt, B. Allen, and G. S. Allen, “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962 (2011).
    [Crossref]
  6. S. M. Roy and S. L. Braunstein, “Exponentially enhanced quantum metrology,” Phys. Rev. Lett. 100, 220501 (2008).
    [Crossref] [PubMed]
  7. J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777 (2012).
    [Crossref]
  8. S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled Fock states for robust quantum optical metrology, imaging, and sensing,” Phys. Rev. A 78, 063828 (2008).
    [Crossref]
  9. M. V. Chekhova and Z. Y. Ou, “Nonlinear interferometers in quantum optics,” Adv. Opt. Photonics 8, 104 (2016).
    [Crossref]
  10. M. Hillery and L. Mlodinow, “Interferometers and minimum-uncertainty states,” Phys. Rev. A 48, 1548 (1993).
    [Crossref] [PubMed]
  11. G. M. D’Ariano and M. G. A. Paris, “Lower bounds on phase sensitivity in ideal and feasible measurements,” Phys. Rev. A 49, 3022 (1994).
    [Crossref]
  12. M. Xiao, L. A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59, 278 (1987).
    [Crossref] [PubMed]
  13. P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-light-enhanced polarization interferometer,” Phys. Rev. Lett. 59, 2153 (1987).
    [Crossref] [PubMed]
  14. H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura, D. W. Berry, and T. C. Ralph, “Quantum-enhanced optical-phase tracking,” Science 337, 1514 (2012).
    [Crossref] [PubMed]
  15. Q. S. Tan, J. Q. Liao, X. G. Wang, and F. Nori, “Enhanced interferometry using squeezed thermal states and even or odd states,” Phys. Rev. A 89, 053822 (2014).
    [Crossref]
  16. A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733 (2000).
    [Crossref] [PubMed]
  17. Y. M. Zhang, X. W. Li, W. Yang, and G. R. Jin, “Quantum Fisher information of entangled coherent states in the presence of photon loss,” Phys. Rev. A 88, 043832 (2013).
    [Crossref]
  18. S. Boixo, A. Datta, M. J. Davis, S. T. Flammia, A. Shaji, and C. M. Caves, “Quantum metrology: dynamics versus entanglement,” Phys. Rev. Lett. 101, 040403 (2008).
    [Crossref] [PubMed]
  19. K. P. Seshadreesan, P. M. Anisimov, H. Lee, and J. P. Dowling, “Parity detection achieves the Heisenberg limit in interferometry with coherent mixed with squeezed vacuum light,” New J. Phys. 13, 083026 (2011).
    [Crossref]
  20. M. J. Holland and K. Burnett, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. 71, 1355 (1993).
    [Crossref] [PubMed]
  21. L. Pezzé and A. Smerzi, “Mach-Zehnder interferometry at the Heisenberg limit with coherent and squeezed-vacuum light,” Phys. Rev. Lett. 100, 073601 (2008).
    [Crossref] [PubMed]
  22. B. Yurke, S. L. McCall, and J. R. Klauder, “SU (2) and SU (1, 1) interferometers,” Phys. Rev. A 33, 4033 (1986).
    [Crossref]
  23. C. Sparaciari, S. Olivares, and M. G. A. Paris, “Gaussian-state interferometry with passive and active elements,” Phys. Rev. A 93, 023810 (2016).
    [Crossref]
  24. F. Hudelist, J. Kong, C. J. Liu, J. T. Jing, Z. Y. Ou, and W. P. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
    [Crossref] [PubMed]
  25. W. N. Plick, J. P. Dowling, and G. S. Agarwal, “Coherent-light-boosted, sub-shot noise, quantum interferometry,” New J. Phys. 12, 083014 (2010).
    [Crossref]
  26. A. M. Marino, N. V. Corzo Trejo, and P. D. Lett, “Effect of losses on the performance of an SU (1, 1) interferometer,” Phys. Rev. A 86, 023844 (2012).
    [Crossref]
  27. Z. Y. Ou, “Enhancement of the phase-measurement sensitivity beyond the standard quantum limit by a nonlinear interferometer,” Phys. Rev. A 85, 023815 (2012).
    [Crossref]
  28. C. Gross, T. Zibold, E. Nicklas, J. Estève, and M. K. Oberthaler, “Nonlinear atom interferometer surpasses classical precision limit,” Nature 464, 1165 (2010).
    [Crossref] [PubMed]
  29. J. Kong, F. Hudelist, Z. Y. Ou, and W. P. Zhang, “Cancellation of internal quantum noise of an amplifier by quantum correlation,” Phys. Rev. Lett. 111, 033608 (2013).
    [Crossref] [PubMed]
  30. B. Chen, C. Qiu, S. Y. Chen, J. X. Guo, L. Q. Chen, Z. Y. Ou, and W. P. Zhang, “Atom-Light Hybrid Interferometer,” Phys. Rev. Lett. 115, 043602 (2015).
    [Crossref] [PubMed]
  31. D. Li, B. T. Gard, Y. Gao, C. H. Yuan, W. P. 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, 063840 (2016).
    [Crossref]
  32. D. Li, C. H. Yuan, Z. Y. Ou, and W. P. Zhang, “The phase sensitivity of an SU (1, 1) interferometer with coherent and squeezed-vacuum light,” New J. Phys. 16, 073020 (2014).
    [Crossref]
  33. C. W. Helstrom, “Quantum detection and estimation theory,” J. Stat. Phys 1, 231 (1969).
    [Crossref]
  34. S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439 (1994).
    [Crossref] [PubMed]
  35. R. Carranza and C. C. Gerry, “Photon-subtracted two-mode squeezed vacuum states and applications to quantum optical interferometry,” J. Opt. Soc. Am. B 29, 2581 (2012).
    [Crossref]
  36. Y. Ouyang, S. Wang, and L. J. Zhang, “Quantum optical interferometry via the photon-added two-mode squeezed vacuum states,” J. Opt. Soc. Am. B 33, 1373 (2016).
    [Crossref]
  37. C. C. Gerry and J. Mimih, “Heisenberg-limited interferometry with pair coherent states and parity measurements,” Phys. Rev. A 82, 013831 (2010).
    [Crossref]
  38. R. Birrittella and C. C. Gerry, “Quantum optical interferometry via the mixing of coherent and photon-subtracted squeezed vacuum states of light,” J. Opt. Soc. Am. B 31, 586 (2014).
    [Crossref]
  39. Q. K. Gong, X. L. Hu, D. Li, C. H. Yuan, Z. Y. Ou, and W. P. Zhang, “Intramode-correlation-enhanced phase sensitivities in an SU (1, 1) interferometer,” Phys. Rev. A 96, 033809 (2017).
    [Crossref]
  40. L. Y. Hu and H. Y. Fan, “Nonclassicality of photon-added squeezed vacuum and its decoherence in thermal environment,” J. Mod. Optic 57, 1344 (2010).
    [Crossref]
  41. Z. Zhang and H. Fan, “Properties of states generated by excitations on a squeezed vacuum state,” Phys. Lett. A 165, 14 (1992).
    [Crossref]
  42. H. Y. Fan, X. G. Meng, and J. S. Wang, “New form of Legendre polynomials obtained by virtue of excited squeezed state and IWOP technique in quantum optics,” Commun. Theor. Phys. 46, 845 (2006).
    [Crossref]
  43. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integration Series and Products (Academic, 2014).
  44. M. Dakna, L. Knöll, and D. G. Welsch, “Photon-added state preparation via conditional measurement on a beam splitter,” Opt. Commun. 145, 309 (1998).
    [Crossref]
  45. A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660 (2004).
    [Crossref] [PubMed]
  46. A. Zavatta, S. Viciani, and M. Bellini, “Single-photon excitation of a coherent state: Catching the elementary step of stimulated light emission,” Phys. Rev. A 72, 023820 (2005).
    [Crossref]
  47. A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106 (2007).
    [Crossref]
  48. K. Si, X. H. Ji, and H. Y. Jia, “Nonclassicality of photon-added squeezed vacuum states,” Chin. Phys. B 19, 064205 (2010).
    [Crossref]
  49. S. Xu, X. X. Xu, C. J. Liu, H. L. Zhang, and L. Y. Hu, “Hermite polynomial excited squeezed vacuum state: Generation and nonclassical properties,” Optik 144, 664 (2017).
    [Crossref]
  50. S. Y. Liu, Y. Z. Li, L. Y. Hu, J. H. Huang, X. X. Xu, and X. Y. Tao, “Nonclassical properties of Hermite polynomial excitation on squeezed vacuum and its decoherence in phase-sensitive reservoirs,” Laser Phys. Lett. 12, 045201 (2015).
    [Crossref]
  51. C. Oh, S. Y. Lee, H. Nha, and H. Jeong, “Practical resources and measurements for lossy optical quantum metrology,” Phys. Rev. A 96, 062304 (2017).
    [Crossref]
  52. 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, 752 (2017).
    [Crossref]
  53. S. S. Szigeti, R. J. Lewis-Swan, and S. A. Haine, “Pumped-up SU (1, 1) interferometry,” Phys. Rev. Lett. 118, 150401 (2017).
    [Crossref]
  54. B. E. Anderson, B. L. Schmittberger, P. Gupta, K. M. Jones, and P. D. Lett, “Optimal phase measurements with bright-and vacuum-seeded SU (1, 1) interferometers,” Phys. Rev. A 95, 063843 (2017).
    [Crossref]
  55. X. X. Jing, J. Liu, W. Zhong, and X. G. Wang, “Quantum Fisher Information of Entangled Coherent States in a Lossy Mach-Zehnder Interferometer,” Commun. Theor. Phys. 61, 115 (2014).
    [Crossref]
  56. M. Jarzyna and R. Demkowicz-Dobrzański, “Quantum interferometry with and without an external phase reference,” Phys. Rev. A 85, 011801 (2012).
    [Crossref]
  57. X. Yu, X. Zhao, L. Y. Shen, Y. Y. Shao, J. Liu, and X. G. Wang, “Maximal quantum Fisher information for phase estimation without initial parity,” Opt. Express 26, 16292 (2018).
    [Crossref] [PubMed]
  58. R. Demkowicz-Dobrzański, M. Jarzyna, and J. Kołodyński, “Quantum limits in optical interferometry,” Prog. Optics (Elsevier, 2015), pp. 345-435.
    [Crossref]
  59. G. Tóth and I. Apellaniz, “Quantum metrology from a quantum information science perspective,” J. Phys. A: Math. Theor. 47, 424006 (2014).
    [Crossref]

