Heng-Mei Li(李恒梅), Bao-Hua Yang(楊保華), Hong-Chun Yuan(袁洪春), and Ye-Jun Xu(許業(yè)軍)

1School of Information Engineering,Changzhou Vocational Institute of Mechatronic Technology,Changzhou 213164,China

2School of Electrical and Information Engineering,Changzhou Institute of Technology,Changzhou 213032,China

3School of Mechanical and Electronic Engineering,Chizhou University,Chizhou 247000,China

Keywords: optomechanical system,quantum scissors device,quantum state engineering,linear entropy

1. Introduction

In recent decades,physicists have devoted themselves to preparation and manipulation of highly non-classical quantum states in order to meet requirements of quantum information processing.[1–3]Among them, many protocols using non-Gaussian operations such as photon addition,[4–8]photon substraction,[9–11]and various combinations[12–14]have been proposed to achieve novel non-classical quantum states,which is one of most effective methods. Agarwal and Tara first introduced the photon-added coherent state obtained by repeated applications of the creation operator on a coherent state,exhibiting both higher-order squeezing and sub-Poissonian characteristics.[4]In addition, these non-Gaussian operations can also enhance the nonlocality and increase the degree of entanglement of quantum states,[15,16]which was then used to improve the fidelity of quantum teleportation,[17]the performance of the quantum key distribution,[18]and quantum metrology.[19]

Another effective scheme is known to quantum truncation based on quantum scissors device(QSD)proposed by Pegget al.,[20]which enables generation of superposition of vacuum and single-photon states by truncating a coherent state. In the above scheme, beam splitters (BSs) and condition measurements were mainly utilized. Indeed, QSD is just a non-local phenomenon depending on entanglement where there was no light traveled from the input mode to the output mode.The basic idea of QSD had been extended to generate not only qubits but also qutrits and qudits of any dimension.[21–26]Later, the QSD scheme was generalized to a non-deterministic noiseless amplifier[27]and also applied as a teleportation device for superposition states.[28,29]To extend the above-mentioned works,thermal state truncation was proposed via QSD to generate a mixture of vacuum and single-photon states.[30]By applying QSDs to both modes of the two-mode squeezed vacuum state(TMSVS),the resulting state was the entangled state composed of the twin vacuum and the twin single-photon,which can realize the enhance of entanglement and the improvement of the fidelity of teleportation.[31]Zhanget al.presented a slight modification to propose a catalytic QSD[32]and a displacement-based QSD[33]to truncate the input coherent state, indicating that the output state is a highly nonclassical quantum state. In addition, we adopted the basic idea of QSD to generate the finite-dimensional quantum states including the single-mode case and the two-mode case.[34]More interestingly,QSD can be placed at the receiving end of the continuous-variable quantum key distribution in order to lengthen the transmission distance.[35]These truncated states exhibit various interesting properties that may be a subject of a broad range of investigation in the field of quantum optics.

On the other hand, it is well known that an optomechanical system is a very useful platform for quantum state engineering.[36–40]For example, Manciniet al. synthesized Schr¨odinger-cat states of cavity field by means of the optomechanical system.[41]In Ref. [42], the authors utilized an optomechanical system to generate a large variety of nonclassical states of both the cavity field and the mechanical mode,including entangled states of two or more cavity modes.[43]A coupled optomechanical system can also be used to obtain squeezing states of mechanical mode.[44]A question arises naturally:How do the resulting nonclassical states effect when operating QSD on the dynamics of the optomechanical system? So far as we know,this question has not been clearly investigated in the literature to date.

In order to clearly understand the above influence, here we propose a scheme to generate new nonclassical quantum states by operating QSD on the cavity mode of the optomechanical system, where two single-photons are used to be in input modes of BS1 and perfect photondetector in output modes of BS2 registers one photoncount(see Fig.1 in detail).The organization of this paper is as follows. In Section 2,we first briefly review the theoretical models of the optomechanical system and the catalytic QSD with single photon input and single photon detection. In addition,we also give the corresponding equivalent operator of catalytic QSD depending upon the transmission coefficients of BSs. In Section 3, under the catalytic QSD, the main focus is the fidelity between the input state and the output state to assess the influence of the optomechanical system. In Section 4, we mainly discuss linear entropy of the cavity mode to numerically evaluate how QSD operation can effect the degree of entanglement between the cavity field and the mechanical oscillator. In Section 5,the nonclassical characteristics of the output state are analyzed in detail by means of the Wigner function(WF).Finally,the main results are summarized in Section 6.

2. Theoretical model

We first theoretically introduce a scheme to generate the new nonclassical states by operating QSD on the cavity mode as illustrated in Fig.1.

