Zhong Wang (王众)

Department of Physics,Beijing Normal University,Beijing 100875,China

Abstract Quantum steering has attracted great interest in the last decade,especially in the celebrated optomechanical,cold atom,and quantum optical systems.However,there is still a lack of studies on quantum steering in circuit quantum electrodynamics (QED),which provides a useful experimental platform for revealing novel quantum phenomena.In this work,we investigate the steering of qubit by continuous weak measurement in a circuit QED system and establish a set of multiplicative steering inequalities based on the Heisenberg uncertainty principle.Different from the widely studied systems mentioned above,multiplicative steering inequalities in the circuit QED system are in various forms.We find that only a portion of them can be used to show the detection dependence of the qubit state and we also analyze the reason.Furthermore,we discuss several conditions for the violation of a typical steering inequality,including the measurement strength and methods in detecting the cavity field as well as the quantum efficiency of the detector.This preliminary work could be helpful to quantum steering experiments in circuit QED systems.

Key words: quantum steering,circuit quantum electrodynamics,quantum weak measurement

1 Introduction

With further understanding of quantum nonlocality,quantum steering,a former casual theoretical concept,has attracted renewed attentions in the past decade.[1−2]Unlike the previously widely discussed description on quantum entanglement such as the density matrix inseparability and Bell nonlocality,quantum steering focuses on the state of local subsystem under the condition of measuring another distant subsystem.In the hierarchy of quantum nonlocality,steering lies between inseparability and Bell nonlocality with unique asymmetric properties.

An attractive research field on quantum steering is its verification and quantification.[3−8]Steering inequality,proposed as an experimental criterion analogous to Bell inequality,[3]is utilized in various systems,including optomechanics,[9−13]Bose-Einstein condensates[14−16]and photons.[17−19]The studies in these systems shed light on the unique prosperities on quantum steering,as well as promote its practical applications such as high precision measurement[20]and quantum information processing.[21]Furthermore,plenty of experiments on steering are also conducted,for instance,experimental realization of the criteria,[22−24]steering with Bell local state,[25]one-way steering,[26−27]loophole free steering,[28−30]one-side device independent quantum key distribution.[31]However,These experiments are mostly limited in photon systems.

As a state-of-art experimental setup,the circuit quantum electro-dynamics(QED)system is not only a promising scheme in quantum information processing,but also a fascinating platform in exhibiting unusual quantum phenomena,such as the preparation of Schrödinger’s cat state,[32]the violation of Bell inequality,[33]and Leggett-Garg inequality.[34]For now,however there is still an absence of research on quantum steering conducted on a circuit QED system.In this work,we scrutinize the steering of the qubit state by continuous weak measurement on the cavity field and obtain a set of multiplicative steering inequalities based on the Heisenberg uncertainty principle.Unlike the position-momentum uncertainty relation in optomechanics,BEC and photon systems,there are three non-commutative components in the spin angular momentum of a qubit.We find that not every non-commutation relation can derive a usable steering inequality and then we analyze the reason.Furthermore,we discuss some important conditions of violating the steering inequality,including measurement strength,measurement method,and detector efficiency.

The rest of this paper is organized as follows.In Sec.2 we briefly introduce the circuit QED system and the continuous weak measurement scheme in it.In Sec.3,on the foundation of introducing the concept of steering,we get a set of multiplicative steering inequality and find that not every inequality can work as a criterion.In Sec.4 we give an explanation and discuss several conditions for steering.We present a brief summary in Sec.5.