2018 (1)

2017 (6)

S. Xu, X. X. Xu, C. J. Liu, H. L. Zhang, and L. Y. Hu, “Hermite polynomial excited squeezed vacuum state: Generation and nonclassical properties,” Optik 144, 664 (2017).
[Crossref]

C. Oh, S. Y. Lee, H. Nha, and H. Jeong, “Practical resources and measurements for lossy optical quantum metrology,” Phys. Rev. A 96, 062304 (2017).
[Crossref]

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, 752 (2017).
[Crossref]

S. S. Szigeti, R. J. Lewis-Swan, and S. A. Haine, “Pumped-up SU (1, 1) interferometry,” Phys. Rev. Lett. 118, 150401 (2017).
[Crossref]

B. E. Anderson, B. L. Schmittberger, P. Gupta, K. M. Jones, and P. D. Lett, “Optimal phase measurements with bright-and vacuum-seeded SU (1, 1) interferometers,” Phys. Rev. A 95, 063843 (2017).
[Crossref]

Q. K. Gong, X. L. Hu, D. Li, C. H. Yuan, Z. Y. Ou, and W. P. Zhang, “Intramode-correlation-enhanced phase sensitivities in an SU (1, 1) interferometer,” Phys. Rev. A 96, 033809 (2017).
[Crossref]

2016 (4)

M. V. Chekhova and Z. Y. Ou, “Nonlinear interferometers in quantum optics,” Adv. Opt. Photonics 8, 104 (2016).
[Crossref]

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Gaussian-state interferometry with passive and active elements,” Phys. Rev. A 93, 023810 (2016).
[Crossref]

D. Li, B. T. Gard, Y. Gao, C. H. Yuan, W. P. 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, 063840 (2016).
[Crossref]

Y. Ouyang, S. Wang, and L. J. Zhang, “Quantum optical interferometry via the photon-added two-mode squeezed vacuum states,” J. Opt. Soc. Am. B 33, 1373 (2016).
[Crossref]

2015 (2)

B. Chen, C. Qiu, S. Y. Chen, J. X. Guo, L. Q. Chen, Z. Y. Ou, and W. P. Zhang, “Atom-Light Hybrid Interferometer,” Phys. Rev. Lett. 115, 043602 (2015).
[Crossref] [PubMed]

S. Y. Liu, Y. Z. Li, L. Y. Hu, J. H. Huang, X. X. Xu, and X. Y. Tao, “Nonclassical properties of Hermite polynomial excitation on squeezed vacuum and its decoherence in phase-sensitive reservoirs,” Laser Phys. Lett. 12, 045201 (2015).
[Crossref]

2014 (6)

G. Tóth and I. Apellaniz, “Quantum metrology from a quantum information science perspective,” J. Phys. A: Math. Theor. 47, 424006 (2014).
[Crossref]

X. X. Jing, J. Liu, W. Zhong, and X. G. Wang, “Quantum Fisher Information of Entangled Coherent States in a Lossy Mach-Zehnder Interferometer,” Commun. Theor. Phys. 61, 115 (2014).
[Crossref]

R. Birrittella and C. C. Gerry, “Quantum optical interferometry via the mixing of coherent and photon-subtracted squeezed vacuum states of light,” J. Opt. Soc. Am. B 31, 586 (2014).
[Crossref]

D. Li, C. H. Yuan, Z. Y. Ou, and W. P. Zhang, “The phase sensitivity of an SU (1, 1) interferometer with coherent and squeezed-vacuum light,” New J. Phys. 16, 073020 (2014).
[Crossref]

F. Hudelist, J. Kong, C. J. Liu, J. T. Jing, Z. Y. Ou, and W. P. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
[Crossref] [PubMed]

Q. S. Tan, J. Q. Liao, X. G. Wang, and F. Nori, “Enhanced interferometry using squeezed thermal states and even or odd states,” Phys. Rev. A 89, 053822 (2014).
[Crossref]

2013 (2)

Y. M. Zhang, X. W. Li, W. Yang, and G. R. Jin, “Quantum Fisher information of entangled coherent states in the presence of photon loss,” Phys. Rev. A 88, 043832 (2013).
[Crossref]

J. Kong, F. Hudelist, Z. Y. Ou, and W. P. Zhang, “Cancellation of internal quantum noise of an amplifier by quantum correlation,” Phys. Rev. Lett. 111, 033608 (2013).
[Crossref] [PubMed]

2012 (6)

R. Carranza and C. C. Gerry, “Photon-subtracted two-mode squeezed vacuum states and applications to quantum optical interferometry,” J. Opt. Soc. Am. B 29, 2581 (2012).
[Crossref]

A. M. Marino, N. V. Corzo Trejo, and P. D. Lett, “Effect of losses on the performance of an SU (1, 1) interferometer,” Phys. Rev. A 86, 023844 (2012).
[Crossref]

Z. Y. Ou, “Enhancement of the phase-measurement sensitivity beyond the standard quantum limit by a nonlinear interferometer,” Phys. Rev. A 85, 023815 (2012).
[Crossref]

H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura, D. W. Berry, and T. C. Ralph, “Quantum-enhanced optical-phase tracking,” Science 337, 1514 (2012).
[Crossref] [PubMed]

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777 (2012).
[Crossref]

M. Jarzyna and R. Demkowicz-Dobrzański, “Quantum interferometry with and without an external phase reference,” Phys. Rev. A 85, 011801 (2012).
[Crossref]

2011 (3)

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222 (2011).
[Crossref]

J. Abadie, B. P. Abbott, R. Abbott, T. D. Abbott, M. Abernathy, C. Adams, R. Adhikari, C. Affeldt, B. Allen, and G. S. Allen, “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962 (2011).
[Crossref]

K. P. Seshadreesan, P. M. Anisimov, H. Lee, and J. P. Dowling, “Parity detection achieves the Heisenberg limit in interferometry with coherent mixed with squeezed vacuum light,” New J. Phys. 13, 083026 (2011).
[Crossref]

2010 (6)

C. Gross, T. Zibold, E. Nicklas, J. Estève, and M. K. Oberthaler, “Nonlinear atom interferometer surpasses classical precision limit,” Nature 464, 1165 (2010).
[Crossref] [PubMed]

W. N. Plick, J. P. Dowling, and G. S. Agarwal, “Coherent-light-boosted, sub-shot noise, quantum interferometry,” New J. Phys. 12, 083014 (2010).
[Crossref]

C. C. Gerry and J. Mimih, “Heisenberg-limited interferometry with pair coherent states and parity measurements,” Phys. Rev. A 82, 013831 (2010).
[Crossref]

M. Zwierz, C. A. Pérez-Delgado, and P. Kok, “General optimality of the Heisenberg limit for quantum metrology,” Phys. Rev. Lett. 105, 180402 (2010).
[Crossref]

L. Y. Hu and H. Y. Fan, “Nonclassicality of photon-added squeezed vacuum and its decoherence in thermal environment,” J. Mod. Optic 57, 1344 (2010).
[Crossref]

K. Si, X. H. Ji, and H. Y. Jia, “Nonclassicality of photon-added squeezed vacuum states,” Chin. Phys. B 19, 064205 (2010).
[Crossref]

2008 (4)

S. Boixo, A. Datta, M. J. Davis, S. T. Flammia, A. Shaji, and C. M. Caves, “Quantum metrology: dynamics versus entanglement,” Phys. Rev. Lett. 101, 040403 (2008).
[Crossref] [PubMed]

S. M. Roy and S. L. Braunstein, “Exponentially enhanced quantum metrology,” Phys. Rev. Lett. 100, 220501 (2008).
[Crossref] [PubMed]

S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled Fock states for robust quantum optical metrology, imaging, and sensing,” Phys. Rev. A 78, 063828 (2008).
[Crossref]

L. Pezzé and A. Smerzi, “Mach-Zehnder interferometry at the Heisenberg limit with coherent and squeezed-vacuum light,” Phys. Rev. Lett. 100, 073601 (2008).
[Crossref] [PubMed]

2007 (1)

A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106 (2007).
[Crossref]

2006 (2)

H. Y. Fan, X. G. Meng, and J. S. Wang, “New form of Legendre polynomials obtained by virtue of excited squeezed state and IWOP technique in quantum optics,” Commun. Theor. Phys. 46, 845 (2006).
[Crossref]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
[Crossref] [PubMed]

2005 (1)

A. Zavatta, S. Viciani, and M. Bellini, “Single-photon excitation of a coherent state: Catching the elementary step of stimulated light emission,” Phys. Rev. A 72, 023820 (2005).
[Crossref]

2004 (1)

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660 (2004).
[Crossref] [PubMed]

2000 (1)

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733 (2000).
[Crossref] [PubMed]

1998 (1)

