Fei Yao,Yi-Mu Du,Haijun Xingand Libin Fu

Graduate School of China Academy of Engineering Physics,Beijing 100193,China

Abstract We propose a scheme to realize two-parameter estimation via Bose–Einstein condensates confined in a symmetric triple-well potential.The three-mode NOON state is prepared adiabatically as the initial state.The two parameters to be estimated are the phase differences between the wells.The sensitivity of this estimation scheme is studied by comparing quantum and classical Fisher information matrices.As a result,we find an optimal particle number measurement method.Moreover,the precision of this estimation scheme means that the Heisenberg scaling behaves under the optimal measurement.

Keywords:multi-parameter estimation,Heisenberg scaling precision,three-mode NOON state,symmetric triple-well potential

1.Introduction

Quantum metrology[1–3]has attracted considerable interest in recent years due to its wide applications in both fundamental sciences and applied technologies.As crucial tools in quantum metrology,the quantum parameter estimation theory[4,5]and Fisher information provide the theoretical bases for enhancing the precision of parameter estimation with quantum resources.In the previous researches,the single-parameter estimation has been well studied and a series of achievements have been made[6–8],such as gravitational wave detection[9],magnetometry[10–12],atomic clocks[13,14],and quantum gyroscope[15–17].

Although the single parameter estimation plays a significant role in many applications,it is often necessary to estimate multiple parameters simultaneously in practical problems,e.g.quantum imaging[18–20],waveform estimation[21],measurements of multidimensional fields[22],joint estimation of phase and phase diffusion[23,24].Studying the multi-parameter estimation is thus an urgent need for effectively solving the practical parameter estimation problems.It has attracted lots of attention[22–38]in recent years.Most of these works aim to propose a general theory and framework for multi-parameter estimation.Few concrete schemes are proposed for realizing practical highprecision multi-parameter estimations.In this article,we will propose a scheme to estimate multiple parameters simultaneously with Heisenberg scaling sensitivity.

The Bose–Josephson junction,formed by confining Bose–Einstein condensate in the double-well potential(in spatial freedom[39]or internal freedom[40,41]equivalently)is a wellestablished model[6,42–44].It is widely used in quantum parameter estimation as interferometries for its high controllability[6,42–44].Especially in some of the schemes[41,45–48],one can prepare condensate into the two-mode NOON state[49](also known as GHZ state[50]and Schr?dinger cat state[51]),which can perform single parameter estimation in Heisenberg limit precision.As an extension of the double-well interferometry,we will confine Bose condensate in the symmetrical triple-well potential[52–56]to realize high precision twoparameter estimation.

Figure 1.(a)The schematic diagram of a symmetric triple-well trapped Bose condensates.(b)Framework of multi-parameter estimation.

Our measurement scheme consists of four stages:initialization,parameterization,rotation,and measurement.We prepare the condensate into the three-mode NOON state adiabatically as the initial state.The parameters to be estimated are two-phase differences between the wells caused by the external field.The parameterized state is read via the particle number measurement.In order to study the precision of the measurement scheme,we calculate the quantum Fisher information matrix(QFIM)and classical Fisher information matrix(CFIM)on the two parameters.By comparing the CFIM and QFIM,we find that the measurement rotation time significantly affects the measurement precision,and the optimal rotation time is given.In addition,the result shows that the optimal measurement precision of our scheme can approach the Heisenberg scaling.

The paper is organized as follows.In section 2,the model and basic measurement theory are introduced.In section 3,we give the scheme of estimating two parameters with the triplewell system,including initial state preparation,parameterization,rotation,and measurement.The optimal precision and measurement conditions are given by analyzing the CFIM and QFIM.At last,we summarize this article in section 4.

2.Model and basic theory

3.Two-parameter estimation with the STWS

3.1.The initial state preparation

Figure 2.Energy spectrum versus J.Here we set N=30.Only the lowest 30 energy levels are given.

3.2.Parameterization and the QFI

Figure 3.State preparation.(U=-0.5 and N=30)(a)The fidelity versus J,where with J0=10 and v=0.2.|ψ(t)〉 is the evolving state from the ground state at J=10.Three-mode NOON stateis the target initial state.are the orthogonal basis of the there-dimensional ground state manifolds NH at J=0.(b)–(c)Distribution of quantum states in basis |n1, n2, n3〉.(b)Ground state at J=10.(c)|ψ(t)〉 at J=0,with v=0.2.It isapproximately.(d)The inset shows the speed dependence of the state preparation via the final state fidelity Fid0 at J=0 versus the speed v.

3.3.Projective measurement

3.4.The optimal measurement precision

In this section,we will analyze the measurement precision with the gap Δ defined in equation(9).The optimal precision can be given by optimizing both the encoded parameters θ and rotation time τ.

Figure 4.ln(Δ)versus(θ1,θ2)with a given τ.(a)τ=0.2π.(b)τ=τO=2π/9.The maximum precision(minimum of ln()Δ)for a given τ is marked as the black dots‘?’.θG=(2π/3,2π/3).Here,we set N=30.