2 Continuous Weak Measurement in Circuit QED System

A circuit QED setup consists of a one-dimensional superconducting transmission line and an artificial atom made of superconducting Josephson junctions.The transmission line driven by alternating-current (AC) voltage is treated as a harmonic oscillating cavity and the artificial atom is treated as a qubit.The Hamiltonian of the entire system is described by the Jaynes-Cummings (JC) model

whereωqis the intrinsic frequency of the qubit,ωrthe frequency of the cavity field,gthe coupling strength between the cavity and the qubit,σ+(a†) andσ−(a) the raising and lowering operator of the qubit(cavity),respectively.E=ϵm(e−iωmt+c.c.),in whichϵmis the driving strength andωmthe driving frequency.A circuit QED system is usually measured in the dispersive regime,i.e.∆≫g,in which ∆=|ωr−ωq|is the detuning betweenωrandωq.Under this approximation,an effective Hamiltonian can be derived through unity transformation as

The cavity field is usually detected through homodyne detection.System state at timetwithout additional decay can be written asin whichare stochastic coherent cavity states corresponding to the excited and ground state of the qubit|e〉and|g〉,respectively.It is difficult to solve the exact cavity field states,but we can use their ensemble average instead and it is a good approximation.[35]The evolution equation of the approximated cavity field is

whereκis the cavity photon leaking rate inversely proportional to the cavity Q-factor.

In circuit QED system,continuous weak measurement such as the homodyne detection is usually described by the quantum trajectory equation (QTE).When considering the steering of the qubit by the cavity field,we do not need to know the cavity field state (the rationale will be explained there-in-after),so it is possible to use the polaron transformation to eliminate the degrees of freedom(DOF) of the cavity field and obtain an effective QTE with only the qubit DOF.[36]The advantage of the effective QTE approach is that,as a harmonic oscillator,there are infinitely many cavity field energy levels,which make calculation very difficult.By omitting these additional DOF from consideration the calculation can be greatly simplified.Under a continuous homodyne measurement,the evolution of the qubit density matrix is

whereωac=ωq+χ+B(t) is the effective frequency of the qubit with AC Stark effect,in whichB(t)=2χRe[αg(t)α∗e(t)].For an arbitrary operatorc,the Lindblad superoperator isD[c]ρ=cρc†−(1/2){c†c,ρ}and the measurement superoperatorΓciis the rate of extracting information from the qubit,i.e.,the “spooky” back-action caused by entanglement.Γbais the “realistic” back-action caused by the AC Stark effect.

whereηis the detector efficiency,φis the local oscillator (LO) phase of homodyne measurement andβ(t)=αe(t)−αg(t)=|β|eiθβ.The overall measurement rate is Γm=(Γba+Γci)/η=κ|β(t)|2,and the measurement strength can be characterized as Γmt.The overall qubit decay rate is Γd=2χIm[αg(t)α∗e(t)],which gives a limit to the upper bound of Γm.Both Γmand Γdare determined by the system parametersχ,κ,andϵm.When the system parameters are specified,the measurement strength is proportional to measurement timet.Moreover,after the system evolutes into steady state,there is Γd=Γm/2,thus the quantum measurement efficiency can be expressed asη=(Γci+Γba)/2Γd.

The output current of the homodyne measurement is

in whichξ(t)is Gaussian white noise satisfyingE[ξ(t)]=0 andE[ξ(t)ξ(t′)]=δ(t −t′).It reflects the randomness of quantum measurement.Under homodyne detection,the weakly measured qubit evolves gradually and stochastically.

3 Quantum Steering and Multiplicative Criteria in Circuit QED System

Here we briefly introduce the concept of quantum steering,then derive a set of multiplicative steering inequalities based on the Heisenberg uncertainty relation as criteria for experiments in the circuit QED system.

Firstly,we review two frequently utilized viewpoints in describing entanglement,i.e.,the inseparability of density matrix and Bell nonlocality.For a pair of particles separated in space,inseparability means that the total density matrix cannot be decomposed into the sum of the direct product of each particle’s density matrix.It focuses on the quantum state of the system.Bell nonlocality means that the correlation of measurement results of each particle can exceed the upper bound predicted by a local hidden variable(LHV)theory.It focuses on the measurement results as c-numbers.For pure states,these two viewpoints are equivalent.However for mixed states,Werner proved in 1989 that the latter is a subset of the former.[37]

