Xiao Xiao(肖驍),Hong-Bin Liang(梁宏賓),Guo-Long Li(李國龍),and Xiao-Guang Wang(王曉光)

Zhejiang Institute of Modern Physics,Department of Physics,Zhejiang University,Hangzhou 310027,China

Abstract We derive a general phase-matching condition(PMC)for enhancement of sensitivity in SU(1,1)interferometers.Under this condition,the quantum Fisher information(QFI)of two-mode SU(1,1)interferometry becomes maximal with respect to the relative phase of two modes,for the case of an arbitrary state in one input port and an even(odd)state in the other port,and the phase sensitivity is enhanced.We also find that optimal parameters can let the Qfiin some areas achieve the Heisenberg limit for both pure and mixed initial states.As examples,we consider several input states:coherent and even coherent states,squeezed vacuum and even coherent states,squeezed thermal and even coherent states.Furthermore,in the realistic scenario of the photon loss channel,we investigate the effect of photon losses on Qfiwith numerical studies.We find the PMC remains unchanged and is not affected by the transmission coefficients for the above input states.Our results suggest that the PMC can exist in various kinds of interferometers and the phase-matching is robust to even strong photon losses.

Key words:quantum Fisher information,SU(1,1),parameter estimation,phase-matching condition

1 Introduction

Nowadays quantum metrology is becoming more and more widely used in many areas due to the development of quantum information theory[1?16]and quantum technology.[16?25]The ultimate aim of quantum metrology is to achieve strong sensitivity of parameter estimation.Experimentally,phase estimation includes measurement of gravity,temperature,weak magnetic strength,and many other parameters.[27?28]In usual,high-precision measurement is the optical Mach-Zehnder interferometer,which typically contains two beam splitters.The phase shift ? is emerged from interferometer,and can be measured from the output light.The precision can beat the the standard quantum limit(also called shotnoise limit),i.e.,1(N is the total photon number)[29]due to input states of a Mach-Zehnder interferometer by exploiting a high-intensity coherent state and a lowintensity squeezed vacuum state.The uncertainty can be achieved or surpassed a scaling 1/N known as the Heisenberg limit,[30?31]with the quantum signature of highly nonclassical states,such as NOON states,[32?33]entangled coherent states,[34]two-mode squeezed states,[35]and number squeezed states.[36]

There is another possibility to beat the SQL and the example is the SU(1,1)interferometer,which is configured as a Mach-Zehnder interferometer with the passive beam splitters replaced by the active nonlinear beam splitters that create or annihilate pairs of photons.[37?38]The active beam splitters can be optical parameter amplifiers or four-wave mixers.The SU(1,1)interferometer,different with the SU(2)Mach-Zehnder one,first proposed by Yurke et al.,[37]is described by the SU(1,1)group.Another key difference is that the total photon number is not conserved,unlike the SU(2)interferometer.Both the passive and active beam splitters can generate high entanglement between two modes,which may lead to phase measurement with high precision.

In the previous paper of Caves,[29]to enhance the precision,the phases of the two input states need to satisfy a relation.This relation can be considered as a kind of PMC.[26]A more general PMC in an SU(2)Mach-Zehnder interferometer in order to enhance the phase sensitivity has been discussed by Liu et al.[26]Under the PMC,the Qfibecomes maximal and the phase sensitivity is enhanced.Thus,to enhance the sensitivity,we can have two steps.The first step is to adjust phases of two modes to meet the PMC and the second is to change the intensities of the two modes to get higher sensitivity.It is interesting and important to know whether there exists similar PMC in other interferometers.To derive the PMC in SU(1,1)interferometer is the major motivation of us.

In this paper,by examining the general analytic expression of the QFI,we give a general PMC for enhancement of sensitivity in SU(1,1)interferometers.Then,we investigate two-mode SU(1,1)interferometry with either pure initial state or mixed initial state.As examples,we consider several input states:coherent and even coherent states,squeezed vacuum and even coherent states,squeezed thermal and even coherent states.In this sce-nario,we find that the Qfiis determined by the average photon numbers of the two modes and the corresponding expectation values of the square of the annihilation operators,and is related to transition matrix elements between eigenstates of the initial density matrix.

We also find that optimal parameters can let the Qfiin some areas achieve the Heisenberg limit.Further,for the case where photon losses occur in both arms with the same transmission coefficients,we obtain numerical studies to prove that the PMC remains unchanged for any transmission coefficient.