M. Dakna, L. Knöll, and D. G. Welsch, “Photon-added state preparation via conditional measurement on a beam splitter,” Opt. Commun. 145, 309 (1998).
[Crossref]

1994 (2)

G. M. D’Ariano and M. G. A. Paris, “Lower bounds on phase sensitivity in ideal and feasible measurements,” Phys. Rev. A 49, 3022 (1994).
[Crossref]

S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439 (1994).
[Crossref] [PubMed]

1993 (2)

M. J. Holland and K. Burnett, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. 71, 1355 (1993).
[Crossref] [PubMed]

M. Hillery and L. Mlodinow, “Interferometers and minimum-uncertainty states,” Phys. Rev. A 48, 1548 (1993).
[Crossref] [PubMed]

1992 (1)

Z. Zhang and H. Fan, “Properties of states generated by excitations on a squeezed vacuum state,” Phys. Lett. A 165, 14 (1992).
[Crossref]

1987 (2)

M. Xiao, L. A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59, 278 (1987).
[Crossref] [PubMed]

P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-light-enhanced polarization interferometer,” Phys. Rev. Lett. 59, 2153 (1987).
[Crossref] [PubMed]

1986 (1)

B. Yurke, S. L. McCall, and J. R. Klauder, “SU (2) and SU (1, 1) interferometers,” Phys. Rev. A 33, 4033 (1986).
[Crossref]

1981 (1)

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693 (1981).
[Crossref]

1969 (1)

C. W. Helstrom, “Quantum detection and estimation theory,” J. Stat. Phys 1, 231 (1969).
[Crossref]

Abadie, J.

J. Abadie, B. P. Abbott, R. Abbott, T. D. Abbott, M. Abernathy, C. Adams, R. Adhikari, C. Affeldt, B. Allen, and G. S. Allen, “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962 (2011).
[Crossref]

Abbott, B. P.

J. Abadie, B. P. Abbott, R. Abbott, T. D. Abbott, M. Abernathy, C. Adams, R. Adhikari, C. Affeldt, B. Allen, and G. S. Allen, “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962 (2011).
[Crossref]

Abbott, R.

J. Abadie, B. P. Abbott, R. Abbott, T. D. Abbott, M. Abernathy, C. Adams, R. Adhikari, C. Affeldt, B. Allen, and G. S. Allen, “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962 (2011).
[Crossref]

Abbott, T. D.

J. Abadie, B. P. Abbott, R. Abbott, T. D. Abbott, M. Abernathy, C. Adams, R. Adhikari, C. Affeldt, B. Allen, and G. S. Allen, “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962 (2011).
[Crossref]

Abernathy, M.

J. Abadie, B. P. Abbott, R. Abbott, T. D. Abbott, M. Abernathy, C. Adams, R. Adhikari, C. Affeldt, B. Allen, and G. S. Allen, “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962 (2011).
[Crossref]

Abrams, D. S.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733 (2000).
[Crossref] [PubMed]

Adams, C.

J. Abadie, B. P. Abbott, R. Abbott, T. D. Abbott, M. Abernathy, C. Adams, R. Adhikari, C. Affeldt, B. Allen, and G. S. Allen, “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962 (2011).
[Crossref]

Adhikari, R.

J. Abadie, B. P. Abbott, R. Abbott, T. D. Abbott, M. Abernathy, C. Adams, R. Adhikari, C. Affeldt, B. Allen, and G. S. Allen, “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962 (2011).
[Crossref]

Affeldt, C.

J. Abadie, B. P. Abbott, R. Abbott, T. D. Abbott, M. Abernathy, C. Adams, R. Adhikari, C. Affeldt, B. Allen, and G. S. Allen, “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962 (2011).
[Crossref]

Agarwal, G. S.

W. N. Plick, J. P. Dowling, and G. S. Agarwal, “Coherent-light-boosted, sub-shot noise, quantum interferometry,” New J. Phys. 12, 083014 (2010).
[Crossref]

Allen, B.

J. Abadie, B. P. Abbott, R. Abbott, T. D. Abbott, M. Abernathy, C. Adams, R. Adhikari, C. Affeldt, B. Allen, and G. S. Allen, “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962 (2011).
[Crossref]

Allen, G. S.

J. Abadie, B. P. Abbott, R. Abbott, T. D. Abbott, M. Abernathy, C. Adams, R. Adhikari, C. Affeldt, B. Allen, and G. S. Allen, “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962 (2011).
[Crossref]

Anderson, B. E.

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, 752 (2017).
[Crossref]

B. E. Anderson, B. L. Schmittberger, P. Gupta, K. M. Jones, and P. D. Lett, “Optimal phase measurements with bright-and vacuum-seeded SU (1, 1) interferometers,” Phys. Rev. A 95, 063843 (2017).
[Crossref]

Anisimov, P. M.

K. P. Seshadreesan, P. M. Anisimov, H. Lee, and J. P. Dowling, “Parity detection achieves the Heisenberg limit in interferometry with coherent mixed with squeezed vacuum light,” New J. Phys. 13, 083026 (2011).
[Crossref]

Apellaniz, I.

G. Tóth and I. Apellaniz, “Quantum metrology from a quantum information science perspective,” J. Phys. A: Math. Theor. 47, 424006 (2014).
[Crossref]

Arao, H.

H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura, D. W. Berry, and T. C. Ralph, “Quantum-enhanced optical-phase tracking,” Science 337, 1514 (2012).
[Crossref] [PubMed]

Bellini, M.

A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106 (2007).
[Crossref]

A. Zavatta, S. Viciani, and M. Bellini, “Single-photon excitation of a coherent state: Catching the elementary step of stimulated light emission,” Phys. Rev. A 72, 023820 (2005).
[Crossref]

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660 (2004).
[Crossref] [PubMed]

Berry, D. W.

H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura, D. W. Berry, and T. C. Ralph, “Quantum-enhanced optical-phase tracking,” Science 337, 1514 (2012).
[Crossref] [PubMed]

Birrittella, R.

Boixo, S.

S. Boixo, A. Datta, M. J. Davis, S. T. Flammia, A. Shaji, and C. M. Caves, “Quantum metrology: dynamics versus entanglement,” Phys. Rev. Lett. 101, 040403 (2008).
[Crossref] [PubMed]

Boto, A. N.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733 (2000).
[Crossref] [PubMed]

Braunstein, S. L.

S. M. Roy and S. L. Braunstein, “Exponentially enhanced quantum metrology,” Phys. Rev. Lett. 100, 220501 (2008).
[Crossref] [PubMed]

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733 (2000).
[Crossref] [PubMed]

S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439 (1994).
[Crossref] [PubMed]

Burnett, K.

M. J. Holland and K. Burnett, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. 71, 1355 (1993).
[Crossref] [PubMed]

Carranza, R.

Caves, C. M.

S. Boixo, A. Datta, M. J. Davis, S. T. Flammia, A. Shaji, and C. M. Caves, “Quantum metrology: dynamics versus entanglement,” Phys. Rev. Lett. 101, 040403 (2008).
[Crossref] [PubMed]

S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439 (1994).
[Crossref] [PubMed]

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693 (1981).
[Crossref]

Chekhova, M. V.

M. V. Chekhova and Z. Y. Ou, “Nonlinear interferometers in quantum optics,” Adv. Opt. Photonics 8, 104 (2016).
[Crossref]

Chen, B.

B. Chen, C. Qiu, S. Y. Chen, J. X. Guo, L. Q. Chen, Z. Y. Ou, and W. P. Zhang, “Atom-Light Hybrid Interferometer,” Phys. Rev. Lett. 115, 043602 (2015).
[Crossref] [PubMed]

Chen, L. Q.

B. Chen, C. Qiu, S. Y. Chen, J. X. Guo, L. Q. Chen, Z. Y. Ou, and W. P. Zhang, “Atom-Light Hybrid Interferometer,” Phys. Rev. Lett. 115, 043602 (2015).
[Crossref] [PubMed]

Chen, S. Y.

B. Chen, C. Qiu, S. Y. Chen, J. X. Guo, L. Q. Chen, Z. Y. Ou, and W. P. Zhang, “Atom-Light Hybrid Interferometer,” Phys. Rev. Lett. 115, 043602 (2015).
[Crossref] [PubMed]

Chen, Z. B.

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777 (2012).
[Crossref]

D’Ariano, G. M.

G. M. D’Ariano and M. G. A. Paris, “Lower bounds on phase sensitivity in ideal and feasible measurements,” Phys. Rev. A 49, 3022 (1994).
[Crossref]

Dakna, M.

M. Dakna, L. Knöll, and D. G. Welsch, “Photon-added state preparation via conditional measurement on a beam splitter,” Opt. Commun. 145, 309 (1998).
[Crossref]

Datta, A.

S. Boixo, A. Datta, M. J. Davis, S. T. Flammia, A. Shaji, and C. M. Caves, “Quantum metrology: dynamics versus entanglement,” Phys. Rev. Lett. 101, 040403 (2008).
[Crossref] [PubMed]

Davis, M. J.

S. Boixo, A. Datta, M. J. Davis, S. T. Flammia, A. Shaji, and C. M. Caves, “Quantum metrology: dynamics versus entanglement,” Phys. Rev. Lett. 101, 040403 (2008).
[Crossref] [PubMed]

Demkowicz-Dobrzanski, R.

M. Jarzyna and R. Demkowicz-Dobrzański, “Quantum interferometry with and without an external phase reference,” Phys. Rev. A 85, 011801 (2012).
[Crossref]

R. Demkowicz-Dobrzański, M. Jarzyna, and J. Kołodyński, “Quantum limits in optical interferometry,” Prog. Optics (Elsevier, 2015), pp. 345-435.
[Crossref]

Dowling, J. P.

