Controlling stationary one-way steering in a three-level atomic ensemble

Jie Peng(彭洁), Jun Xu(徐俊), Hua-Zhong Liu(刘华忠), and Zhang-Li Lai(赖章丽)

1Department of Basic Courses,Wuhan Donghu University,Wuhan 430071,China

2College of Physical Science and Technology,Central China Normal University,Wuhan 430079,China

3College of Mathematics and Physics,Jinggangshan University,Ji’an 343009,China

Keywords: one-way quantum steering,entanglement,quantum correlation,reservoir-engineered method

1.Introduction

Ever since Einstein, Podolsky and Rosen (EPR) demonstrated that measuring one of a pair of quantum-entangled particles instantaneously affects the other, regardless of their physical separation, it has been observed that these particles can exist in two distinct states simultaneously.[1]In 1935,Schr¨odinger introduced the concept of steering to provide a broader framework for understanding the strange phenomenon of action at a distance.[2]Wisemanet al.showed that EPR steering can be described as a new kind of quantum correlation, which is stronger than entanglement but weaker than Bell nonlocality.[3]This suggests that quantum steering can transition from being a conceptual idea to a quantifiable theoretical and experimental reality.A violation of the Bell inequality means that a quantum bipartite state can exhibit steering in both directions.EPR steering is inherently distinct from entanglement and Bell nonlocality, i.e., one-way EPR steering.[4-8]Asymmetric one-way quantum steering is demonstrated as a situation in which Alice can steer Bob but Bob cannot steer Alice,or vice versa.[9,10]This unique feature has attracted significant attention and is widely used in practical applications, such as directional recessive conduction,one-side device-independent quantum cryptography,quantum teleportation and high-fidelity heralded teleportation.[11-14]Many different criteria based on the uncertainty principle have been proposed to decide whether a quantum state is steerable.Reid put forward the criterion of a continuous variable Gaussian state.[15]Wisemanet al.raised a criterion applicable to discrete and continuous variables.[3]Walbornet al.presented a generalized entropy criterion for the continuous variable non-Gaussian state.[16-18]Ioannis Kogiaset al.introduced a computable measure of steering for arbitrary bipartite Gaussian states.[19]In addition to the relevant theoretical advances,[15-21]quantum steering experiments have also made great progress.[5,22-29]For example,Handchenet al.observed one-way EPR steering of two-mode Gaussian squeezed states for the first time.[27]Armstronget al.showed multipartite EPR steering and genuine tripartite entanglement with optical networks.[22]Xiaoet al.experimentally demonstrated one-way EPR steering with multimeasurement settings for a class of two-qubit states, which are still one-way steerable even with infinite settings,[29]and so on.

One-way EPR steering plays an important role in quantum information and fundamental physics.Over the past decades, much work has been devoted to finding effective ways to test the quantum effect in atomic systems,[30-35]such as quantum jumps in atomic systems,[30]strong coupling of optical nanoantennas and atomic systems,[32]and weak-interaction effects in heavy atomic systems.[33]Many schemes for quantum steering based on atomic systems have been proposed.[36-42]For instance,Xuet al.employed a twolevel atomic ensemble driven by a strong laser field inside a two-mode cavity and showed a significant enhancement of the output steady-state quantum correlations via a dissipative atomic reservoir.On the condition of an unbalanced coupling strength,asymmetrical output one-way EPR steering has been achieved.[37]Tanet al.proposed a scheme for realizing hybrid atom-mechanical quantum steering in the steady-state regime.An optomechanical two-mode cavity was coupled to a distant ensemble of double-Λ atoms in a cascade setup, leading to a dissipative interaction between the mechanics and the internal atomic states.[38]Heet al.studied multipartite entanglement,the generation of EPR states and quantum steering in a threemode optomechanical system composed of an atomic ensemble located inside a single-mode cavity with a movable mirror.Two-way EPR steering was found between the mirror and the atomic ensemble despite the fact that they were not directly coupled to each other.[39]

