Tian-Le Yang(杨天乐), Chen-Long Zhu(朱陈龙), Sheng Liu(刘声), and Ye-Jun Xu(许业军)

International Research Center of Quantum Information and Photoelectric Information,School of Mechanical and Electronic Engineering,Chizhou University,Chizhou 247000,China

Keywords: entanglement,Kerr interaction,Coulomb interaction,optomechanics

1. Introduction

Quantum entanglement is one of inherently physical phenomena that it is considered to be important resource not only in quantum information processing,[1–6]e.g.,performing communication and computation tasks with an efficiency which is unachievable in classical world,[7,8]but also in setting the transition between classical and quantum world.[9–11]As a basic feature of quantum theory,quantum entanglement is a significant resource of quantum network that allows the inseparable quantum correlation to be shared between distant parties.[12,13]The generation of quantum entanglement is a crucial task in quantum information processing. Many methods for generating quantum entanglement have been proposed,[14–21]in which the principle of Schr¨odinger’s cat is often the basis of experimental proposals to observe macrosopic quantum entanglement.[22]More recently entanglement has been experimentally achieved in microscopic objects such as photons,phonons and atoms.[23,24]

A cavity optomechanical system is an ideal platform for preparation of entangled states. Over the past few years, due to its unique applications,the research on entangled states has attracted considerable interest,such as the high-sensitivity detection of small masses or forces with the precision of quantum limits.[25–30]A standard optomechanical system is composed of a movable mirror coupling to a Fabry–P´erot cavity by the radiation pressure force.The variant based on the standard optomechanics is abundant,for example,nanoelectromechanical systems formed by a microwave cavity capacitively coupling to a nanoresonator,[31]cavity optomechanics with stoichiometric SiN films,[32]and a semitransparent membrane within a standard Fabry–P´erot cavity.[33,34]More recently,many hybrid optomechanical systems have emerged, including a hybrid cavity magnomechanical system,[23]an atomic ensemble placed within an optical Fabry–P´erot cavity,[35,36]a nonlinear Kerr medium inside an optomechanical cavity, and so on.[33]Especially, the effect of Kerr medium on the dynamics of optomechanical systems was extensively investigated.[37–40]Generally,the entanglement in optomechanical systems is limited by photon blockade mechanism.[41–44]

Here, we consider an optomechanical cavity coupling to an external nanomechanical oscillator (NMO) and the cavity is additionally filled with a Kerr medium.[45–47]The range of interaction distance is from nanometer to meter due to the Coulomb coupling,[48,49]and the strength can be adjusted by the bias voltage.[50,51]We primarily discuss the effect of Kerr interaction and Coulomb coupling strength on the stationary optomechanical entanglement. The entanglement is enhanced with increasing the Coulomb coupling strength. In consideration of Kerr interaction, this leads to the photon blockade,which will lower the entanglement. Utilizing the adjustability of Kerr interaction and Coulomb coupling strength, it is easy to accomplish the transformation between different degrees of entanglement. All entanglements are robust against environmental temperature. We use the standard Langevin formalism to model the system,solve the linearized dynamics and quantify the entanglement in the stationary state. Compared with a related research in Ref.[51],which models a hybrid optomechanical system with a Fabry–P´erot cavity coupling an external NMO through Coulomb force for studying the entanglements between two NMOs, here our work considers that the cavity is additionally filled with a Kerr medium. We mainly demonstrate the influences of Kerr interaction,Coulomb coupling strength and cavity decay rate,i.e.,on the entanglement between two NMOs. We also show the effects of Kerr interaction and Coulomb coupling on the entanglement among cavity and NMO1. Even in the absence of Kerr interaction, weak degree of entanglement can be detected among them.

2. Model and steady state solutions

We consider an optomechanical cavity filling with the Kerr medium coupling to an external nanomechanical oscillator via Coulomb interaction,as shown in Fig.1. One strong pump field with frequencyωpis injected into a high quality Fabry–P´erot cavity with frequencyωc. The cavity is composed of a fixed mirror and a movable mirror NMO1,which is charged by the bias gate voltageV1and subject to the Coulomb force due to another charged NMO2with the bias gate voltage−V2. Herer0is the equilibrium distance between the two NMOs.

The Hamiltonian of the system reads

Fig.1. The sketch of the optomechanical system. The cavity contains the Kerr medium.

Using a standard method in quantum optics, we take the expectation with respect to the steady state of Eqs. (2)–(6).[54–57]Neglecting all the fluctuations,we set

3. Entanglement

Equation(15)is a liner equation forVand can be straightforwardly solved,whereas the explicit expression is too cumbersome and will not be reported here.

To investigate bipartite entanglement of the system, we can trace over the remaining degrees of freedom. Here we adopt the logarithmic negativityENas a convenient measure of entanglement,[59,60]which is defined as

The stationary entanglement of the system in the steady state is guaranteed when all eigenvalues of the drift matrixApossess negative real parts,which can be derived using Routh–Hurwitz criteria.[62]For numerically illustrating the effect of Kerr medium in a hybrid cavity optomechanical system,a set of experimentally accessible parameters have been taken into account here:m=5 ng,L=1 mm,andκ=88.1 MHz.[63,64]For simplicity, we have taken equal parameters of the two NMOs,i.e.,ω1=ω2=ωm=2π×100 MHz,γ1=γ2=γm=2π×100 Hz,and the cavity-field wavelengthλ0is 810 nm.