Therefore a natural but long neglected question is,how to describe the measurement result of particleAand the state of particleB? There was no answer to this question until 2007,when Wisemanet al.formulized the effect on particleBcaused by measurement on particleAand named it as steering,a word first proposed by Schrödinger in 1935.[1−2]For a pair of entangled particlesAandB,the corresponding state of particleB,when measuring the observableof particleAand obtaining a resulta,is called the conditional state ofB(under the condition of resulta).Considering an ensemble ofAandBparticle pairs,if we obtain different conditional states ofBwhen measuring different observables ofA,can we confirm that the measurement ofAhas affected the state ofB? For particles in mixed states,the answer is no.Because a possible explanation is that there is a set of local hidden states(LHS)ofBand the differences between conditional states come from different combinations of these LHS.With the LHS set{ρλ},the conditional state ofBunder the condition of obtaining resultawhen measuring ˆAcould be written as

whereP(λ|a)is a pre-existed probability which is independent of measurement.In this way,different conditional states are made up only from different combinations of LHS without any nonlocal effect on the very state ofB,i.e.,without any steering.

To determine whether the conditional states of particleBare steered by the measurements onA,or just follow some kind of LHS model,we need a criterion.As with the Bell inequality for the LHV model,we can use a steering inequality to check the LHS model.In optomechanics,BEC systems and photon systems,a multiplicative steering inequality can be derived from the positionmomentum uncertainty relation ∆x∆p ≥1/2.However in a circuit QED system,there are three non-commutative observables and the measurement methods are quite different,so the situation becomes relatively more complex.We next investigate this issue.

For the Pauli operators of a qubit,there is commutation relation

in whichi,j,andkare cycles ofx,y,andz,respectively.Then for arbitrary state,there exists the Heisenberg uncertainty relation

In circuit QED system,a steering task could be schemed as follows: Prepare the system into an initial state and choose an LO phaseφm(corresponding measurement method noted asm).Measure the qubit continuously and weakly.Meanwhile,record the homodyne output currentI(t).At the end of the evolution,measure a qubit observableσiprojectively and obtain the results=±1.Repeat the above procedure many times.Then we can obtain the conditional absolute valueand conditional variance ∆2(σi|I) of observableσiunder conditionI(t).There is no two identical stochastic currents.However,technically we can pick out current close enough to instead approximately.For a specific measurement resultI(t),the conditional variance and absolute value are

Taking an average over all currents under the measurement methodm,we can obtain the conditional absolute value|σmi|and variance ∆2σmiunder the conditionφm.For observableσiand measurement methodm,we have

in which ∆2σmiis the conditional variance of observableσiaveraged over all resultsI(t) under measurement methodm,and|σmi|is the conditional absolute value under the same condition.

It is noteworthy that in the above procedures,the measurement results are only a mark with which to classify the conditional states.Their exact values do not affect the judgement of the LHS model.Thus it is reasonable to eliminate the cavity field DOF in QTE as we did in Sec.2.This is the asymmetry that makes quantum steering different from conventional density matrix inseparability and Bell nonlocality.

Through some simple derivations (see the Appendix),we can obtain a multiplicative steering inequality in circuit QED system This is similar to the Heisenberg uncertainty relation in form,but different in intrinsic physical meaning.We then define a steering parameteras wherei,j,kare cycles ofx,y,zandm,n,lare labels of measurement methods.By comparing the steering parameter with 0,i.e.,checking the violation of steering inequality,we can rule out LHS models of the qubit.

Table 1 Steering parameters are corresponding to different observables and different measurement methods.“Y” and “N”indicates whether a steering parameter can be used to show the violation of the steering inequality or not.

Consider two frequently used LO phasesφ1=0 andφ2=π/2.For 3 subscripts corresponding toσx,σy,σzand 2 superscripts corresponding toφ1,φ2in steering parameterthere are 18 different combinations.We find through simulation that not every steering parameter can be used to show the violation of steering inequality.This is different from the widely discussed positionmomentum type steering inequalities in other systems.In fact,there are only 9 different combinations can be used to show steering; see Table 1.