D. Li, B. T. Gard, Y. Gao, C. H. Yuan, W. P. 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, 063840 (2016).
[Crossref]

K. P. Seshadreesan, P. M. Anisimov, H. Lee, and J. P. Dowling, “Parity detection achieves the Heisenberg limit in interferometry with coherent mixed with squeezed vacuum light,” New J. Phys. 13, 083026 (2011).
[Crossref]

W. N. Plick, J. P. Dowling, and G. S. Agarwal, “Coherent-light-boosted, sub-shot noise, quantum interferometry,” New J. Phys. 12, 083014 (2010).
[Crossref]

S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled Fock states for robust quantum optical metrology, imaging, and sensing,” Phys. Rev. A 78, 063828 (2008).
[Crossref]

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733 (2000).
[Crossref] [PubMed]

Estève, J.

C. Gross, T. Zibold, E. Nicklas, J. Estève, and M. K. Oberthaler, “Nonlinear atom interferometer surpasses classical precision limit,” Nature 464, 1165 (2010).
[Crossref] [PubMed]

Fan, H.

Z. Zhang and H. Fan, “Properties of states generated by excitations on a squeezed vacuum state,” Phys. Lett. A 165, 14 (1992).
[Crossref]

Fan, H. Y.

L. Y. Hu and H. Y. Fan, “Nonclassicality of photon-added squeezed vacuum and its decoherence in thermal environment,” J. Mod. Optic 57, 1344 (2010).
[Crossref]

H. Y. Fan, X. G. Meng, and J. S. Wang, “New form of Legendre polynomials obtained by virtue of excited squeezed state and IWOP technique in quantum optics,” Commun. Theor. Phys. 46, 845 (2006).
[Crossref]

Flammia, S. T.

S. Boixo, A. Datta, M. J. Davis, S. T. Flammia, A. Shaji, and C. M. Caves, “Quantum metrology: dynamics versus entanglement,” Phys. Rev. Lett. 101, 040403 (2008).
[Crossref] [PubMed]

Gao, Y.

D. Li, B. T. Gard, Y. Gao, C. H. Yuan, W. P. 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, 063840 (2016).
[Crossref]

Gard, B. T.

D. Li, B. T. Gard, Y. Gao, C. H. Yuan, W. P. 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, 063840 (2016).
[Crossref]

Gerry, C. C.

Giovannetti, V.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222 (2011).
[Crossref]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
[Crossref] [PubMed]

Gong, Q. K.

Q. K. Gong, X. L. Hu, D. Li, C. H. Yuan, Z. Y. Ou, and W. P. Zhang, “Intramode-correlation-enhanced phase sensitivities in an SU (1, 1) interferometer,” Phys. Rev. A 96, 033809 (2017).
[Crossref]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integration Series and Products (Academic, 2014).

Grangier, P.

P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-light-enhanced polarization interferometer,” Phys. Rev. Lett. 59, 2153 (1987).
[Crossref] [PubMed]

Gross, C.

C. Gross, T. Zibold, E. Nicklas, J. Estève, and M. K. Oberthaler, “Nonlinear atom interferometer surpasses classical precision limit,” Nature 464, 1165 (2010).
[Crossref] [PubMed]

Guo, J. X.

B. Chen, C. Qiu, S. Y. Chen, J. X. Guo, L. Q. Chen, Z. Y. Ou, and W. P. Zhang, “Atom-Light Hybrid Interferometer,” Phys. Rev. Lett. 115, 043602 (2015).
[Crossref] [PubMed]

Gupta, P.

B. E. Anderson, B. L. Schmittberger, P. Gupta, K. M. Jones, and P. D. Lett, “Optimal phase measurements with bright-and vacuum-seeded SU (1, 1) interferometers,” Phys. Rev. A 95, 063843 (2017).
[Crossref]

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, 752 (2017).
[Crossref]

Haine, S. A.

S. S. Szigeti, R. J. Lewis-Swan, and S. A. Haine, “Pumped-up SU (1, 1) interferometry,” Phys. Rev. Lett. 118, 150401 (2017).
[Crossref]

Helstrom, C. W.

C. W. Helstrom, “Quantum detection and estimation theory,” J. Stat. Phys 1, 231 (1969).
[Crossref]

Hermann-Avigliano, C.

Hillery, M.

M. Hillery and L. Mlodinow, “Interferometers and minimum-uncertainty states,” Phys. Rev. A 48, 1548 (1993).
[Crossref] [PubMed]

Holland, M. J.

M. J. Holland and K. Burnett, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. 71, 1355 (1993).
[Crossref] [PubMed]

Horrom, T.

Hu, L. Y.

S. Xu, X. X. Xu, C. J. Liu, H. L. Zhang, and L. Y. Hu, “Hermite polynomial excited squeezed vacuum state: Generation and nonclassical properties,” Optik 144, 664 (2017).
[Crossref]

S. Y. Liu, Y. Z. Li, L. Y. Hu, J. H. Huang, X. X. Xu, and X. Y. Tao, “Nonclassical properties of Hermite polynomial excitation on squeezed vacuum and its decoherence in phase-sensitive reservoirs,” Laser Phys. Lett. 12, 045201 (2015).
[Crossref]

L. Y. Hu and H. Y. Fan, “Nonclassicality of photon-added squeezed vacuum and its decoherence in thermal environment,” J. Mod. Optic 57, 1344 (2010).
[Crossref]

Hu, X. L.

Q. K. Gong, X. L. Hu, D. Li, C. H. Yuan, Z. Y. Ou, and W. P. Zhang, “Intramode-correlation-enhanced phase sensitivities in an SU (1, 1) interferometer,” Phys. Rev. A 96, 033809 (2017).
[Crossref]

Huang, J. H.

S. Y. Liu, Y. Z. Li, L. Y. Hu, J. H. Huang, X. X. Xu, and X. Y. Tao, “Nonclassical properties of Hermite polynomial excitation on squeezed vacuum and its decoherence in phase-sensitive reservoirs,” Laser Phys. Lett. 12, 045201 (2015).
[Crossref]

Hudelist, F.

F. Hudelist, J. Kong, C. J. Liu, J. T. Jing, Z. Y. Ou, and W. P. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
[Crossref] [PubMed]

J. Kong, F. Hudelist, Z. Y. Ou, and W. P. Zhang, “Cancellation of internal quantum noise of an amplifier by quantum correlation,” Phys. Rev. Lett. 111, 033608 (2013).
[Crossref] [PubMed]

Huver, S. D.

S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled Fock states for robust quantum optical metrology, imaging, and sensing,” Phys. Rev. A 78, 063828 (2008).
[Crossref]

Iwasawa, K.

H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura, D. W. Berry, and T. C. Ralph, “Quantum-enhanced optical-phase tracking,” Science 337, 1514 (2012).
[Crossref] [PubMed]

Jarzyna, M.

M. Jarzyna and R. Demkowicz-Dobrzański, “Quantum interferometry with and without an external phase reference,” Phys. Rev. A 85, 011801 (2012).
[Crossref]

R. Demkowicz-Dobrzański, M. Jarzyna, and J. Kołodyński, “Quantum limits in optical interferometry,” Prog. Optics (Elsevier, 2015), pp. 345-435.
[Crossref]

Jeong, H.

C. Oh, S. Y. Lee, H. Nha, and H. Jeong, “Practical resources and measurements for lossy optical quantum metrology,” Phys. Rev. A 96, 062304 (2017).
[Crossref]

Ji, X. H.

K. Si, X. H. Ji, and H. Y. Jia, “Nonclassicality of photon-added squeezed vacuum states,” Chin. Phys. B 19, 064205 (2010).
[Crossref]

Jia, H. Y.

K. Si, X. H. Ji, and H. Y. Jia, “Nonclassicality of photon-added squeezed vacuum states,” Chin. Phys. B 19, 064205 (2010).
[Crossref]

Jin, G. R.

Y. M. Zhang, X. W. Li, W. Yang, and G. R. Jin, “Quantum Fisher information of entangled coherent states in the presence of photon loss,” Phys. Rev. A 88, 043832 (2013).
[Crossref]

Jing, J. T.

F. Hudelist, J. Kong, C. J. Liu, J. T. Jing, Z. Y. Ou, and W. P. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
[Crossref] [PubMed]

Jing, X. X.

X. X. Jing, J. Liu, W. Zhong, and X. G. Wang, “Quantum Fisher Information of Entangled Coherent States in a Lossy Mach-Zehnder Interferometer,” Commun. Theor. Phys. 61, 115 (2014).
[Crossref]

Jones, K. M.

B. E. Anderson, B. L. Schmittberger, P. Gupta, K. M. Jones, and P. D. Lett, “Optimal phase measurements with bright-and vacuum-seeded SU (1, 1) interferometers,” Phys. Rev. A 95, 063843 (2017).
[Crossref]

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, 752 (2017).
[Crossref]

Kimble, H. J.

M. Xiao, L. A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59, 278 (1987).
[Crossref] [PubMed]

Klauder, J. R.

B. Yurke, S. L. McCall, and J. R. Klauder, “SU (2) and SU (1, 1) interferometers,” Phys. Rev. A 33, 4033 (1986).
[Crossref]

Knöll, L.

M. Dakna, L. Knöll, and D. G. Welsch, “Photon-added state preparation via conditional measurement on a beam splitter,” Opt. Commun. 145, 309 (1998).
[Crossref]

Kok, P.

M. Zwierz, C. A. Pérez-Delgado, and P. Kok, “General optimality of the Heisenberg limit for quantum metrology,” Phys. Rev. Lett. 105, 180402 (2010).
[Crossref]

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733 (2000).
[Crossref] [PubMed]

Kolodynski, J.