Fig. 2. Logarithmic negativity EN with respect to the normalized detuning ∆/ωm at temperature T =4 mK and driving power P=50 mW.Here we use λ =0.9ωm in(a)and χ =3 Hz in(b). See text for details of the other parameters.

Generally speaking,ENis repressed by the large Kerr parameterχ. The results on the behavior ofENwith varyingχare shown in Fig. 2(a), it clearly demonstrates thatENis repressed with increasingχ. In the case ofχ= 0, the significant entanglement remains near∆/ωm= 0.6 and forχ=5 Hz it is shifted towards larger detuning value around∆/ωm=0.8. More close to the point, with increasingχ, the range of entanglement between two NMOs can be narrowed,i.e.,from∆/ωm∈[0.2,1.6](solid line)to∆/ωm∈[0.45,1.4](dot-dashed line). This phenomenon indicates that the photon blockade greatly lowers the number of the cavity photon,which is induced by the Kerr interactionχ, thusENcan be spoilt when the effective coupling between the two NMOs is weakened, as is shown in Fig. 3, whereNis the mean intracavity photon number and is defined asN=|cs|2.[65]In Fig.3(a), we can see that negative values of cavity-pump detuning are necessary to observe the optical bistable behavior with the increasingχ. Therefore, we choose∆c=−3ωmin Fig. 3(b), and we can observe thatNdecreases with largeχ.In contrast,Fig.2(b)reflects thatENis enhanced via increasing the Coulomb coupling parameterλ.The optimal entanglement is shifted from∆/ωm=0.9(solid line)to lower detuning value around∆/ωm=0.7(dot-dashed line). The effective coupling between the two NMOs is enhanced with increasing the Coulomb coupling strengthλ. Simultaneously, the existence region ofENbetween the two NMOs can be enlarged,i.e., from∆/ωm∈[0.6,1.2] (solid line) to∆/ωm∈[0.3,1.5](dot-dashed line). The wider the effective detuning range is,the easier it is to achieve in experiments. These two figures imply that Kerr interactionχand Coulomb coupling strengthλare the two main factors affecting the entanglement between the two NMOs.ENwill disappear with large Kerr interactionχ=10 Hz or small Coulomb coupling strengthλ=0 (not shown here),respectively.

Fig.3.(a)Mean intracavity photon number as a function of ∆c with different χ. (b) Mean intracavity photon number as a function of χ with∆c=−3ωm.

For further investigation about the effect of Kerr interactionχand Coulomb coupling strengthλonEN, we plot in Fig. 4(a)ENversusλ/ωmand in Fig. 4(b)ENversusχrespectively. In Fig. 4(a), it clearly displays that the two NMOs are unentangled below a critical Coulomb coupling strengthλcandENmonotonically increases withλin the rangeλ ∈[λc,ωm]. Critical valuesλc1=0.05ωmforχ=0(solid line),λc2=0.13ωmforχ=3 Hz (dashed line), andλc3=0.29ωmforχ=5 Hz(dot-dashed line)are observed. It is obvious from Fig.4(b)that the two NMOs are disentangled above a critical Kerr interactionχcandENmonotonically decreases withχin the rangeχ ∈[0,χc].We find critical Kerr interactionsχc1=5 Hz forλ=0.3ωm(solid line),χc2=7.2 Hz forλ=0.6ωm(dashed line), andχc3=9 Hz forλ=0.9ωm(dot-dashed line).ENdecreases with increasing the Kerr interaction due to the photon blockade mechanism.It is worth mentioning that the critical value of Coulomb coupling strengthλcincreases with Kerr interactionχ. Therefore, we infer that whenχincreases,λcshould also increase. Only in this limit can the entanglement between the two NMOs be guaranteed.

Fig. 4. The logarithmic negativity versus (a) λ/ωm, and (b) Kerr parameter χ. The parameters are the same as in Fig.2 but for ∆=0.9ωm.

The dependence ofENon driving powerP, Kerr interactionχand coupling strengthλare presented in Figs. 5(a)and 5(b), respectively. Apparently,ENhas two distinct behaviors with different driving powersP. In the rangeP ∈[2 mW,15 mW],ENincreases very quickly with respect toP.If we continue to increase the driving powerP,ENwill decrease gradually. In particular,in Fig.5(a),ENincreases with Kerr interaction beingχ=0.It is crucial to analyze the robustness ofENas a function of the environmental temperatureT.As shown in Figs.6(a)and 6(b),we can see the expected decay ofENfor increasing temperatureT.In Fig.6(a),in the absence of Kerr medium,ENis very robust as a function ofT.With the increase ofχ, the value and the critical value of temperatureTcdecrease (Tcis defined asT>Tc,EN=0). This explains that the robustness ofENis inversely correlated with Kerr interactionχ.We can likewise explain the phenomenon in terms of photon blockade. In contrast,it is found from Fig.6(b)that the value and the critical value of temperatureTcincrease with increasingλ. Therefore,the robustness ofENis enlarged with increasingλ.