Here, we present a scheme for generating tunable asymmetric quantum steering in a three-level Λ-type atomic system.The cavity modes are generated from two atomic dipoleallowed transitions,which are driven by two external classical fields, respectively.When two cavity modes are tuned to be resonant with Rabi sidebands of the dressed atom,two collective Bogoliubov modes formed by the original cavity modes interact effectively with the dressed atom.The atomic ensemble, which serves as an engineered reservoir, can cool two Bogoliubov modes into a vacuum state.At this moment, the original cavity modes reach a squeezing state.It is because of the two Bogoliubov dissipation pathways that entanglement is greatly enhanced.The best obtained state tends to be the original EPR entangled state.By adjusting the normalized detuning and the cavity damping rates,steadystate one-way quantum steering of the intracavity and output fields can be achieved.To describe the physical process more clearly, the dressed state representation[46]and Bogoliubov transformation[47]have been used.Compared with other schemes, our scheme has the following features: (i) A fourwave mixing process in the three-level atomic system is used to generate the entangled light.[43-45]This is very easy to implement experimentally.(ii) The obtained asymmetric quantum steering by two Bogoliubov dissipation pathways is robust against environmental noise and does not require the initial preparation of nonclassical states.(iii)The direction of the one-way quantum steering can be easily controlled by adjusting the normalized detuning and the cavity damping rates.

The rest of this article is organized as follows.In Section 2,we describe the system model and present the interaction equations.In Section 3,we discuss one-way EPR steering and entanglement of intracavity fields, and then analyze the physical mechanism.The output steering and entanglement are given in Section 4.The article ends with a conclusion in Section 5.

2.Model and equations

As sketched in Fig.1(a), an ensemble of three-level Λ atoms placed at the intersection of two cavities is driven by two external pumping fields with Rabi frequenciesΩ1andΩ2.This atomic ensemble has a ground state|1〉,a metastable state|2〉 and an excited state|3〉, as shown in Fig.1(b).A large number of atomic structures can be used as candidates,for example, the rubidium 87 D1transition,|1〉=|52S1/2,F=1〉,|2〉=|52S1/2,F=2〉and|3〉=|52P1/2,F=1〉.The pumping field withΩ1drives atoms from the ground state|1〉to the virtual energy level below the excited state|3〉.Since the virtual level is an unstable state, it will emit a photon ofa2back to the virtual energy level above the ground state|2〉.Then it absorbs a pumping photon ofΩ2and emits a photon ofa1.This is commonly known as the four-wave mixing process.In this case,energy and momentum conservation conditions must be satisfied, i.e.,ω1+ω2=ν1+ν2andkω1+kω2=kν1+kν2.ωlis the corresponding driving field frequency,νlis the cavity field frequency andkωl,νlis the corresponding wave vector.

Fig.1.(a) An ensemble of Λ-type atoms are placed at the intersection of two cavities driven by two external coherent fields with half Rabi frequencies Ω1,2 (b)The bare atomic states and transitions in Λ configuration.Ω1,2 are applied to the atomic transitions|1,2〉→|3〉,and separately tuned from the atomic transitions by ∆1 =-∆2 =∆.Two cavity fields a1,2 are detuned from the driving field frequencies by ˜Ω,where ˜Ω is the spacing between adjacent dressed states.

In the dipole approximation and an appropriate rotating frame,the master equation for the atom-field density operatorρis written as

Next, we transform the atomic variables into a dressedstate representation.For simplicity, we take the antisymmetrical tuning∆1=-∆2=∆and the same Rabi sidebandsΩ1=Ω2=Ω.We assume that the Rabi frequency is real and much stronger than the atomic decays and cavity lossesΩ ≫gl〈al〉,γl,κl.By diagonalizing the HamiltonianH0,the dressed states are expressed as

withg11=-g1cos2θ/2,g12=g1sinθ(1-sinθ)/2,g21=-g2cos2θ/2,g22=g2sinθ(1-sinθ)/2.Physically,it is easy to find that the transition|0〉→|+〉is accompanied by the absorption of ana1photon and the emission of ana2photon.Correspondingly, the transition|0〉→|-〉is accompanied by the emission of ana1photon and the absorption of ana2photon.

Quantum correlations can be calculated by following the standard technique.[48]From the master equation, we derive the set of linearized Langevin equations for the operatorsδo=o-〈o〉as follows:

whereγ′c1=γc1N0+γc2N-andγ′c2=γc1N++γc2N0.The steady-state solution of the second-order moments can be obtained from Eq.(9).

In order to calculate the quantum correlations between the two modes, it is desirable that the system should be stable.According to the Routh-Hurwitz criterion,[50-52]the stability conditions of the system can be calculated to be

3.Quantum steering and entanglement of intracavity fields

and the steering measurement in the directiona2→a1can be obtained by swappinga1anda2,

Here we introduce the steering asymmetry defined as|GAGB| in order to check the asymmetric steerability of the twomode Gaussian states.