R. Demkowicz-Dobrzański, M. Jarzyna, and J. Kołodyński, “Quantum limits in optical interferometry,” Prog. Optics (Elsevier, 2015), pp. 345-435.
[Crossref]

Kong, J.

F. Hudelist, J. Kong, C. J. Liu, J. T. Jing, Z. Y. Ou, and W. P. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
[Crossref] [PubMed]

J. Kong, F. Hudelist, Z. Y. Ou, and W. P. Zhang, “Cancellation of internal quantum noise of an amplifier by quantum correlation,” Phys. Rev. Lett. 111, 033608 (2013).
[Crossref] [PubMed]

LaPorta, A.

P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-light-enhanced polarization interferometer,” Phys. Rev. Lett. 59, 2153 (1987).
[Crossref] [PubMed]

Lee, H.

D. Li, B. T. Gard, Y. Gao, C. H. Yuan, W. P. 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, 063840 (2016).
[Crossref]

K. P. Seshadreesan, P. M. Anisimov, H. Lee, and J. P. Dowling, “Parity detection achieves the Heisenberg limit in interferometry with coherent mixed with squeezed vacuum light,” New J. Phys. 13, 083026 (2011).
[Crossref]

Lee, S. Y.

C. Oh, S. Y. Lee, H. Nha, and H. Jeong, “Practical resources and measurements for lossy optical quantum metrology,” Phys. Rev. A 96, 062304 (2017).
[Crossref]

Lett, P. D.

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, 752 (2017).
[Crossref]

B. E. Anderson, B. L. Schmittberger, P. Gupta, K. M. Jones, and P. D. Lett, “Optimal phase measurements with bright-and vacuum-seeded SU (1, 1) interferometers,” Phys. Rev. A 95, 063843 (2017).
[Crossref]

A. M. Marino, N. V. Corzo Trejo, and P. D. Lett, “Effect of losses on the performance of an SU (1, 1) interferometer,” Phys. Rev. A 86, 023844 (2012).
[Crossref]

Lewis-Swan, R. J.

S. S. Szigeti, R. J. Lewis-Swan, and S. A. Haine, “Pumped-up SU (1, 1) interferometry,” Phys. Rev. Lett. 118, 150401 (2017).
[Crossref]

Li, D.

Q. K. Gong, X. L. Hu, D. Li, C. H. Yuan, Z. Y. Ou, and W. P. Zhang, “Intramode-correlation-enhanced phase sensitivities in an SU (1, 1) interferometer,” Phys. Rev. A 96, 033809 (2017).
[Crossref]

D. Li, B. T. Gard, Y. Gao, C. H. Yuan, W. P. 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, 063840 (2016).
[Crossref]

D. Li, C. H. Yuan, Z. Y. Ou, and W. P. Zhang, “The phase sensitivity of an SU (1, 1) interferometer with coherent and squeezed-vacuum light,” New J. Phys. 16, 073020 (2014).
[Crossref]

Li, X. W.

Y. M. Zhang, X. W. Li, W. Yang, and G. R. Jin, “Quantum Fisher information of entangled coherent states in the presence of photon loss,” Phys. Rev. A 88, 043832 (2013).
[Crossref]

Li, Y. Z.

S. Y. Liu, Y. Z. Li, L. Y. Hu, J. H. Huang, X. X. Xu, and X. Y. Tao, “Nonclassical properties of Hermite polynomial excitation on squeezed vacuum and its decoherence in phase-sensitive reservoirs,” Laser Phys. Lett. 12, 045201 (2015).
[Crossref]

Liao, J. Q.

Q. S. Tan, J. Q. Liao, X. G. Wang, and F. Nori, “Enhanced interferometry using squeezed thermal states and even or odd states,” Phys. Rev. A 89, 053822 (2014).
[Crossref]

Liu, C. J.

S. Xu, X. X. Xu, C. J. Liu, H. L. Zhang, and L. Y. Hu, “Hermite polynomial excited squeezed vacuum state: Generation and nonclassical properties,” Optik 144, 664 (2017).
[Crossref]

F. Hudelist, J. Kong, C. J. Liu, J. T. Jing, Z. Y. Ou, and W. P. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
[Crossref] [PubMed]

Liu, J.

X. Yu, X. Zhao, L. Y. Shen, Y. Y. Shao, J. Liu, and X. G. Wang, “Maximal quantum Fisher information for phase estimation without initial parity,” Opt. Express 26, 16292 (2018).
[Crossref] [PubMed]

X. X. Jing, J. Liu, W. Zhong, and X. G. Wang, “Quantum Fisher Information of Entangled Coherent States in a Lossy Mach-Zehnder Interferometer,” Commun. Theor. Phys. 61, 115 (2014).
[Crossref]

Liu, S. Y.

S. Y. Liu, Y. Z. Li, L. Y. Hu, J. H. Huang, X. X. Xu, and X. Y. Tao, “Nonclassical properties of Hermite polynomial excitation on squeezed vacuum and its decoherence in phase-sensitive reservoirs,” Laser Phys. Lett. 12, 045201 (2015).
[Crossref]

Lloyd, S.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222 (2011).
[Crossref]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
[Crossref] [PubMed]

Lu, C. Y.

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777 (2012).
[Crossref]

Maccone, L.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222 (2011).
[Crossref]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
[Crossref] [PubMed]

Marino, A. M.

A. M. Marino, N. V. Corzo Trejo, and P. D. Lett, “Effect of losses on the performance of an SU (1, 1) interferometer,” Phys. Rev. A 86, 023844 (2012).
[Crossref]

McCall, S. L.

B. Yurke, S. L. McCall, and J. R. Klauder, “SU (2) and SU (1, 1) interferometers,” Phys. Rev. A 33, 4033 (1986).
[Crossref]

Meng, X. G.

H. Y. Fan, X. G. Meng, and J. S. Wang, “New form of Legendre polynomials obtained by virtue of excited squeezed state and IWOP technique in quantum optics,” Commun. Theor. Phys. 46, 845 (2006).
[Crossref]

Mimih, J.

C. C. Gerry and J. Mimih, “Heisenberg-limited interferometry with pair coherent states and parity measurements,” Phys. Rev. A 82, 013831 (2010).
[Crossref]

Mlodinow, L.

M. Hillery and L. Mlodinow, “Interferometers and minimum-uncertainty states,” Phys. Rev. A 48, 1548 (1993).
[Crossref] [PubMed]

Nakane, D.

H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura, D. W. Berry, and T. C. Ralph, “Quantum-enhanced optical-phase tracking,” Science 337, 1514 (2012).
[Crossref] [PubMed]

Nha, H.

C. Oh, S. Y. Lee, H. Nha, and H. Jeong, “Practical resources and measurements for lossy optical quantum metrology,” Phys. Rev. A 96, 062304 (2017).
[Crossref]

Nicklas, E.

C. Gross, T. Zibold, E. Nicklas, J. Estève, and M. K. Oberthaler, “Nonlinear atom interferometer surpasses classical precision limit,” Nature 464, 1165 (2010).
[Crossref] [PubMed]

Nori, F.

Q. S. Tan, J. Q. Liao, X. G. Wang, and F. Nori, “Enhanced interferometry using squeezed thermal states and even or odd states,” Phys. Rev. A 89, 053822 (2014).
[Crossref]

Oberthaler, M. K.

C. Gross, T. Zibold, E. Nicklas, J. Estève, and M. K. Oberthaler, “Nonlinear atom interferometer surpasses classical precision limit,” Nature 464, 1165 (2010).
[Crossref] [PubMed]

Oh, C.

C. Oh, S. Y. Lee, H. Nha, and H. Jeong, “Practical resources and measurements for lossy optical quantum metrology,” Phys. Rev. A 96, 062304 (2017).
[Crossref]

Ohki, K.

H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura, D. W. Berry, and T. C. Ralph, “Quantum-enhanced optical-phase tracking,” Science 337, 1514 (2012).
[Crossref] [PubMed]

Olivares, S.

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Gaussian-state interferometry with passive and active elements,” Phys. Rev. A 93, 023810 (2016).
[Crossref]

Ou, Z. Y.

Q. K. Gong, X. L. Hu, D. Li, C. H. Yuan, Z. Y. Ou, and W. P. Zhang, “Intramode-correlation-enhanced phase sensitivities in an SU (1, 1) interferometer,” Phys. Rev. A 96, 033809 (2017).
[Crossref]

M. V. Chekhova and Z. Y. Ou, “Nonlinear interferometers in quantum optics,” Adv. Opt. Photonics 8, 104 (2016).
[Crossref]

B. Chen, C. Qiu, S. Y. Chen, J. X. Guo, L. Q. Chen, Z. Y. Ou, and W. P. Zhang, “Atom-Light Hybrid Interferometer,” Phys. Rev. Lett. 115, 043602 (2015).
[Crossref] [PubMed]

D. Li, C. H. Yuan, Z. Y. Ou, and W. P. Zhang, “The phase sensitivity of an SU (1, 1) interferometer with coherent and squeezed-vacuum light,” New J. Phys. 16, 073020 (2014).
[Crossref]

F. Hudelist, J. Kong, C. J. Liu, J. T. Jing, Z. Y. Ou, and W. P. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
[Crossref] [PubMed]

J. Kong, F. Hudelist, Z. Y. Ou, and W. P. Zhang, “Cancellation of internal quantum noise of an amplifier by quantum correlation,” Phys. Rev. Lett. 111, 033608 (2013).
[Crossref] [PubMed]

Z. Y. Ou, “Enhancement of the phase-measurement sensitivity beyond the standard quantum limit by a nonlinear interferometer,” Phys. Rev. A 85, 023815 (2012).
[Crossref]

Ouyang, Y.

