Phase sensitivity with a coherent beam and twin beams via intensity difference detection

Jun Liu(刘俊), Tao Shao(邵涛), Chenlu Li(李晨露), Minyang Zhang(张敏洋),Youyou Hu(胡友友), Dongxu Chen(陈东旭), and Dong Wei(卫栋)

1School of Science,Jiangsu University of Science and Technology,Zhenjiang 212003,China

2Quantum Information Research Center,Shangrao Normal University,Shangrao 334001,China

3School of Physics,Xi’an Jiaotong University,Xi’an 710049,China

Keywords: Mach–Zehnder interferometer,phase sensitivity,quantum squeezing

1.Introduction

Interferometers play an important role in the field of precision measurement.[1–11]One of the most typical interferometers is the Mach–Zehnder interferometer (MZI).A general MZI is composed of two beam splitters (BS) and is used to measure phase shift variations in the two paths.The measurement sensitivity of such phase shifts can not surpasswhen the inputs are the combination of a coherent state and an vacuum state, whereNis the total photon number of the inputs, andis named as the shot noise limit (SNL) or the standard quantum limit.[12]

In order to beat SNL, non-classical states such as the single-mode squeezed vacuum state and two-mode squeezed vacuum state are employed.[13–16]Then the phase sensitivity can reach 1/N, which is also named as the Heisenberg limit(HL).When inputs are Fock state, N00N state and entangled coherent state, the phase sensitivity can also reach sub-SNL and HL.[3,17,18]However, there are many potential problems.For the N00N state,this input state has been shown to be very sensitive to losses, and the maximum photon numberNremains very low in experiments.

Recently, due to the good performance in phase estimation, a coherent state and a single-mode squeezed vacuum state have been investigated by many groups.[10]Caveset al.claimed that when one of the inputs is a coherent state,the optimal state of the other input is the single-mode squeezed vacuum state,[15]which has been employed in LIGO.[16]When the inputs are the combination of a single-mode squeezed vacuum state and a coherent state, several detection methods are proposed,such as intensity detection,balanced homodyne detection and parity detection.[8]Different detection methods can lead to different optimal phase sensitivities.In order to achieve the optimal phase sensitivity,all the possible measurement strategies need to be taken into account,which is impossible.Fortunately, quantum Fisher information (QFI) and its related quantum Cram´er–Rao bound (QCRB) are introduced to find the optimal theoretical phase sensitivity without considering the specific measurement strategies.

The QFI is an effective tool in the estimation process when the inputs are the coherent state, Fock state, singlemode squeezed vacuum state, and single-mode squeezed coherent state,etc.[19–28]However,with an external phase reference beam in the detection process, the QFI can be different,which is observed by Jarzynaet al.[19]In addition,Takeokaet al.used a phase-averaging method to solve this problem.[23]Later, Youet al.claimed that the two-parameter phase estimation and the phase-averaging method is equivalent with specific inputs.[24]They also pointed out that it needs to take the reference beam into consideration when a phase reference beam is employed.Then,the detection strategy with external beams has aroused much concern.

Recently, squeezing and entanglement-assisted input states are proposed.[17,29–33]Under the condition of the squeezing-assisted input,[17,29]the intensity difference detection with an external power reference beam is employed.In the detection process, the conjugate beam offers an external power reference and the phase sensitivity can reach sub-SNL.However, they only consider the condition that one of the inputs is the vacuum beam and the advantage of the external power reference beam is not shown.Meanwhile, the optimal phase sensitivity can only reach sub-SNL.In this paper,we focus on the phase sensitivity with one coherent beam plus one of the bright entangled twin beams(BETB).Moreover,we aim to show that when the inputs of the MZI are a coherent beam plus one of the BETB,the optimal phase sensitivity can reach sub-HL and approach QCRB.

This paper is structured as follows.In Section 2, the scheme with intensity difference detection is introduced.The QFI and the QCRB are introduced.In Section 3, we analyze how the phase sensitivity is affected by the factors.The impacts of the detection efficiencies of the photon detectors are also studied.Then, detailed comparisons are shown in Section 4.Finally,the conclusions are drawn in Section 5.