Figure 5.(a)δ(τ)versus τ.(b)Λ versus Δτ.Λ=δ(τ)-δ(τO),‘Δ±’denote the line for τ=τO±Δτ,with τO=2π/9.Λ decreases to zero from above as Δτ approaches 0.Here,we set N=30.

Firstly,we discuss the dependence of Δ on θ with a given rotation time τ numerically.The results are shown in figure 4,where only one period is given.We observe that Δ highly depends on θ1and θ2.For a given τ,the maximal precision can only be achieved in several points ‘?’.Luckily,with enough prior information provided,we can shift the estimand to the vicinity of these points to approach the maximal precision.

Secondly,by comparing figures 4(a)and(b),we find that the maximal precision over parameters θ highly depends on the rotation time τ.Specifically,we define

to study the effect of rotation time τ on the optimal precision over θ.As shown in figure 5(a),δ(τ)varies periodically with τ.There are three short periods with duration 2π/3 in a long period with duration 2π.More importantly,δ(τ)takes the minimum value at points

withk?N .We further show its validity numerically in figure 5(b),which indicates that τO=2π/9 is one of the optimal rotation times.

Now,one can acquire the optimal precision of our scheme by choosing the optimal phases(θ1,θ2)at an optimal time τ given in equation(21).It can be done numerically.An example with τ=τO=2π/9 is given in figure 4(b),where the point ‘H’ is one of the optimal sets of phases.

To evaluate the quality of the optimal precision,we study the scaling of CFIM with particle numberNat τ=τO=2π/9 both numerically and analytically.The numerical result is shown as the red line in figure 6.By searching the minimum of tr[(Fc)-1]over the phase parameter θ at τO=2π/9,we find the optimal precision satisfies the following linear relationship

Figure 6.The precision versus N2.Here,we set τ=τO=2π/9.The red line denotes the numerical optimal precision.The blue line is analytical result atwhich reads.

Figure 7.The precision versus ,with τO=2π/9.is the optimal precision acquired via the imperfect rotationHere,we take J=10, N=30.

It indicates a Heisenberg scaling precision.To show it more concretely,we further give a lower bound of the optimal precision analytically.The precision for the optimal phase point at τOis challenging to be given analytically.Hence,we calculate the precision for point ‘G’ instead,which is located near the optimal point ‘H’.The phase parameterθGNof the point ‘G’ is given byNθ1=Nθ2=2(N+1)π/3,withNdenoting the particle number.The CFIM of the state ‘G’ is(see appendix B)

And the corresponding precision is

The result is shown as the blue line in figure 6 in comparison with the optimal precision given by equation(22).It shows that the scaling of the two precisions is very close.

We mention that equation(24)is valid for all particle numbersN.It indicates that,with the proposed measurement scheme,the optimal measurement precision of θ can always show a better Heisenberg scaling than equation(24).Furthermore,as indicated by figures 4(b)and 6,the precision is robust around the optimal parameters,e.g.|ψ(θG)〉 with parameters θGstill have relatively high precision with a Heisenberg scaling.It significantly reduces the demands for practical studies,which relieves the working parameters’constrain from a point ‘H’ to,e.g.the white zoom in figure 4(b).With a relatively larger acceptance zoom of the parameter shifts,the demands for the prior information about the estimand θ are also highly reduced.

We have discussed the optimal precision under the projection measurement.However,the result is acquired under the approximation equation(17).If the on-site interaction between atoms cannot be tuned to zero precisely in the rotation operation,the total Hamiltonian reads

4.Conclusion and discussion

In this work,we have proposed a scheme for two-parameter estimation via a Bose–Einstein condensate confined in a symmetric triple-well potential.The three-mode NOON state has been prepared adiabatically as the initial state.The two parameters to be estimated are the two-phase differences between the wells,which are encoded into the initial state via the external fields.We perform the particle number measurement in each well to read the parameterized state.Moreover,a rotation operation is adopted on the output state before the measurement.We optimize both the parameters and rotation time to maximize the estimation precision.As a result,we have approached the Heisenberg scaling precision on simultaneous estimating two parameters under the optimal measurement.

We mention that our scheme is discussed in the ideal scenario in this article.To study it more rigorously,one should build an open quantum system model and introduce noise analysis based on practical experiments.We will advance this research in further studies.We expect to realize the high precision estimation of the two-dimensional fields,such as the magnetic field and gravity field,via ongoing research in this triple-well system.

Acknowledgments

F Y thanks Peng Wang for the helpful discussion.H X thanks Prof.Chang-Pu Sun for providing the valuable opportunity of visiting GSCAEP.This work was supported by the National Natural Science Foundation of China(NSFC)(Grant Nos.12 088 101,11 725 417,and U1930403)and Science Challenge Project(Grant No.TZ2018005).

Appendix A.The derivation of the rotated final state

Appendix B.Scaling of the CFIM entries

Table B1.Classification of fμν(n)with n mod 3.