4 Results and Discussion on Steering Conditions

By theoretically and numerically analyzing the evolution of qubit under continuous weak measurement,we explain the reason behind which specified steering parameters can be used to show the violation of steering inequality.We also discuss some conditions that may need to be considered in experiment,including measurement strength of the qubit,measurement method (characterized by the LO phase) and detector efficiency.

For simplicity,we set the initial state of the system as(not normalized)and consider the situation without additional decay.

When the LO phaseφ=0,it has Γci=Γmand Γba=0,which means the qubit state information is gradually obtained through continuous weak measurement.As the information is obtained,the qubit state gradually and stochastically collapse to eitheror.Without loss of generality,we assume the qubit final state is|g〉.When measuringσxwe haveP(x=+1)=P(x=−1)=1/2,which means it is completely uncertain.Thusreaches a minimum of 0 and ∆(σx|I) reaches a maximum of 1.The same argument applies forσy.When measuringσzwe haveP(z=−1)=1.Thusreaches a maximum of 1 and ∆(σz|I) reaches a minimum of 0.

Fig.1 (Color online) The evolution of terms in the steering parameter.System parameters are χ=0.1 MHz,κ=100MHz,ϵm=400 MHz,∆=2.5 GHz.(a) |σi(1)| when φ=0.(b) ∆σi(1)when φ=0.(c) |σi(2)| when φ=π/2.(d) ∆σi(2)when φ=π/2.In all sub-figures blue,green,and red represents x,y,and z respectively.

Whenφ=π/2,Γci=0,and Γba=Γm,the measurement does not obtain information of the qubit state,but only drive the qubit coherently through AC stark effect.Theσxandσycomponents oscillate stochastically and theσzcomponent keeps unchanged because no information is extracted.For stochastically oscillation,the variance and absolute value will evolute into a steady state.Through numerical simulation we find that it is a value decided by system parameters between 0 and 1.Forσz,P(z=+1)=P(z=−1)=1/2,thus ∆(σz|I) stays at a maximum of 1 andstays as a minimum of 0.

According to the above analysis of the conditional values under specific measurement resultI(t),we can have an understanding on each term in steering parameter by averaging over allI(t)withP(I|m).The evolution of each term is shown in Fig.1.We can find that|σ(1)x|and|σ(1)y|decrease to a minimum of 0 with oscillating,in which the frequency is very high because of largeωqwhich is usually at GHz in practical experiment.|σ(1)z|increases to a maximum of 1.∆σ(1)xand ∆σ(1)yincrease to a maximum of 1 with oscillating while ∆σ(1)zdecreases to a minimum of 0.|σ(2)x|and|σ(2)y|evolve into steady value while ∆σ(2)xstays at 0.∆σ(2)xand ∆σ(2)yevolve into steady value while∆σ(2)zkeeps in 1.The numerical evolution under specifiedmagrees with our analysis above under specificI(t).

Fig.2 (Color online) Evolution of several typical steering parametersSubscripts for observables x,y and z,superscripts for measurement methods φ1 and φ2.(a)cannot violate the steering inequality.(b) violates the steering inequalities after a period of evolution.The minimum is −0.31.(c) violates the inequality only when oscillating to valleys before getting into steady state.The minimum is −0.11.Evolution times in (a),(b) and (c) are from 0 to 5 µs.(d) from 0.3 µs to 0.7 µs,i.e.,the vertical black lines in (c),in which the steering inequality is violated intermittently.System parameters are the same as Fig.1.