2.The proposed model and phase sensitivity estimation

2.1.The scheme and operator transformation

In the first part of Fig.1, we show the generation process of the BETB,which is named as the two-mode squeezed process.[34–37]This two-mode squeezed process can be accomplished by the FWM experiments.[38–41]The FWM transformation can be expressed as

Fig.1.The scheme for phase sensitivity measurement based on the Mach–Zehnder interferometer (MZI).For the MZI, the inputs are one of the bright entangled twin beams and one coherent beam.The intensity difference is employed for the measurement process.M: mirror;FWM: four-wave mixing; BS: beam splitter; PD: photon detector; B:beam block;dashed line: the vacuum beam.

2.2.Phase sensitivity estimation with intensity difference detection

According to the error propagation formula, the phase sensitivity Δφas the uncertainty in estimating the phase shiftφis

The intensity difference detection signal can be expressed asThen the slope can be expressed as

while the variance of intensity difference is

The photon number of the inputs is given by

Therefore,the SNL and HL of the scenario in Fig.1 isand 1/NB.

2.3.The QFI and QCRB in two-parameter phase estimation

For the parameter estimation process, the QCRB is(Δφ)2≥F(φ)−1.The inputs in Fig.1 are pure states.Under this constraint,the QFIF(φ)is

and∂φψ=∂ψ/∂φ,ψis the state after the phase shift.In order to avoid the potential unavailable measurement strategy,twoparameter phase estimation is necessary.As shown in Fig.1,there are two phasesφ1andφ2in the two arms of the MZI,and we define the two-parameter QFI as

The QCRB can be better with a largerF.Then the QFIFcan be maximized withF−+=F+−=0.

3.Analysis of the phase sensitivity

3.1.Results

According to Eqs.(3)and(11),figure 2 shows the phase sensitivity versus phase shift and phase difference withr=0.65,Na0=0.1, andNc=0.5.The lower value Δφrepresents the better phase sensitivity.In this scenario,with one of the BETB entering the MZI and the other one being employed for detection, the phase sensitivity can beat HL in Fig.2(a),and the optimal phase sensitivity approaches the QCRB.In the inset of Fig.2(a),it is apparent that the optimal phase sensitivity can approach QCRB.In Fig.2(b), with the variation of the phase difference, the phase sensitivity can be optimal withφd=kπ(kis an integer).Whenφd=kπ+π/2,both the QCRB and the optimal phase sensitivity with intensity difference detection become the worst.The optimal phase sensitivity with intensity difference detection can only approach the QCRB whenφd=kπ.Under this constraint,in the following part,we takeφd=0 for simplification.

Fig.2.Phase sensitivity versus phase shift (a) and phase difference(b).Others parameters are r=0.65, Na0 =0.1, and Nc =0.5.In (a),φd =0.BETB: bright entangled twin beams; SNL: shot noise limit;HL:Heisenberg limit;QCRB:quantum Cram´er–Rao bound.The inset of(a)is the zoom of the phase sensitivity and it shows that the optimal phase sensitivity can not achieve the QCRB.

As shown in Fig.3, the phase sensitivities vary with the squeezing parameterr, photon numberNa0, andNc.WhenNa0=0.1 andNc=0.5, the phase sensitivity and the QCRB can beat HL withr ≤0.8 in Fig.3(a).Whenrincreases, the phase sensitivity with intensity difference detection and the QCRB can only reach sub-HL.The optimal phase sensitivity always approaches QCRB with the increase ofr.In Fig.3(b),the optimal phase sensitivity and the QCRB beat HL whenNa0is less than 0.2.WhenNa0becomes larger,they can reach sub-SNL.The optimal phase sensitivity approaches SNL whenNa0≥1.Figure 3(c) shows that both the optimal phase sensitivity and QCRB can reach sub-HL whenNc ≤0.9.They become worse than HL whenNcbecomes larger.In Fig.3,only the photon numberNB≥1 is considered.

Fig.3.Phase sensitivity versus parametric strength r(a),photon number Na0(b),photon number Nc(c),with r=0.65,Na0=0.1,and Nc=0.5.The other parameters are the same as those in Fig.2.

Fig.4.Phase sensitivity versus parametric strength r (a) and photon number Nc (b). Na0 =0.TSVB: two-mode squeezed vacuum beam.The other parameters are the same as those in Fig.2.

