Improving the phase sensitivity of an SU(1,1) interferometer with photon-added squeezed vacuum light

Li-Li Guo; Ya-Fei Yu; Zhi-Ming Zhang

doi:10.1364/OE.26.029099

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 U_OPA(ξ) = exp[−ξa⁺b ⁺ + ξ^∗ab] [23], here a and b are the annihilation operators for the two modes and the coupling constant $ξ = g e^{i θ_{g}}$ 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 $U_{ϕ} = e^{i ϕ a^{+} a}$ , then the two beams recombine in the second OPA. The relation between the output operators and the input operators is given by [25, 26]

(\begin{matrix} a_{2} \\ b_{2}^{+} \end{matrix}) = S (\begin{matrix} a_{0} \\ b_{0}^{+} \end{matrix}),

where

S = S_{O P A 2} S_{ϕ} S_{O P A 1},

with

S_{O P A 1} = (\begin{matrix} \cosh g_{1} & - e^{i θ_{1}} \sinh g_{1} \\ - e^{- i θ_{1}} \sinh g_{1} & \cosh g_{1} \end{matrix}),

S_{ϕ} = (\begin{matrix} e^{i ϕ} & 0 \\ 0 & 1 \end{matrix}),

S_{O P A 2} = (\begin{matrix} \cosh g_{2} & - e^{i θ_{2}} \sinh g_{2} \\ - e^{- i θ_{2}} \sinh g_{2} & \cosh g_{2} \end{matrix}),

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 (θ₂ − θ₁ = π) and the same gain factor (g₁ = g₂ = g).

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; D_a and D_b: detectors.

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

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}^{+} .

We consider an SU(1,1) interferometer with a coherent state ${| α 〉}_{a} (α = | α | e^{i θ_{α}})$ and an m-photon-added squeezed vacuum state |r, m〉_b as inputs. The m-PA-SVS is defined as [41, 42]

{| r, m 〉}_{b} = N_{m} b^{+ m} {| r, 0 〉}_{b},

where |r, 0〉_b = S_b(r ) |0〉_b is the squeezed vacuum state,

S_{b} (r) = \exp [\frac{r}{2} (b^{+ 2} - b^{2})]

is the single-mode squeezing operator with r the squeezing parameter, and the normalization constant N_m is given by [40, 41]

N_{m}^{- 2} = m! \cosh^{m} r P_{m} (\cosh r),

where P_m 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 = A_{a} N_{a} + A_{b} N_{b},

where

N_{a} = a_{2}^{+} a_{2}

and

N_{b} = b_{2}^{+} b_{2}

are the photon-number operators at the output ports a and b, respectively. A_a (A_b) has a value of 1 or 0 depending on whether the detector D_a (D_b) 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]

Δ^{2} ϕ = \frac{〈 N^{2} 〉 - {〈 N 〉}^{2}}{{| \partial_{ϕ} 〈 N 〉 |}^{2}} .

In this paper we consider two kinds of intensity detection: I. A_a = 1, A_b = 0, i.e. only detector D_a is on; II. A_a = 1, A_b = 1, i.e. both detectors D_a and D_b are on. In following calculations, for convenience, we take the phase θ_α = 0 and θ₁ = 0. For an ideal SU(1,1) interferometer with inputting a coherent light |α〉_a and an m-photon-added squeezed vacuum light |r, m 〉_b, we obtain

\begin{matrix} {(Δ^{2} ϕ)}_{I} = \frac{1}{8 {(1 + {\bar{n}}_{m} + α^{2})}^{2}} {- 8 [1 + B_{m} + 2 {\bar{n}}_{m} - {\bar{n}}_{m}^{2} + 2 (1 + A_{m} + {\bar{n}}_{m}) α^{2} \\ + 4 A_{m} α^{2} {csch}^{2} 2 g] + {csch}^{4} 2 g [8 α^{2} \csc^{2} \frac{ϕ}{2} + [3 B_{m} - 1 + 2 {\bar{n}}_{m} - 3 {\bar{n}}_{m}^{2} \\ - 2 ({\bar{n}}_{m} + A_{m} - 1) α^{2} + 4 (α^{2} - B_{m} - {\bar{n}}_{m} + {\bar{n}}_{m}^{2}) \cosh 4 g + (1 + B_{m} + 2 {\bar{n}}_{m} \\ - {\bar{n}}_{m}^{2} + 2 (1 + A_{m} + {\bar{n}}_{m}) α^{2}) \cosh 8 g] \sec^{2} \frac{ϕ}{2}]}, \end{matrix}