To violate the steering inequality,small ∆σmiand large|σlk|should be chosen to make the steering parameterless than 0.Three typical combinations of measurement methods and observables are presented in Fig.2.in Fig.2(a) is an instance that can not violate the steering inequality.According to the above discussion,we know that ∆σ(1)xand ∆σ(1)ygradually increase to 1 and|σ(2)z|stays at 0.Thusincreases to 1 and will never be less than 0.Figure 2(b) shows,which is an instance of violating the inequality.In this case,∆σ(1)xgradually increases to 1,∆σ(2)yoscillates into a finite steady value between 0 and 1,and|σ(1)z|stays at 1.Thus after a period of evolution,the entire parameter value is less than 0 before the system completely collapses.in Fig.2(c)is more interesting.In the stage of establishing steady state,∆σ(1)zkeeps at 1,∆σ(2)xand|σ(1)y|both oscillate.A numerical “interference” between ∆σ(2)xand|σ(1)y|will lead to a large oscillation amplitude of.Thus when the measurement strength is small,the violation of steering inequality can be shown.However,it disappears after the steady state is established.This is a direct example which shows the effect of weak measurement strength in multiplicative steering inequality.To see this more clear,on the time period from 300 ns to 700 ns is plotted in Fig.2(d).

The oscillations in Figs.1 and 2 are originated from the AC Stark shift of the qubit energy caused by the dispersive homodyne readout.To demonstrate the violation of the steering inequality in experiment,one only needs to choose a specific time,i.e.,a proper measurement strength.Moreover,the measurement results is Gaussian stochastic currents and the oscillations do not directly appear in the measurement results.In an experiment,if a specified currentI(t) is obtained,and the system initial state and system parameters are already known,one could estimate the system stateρ(t)and its evolution trajectory,then calculate the steering parameter.Furthermore,it is worth noting that unlike the Bell inequality of LHV model,the violation of a steering inequality is only a sufficient condition rather than a necessary condition for ruling out LHS model.However even a sufficient criterion is valuable in practical applications,for instance,in one side device independent QKD.[31]In fact,it is an attractive field in recent years to search more universal and tighter steering criteria.[38−39]

The analysis above is based on two typical measurement methodsφ=0 andπ/2.In circuit QED system,the LO phase of a homodyne measurement can be adjusted continuously to realize different measurement methods.We consider a steering parameterin which LO phase corresponding to|σ(l)x|is fixed onπ/2 while LO phase corresponding to ∆σ(m)yand ∆σ(n)zchange from 0 toπ/2 continuously.When the phase difference ∆φ=0,it degenerates into the trivial situation with same measurement method and cannot show any steering.When the phase difference ∆φ=π/2,it is the situation withφ1andφ2discussed above.For ∆φbetween 0 andπ/2,noticing that Γbaand Γciare monotonic functions ofφ,we can infer thatis between the two situations of ∆φ=0 andπ/2.The surface plot in Fig.3 confirms our analysis.It can be seen when the measurement is very weak,there is no proper ∆φthat can show steering.As the measurement becomes stronger,the critical ∆φbecomes smaller.A violation bound is plotted to show this directly.Generally speaking,showing steering requires a long measurement time and a large phase difference.

In practical circuit QED system,the detector efficiencyηis always an important restriction and be treated as an indicator of the “detector dependency” of a quantum system.[40−41]We now discuss the effect on steering by a non-ideal detector.Withηchanging from 1.0 to 0.5,the violation bounds are plotted in Fig.4.As the detector efficiency decreases,the violation region narrows.When the efficiency is less than 0.7,the violation region disappears and we cannot check whether steering exists through multiplicative inequality any more.To see this more clearly,we show the minimum steering parameter versus efficiency in the inset of Fig.4.Comparing with an ideal detector,low detector efficiency will bring about more mixed components in the system state.Steering is a quantum phenomenon just like the density matrix inseparability and Bell nonlocality,while the mixed state is described by classical probability.Thus it is not surprising that low detector efficiency will depress quantum steering.Daryanooshet al.proved that for all processes described by QTE,the lowest efficiency to show steering is 0.5.Our results do not violate their prediction.[41]We also notice that,in the state-of-the-art experimental setup,a quantum efficiency ofη=0.67 is reported,[42]which almost satisfies the steering requirement.