Fig.1.Schematic diagram of generating the non-classical state|ψ〉o via QSD based on BSs and conditional measurements, where two singlephotons|1〉are used to be in input modes of BS1 and perfect photodetector in output modes of BS2 all registers one photocount.

2.1. The optomechanical system

To begin with,we briefly review the isolated optomechanical system of a cavity field and a movable mirror. As shown in Fig. 1, the movable mirror is a quantum harmonic oscillator with frequencyωmand annihilation operator denoted byb, interacting with a cavity field mode of frequencyω0and annihilation operator denoted byc. The relevant Hamiltonian is[41]

It is easily obtained from Eq. (2) that whent=2πandη=0, the mirror returns to its original state and the state of cavity field turns into

In addition, depending on the value of the parameterk, the state|ξ〉ccan be made equivalent to a variety of multicomponent cat states at timet=2π. For example, fork=1/2, we obtain the two-component Schr¨odinger cat state

2.2. Quantum scissors device

The schematic diagram of QSD proposed by Pegget al.[20]consists of two BSs and two photon counting detectors,as shown in Fig.1.The input modes of the setup are denoted asc,d,andf,and their corresponding creation operators arec?,d?,andf?,respectively. Two single-photon states at auxiliary input modesdandfsuccessively interact with two BSsB1=expandB2= exp, wheretj=cosθjandrj=sinθj(j=1,2) are transmission coefficient and reflection coefficient of BSs. When perfect photodetectors at output modescandfof BS2register only one photocount, respectively, the output state|ψ〉oat output modedof BS1is given by

where the normalization factorpsrepresents the success probability heralded by the detection of one photoncount at the modescandfof BS2, and ?O=c〈1|f〈1|B2B1|1〉d|1〉fis derived as

Such state is modulated by adjusting the parameterkand timetof the optomechanical system as well as the transmittance of BSs, generating a wide range of nonclassical phenomena.Especially,at timet=2π,the nonclassical state of the cavity field at output modedis given by

Whenk=0.5,the Schr¨odinger cat state was operated by quantum scissor,discussed in detail in Ref.[26].

2.3. Detection probability

Next,before analyzing the nonclassical characteristics of the output state|ψ〉o, it is necessary to derive the explicit expression of the normalization factorps,i.e.,the success probability of such a state in Eq.(9). According to the completeness of quantum state,pscan be directly calculated by

which is just the specific expression of the detection probability of the output state|ψ〉o. It is seen from Eq. (11) that the detection probabilitypsis related to the transmittanceT1,T2of two BSs and the amplitude|α|of initial coherent state, independent of parameterkand timet. According to Eq.(11),we plot the detection probability of|ψ〉oin Fig.2 as a function of the coherent amplitude|α| and the transmissivityT2=t22for several different cases. WhenT1=1 andT2=1, leading toΓ0=0 andΓ1=1,ps= e-|α|2|α|2(see the blue solid line in Fig. 2(a)). We see that for the case ofT1/=1 andT2/=1 the success probability all decreases with the increasing amplitude|α|in Fig.2(a)and for a given|α|=0.5 andT1,the curves in Fig.2(b)are symmetric aboutT2=0.5,and the corresponding maximum value increases with the increasingT1(T1＜0.5).

Fig.2. Success probability of such a state|ψ〉o as a function of(a)the coherent amplitude |α| for several different T1 and T2 and (b) T2 for several different T1 at fixed|α|=0.5,respectively.

3. Fidelity

In this section,under the catalytic QSD,we introduce the fidelity between the input state|φ〉inand the output state|ψ〉oto assess the influence of the optomechanical system, which is defined asF=|in〈φ|ψ〉o|2.[26,45]Thus,using Eqs.(2)and(9),the fidelityFcan be calculated by

It is found from Eq. (12) that the fidelity is also independent of the scaled coupling parameterkand timet.

To see clearly the variation of the fidelity,using Eq.(12)we plot the fidelityFbetween the output state and input state in the QSD as a function of amplitude|α| or transmissivitiesT1andT2for several different cases,respectively,as shown in Fig.3. It is clear in Fig.3(a)that the fidelity can be close to 1 for a small amplitude|α|with given transmissivitiesT1andT2,and monotonously decreases with the increasing|α| for high transmissivitiesT2(e.g.,T2=0.4 orT2=0.5). However, for low transmissivitiesT2(e.g.,T2=0.1 orT2=0.15) it firstly decreases from 1 to zero and then increases to the maximum value and subsequently tends to zero with the increasing|α|.For the case ofT1=T2(see Fig.3(b)),the fidelity is symmetrically distributed withT1=T2=0.5(i.e.,50:50 beam splitter)and the maximum value decreases with the increasing|α|. In fact, whenT2=0.5 (orT1=0.5), leading toΓ1（ =0, we ob）-tain that the fdielity can be simplifeid to e-|α|2|α|4+1,which is independent ofT1(orT2),as shown in Figs.3(c)and 3(d). When|α|=1.0 and 1.5,Fis approximately equal to 0.55 and 0.37,respectively. In addition,for a given|α|,there exists a common intersection atT1=0.5 (T2=0.5) for several differentT2(T1), andT1dependence of the fidelity is the same asT2. These results illustrate that the high fidelity can be achieved by adjusting the transmissivity and amplitude.