Fig.5.The device for phase sensitivity measurement with transmissivity efficiencies T1 and T2 of the photon detectors.Fictitious beam splitters are employed to represent the losses of the photon detectors.The other parameters are the same as those in Fig.2.

In the above subsections, we show the phase sensitivity with the inputs of a coherent beam and one of the BETB.In fact,when the photon number of the coherent beamNcis zero,this scheme becomes the same as the scenario in Ref.[17].If the squeezing parameterris zero, for the MZI, the inputs are two coherent beams without the reference beam.WithNa0=0,the inputs in Fig.1 become a coherent beam and one of the two-mode squeezed vacuum beams(TSVB).The other one of the TSVB is employed as the external power reference beam.The phase sensitivities versus parametric strengthrand photon numberNcare shown in Fig.4.We immediately notice that the QCRB can beat HL and the optimal phase sensitivity with intensity difference detection is worse than SNL.In Fig.4(a),the QCRB can only reach sub-SNL whenris more than 0.95 withNc=0.5.Withr=0.65, the QCRB can beat HL whenNc ≤1.1 in Fig.4(b).Therefore, for the input of a coherent beam plus one of TSVB based on the MZI,the intensity difference detection is not preferred.

3.2.Non-unit photon detection efficiency

In this subsection,we consider the condition that the detection efficiency of the photon detectors is not ideal in Fig.5.Then the transformation of the operators in the scheme is expressed as

where ˆυ2(ˆυ1)and ˆaoutT(ˆboutT)are the annihilation operators of the two input-output modes of the fictitious BS, respectively.T1andT2represent the transmissivity of the detector.Only the losses of the photon detectors are considered and we assume that there are no losses inside the interferometer.The slope is given by

and the variance of intensity difference is Δ2ˆI−BT(Details can be seen in Appendix B).As displayed in Fig.6(a), withT2= 0.2 or 0.5, the phase sensitivity is always worse than SNL.WithT2=0.8,the phase sensitivity can beat SNL whenT1is larger than 0.73.When the photon detector has no loss(T2=1),the phase sensitivity is better than SNL withT1larger than 0.58.In this case, the better phase sensitivity can be achieved by the lower loss.According to Eq.(13), the transmissivityT1has no effects on the slope.The phase sensitivity is worse than SNL when the external power reference beam is absent (T1=0).The external beam can boost the phase sensitivity by reducing the variance and keeping the slope unchanged.Note that the intensity difference detection will become intensity detection withT1=0.The phase sensitivity can not beat SNL in this case.In Fig.6(b),the larger the valueT2, the better the sensitivity.The phase sensitivity can beat SNL withT2larger than 0.76 withT1=0.8.WhenT1=1,the phase sensitivity can still reach SNL withT2=0.7,which shows the good robustness of this scenario.In Fig.6, withr=1,Na0=10 andNc=102, the optimal phase sensitivity with intensity difference detection can not surpass the HL withT1=T2=1.

Fig.6.Phase sensitivity with the increase of transmissivity T1 (a) and T2 (b),for r=1,Na0=10,and Nc=102.The other parameters are the same as those in Fig.2.

4.Comparison

4.1.Two coherent beams with an external coherent beam

and the variance becomes

Nco=.We also imposeφ1=0 andφ2=φin the error propagation formula.As displayed in Fig.8, with the employment of the external beam,the QCRB can not surpass SNL.In Figs.8(a)and 8(b),when the inputs are two coherent beams and another coherent beam is employed in the intensity difference detection, the phase sensitivity is worse than SNL.As displayed by Eqs.(14)and (15), when the external coherent beam is employed, the slope is unchanged and the variance becomes larger.BecauseNco≥0 andNb2≥0,andNco+Nb2≥Nco.Hence the external coherent beam can not boost the phase sensitivity and makes it worse.As shown in Fig.8(a), with the increase ofNa0,the photon number of the external coherent beam is larger and the phase sensitivities with the intensity difference detection can never approach SNL.Meanwhile, the QCRB can only reach SNL.In Fig.8(b),the phase sensitivity with the intensity difference detection is worse than SNL with the increase of the photon numberNc2.

Fig.7.The device for phase sensitivity measurement based on the Mach–Zehnder interferometer with two coherent beams and another coherent beam as the external beam.The other parameters are the same as those in Fig.1.