1.1 SU(1,1)Interferometry

The SU(2)interferometer is a well-known optical device in quantum metrology,which is constructed with two beam splitters and one or two phase shifts.The model of an SU(1,1)interferometer in which the optical parameter amplifiers(OPAs)replace the 50:50 beam splitters in a traditional MZI.The unitary transformation of SU(1,1),associated with this interferometer can be written as

where ? is the phase to be estimated.The generators of SU(1,1)algebra satisfy the commutation relations

One of the key differences is that the SU(2)interferometry keeps the total photon number unchanged,while the photon number is not conserved for the SU(1,1)interferometry.

From the above commutation relations,we can rewrite the whole unitary operator as

with

This operator is a Hermitian generator and will play a key role in the following discussions.One important difference of this operator from its SU(2)counterpart is that one cannot cancel z-term by varying θ here.

1.2 Quantum Fisher Information

Qfiis a central concept in quantum metrology,and it is defined as[39?40]F :=Tr(ρL2),where L is the so-called symmetric logarithmic derivative determined by?θρθ=(ρθL+Lρθ)/2.Utilizing the spectral decomposition of initial density matrixthe Qfican be written as[39?40]

For states satisfying

Eq.(5)reduces to

Next section we will give some examples which satisfy Eq.(6).

If we consider a pure initial state,the above equation further reduces to

Recalling that the annihilation and creation operators for SU(1,1)algebra are given by K±=Kx±iKy,we can rewrite the QfiEq.(8)as

under which the Qfibecomes maximal.

To get maximal QFI,one can first apply a unitary transformation to the initial state

As seen from Eq.(11),only the term containing Θ becomes maximal and other terms are unchanged.Thus,we can only make a phase shift to the initial state for getting maximal QFI.The above discussions can be directly applied to the SU(2)interferometry with unbalanced beam splitters.

2 Phase-Matching Condition for QFI

Now,we consider the two-mode SU(1,1)interferometry with the bosonic-mode annihilation operators of the two ports as a and b,and the two-mode realization of SU(1,1)algebra is given by

The operator for input photon number of two modes is denoted as

The creation and annihilation operators are given by K+=a?b?,K?=ab,respectively.

We consider a separable input state ρin= ρA?ρB.Here ρAis an arbitrary state withandis an even(odd)state,satisfyingFor such initial states,all the conditions given by Eq.(6)are satisfied.And from Eq.(11),the PMC for this system becomes

Under this condition,from Eq.(11)the Qfican be written as

The second term of the equation above depends on the variance of photon numbers Var(In the following,we will give some examples of the PMC.

2.1 Pure Initial States

(i)Product of Coherent and Even Coherent States

We now choose ρAto be a coherent stateand ρBto be an even coherent statewhereand=1/(2+2e?2|α|2).Here,we denote α =|α|exp(iΦα)and β =|β|exp(iΦβ).For this case where argand argfrom Eq.(15)the PMC can be specifically written as

The Qfican be expressed by

where

Here,tanh|α|2is a monotonic function and very close to 1 for|α|2≥2,and in this situation=|α|2.Then Qfireduces to Fm=(+1)sinh2(θ)++(cosh2(θ)+sinh2(θ)).It is not difficult to obtain that Fm≤ N2sinh2(θ)+N(cosh2(θ)+sinh2(θ))+sinh2(θ).The equality above can be achieved when=

(ii)Product of Squeezed Vacuum and Even Coherent States

The squeezed vacuum state,which is defined as[17]is another well-known state.The squeezing operator is given by S(ξ)=exp[(ξ?a2? ξa?2)/2]with the squeezing factor ξ=reiΦξ.For convenience,we still choose the input state in port B to be an even coher-ent stateIn this case,we have argHere we use unitary transformation properties of the squeeze operator S?(ξ)aS(ξ)=acoshr ? a?eiΦξsinhr,S?(ξ)a?S(ξ)=a?coshr ? ae?iΦξsinhr.From Eq.(15),PMC can be specifically written as

And from Eqs.(16)and(17),the Qfican be expressed by

where

2.2 Mixed Initial State

Now we consider a mixed state. We choose ρAa squeezed thermal state,[41]

with the average thermal photon numberThe port B is in the even coherent state.

From Eq.(16),the Qfican be obtained as

This term is the contribution of the transition matrix elements between eigenstates of the initial density matrix.When=0,ρAchanges into a pure initial state of a squeezed vacuum state.Then Eq.(27)reduces to Eq.(23).