Pan, J. W.

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777 (2012).
[Crossref]

Parigi, V.

A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106 (2007).
[Crossref]

Paris, M. G. A.

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Gaussian-state interferometry with passive and active elements,” Phys. Rev. A 93, 023810 (2016).
[Crossref]

G. M. D’Ariano and M. G. A. Paris, “Lower bounds on phase sensitivity in ideal and feasible measurements,” Phys. Rev. A 49, 3022 (1994).
[Crossref]

Pérez-Delgado, C. A.

M. Zwierz, C. A. Pérez-Delgado, and P. Kok, “General optimality of the Heisenberg limit for quantum metrology,” Phys. Rev. Lett. 105, 180402 (2010).
[Crossref]

Pezzé, L.

L. Pezzé and A. Smerzi, “Mach-Zehnder interferometry at the Heisenberg limit with coherent and squeezed-vacuum light,” Phys. Rev. Lett. 100, 073601 (2008).
[Crossref] [PubMed]

Plick, W. N.

W. N. Plick, J. P. Dowling, and G. S. Agarwal, “Coherent-light-boosted, sub-shot noise, quantum interferometry,” New J. Phys. 12, 083014 (2010).
[Crossref]

Qiu, C.

B. Chen, C. Qiu, S. Y. Chen, J. X. Guo, L. Q. Chen, Z. Y. Ou, and W. P. Zhang, “Atom-Light Hybrid Interferometer,” Phys. Rev. Lett. 115, 043602 (2015).
[Crossref] [PubMed]

Ralph, T. C.

H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura, D. W. Berry, and T. C. Ralph, “Quantum-enhanced optical-phase tracking,” Science 337, 1514 (2012).
[Crossref] [PubMed]

Roy, S. M.

S. M. Roy and S. L. Braunstein, “Exponentially enhanced quantum metrology,” Phys. Rev. Lett. 100, 220501 (2008).
[Crossref] [PubMed]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integration Series and Products (Academic, 2014).

Schmittberger, B. L.

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, 752 (2017).
[Crossref]

B. E. Anderson, B. L. Schmittberger, P. Gupta, K. M. Jones, and P. D. Lett, “Optimal phase measurements with bright-and vacuum-seeded SU (1, 1) interferometers,” Phys. Rev. A 95, 063843 (2017).
[Crossref]

Seshadreesan, K. P.

K. P. Seshadreesan, P. M. Anisimov, H. Lee, and J. P. Dowling, “Parity detection achieves the Heisenberg limit in interferometry with coherent mixed with squeezed vacuum light,” New J. Phys. 13, 083026 (2011).
[Crossref]

Shaji, A.

S. Boixo, A. Datta, M. J. Davis, S. T. Flammia, A. Shaji, and C. M. Caves, “Quantum metrology: dynamics versus entanglement,” Phys. Rev. Lett. 101, 040403 (2008).
[Crossref] [PubMed]

Shao, Y. Y.

Shen, L. Y.

Si, K.

K. Si, X. H. Ji, and H. Y. Jia, “Nonclassicality of photon-added squeezed vacuum states,” Chin. Phys. B 19, 064205 (2010).
[Crossref]

Slusher, R. E.

P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-light-enhanced polarization interferometer,” Phys. Rev. Lett. 59, 2153 (1987).
[Crossref] [PubMed]

Smerzi, A.

L. Pezzé and A. Smerzi, “Mach-Zehnder interferometry at the Heisenberg limit with coherent and squeezed-vacuum light,” Phys. Rev. Lett. 100, 073601 (2008).
[Crossref] [PubMed]

Sparaciari, C.

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Gaussian-state interferometry with passive and active elements,” Phys. Rev. A 93, 023810 (2016).
[Crossref]

Szigeti, S. S.

S. S. Szigeti, R. J. Lewis-Swan, and S. A. Haine, “Pumped-up SU (1, 1) interferometry,” Phys. Rev. Lett. 118, 150401 (2017).
[Crossref]

Takeda, S.

H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura, D. W. Berry, and T. C. Ralph, “Quantum-enhanced optical-phase tracking,” Science 337, 1514 (2012).
[Crossref] [PubMed]

Tan, Q. S.

Q. S. Tan, J. Q. Liao, X. G. Wang, and F. Nori, “Enhanced interferometry using squeezed thermal states and even or odd states,” Phys. Rev. A 89, 053822 (2014).
[Crossref]

Tao, X. Y.

S. Y. Liu, Y. Z. Li, L. Y. Hu, J. H. Huang, X. X. Xu, and X. Y. Tao, “Nonclassical properties of Hermite polynomial excitation on squeezed vacuum and its decoherence in phase-sensitive reservoirs,” Laser Phys. Lett. 12, 045201 (2015).
[Crossref]

Tóth, G.

G. Tóth and I. Apellaniz, “Quantum metrology from a quantum information science perspective,” J. Phys. A: Math. Theor. 47, 424006 (2014).
[Crossref]

Trejo, N. V. Corzo

A. M. Marino, N. V. Corzo Trejo, and P. D. Lett, “Effect of losses on the performance of an SU (1, 1) interferometer,” Phys. Rev. A 86, 023844 (2012).
[Crossref]

Tsumura, K.

H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura, D. W. Berry, and T. C. Ralph, “Quantum-enhanced optical-phase tracking,” Science 337, 1514 (2012).
[Crossref] [PubMed]

Viciani, S.

A. Zavatta, S. Viciani, and M. Bellini, “Single-photon excitation of a coherent state: Catching the elementary step of stimulated light emission,” Phys. Rev. A 72, 023820 (2005).
[Crossref]

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660 (2004).
[Crossref] [PubMed]

Wang, J. S.

H. Y. Fan, X. G. Meng, and J. S. Wang, “New form of Legendre polynomials obtained by virtue of excited squeezed state and IWOP technique in quantum optics,” Commun. Theor. Phys. 46, 845 (2006).
[Crossref]

Wang, S.

Wang, X. G.

X. Yu, X. Zhao, L. Y. Shen, Y. Y. Shao, J. Liu, and X. G. Wang, “Maximal quantum Fisher information for phase estimation without initial parity,” Opt. Express 26, 16292 (2018).
[Crossref] [PubMed]

X. X. Jing, J. Liu, W. Zhong, and X. G. Wang, “Quantum Fisher Information of Entangled Coherent States in a Lossy Mach-Zehnder Interferometer,” Commun. Theor. Phys. 61, 115 (2014).
[Crossref]

Q. S. Tan, J. Q. Liao, X. G. Wang, and F. Nori, “Enhanced interferometry using squeezed thermal states and even or odd states,” Phys. Rev. A 89, 053822 (2014).
[Crossref]

Weinfurter, H.

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777 (2012).
[Crossref]

Welsch, D. G.

M. Dakna, L. Knöll, and D. G. Welsch, “Photon-added state preparation via conditional measurement on a beam splitter,” Opt. Commun. 145, 309 (1998).
[Crossref]

Wheatley, T. A.

H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura, D. W. Berry, and T. C. Ralph, “Quantum-enhanced optical-phase tracking,” Science 337, 1514 (2012).
[Crossref] [PubMed]

Wildfeuer, C. F.

S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled Fock states for robust quantum optical metrology, imaging, and sensing,” Phys. Rev. A 78, 063828 (2008).
[Crossref]

Williams, C. P.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733 (2000).
[Crossref] [PubMed]

Wu, L. A.

M. Xiao, L. A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59, 278 (1987).
[Crossref] [PubMed]

Xiao, M.

M. Xiao, L. A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59, 278 (1987).
[Crossref] [PubMed]

Xu, S.

S. Xu, X. X. Xu, C. J. Liu, H. L. Zhang, and L. Y. Hu, “Hermite polynomial excited squeezed vacuum state: Generation and nonclassical properties,” Optik 144, 664 (2017).
[Crossref]

Xu, X. X.

S. Xu, X. X. Xu, C. J. Liu, H. L. Zhang, and L. Y. Hu, “Hermite polynomial excited squeezed vacuum state: Generation and nonclassical properties,” Optik 144, 664 (2017).
[Crossref]

S. Y. Liu, Y. Z. Li, L. Y. Hu, J. H. Huang, X. X. Xu, and X. Y. Tao, “Nonclassical properties of Hermite polynomial excitation on squeezed vacuum and its decoherence in phase-sensitive reservoirs,” Laser Phys. Lett. 12, 045201 (2015).
[Crossref]

Yang, W.

Y. M. Zhang, X. W. Li, W. Yang, and G. R. Jin, “Quantum Fisher information of entangled coherent states in the presence of photon loss,” Phys. Rev. A 88, 043832 (2013).
[Crossref]

Yonezawa, H.

H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura, D. W. Berry, and T. C. Ralph, “Quantum-enhanced optical-phase tracking,” Science 337, 1514 (2012).
[Crossref] [PubMed]

Yu, X.

Yuan, C. H.

Q. K. Gong, X. L. Hu, D. Li, C. H. Yuan, Z. Y. Ou, and W. P. Zhang, “Intramode-correlation-enhanced phase sensitivities in an SU (1, 1) interferometer,” Phys. Rev. A 96, 033809 (2017).
[Crossref]

D. Li, B. T. Gard, Y. Gao, C. H. Yuan, W. P. 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, 063840 (2016).
[Crossref]

D. Li, C. H. Yuan, Z. Y. Ou, and W. P. Zhang, “The phase sensitivity of an SU (1, 1) interferometer with coherent and squeezed-vacuum light,” New J. Phys. 16, 073020 (2014).
[Crossref]

Yurke, B.