Figure 3 shows the intracavity entanglementEN(black solid line), the intracavity steeringGA(red dashed line),GB(blue dot-dashed line) and the steering asymmetry|GA-GB|(green dotted line) versus cavity loss rate ratioκ1/κ2for different normalized detunings∆/Ω: (a)∆/Ω=-0.7, (b)∆/Ω=-0.5,(c)∆/Ω=-0.4,(d)∆/Ω=-0.1.The other parameters are the same as in Fig.2.The cavity loss rate ratioκ1/κ2determines the asymmetry of the system.It can be seen that the entanglement and the steering all drop rapidly with the increase ofκ1/κ2and thea1→a2steeringGAusually vanishes earlier than thea2→a1steeringGB.Therefore there is a situation where there is only one-waya2→a1steering.Taking Fig.3(a)as an example(∆/Ω=-0.7), bothGAandGBare greater than zero in the region of 0<κ1/κ2<3.2.This means that two-way EPR steering can be obtained.When 3.2<κ1/κ2<7,GA=0 andGB>0 and it means one-waya2→a1steering.Forκ1/κ2>7,GA=0 andGB=0,which indicates that there is no-way quantum steering between two cavity modes.When∆/Ω=-0.1, as shown in Fig.3(d),GA=0 andGB=0 while the entanglement still exists.This illustrates that the quantum steering is more fragile than the entanglement against the cavity loss rate ratio.We can control the normalized detuning∆/Ωand cavity loss rate ratioκ1/κ2to obtain large one-way steering while retaining a relatively high amount of entanglement.

Fig.2.The intracavity entanglement EN (black solid line), the intracavity steering GA (red dashed line), GB (blue dot-dashed line) and the steering asymmetry|GA-GB|(green dotted line)versus the normalized detuning ∆/Ω for different cavity loss rate ratios κ1/κ2: (a)κ1/κ2=1,(b)κ1/κ2=3,(c)κ1/κ2=4,(d)κ1/κ2=16.The other parameters are γ1=γ2=1,κ2=0.1,=5.In order to show the steering directivity clearly,we use the pink area to represent the presence of no-way steering,the green area to the presence of one-way a2 →a1 steering and the purple area to the presence of two-way steering.

Fig.3.The intracavity entanglement EN (black solid line),the intracavity steering GA (red dashed line),GB (blue dot-dashed line)and the steering asymmetry|GA-GB|(green dotted line)versus cavity loss rate ratio κ1/κ2 for different normalized detunings ∆/Ω: (a)∆/Ω =-0.7,(b)∆/Ω =-0.5,(c)∆/Ω =-0.4,(d)∆/Ω =-0.1.The other parameters are the same as in Fig.2.The area colors have the same meaning as in Fig.2.

To understand the internal mechanisms more clearly, we introduce a pair of Bogoliubov modesb1andb2, which are unitary transformations of the cavity modesa1anda2with a two-mode squeezed operator,respectively,

withga=-gcos2θ/2,gb=gsinθ(1-sinθ)/2 by assumingg1=g2=g.Physically,this Hamiltonian implies that the absorption of Bogoliubov modesb1,2is accompanied by the transitions|0〉→|±〉.There are two dissipation channels to generate nonclassical states.If the steady-state population of state|0〉is larger than that of states|±〉,i.e.,N0>N±,the dissipation processes are dominant over the amplification,giving rise to the suppression of quantum fluctuations.That is to say that the dressed atoms can be exploited to cool the Bogoliubov modesb1andb2to near ground states in a long enough time.The joint ground state ofb1andb2is a two-mode squeezed vacuum state of the cavity modesa1anda2, which can be checked by

4.Output steering and entanglement spectra

wheren1(ω)andn2(ω)represent the output spectra for thea1anda2modes,respectively,andnx(ω)represents the correlation between theωcomponent in thea1,out(t) mode and the-ωcomponent in thea2,out(t)mode.Next we analyze the entanglement spectra betweena1,out(ω)anda2,out(-ω).The entanglement of the system can be easily computed numerically using the covariance matrix and the definition of logarithmic negativity.In order to analyze the output quantum correlation more clearly, we map the output state into a two-mode squeezed thermal state.Then,combining with Eqs.(11)-(13),we can plot the entanglement and quantum steering of the output fields.