Fig.5. The logarithmic negativity EN with respect to the driving power P(mW)at temperature T=4 mK,and normalized detuning ∆=0.9ωm.Here we set λ =0.9ωm in(a)and χ =3 Hz in(b).

Fig.6. EN as a function of the environment temperature T (in units of K).The parameters are the same as in Fig.5 but for P=50 mW.

In general,lower cavity decay rateκis needed in an optomechanical system to improve the intracavity photon number. With large cavity decayκ,ENdisappears completely,which can be derived from the steady state solutioncs=εp/(κ+i∆). The biggest entanglement is obtained nearκ/ωm=0.2 in Fig.7(a),and the cavity decayκcorresponding to maximal entanglement increases withλin Fig. 7(b). The range of entanglement can be narrowed, i.e., fromκ/ωm∈[0.03,0.9](solid line)toκ/ωm∈[0.03,0.8](dot-dashed line)with increasingχ. In contrast,the range of entanglement can be enlarged greatly withλ.

Fig. 7. EN versus cavity decay rate κ/ωm. Other parameters are the same as in Fig.5 but for P=50 mW.

Fig. 8. EN with respect to frequency ratio φ when the two charged NMOs have different frequencies(ω2=φω1). The rest of the parameters are P=50 mW and T =4 mK.

In fact,identical frequencies of two NMOs are hard to be guaranteed. Therefore,one important task is to analyzeENin the general case of frequency ratioφ/=1. The maximal entanglement occurs only in the small range aroundφ=1,as shown in Figs.8(a)and 8(b),which means that the difference in frequency values between two NMOs should not be significant.Figure 8(a)partly reflects that without the Kerr interactionχ,the maximal entanglement between the two charged NMOs is obtained whenφ=1. With increasingχ, the value of maximal entanglement decreases, and the maximal entanglement only occurs for higher value ofφ(ω2>ω1). In Fig. 8(b),the maximal entanglement is obtained whenφ=1.1,and the entanglement increases withλ. However, the optimal entanglement corresponds to the identical value ofφwhile varyingλ.On the contrary,the Kerr interactionχcan change the value ofφcorresponding to optimal entanglement.

We plotENin Fig.9 for the NMO-NMO entanglement in the case of different decay rates of two NMOs. The entanglement is anti-correlation withΓ, and therefore the decay rate of NMO1should be larger than the decay rate of NMO2to attain optimal entanglement. In this case,the maximal entanglement appears atΓ=0. The above discussion implies that the NMO-NMO entanglement can be enhanced by the decay rate of NMO1and Coulomb coupling strengthλ. The entanglement will decrease if the Kerr interactionχincreases in terms of photon blockade mechanism.

Fig. 9. EN with respect to decay rate ratio Γ when the two charged NMOs have different decay rates(γ2=Γ γ1). The rest of the parameters are P=50 mW,φ =1 and T =4 mK.

Lastly, we present the entanglement between cavity and NMO1. HereV4should be changed, which is obtained by disregarding the third and fourth rows and columns inV. In this three-mode hybrid system,the entanglement between two NMOs is stronger than that between the cavity and NMO1.Simultaneously,the entanglement between the cavity and NMO2is too weak (10−7in order of magnitude) and will not be demonstrated here. Figure 10 clearly displays that the degree of entanglement between the cavity and NMO1is very small.Here,the effects ofχandλare similar to the cases in Fig.2.It is worth noting thatgis the coupling parameter between NMO1and the cavity field. Heightening the value ofφ, the value ofgwill increase, and then the entanglement between the cavity and NMO1will be enhanced.

Fig.10. EN between the Fabry–P´erot cavity and NMO1 with respect to the normalized detuning ∆/ωm. We adopt χ =0 in(b). Other parameters are consistent with those in Fig.2.

4. Conclusion

In summary, we have presented the NMO-NMO entanglement in an optomechanical system with a Kerr medium and two charged NMOs, where one oscillator is coupling to another oscillator via Coulomb interaction. Using possible strategies to measure the entanglement, we find that the generated entanglement can be controlled by adjusting the Kerr interaction,Coulomb coupling strength and the frequency ratio of two NMOs. The maximal entanglement can be obtained when the decay rate of NMO1is larger than the decay rate of NMO2(γ2<γ1). The NMO-NMO entanglement is spoilt by Kerr interaction due to the photon blockade mechanism. In contrast to the case with Kerr interaction, the Coulomb coupling strength can enhance the NMO-NMO entanglement due to stronger effective coupling between two NMOs. Moreover,Kerr interaction has an important influence on changing the values of cavity decay rate and frequency ratio of two NMOs corresponding to maximal entanglement. Our results indicate that this hybrid optomechanical system could provide a promising platform for research of macroscopic quantum phenomena.