P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-light-enhanced polarization interferometer,” Phys. Rev. Lett. 59, 2153 (1987).
[Crossref] [PubMed]

B. Yurke, S. L. McCall, and J. R. Klauder, “SU (2) and SU (1, 1) interferometers,” Phys. Rev. A 33, 4033 (1986).
[Crossref]

Zavatta, A.

A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106 (2007).
[Crossref]

A. Zavatta, S. Viciani, and M. Bellini, “Single-photon excitation of a coherent state: Catching the elementary step of stimulated light emission,” Phys. Rev. A 72, 023820 (2005).
[Crossref]

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660 (2004).
[Crossref] [PubMed]

Zeilinger, A.

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777 (2012).
[Crossref]

Zhang, H. L.

S. Xu, X. X. Xu, C. J. Liu, H. L. Zhang, and L. Y. Hu, “Hermite polynomial excited squeezed vacuum state: Generation and nonclassical properties,” Optik 144, 664 (2017).
[Crossref]

Zhang, L. J.

Zhang, W. P.

Q. K. Gong, X. L. Hu, D. Li, C. H. Yuan, Z. Y. Ou, and W. P. Zhang, “Intramode-correlation-enhanced phase sensitivities in an SU (1, 1) interferometer,” Phys. Rev. A 96, 033809 (2017).
[Crossref]

D. Li, B. T. Gard, Y. Gao, C. H. Yuan, W. P. 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, 063840 (2016).
[Crossref]

B. Chen, C. Qiu, S. Y. Chen, J. X. Guo, L. Q. Chen, Z. Y. Ou, and W. P. Zhang, “Atom-Light Hybrid Interferometer,” Phys. Rev. Lett. 115, 043602 (2015).
[Crossref] [PubMed]

D. Li, C. H. Yuan, Z. Y. Ou, and W. P. Zhang, “The phase sensitivity of an SU (1, 1) interferometer with coherent and squeezed-vacuum light,” New J. Phys. 16, 073020 (2014).
[Crossref]

F. Hudelist, J. Kong, C. J. Liu, J. T. Jing, Z. Y. Ou, and W. P. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
[Crossref] [PubMed]

J. Kong, F. Hudelist, Z. Y. Ou, and W. P. Zhang, “Cancellation of internal quantum noise of an amplifier by quantum correlation,” Phys. Rev. Lett. 111, 033608 (2013).
[Crossref] [PubMed]

Zhang, Y. M.

Y. M. Zhang, X. W. Li, W. Yang, and G. R. Jin, “Quantum Fisher information of entangled coherent states in the presence of photon loss,” Phys. Rev. A 88, 043832 (2013).
[Crossref]

Zhang, Z.

Z. Zhang and H. Fan, “Properties of states generated by excitations on a squeezed vacuum state,” Phys. Lett. A 165, 14 (1992).
[Crossref]

Zhao, X.

Zhong, W.

X. X. Jing, J. Liu, W. Zhong, and X. G. Wang, “Quantum Fisher Information of Entangled Coherent States in a Lossy Mach-Zehnder Interferometer,” Commun. Theor. Phys. 61, 115 (2014).
[Crossref]

Zibold, T.

C. Gross, T. Zibold, E. Nicklas, J. Estève, and M. K. Oberthaler, “Nonlinear atom interferometer surpasses classical precision limit,” Nature 464, 1165 (2010).
[Crossref] [PubMed]

Zukowski, M.

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777 (2012).
[Crossref]

Zwierz, M.

M. Zwierz, C. A. Pérez-Delgado, and P. Kok, “General optimality of the Heisenberg limit for quantum metrology,” Phys. Rev. Lett. 105, 180402 (2010).
[Crossref]

Adv. Opt. Photonics (1)

M. V. Chekhova and Z. Y. Ou, “Nonlinear interferometers in quantum optics,” Adv. Opt. Photonics 8, 104 (2016).
[Crossref]

Chin. Phys. B (1)

K. Si, X. H. Ji, and H. Y. Jia, “Nonclassicality of photon-added squeezed vacuum states,” Chin. Phys. B 19, 064205 (2010).
[Crossref]

Commun. Theor. Phys. (2)

H. Y. Fan, X. G. Meng, and J. S. Wang, “New form of Legendre polynomials obtained by virtue of excited squeezed state and IWOP technique in quantum optics,” Commun. Theor. Phys. 46, 845 (2006).
[Crossref]

X. X. Jing, J. Liu, W. Zhong, and X. G. Wang, “Quantum Fisher Information of Entangled Coherent States in a Lossy Mach-Zehnder Interferometer,” Commun. Theor. Phys. 61, 115 (2014).
[Crossref]

J. Mod. Optic (1)

L. Y. Hu and H. Y. Fan, “Nonclassicality of photon-added squeezed vacuum and its decoherence in thermal environment,” J. Mod. Optic 57, 1344 (2010).
[Crossref]

J. Opt. Soc. Am. B (3)

J. Phys. A: Math. Theor. (1)

G. Tóth and I. Apellaniz, “Quantum metrology from a quantum information science perspective,” J. Phys. A: Math. Theor. 47, 424006 (2014).
[Crossref]

J. Stat. Phys (1)

C. W. Helstrom, “Quantum detection and estimation theory,” J. Stat. Phys 1, 231 (1969).
[Crossref]

Laser Phys. Lett. (1)

S. Y. Liu, Y. Z. Li, L. Y. Hu, J. H. Huang, X. X. Xu, and X. Y. Tao, “Nonclassical properties of Hermite polynomial excitation on squeezed vacuum and its decoherence in phase-sensitive reservoirs,” Laser Phys. Lett. 12, 045201 (2015).
[Crossref]

Nat. Commun. (1)

F. Hudelist, J. Kong, C. J. Liu, J. T. Jing, Z. Y. Ou, and W. P. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
[Crossref] [PubMed]

Nat. Photonics (1)

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222 (2011).
[Crossref]

Nat. Phys. (1)

J. Abadie, B. P. Abbott, R. Abbott, T. D. Abbott, M. Abernathy, C. Adams, R. Adhikari, C. Affeldt, B. Allen, and G. S. Allen, “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962 (2011).
[Crossref]

Nature (1)

C. Gross, T. Zibold, E. Nicklas, J. Estève, and M. K. Oberthaler, “Nonlinear atom interferometer surpasses classical precision limit,” Nature 464, 1165 (2010).
[Crossref] [PubMed]

New J. Phys. (3)

W. N. Plick, J. P. Dowling, and G. S. Agarwal, “Coherent-light-boosted, sub-shot noise, quantum interferometry,” New J. Phys. 12, 083014 (2010).
[Crossref]

K. P. Seshadreesan, P. M. Anisimov, H. Lee, and J. P. Dowling, “Parity detection achieves the Heisenberg limit in interferometry with coherent mixed with squeezed vacuum light,” New J. Phys. 13, 083026 (2011).
[Crossref]

D. Li, C. H. Yuan, Z. Y. Ou, and W. P. Zhang, “The phase sensitivity of an SU (1, 1) interferometer with coherent and squeezed-vacuum light,” New J. Phys. 16, 073020 (2014).
[Crossref]

Opt. Commun. (1)

M. Dakna, L. Knöll, and D. G. Welsch, “Photon-added state preparation via conditional measurement on a beam splitter,” Opt. Commun. 145, 309 (1998).
[Crossref]

Opt. Express (1)

Optica (1)

Optik (1)

S. Xu, X. X. Xu, C. J. Liu, H. L. Zhang, and L. Y. Hu, “Hermite polynomial excited squeezed vacuum state: Generation and nonclassical properties,” Optik 144, 664 (2017).
[Crossref]

Phys. Lett. A (1)

Z. Zhang and H. Fan, “Properties of states generated by excitations on a squeezed vacuum state,” Phys. Lett. A 165, 14 (1992).
[Crossref]

Phys. Rev. A (17)

A. Zavatta, S. Viciani, and M. Bellini, “Single-photon excitation of a coherent state: Catching the elementary step of stimulated light emission,” Phys. Rev. A 72, 023820 (2005).
[Crossref]

A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106 (2007).
[Crossref]

C. Oh, S. Y. Lee, H. Nha, and H. Jeong, “Practical resources and measurements for lossy optical quantum metrology,” Phys. Rev. A 96, 062304 (2017).
[Crossref]

B. E. Anderson, B. L. Schmittberger, P. Gupta, K. M. Jones, and P. D. Lett, “Optimal phase measurements with bright-and vacuum-seeded SU (1, 1) interferometers,” Phys. Rev. A 95, 063843 (2017).
[Crossref]

M. Jarzyna and R. Demkowicz-Dobrzański, “Quantum interferometry with and without an external phase reference,” Phys. Rev. A 85, 011801 (2012).
[Crossref]

Y. M. Zhang, X. W. Li, W. Yang, and G. R. Jin, “Quantum Fisher information of entangled coherent states in the presence of photon loss,” Phys. Rev. A 88, 043832 (2013).
[Crossref]

Q. S. Tan, J. Q. Liao, X. G. Wang, and F. Nori, “Enhanced interferometry using squeezed thermal states and even or odd states,” Phys. Rev. A 89, 053822 (2014).
[Crossref]

M. Hillery and L. Mlodinow, “Interferometers and minimum-uncertainty states,” Phys. Rev. A 48, 1548 (1993).
[Crossref] [PubMed]

G. M. D’Ariano and M. G. A. Paris, “Lower bounds on phase sensitivity in ideal and feasible measurements,” Phys. Rev. A 49, 3022 (1994).
[Crossref]

A. M. Marino, N. V. Corzo Trejo, and P. D. Lett, “Effect of losses on the performance of an SU (1, 1) interferometer,” Phys. Rev. A 86, 023844 (2012).
[Crossref]