for detection method I, and

\begin{matrix} {(Δ^{2} ϕ)}_{I I} = \frac{- 1}{8 \sinh^{4} 2 g {(e^{2 i ϕ} - 1)}^{2} {(1 + {\bar{n}}_{m} + α^{2})}^{2}} {2 e^{2 i ϕ} [9 B_{m} - 7 + 2 {\bar{n}}_{m} - 9 {\bar{n}}_{m}^{2} + 2 (1 + A_{m} \\ - 7 {\bar{n}}_{m}) α^{2} + (3 + 3 B_{m} + 6 {\bar{n}}_{m} - 3 {\bar{n}}_{m}^{2} + (6 - 10 A_{m} + 6 {\bar{n}}_{m}) α^{2}) \cos 2 ϕ] \\ - 4 {(e^{2 i ϕ} - 1)}^{2} [1 + B_{m} + 2 {\bar{n}}_{m} - {\bar{n}}_{m}^{2} + 2 ({\bar{n}}_{m} + 1 - A_{m}) α^{2}] \cosh 4 g + [1 + B_{m} \\ + 2 {\bar{n}}_{m} - {\bar{n}}_{m}^{2} + 2 (1 + {\bar{n}}_{m} + A_{m}) α^{2}] [8 e^{2 i ϕ} \cos ϕ + {(e^{i ϕ} - 1)}^{4} \cosh 8 g]}, \end{matrix}

for detection method II, in which [40, 41]

{\bar{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 .

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.

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).

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 N_T is, the smaller Δϕ will be (the higher the phase sensitivity). In our scheme, $N_{T} = 〈 a_{1}^{+} a_{1} + b_{1}^{+} b_{1} 〉$ [22]. For different m, we have

N_{T, m} = \cosh 2 g (1 + α^{2} + {\bar{n}}_{m}) - 1 .

For the case of m = 0, the total mean photon number is N_T_,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 N_T as functions of the gain factor g and the squeezing parameter r, respectively, with m as a parameter. It can be clearly seen that N_T 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 N_T inside the interferometer, and in turn, leads to the decrease of Δϕ.

Fig. 4 The total average photon number N_T 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 [〈 ψ_{ϕ}^{'} | ψ_{ϕ}^{'} 〉 - {| 〈 ψ_{ϕ}^{'} | ψ_{ϕ}^{'} 〉 |}^{2}],

where |ψ_ϕ〉 = U_ϕU_OPA |ψ_in〉 is the state vector just before the second OPA, and

| ψ_{ϕ}^{'} 〉 = \partial_{ϕ} ψ_{ϕ}

. Then the QFI can be written as [57]

F = 4 〈 Δ^{2} n_{a} 〉,

where

〈 Δ^{2} n_{a} 〉 = 〈 ψ_{i n} | {(a_{1}^{+} a_{1})}^{2} | ψ_{i n} 〉 - 〈 ψ_{i n} | {(a_{1}^{+} a_{1})}^{2} {| ψ_{i n} 〉}^{2}

. In the present work, the input state |ψ_in〉 = |α〉_a ⊗ |r, m〉_b then for different m, we can obtain the QFI F_m as

F_{m} = 4 [α^{2} \cosh^{4} g + (B_{m} + {\bar{n}}_{m} - {\bar{n}}_{m}^{2}) \sinh^{4} g] + [1 + {\bar{n}}_{m} + (2 A_{m} + 1 + 2 {\bar{n}}_{m}) α^{2}] \sinh^{2} 2 g .

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]

Δ ϕ_{F} \geq \frac{1}{\sqrt{v F_{m}}},

where v is the number of trials and we take v = 1. From Eq. (21), we can see that the larger F_m is, the smaller the lower bound of Δϕ_F 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 Δϕ_F, 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.

Fig. 5 (a) Quantum Fisher information F_m versus gain factor g for α=1 and r =1. Phase sensitivity Δϕ_F 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 Δϕ_HL and the ultimate phase precision Δϕ_F. The corresponding HL is an important indicator to evaluate the performance of an interferometer, which is inversely proportional to the total average photon number N_T inside the interferometer, i.e. $Δ ϕ_{H L} = \frac{1}{N_{T}}$ . Similarly, we can obtain the shot-noise limit $Δ ϕ_{S N L} = \frac{1}{\sqrt{N_{T}}}$ .

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.

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 Δϕ_HL < Δϕ < Δϕ_SNL 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.

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.

Fig. 8 The phase sensitivity Δϕ as a function of g with (a) α = (tanh 2g)e^r/2 and r =0.2, (b) α=0 and r =1, (c) α = (tanh 2g)e^r/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).

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Improving the phase sensitivity of an SU(1,1) interferometer with photon-added squeezed vacuum light

Abstract

1. Introduction

2. Model

3. Phase sensitivity based on intensity detection

4. Phase sensitivity based on QFI and QCRB

5. Comparison and discussion

5.1. Comparison with theoretical limits

5.2. Comparison between different detection methods

6. Conclusion

Funding

References

Cited By

Figures (8)

Equations (21)

Optics Express