Figure 4 shows the output entanglementEN(0) (black solid line),the output steeringGA(0)(red dashed line),GB(0)(blue dot-dashed line) and the steering asymmetry|GA(0)-GB(0)| (green dotted line) at zero frequency versus the normalized detuning∆/Ωfor different cavity loss rate ratiosκ1/κ2: (a)κ1/κ2= 1, (b)κ1/κ2= 3, (c)κ1/κ2= 8, (d)κ1/κ2=15.The other parameters areγ1=γ2=1,κ2=0.1,With the increase of the normalized detuning∆/Ω, the output entanglement and steering first increase and then decrease.When increasing the cavity loss rateκ1,the maximal values of entanglement and steering decrease gradually.These features are similar to the intracavity fields.However, unlike the intracavity steering, the direction of the output steering can be changed by adjusting the normalized detuning.Taking Fig.4(b) as an example, in the region of-0.76<∆/Ω<-0.68,there is one-waya2→a1EPR steering.For-0.68<∆/Ω<-0.5, two-way EPR steering occurs.In the region of-0.5<∆/Ω<-0.3,one-waya1→a2EPR steering arises.Whenκ1/κ2=8,as shown in Fig.4(c),two-way EPR steering disappears and the range of one-way quantum steering becomes larger.Therefore,the output fields have more abundant quantum steering phenomena than the intracavity fields.

Figure 5 shows the output entanglementEN(0) (black solid line),the output steeringGA(0)(red dashed line),GB(0)(blue dot-dashed line) and the steering asymmetry|GA(0)-GB(0)| (green dotted line) at zero frequency versus cavity loss rate ratioκ1/κ2for different normalized detunings∆/Ω:(a)∆/Ω=-0.77, (b)∆/Ω=-0.6, (c)∆/Ω=-0.48,(d)∆/Ω=-0.1.The other parameters are the same as in Fig.4.It can be seen that the entanglement and the steering first increase and then decrease with the increase ofκ1/κ2.The variation trend is different from that of intracavity fields.As shown in Fig.5(b), one-waya2→a1EPR steering can be obtained in the region of 0<κ1/κ2< 0.43 and 5.38<κ1/κ2<15.For 0.43<κ1/κ2<5.38,there is two-way EPR steering.When∆/Ω=-0.48, as shown in Fig.5(c), onewaya1→a2EPR steering can be achieved in the region of 1.12<κ1/κ2<9.24.Therefore, the direction of the output steering can be controlled by adjusting the normalized detuning∆/Ωand cavity loss rate ratioκ1/κ2.

Fig.4.The output entanglement EN(0)(black solid line),the output steering GA(0)(red dashed line),GB(0)(blue dot-dashed line)and the steering asymmetry|GA(0)-GB(0)|(green dotted line)at zero frequency versus the normalized detuning ∆/Ω for different cavity loss rate ratios κ1/κ2:(a)κ1/κ2=1,(b) κ1/κ2 =3, (c) κ1/κ2 =8, (d) κ1/κ2 =15.The other parameters are γ1 =γ2 =1, κ2 =0.1, g1N =g2N =1.The yellow area represents the presence of one-way a1 →a2 steering.The other colors have the same meanings as in Fig.2.

Fig.5.The output entanglement EN(0) (black solid line), the output steering GA(0) (red dashed line), GB(0) (blue dot-dashed line) and the steering asymmetry|GA(0)-GB(0)|(green dotted line)at zero frequency versus cavity loss rate ratio κ1/κ2 for different normalized detunings ∆/Ω: (a)∆/Ω =-0.77,(b)∆/Ω =-0.6,(c)∆/Ω =-0.48,(d)∆/Ω =-0.1.The other parameters are the same as in Fig.4.The colors have the same meaning as in Fig.4.

5.Conclusion

In conclusion,we have used a four-wave mixing process to generate asymmetric EPR steering in a three-level Λ-type atomic ensemble.One-way quantum steering of both the intracavity fields and the output fields is achieved.Furthermore,we find that the output fields have more abundant quantum steering phenomena than the intracavity fields.The direction of the output steering can be switched by adjusting the normalized detuning and the cavity damping rates.However,the intracavity fields only have one-waya2→a1steering.The asymmetric quantum steering based on reservoir dissipation is robust against environmental noise.This is an experimentally feasible scheme,which has potential applications in quantum information tasks for quantum secret sharing protocols.

Appendix A

The damping rates in terms of the dressed states are given as

Acknowledgment

Project supported by Wuhan Donghu University Youth Foundation of Natural science(Grant No.2022dhzk009).