2.3 Quantitive Analysis

The total photon number NT=inside the SU(1,1)interferometer,is different from the traditional SU(2)Mach-Zehnder one.This is due to amplification of the phase-sensing photon number by the first OPA.[42]According to transformation,the total photon number is

Fig.1 (Color online)Variation of Fm/ with the total photon number NT.(a)The input states port A and port B here are coherent and even coherent states,and θ =0.5π.(b)The input states port A and port B here are squeezed thermal and even coherent states,with r=0.25 and θ=0.5π.

Fig.2 (Color online)The quantum Cram′er-Rao bound(QCRB) △? =1/√ as a function of Here and red dashed and black dotted lines display the Heisenberg limit(HL)and standard quantum limit(SQL)for comparison,respectively.(a)The input states port A and port B here are coherent and even coherent states,with θ=0.5π.(b)The input states port A and port B here are squeezed thermal and even coherent states,with r=0.25 and θ =0.5π.

To clearly obtain the relation between Qfiand Heisenberg limit,we compare different parameters.Figure 1 shows the variation of Fm/with the change ofandFrom this plot one can find that the optimal value of the quantum Fisher information for a fixed N is obtained near theline,especially for a large N.Also,from Fig.1,one can see that with the increase of input photon numbers N,the region of Fm>is increasing,which indicates that the high intensity input state is good for the enhancement of the phase sensitivity.As shown in Fig.2,the precision of phase with 1as a function of NT.Here we setand compare 1/Fmwith the Heisenberg limit(HL)and standard quantum limit(SQL).We find thatcan reach the Heisenberg limit with the increase of the total number of photons.

3 PMC with Photon Losses

In this section,we determine the PMC of Qfiin the realistic scenario of the photon losses.Traditionally,the photon losses can be described as two beam splitter transformations characterized by the so-called transmission coefficient T.We consider the scenario of equal losses in both arms of the SU(1,1)interferometer,with T1=T2=T.[39?51]We also define R=1? T as the refl ection coefficient.Obviously,there are no photon losses in the interferometer with T=1(R=0),and all the photons leak out of the interferometer with T=0(R=1).

Fig.3 (Color online)Plot of QfiFmfor losses in both arms of the interferometer as a function of T and Φ with =2.(a)Coherent and even coherent states,θ=0.5π.(b)Squeezed thermal and even coherent states,r=0.25,θ =0.5π.

For convenience,we assume that the leaks in both arms have the same transmission coefficient T with loss modes C and D,and the input state of port A and B is separable,i.e.,ρin= ρA? ρB.The loss operators are added just after the unitary transformation e?iθKx.Then,the reduced density matrixafter the losses reads

where λjand|λjare the eigenvalues and eigenvectorsTo study the effect of losses on PMC of QFI,we consider two cases of input states:coherent and even coherent states,squeezed thermal and even coherent states.Using the eigenvalues and eigenvectors of the density matrixwe obtain numerically Qfiof symmetric loss cases for the above two input states.In Fig.3(a)and Fig.3(b),it is shown that the Qfiis affected by the phase Φ and transmission coefficient T.Here Φ is phase shift the between two input channel A and B.We find the optimal phase Φ for both input states,corresponding maximum QFI,remains unchanged at different transmission coefficient T.It indicates that the PMC is not affected by the photon losses in our SU(1,1)interferometer.

4 Conclusion

In summary,we have considered a general scenario of SU(1,1)interferometer and provided a general PMC for sensitivity enhancement.In the case of two-mode SU(1,1)interferometry the Qfibecomes maximal with respect to the relative phase of two initial states:an arbitrary state in one input port and an even(odd)state in the other port,and the phase sensitivity is enhanced.As examples,we considered both pure and mixed initial state.We also obtain that there are optimal parameters which can let the Qfiachieve the Heisenberg limit.Under the suitable parameters,one can see that with the increase of input photon numbers N,the region where Qfiachieves the Heisenberg limit is increasing,which indicates that the high intensity input state can be good for the enhancement of the phase sensitivity.For the realistic scenario of the photon loss channel,we investigate the effect of photon losses on the Qfiwith numerical studies,and it is surprising to see that the PMC remains unchanged and is not affected by the transmission coefficients for the states we have considered.In other words,the PMC is very robust to photon losses for the SU(1,1)interferometer.Combining the studies of PMC for the SU(2)interferometer with the present investigations,we are optimistic to say that the PMC can exist in various kinds of interferometers and the phase-matching is robust to even strong photon losses.