Fig.3. Fidelity F between the output state and input state in the quantum scissor device as a function of amplitude|α|or transmissivities T1 and T2 for several different cases,where(a)at fixed T1=0.1,(b)T =T1=T2,(c)at fixed|α|=1.0,and(d)at fixed|α|=1.5.

4. Linear entropy

In this section, we discuss whether QSD can affect the degree of entanglement between the cavity field and the mechanical oscillator by evaluating linear entropy[46]

whereρc(t)is the reduced density matrix of the cavity mode.Thus,for the initial optomechanical system,using Eq.(2)the partial trace of density matrixρ(t)=|φ(t)〉〈φ(t)|over the mechanical mode is expressed as

and linear entropySiof the cavity mode can also be obtained by

Here,we have used the following formula:

Similarly, when operating QSD, we obtain from Eq. (9)that

where

Substituting Eq.(18)into Eq.(13)yields

which is just the analytical expression of linear entropy for the cavity mode. Generally, the greater the linear entropy is, the stronger the entanglement is.

Fig.4. Linear entropy Si of the cavity mode without operating quantum scissors as a function of time t for several different values of the scaled coupling parameter k with |α|=0.5, where the initial number state is(a)|1〉m and(b)|4〉m in the mechanical mode.

According to Eq.(15),we plot the linear entropySiof the cavity mode before operating quantum scissors as a function of timetfor several different values of the scaled coupling parameterkin Fig.4.It is seen that the degree of entanglement of the cavity field with the mechanical oscillator all are symmetric aboutt=π. For the case of the initial number state|1〉min Fig. 4(a), the degree of entanglement increases from zero to the maximum value att=π, where the maximum value decreases with the decreasing parameterk, and subsequently decreases to zero att=2π. After a timet=2π,the mirror returns to its original state and the entanglement vanishes gradually,as is expected.However,with the increasing number state the degree of entanglement clearly shows obvious oscillation behavior and their maximum values are not located att=π(see Fig.4(b)).

Fig.5. Linear entropy Sc of the cavity mode with operating quantum scissor(a)as a function of time t for several different transmissivities T1 and T2 with|α|=0.5,k=0.5 and the initial number state|1〉m,(b)as a function of transmissivities T =T1=T2 for several different parameter k with|α|=0.5 and the initial number state|1〉m when t =π/2,(c)as a function of amplitude |α| for several values of k with T1 =T2 =0.2 and the initial number state|1〉m when t=π/2.

Similarly, the linear entropyScafter operating quantum scissor is also plotted in Fig. 5 for several different cases,which shows that quantum scissors can effect the degree of entanglement between the cavity field and the mechanical oscillator. From Fig. 5(a), we see that for a given|α|,kandl,the degree of entanglement is still symmetrically distributed witht=πand the maximum value decreases with the increasing transmissivityT1orT2under the condition ofT1＜0.5 andT2＜0.5. It is interesting from Fig.5(b)that for a given timet(t=π/2)the linear entropyScall appears two maximum values symmetrically atT1=T2=0.5 for several values ofkwith|α|=0.5 andl=1,and the lower the value of the parameterkis,the smaller the maximum value is. The initial amplitude dependence of the entanglement is shown in Fig.5(c)for several values ofkat fixedl=1 whent=π/2. It is clear thatScexists a maximum value at small amplitude|α| range and may be close to zero with the increment of|α|. Moreover,in order to understand whether the entanglement is enhanced by quantum scissor, by comparing Eq. (15) with Eq. (13) we may depict the feasibility region for enhancing entanglement in (|α|,T=T1=T2) space in Fig. 6. Namely, ifSc＞Si,the entanglement is enhanced in principle, or else it is weakened. It is found that the entanglement by evaluating linear entropy turns out to be enhanced within the small amplitude and the low transmissivity or high transmissivity regions(see Fig.6). These results further illustrate that the high entanglement between the cavity field and the mechanical oscillator can be achieved by operating quantum scissor.