Fig.8.Phase sensitivity versus parameter Na0 (a)and Nc (b),for r=1,Na0 =10, and Nc =102.TCB means the phase sensitivity with two coherent beams and the intensity difference detection.The QCRB1 and QCRB2 are the quantum Cram´er–Rao bounds with bright entangled twin beams and two coherent beams.The other parameters are the same as those in Fig.2.

4.2.One coherent beam plus one single-mode squeezed vacuum beam

Ref.[42].In this case,the QCRB and the optimal phase sensitivity can reach HL,respectively.Considering the phase insensitive intensity squeezing degree being less than 10 dB,Gis below 5.5(r ≤1.5).

As shown in Figs.9(a)and 9(b), withNc3=sinh2r3,the QCRB of the coherent beam plus the single-mode squeezed vacuum beam can surpass HL.Withr3=0.8, the QCRB of the coherent beam plus the BETB can beat HL.However,they are still worse than that of the coherent beam plus a singlemode squeezed vacuum beam.Note that the two scenarios have the same photon numbersNBand the BETB have been the TSVB withNc=0.In Figs.9(c) and 9(d), the practical experiments conditionsNc3≫sinh2r3are considered.In this case, both the QCRBs of the Fig.1 and the coherent beam plus the single-mode squeezed vacuum state scenario can only reach sub-SNL.However, with the increase of the squeezing parameterr(r ＞1.86),the QCRB with BETB can beat QCRB of the coherent beam plus a single-mode squeezed vacuum beam in Fig.9(c).In Fig.9(d), the BETB have become the TSVB, and the QCRB1 can still beat the QCRB2 (r ＞1.8).According to Fig.9, in the HL scale, the QCRB of the scenario in Fig.1 is worse than that of the coherent beam plus a single-mode squeezed vacuum beam.However, in the sub-SNL region,with the high squeezing parameterr,the QCRB1 can beat the QCRB2.Though the squeezing parameter cannot be experimentally realized at present, it is hopeful that it can be achieved in the future.

Fig.9.Phase sensitivity versus parameter r.In(a)and(b),r3=0.8,and in(c)and(d)r3=1.5. Na0=0.1 in(a)and(c).In(b)and(d),Na0=0.Nc3 =sinh2r3 in (a) and (b). Nc3+sinh2r3 =NB, and they have the same SNL and HL.QCRB1 and QCRB2 are the quantum Cram´er–Rao bounds of Fig.1 and the scenario with the input of a coherent beam plus a single-mode squeezed vacuum beam.The other parameters are the same as those in Fig.2.

5.Conclusion

In conclusion, this paper presents the phase sensitivity with the inputs of the BETB and coherent beams based on the MZI.The optimal phase sensitivity with intensity difference detection can reach sub-HL and approach QCRB while an external power reference beam is employed.When the inputs are a coherent beam plus one of the TSVB,the QCRB can beat HL and the optimal phase sensitivity with the intensity difference detection is worse than the SNL.We have a detailed discussion about the detection efficiencies of the photon detectors.The results show that the external beam play a vital role in the measurement process and the absence of the external beam can degrade the performance of the phase sensitivity dramatically.The QCRB of the scheme can be better than that of the coherent beam plus a single-mode squeezed vacuum beam input scenario with the high squeezing parameter.Meanwhile,the external coherent beam can not boost the phase sensitivity when the inputs are two coherent beams.This method of employing the external power reference beam offers a novel measurement way for the phase precision measurement.

Appendix A: Exact expression of the QFI elements

The QFI matrix elements can be expressed as

Im(·)represents the imaginary part.

Appendix B:Slope and variance of intensity difference with non-unit photon detection efficiency

The slope is given by

and the variance of intensity difference Δ2ˆI−BTyields

In this part,for simplification,we assume thatφd=0.

Acknowledgments

Project supported by the National Natural Science Foundation of China (Grant Nos.12104190, 12104189,and 12204312), the Natural Science Foundation of Jiangsu Province (Grant No.BK20210874), the Jiangsu Provincial Key Research and Development Program (Grant No.BE2022143); the Jiangxi Provincial Natural Science Foundation (Grant Nos.20224BAB211014 and 20232BAB201042), and the General Project of Natural Science Research in Colleges and Universities of Jiangsu Province(Grant No.20KJB140008).