Fig.3 (Color online) Steering parameterversus measurement time t and LO phse difference ∆φ,in which φm,φn change from 0 to π/2 and φl=π/2.The black line is the bound between keeping and violating inequality.

Fig.4 (Color online)Bound of under different detector efficiencies η,which vary from 0.5 to 1.0.When η <0.7,the violation region disappears.Inset indicates the minimum of under different efficiencies.

5 Summary

In summary,on the foundation of introducing measurement in circuit QED system and EPR steering,we obtained a set of multiplicative steering inequalities as criteria for the steering of a qubit by the measurement of the cavity field.Through simulation,we found that unlike in other widely studied systems,there are only several specified types of steering inequality can be used to show steering,and we also analyzed the reason for this.We then discussed several conditions that may affect steering in practical experiment,including measurement strength,method and detector efficiency.By adjusting the LO phase continuously,we found that for continuous weak measurement,a large LO difference is helpful.High detector efficiency also has benefits and there exists a lower measurement efficiency bound of steering.

Our work may be helpful to EPR steering experiments in circuit QED system.For now,steering experiments in the photon system mostly focus on projective measurement,while the technique of weak measurement of photons has already been applied to the measurement of photon states.[43]This work also sheds light on the steering with weak measurement The comparison and connection of steering with density matrix inseparability and Bell nonlocality in circuit QED system is also an interesting potential work.The preliminary discussion in this work may also be generalized to a multi-qubit circuit QED system.

Appendix A:Derivation of the Multiplicative Steering Inequality

For circuit QED system with the LO phaseφm,the output currentI,the qubit observableσi,and the qubit measurement results,the conditional average ofsunder the currentIis

whereP(s|I) is the probability of gettingsunder condition of gettingI,and the conditional probabilities therein-after are in similar forms.The conditional variance is

When measuring the cavity field,if there exists an LHS model for the qubit,the conditional qubit state underIcan be written as

where the subscriptqinρ(q|I)means “qubit”.According to the quantum measurement theory,

whereMsis the measurement operator of obtaining resultswhen measuring the qubit.For arbitraryaandb,according to probability theory we have the following relation:

Inserting Eq.(A3) into Eq.(A5) we obtain Defining the variance ofσiunder measurement methodmas ∆2σmi=∑

I P(I|m)∆2(σi|I),considering Eq.(A6) we obtain

For a quantum stateρλ,the Heisenberg uncertainty relation is

Here we use Cauchy-Schwarz inequality:letandbe

Noticing thatλis independent from measurement method,i.e.P(λ|m)=P(λ|n)=P(λ),then we obtain

we obtain

Insert Eq.(A12) into Eq.(A10) we finally have

Appendix B: More System Parameters

Fig.5 (Color online) Γm vs χ,κ and ϵm.

The system parametersχ,κ,andϵmdetermine the measurement rate Γm,as well as the qubit evolution,i.e.,they affect the qubit indirectly through the measurement rate.Whenχandϵmincrease,the coupling between cavity field and qubit,and the cavity driving amplitude is strong,which lead to a large Γm.However whenκincreases,Γmdecreases.It is because even though the photon leaking rate is large,the cavity field difference becomes small.Thus the information carried out by the photons in unit time decreases.For more results with different system parameters are given below.The multiplicative steering inequality is valid in a typical range of experimental system parameters.

The dependence of Γmunder steady state and the system parameters are plotted in Fig.5.More over,steering parameterunderχ=(0.05,0.1,0.5,1.0) MHz,κ=(50,100,150,200) MHz,andϵm=(100,200,400,800) MHz are plotted in Fig.6.It can be seen that our result is valid in a typical range of experimental system parameters.

Fig.6 (Color online)with different parameters.

Acknowledgement

The author thanks Prof.Xin-Qi Li of the Center for Joint Quantum Studies,Tianjin University,and Prof.Qiong-Yi He of the State Key Laboratory of Mesoscopic Physics,Peking University,for helpful discussion.