Fig.6. The feasibility region for enhancing entanglement by evaluating linear entropy Sc of the cavity mode in (|α|, T =T1 = T2) space with k=0.5 and the initial number state|1〉m when t=π/2.

5. Wigner function

In the above section,to evaluate the quantum scissor device, we have used fidelity to quantify how close the output state and the input state are. However,fidelity is just a single number and does not give the complete information.[42]Next,we turn our attention to the WF of the output state as a tool which can give complete information on the phase and amplitude of this state. In addition,the negative values in WF are an indicator of the nonclassical property of a state.[47,48]

Firstly,for a given density operator of Fock states,ρnm=|n〉〈m|,using the Wigner function in the coherent state representation|z〉[49,50]

and the expression form of Fock states in the un-normalized coherent state representation(‖γ〉=exp（γa?）|0〉)

we may calculateWn,m(β)by

whereHn,m(x,y)is two-variable Hermite polynomials with

Then,based on the results of Eqs.(18)and(23),we may obtain the analytical expression of WF for the cavity mode after operating quantum scissor,i.e.,

When the mirror returns to its original state, namely,t=2π,the WF of the output state in Eq. (10) for the cavity mode yields

which is just the WF of multicomponent cat states depending on the value of the parameterk. That is to say,k=1/2,1/and 1/correspond to two-component, three-component and four-component cat states,respectively.

In Fig. 7, we plot the WF distributions of(β) of Eq. (25) in phase space withβ= (q+ ip)/for several different transmissivitiesT1=T2and parametersk=0,1/2,1/at fxiedα=1+i,respectively. It is clearly seen that for the given transmissivitiesT1=T2=0.2 or 0.3, the negative region of WF gradually increases with the increase of parametersk,which means that three-component cat state exists the stronger noclassicality than two-component cat state.On the other hand, in comparison of Fig.7(a) with Fig. 7(d),the structure of WF will be affected by the transmissivitiesT1=T2,which leads to the generation of quantum states with the stronger nonclassical properties via quantum scissors device. In a word, by modulating the transmissivities and parametersk,the negative volume of the WF can be realized and enhanced.

Fig.7.Wigner function distribution(β)of the output state in phase space with β =(q+ip)/ for several different cases at fxied α=1+i,where we actually consider the multicomponent cat states for the output mode at time t=2π due to dynamics alone. (a)T1=T2=0.2,k=0,(b)T1=T2=0.2,k=1/2,(c)T1=T2=0.2,k=1/,(d)T1=T2=0.3,k=0,(e)T1=T2=0.3,k=1/2,(f)T1=T2=0.3,k=1/

6. Conclusions

In summary, we have studied theoretically the influence of the non-classical states generated by QSD operation on the the cavity mode of an optomechanical system,where the cavity field is initially in a coherent state and the mechanical resonator is in a number state. It is found that at timet=2πthe mirror recovers its original state and the cavity field returns to a variety of multicomponent cat states by adjusting the value of the parameterk. When QSD with two single photons inputs and two single-photon detections act on the cavity mode of the optomechanical system, the resulted state in Eq. (6) contains only the vacuum, single-photon and two-photon states depending upon the parameterkas well as the transmission coefficients of BSs. We have discussed the success probability of such a state, which all decreases with the increasing amplitude|α|and the curves are symmetric aboutT2=0.5 for a given|α| andT1with the corresponding maximum value increasing with the increment ofT1(T1＜0.5). In addition,the fidelity between the output state and input state via QSD can be close to 1 for a small amplitude|α|for given transmissivitiesT1andT2and monotonously decreases with the increasing|α|for a high transmissivityT2,which indicates that QSD operation can regulate features of optomechanical states. Then we mainly pay attention to investigating the entanglement between the two subsystems by evaluating linear entropy.It turns out that QSD operation can enhance their entanglement degree and the enhanced region is located at the small amplitude and the low transmissivity or high transmissivity regions. Furthermore, we also derive the analytical expression of WF for the cavity mode via QSD and numerically discuss the WF distribution in phase space at timet=2π. The results show that the negative region of WF gradually increases with the increase of parametersk,which means that three-component cat state exists the stronger noclassicality than two-component cat state.Our work further demonstrates that the optomechanical system with QSD operation is an efficient method in manipulating the nonclassical state,which may become an important source in quantum optics and quantum information processing.

Acknowledgements

Project supported by the National Natural Science Foundation of China(Grant No.11704051),the Qinglan Project of the Jiangsu Education Department and the Research Foundation of Six Talents Peaks Project in Jiangsu Province, China(Grant No.XNY-093).