Z. Y. Ou, “Enhancement of the phase-measurement sensitivity beyond the standard quantum limit by a nonlinear interferometer,” Phys. Rev. A 85, 023815 (2012).
[Crossref]

B. Yurke, S. L. McCall, and J. R. Klauder, “SU (2) and SU (1, 1) interferometers,” Phys. Rev. A 33, 4033 (1986).
[Crossref]

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Gaussian-state interferometry with passive and active elements,” Phys. Rev. A 93, 023810 (2016).
[Crossref]

S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled Fock states for robust quantum optical metrology, imaging, and sensing,” Phys. Rev. A 78, 063828 (2008).
[Crossref]

D. Li, B. T. Gard, Y. Gao, C. H. Yuan, W. P. 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, 063840 (2016).
[Crossref]

Q. K. Gong, X. L. Hu, D. Li, C. H. Yuan, Z. Y. Ou, and W. P. Zhang, “Intramode-correlation-enhanced phase sensitivities in an SU (1, 1) interferometer,” Phys. Rev. A 96, 033809 (2017).
[Crossref]

C. C. Gerry and J. Mimih, “Heisenberg-limited interferometry with pair coherent states and parity measurements,” Phys. Rev. A 82, 013831 (2010).
[Crossref]

Phys. Rev. D (1)

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693 (1981).
[Crossref]

Phys. Rev. Lett. (13)

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
[Crossref] [PubMed]

M. Zwierz, C. A. Pérez-Delgado, and P. Kok, “General optimality of the Heisenberg limit for quantum metrology,” Phys. Rev. Lett. 105, 180402 (2010).
[Crossref]

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733 (2000).
[Crossref] [PubMed]

S. Boixo, A. Datta, M. J. Davis, S. T. Flammia, A. Shaji, and C. M. Caves, “Quantum metrology: dynamics versus entanglement,” Phys. Rev. Lett. 101, 040403 (2008).
[Crossref] [PubMed]

M. J. Holland and K. Burnett, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. 71, 1355 (1993).
[Crossref] [PubMed]

L. Pezzé and A. Smerzi, “Mach-Zehnder interferometry at the Heisenberg limit with coherent and squeezed-vacuum light,” Phys. Rev. Lett. 100, 073601 (2008).
[Crossref] [PubMed]

M. Xiao, L. A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59, 278 (1987).
[Crossref] [PubMed]

P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-light-enhanced polarization interferometer,” Phys. Rev. Lett. 59, 2153 (1987).
[Crossref] [PubMed]

S. M. Roy and S. L. Braunstein, “Exponentially enhanced quantum metrology,” Phys. Rev. Lett. 100, 220501 (2008).
[Crossref] [PubMed]

S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439 (1994).
[Crossref] [PubMed]

J. Kong, F. Hudelist, Z. Y. Ou, and W. P. Zhang, “Cancellation of internal quantum noise of an amplifier by quantum correlation,” Phys. Rev. Lett. 111, 033608 (2013).
[Crossref] [PubMed]

B. Chen, C. Qiu, S. Y. Chen, J. X. Guo, L. Q. Chen, Z. Y. Ou, and W. P. Zhang, “Atom-Light Hybrid Interferometer,” Phys. Rev. Lett. 115, 043602 (2015).
[Crossref] [PubMed]

S. S. Szigeti, R. J. Lewis-Swan, and S. A. Haine, “Pumped-up SU (1, 1) interferometry,” Phys. Rev. Lett. 118, 150401 (2017).
[Crossref]

Rev. Mod. Phys. (1)

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777 (2012).
[Crossref]

Science (2)

H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura, D. W. Berry, and T. C. Ralph, “Quantum-enhanced optical-phase tracking,” Science 337, 1514 (2012).
[Crossref] [PubMed]

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660 (2004).
[Crossref] [PubMed]

Other (2)

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integration Series and Products (Academic, 2014).

R. Demkowicz-Dobrzański, M. Jarzyna, and J. Kołodyński, “Quantum limits in optical interferometry,” Prog. Optics (Elsevier, 2015), pp. 345-435.
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 Schematic diagram of an SU(1,1) interferometer. The two input beams are in a coherent state and a photon-added squeezed vacuum state, respectively. OPA : optical parametric amplifier; Da and Db: detectors.
Fig. 2
Fig. 2 Phase sensitivity based on the intensity detection as a function of ϕ with g=1, r =1, and α=1. The solid lines correspond to the detection method I, while the dashed lines are for the detection method II.
Fig. 3
Fig. 3 Phase sensitivity based on the intensity detection as a function of (a) g with α=1 and r =1, (b) r with g=1 and α=1. The solid lines correspond to the detection method I, while the dashed lines are for the detection method II.
Fig. 4
Fig. 4 The total average photon number NT as a function of (a) g with α=1 and r =1. (b) r with α=1 and g=1.
Fig. 5
Fig. 5 (a) Quantum Fisher information Fm versus gain factor g for α=1 and r =1. Phase sensitivity ΔϕF as a function of (b) g with α=1 and r =1.(c) r with α=1 and g=1.(d) α with r =1 and g=1.
Fig. 6
Fig. 6 Pase sensitivity based on the intensity detection against g for different m, (a) m=0, (b) m=1, (c) m=2, (d) m=3. The subscript 1(2) of m corresponds to the detection method I(II). The orange dashed line is for the shot-noise limit, the purple dashed line is for the QCRB, the cyan dashed line is for the Heisenberg limit, where α=1 and r =1.
Fig. 7
Fig. 7 The same as Fig. 6 but α = 0.
Fig. 8
Fig. 8 The phase sensitivity Δϕ as a function of g with (a) α = (tanh 2g)er/2 and r =0.2, (b) α=0 and r =1, (c) α = (tanh 2g)er/2 and r =0, (d) α=1 and r =0. The subscript 1(2) of m corresponds to the intensity detection method I (II).

Equations (21)

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( a 2 b 2 + ) = S ( a 0 b 0 + ) ,
S = S O P A 2 S ϕ S O P A 1 ,
S O P A 1 = ( cosh g 1 e i θ 1 sinh g 1 e i θ 1 sinh g 1 cosh g 1 ) ,
S ϕ = ( e i ϕ 0 0 1 ) ,
S O P A 2 = ( cosh g 2 e i θ 2 sinh g 2 e i θ 2 sinh g 2 cosh g 2 ) ,
a 2 = ( e i ϕ cosh 2 g sinh 2 g ) a 0 [ e i θ 1 ( e i ϕ 1 ) sinh g cosh g ] b 0 + ,
b 2 + = [ e i θ 1 ( e i ϕ 1 ) sinh g cosh g ] a 0 + ( cosh 2 g e i ϕ sinh 2 g ) b 0 + .
| r , m b = N m b + m | r , 0 b ,
N m 2 = m ! cosh m r P m ( cosh r ) ,
N = A a N a + A b N b ,
Δ 2 ϕ = N 2 N 2 | ϕ N | 2 .
( Δ 2 ϕ ) I = 1 8 ( 1 + n ¯ m + α 2 ) 2 { 8 [ 1 + B m + 2 n ¯ m n ¯ m 2 + 2 ( 1 + A m + n ¯ m ) α 2 + 4 A m α 2 csch 2 2 g ] + csch 4 2 g [ 8 α 2 csc 2 ϕ 2 + [ 3 B m 1 + 2 n ¯ m 3 n ¯ m 2 2 ( n ¯ m + A m 1 ) α 2 + 4 ( α 2 B m n ¯ m + n ¯ m 2 ) cosh 4 g + ( 1 + B m + 2 n ¯ m n ¯ m 2 + 2 ( 1 + A m + n ¯ m ) α 2 ) cosh 8 g ] sec 2 ϕ 2 ] } ,
( Δ 2 ϕ ) I I = 1 8 sinh 4 2 g ( e 2 i ϕ 1 ) 2 ( 1 + n ¯ m + α 2 ) 2 { 2 e 2 i ϕ [ 9 B m 7 + 2 n ¯ m 9 n ¯ m 2 + 2 ( 1 + A m 7 n ¯ m ) α 2 + ( 3 + 3 B m + 6 n ¯ m 3 n ¯ m 2 + ( 6 10 A m + 6 n ¯ m ) α 2 ) cos 2 ϕ ] 4 ( e 2 i ϕ 1 ) 2 [ 1 + B m + 2 n ¯ m n ¯ m 2 + 2 ( n ¯ m + 1 A m ) α 2 ] cosh 4 g + [ 1 + B m + 2 n ¯ m n ¯ m 2 + 2 ( 1 + n ¯ m + A m ) α 2 ] [ 8 e 2 i ϕ cos ϕ + ( e i ϕ 1 ) 4 cosh 8 g ] } ,
n ¯ m = r , m | b + b | r , m b = N m 2 N m + 1 2 1 ,
A m = r , m | b 2 | r , m b = N m 2 coth r [ N m + 1 2 ( m + 1 ) N m 2 ] ,
B m = r , m | b + 2 b 2 | r , m b = N m 2 ( N m + 2 2 4 N m + 1 2 ) + 2 .
N T , m = cosh 2 g ( 1 + α 2 + n ¯ m ) 1 .
F = 4 [ ψ ϕ | ψ ϕ | ψ ϕ | ψ ϕ | 2 ] ,
F = 4 Δ 2 n a ,
F m = 4 [ α 2 cosh 4 g + ( B m + n ¯ m n ¯ m 2 ) sinh 4 g ] + [ 1 + n ¯ m + ( 2 A m + 1 + 2 n ¯ m ) α 2 ] sinh 2 2 g .
Δ ϕ F 1 v F m ,

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