Kai-Cai Li (李開才), Xiang-Guo Meng (孟祥國), and Ji-Suo Wang (王繼鎖)

1Shandong Provincial Key Laboratory of Laser Polarization and Information Technology,College of Physics and Engineering,Qufu Normal University,Qufu 273165,China

2School of Physics and Electronic Engineering,Linyi University,Linyi 276000,China

3Shandong Provincial Key Laboratory of Optical Communication Science and Technology,School of Physical Science and Information Engineering,Liaocheng University,Liaocheng 252059,China

Abstract We theoretically analyze the nonclassicality and entanglement of two new non-Gaussian entangled states generated by applying multiple-photon addition and subtraction to a two-mode binomial state.The nonclassical properties are investigated in terms of the partial negativity of the Wigner functions,whose results show that their nonclassicality can be enhanced via one-mode even-number photon operations and two-mode symmetrical operations for the initial two-mode binomial state.We also find that there exists some enhancement in the entanglement properties in certain parameter ranges via one-mode photon-addition and two-mode symmetrical operations.

Key words: two-mode binomial state,photon addition and subtraction,nonclassicality,entanglement

1 Introduction

Non-Gaussian entangled states as a new information resource have been drawn much attention,which are mainly because that such states are indispensable for completing a number of tasks of quantum information processing,such as quantum teleportation,[1]quantum key distribution,[2]and quantum metrology,[3]and can be further distilled to generate higher-quality quantum entanglement,[4?6]which can be useful in improving the quality of some certain real quantum information tasks.

Recently,photon addition,photon subtraction and their coherent superpositions as the typical non-Gaussian operations are usually used to produce non-Gaussian entangled states with highly nonclassical properties.For examples,subtracting or adding photon operations on traditional Gaussian states including two-mode squeezed vacuum thermal states are widely studied from both experimental and theoretical aspects.[7?14]The achieved results presented how do these non-Gaussian operations affect the entanglement,nonlocality and teleportation fidelity in various degree,and the relationship between them.For a review of quantum state engineering with these non-Gaussian operations,we refer to Refs.[15–16]and references therein.

In this paper we shall introduce the multiple-photonadded and -subtracted two-mode binomial states by repeatedly adding or subtracting photons to or from a twomode binomial state (TMBS),and theoretically investigate the influences of the photon-addition and -subtraction on the photon number distributions (PNDs),Wigner functions (WFs),and entanglement of these states.

2 Multiple-Photon-Added and -Subtracted TMBSs

Theoretically,using the Schwinger bosonic realization of the operators,i.e.,J+=a?b,J?=ab?,andJz=(1/2)(a?a?b?b),the exponential operator R=exp(ξJ+?ξ?J?) has the following decomposed form,[17?18]i.e.,

which can result in quantum entanglement by rotating two input states via the unitary transformations

whereξ=(θ/2)e?i?andζ=e?i?tan(θ/2),aandbare two mode annihilation operators,[a,a?]=[b,b?]=1.Indeed,the operatorRcan also be named beamsplitter operator since it can describe the mechanism of an optical beamsplitter.Based on the beamsplitter operatorR,an output state is linked to the input state by=in the Schr?dinger picture.Hence,letting a single-mode photon number stateand another mode vacuum staterespectively enter the two input ports of a beamsplitter described by the operatorRand get overlapped,thus the final output state is[19]

withcq=(q!/[l!(q ?l)!(1+|ζ|2)q])1/2ζl.Thus,Eq.(3) is a TMBS with quantum entanglement since another output has the collapsed state|q?n〉when one measuresrvo-Reyes, M.Rigol, and J.Rubayo-Soneira, Re the number stateat one output port.Using the normal ordering product of the vacuum projection operator,(::refers to normal ordering)and the technique of integration within an ordered product (IWOP) of operators,[20?22]we prove that the set of the statescan form the completeness relation in the whole Fock space.

Further,performing the multiple-photon-added and -subtracted operationsa?mbnandamb?non the TMBS,we obtain the normalized output states as

wherecs,aandca,sare respectively given by

andC,Dare the normalization factors,i.e.,Obviously,whenm=n=0,namely,no photon addition and subtraction case,both the statesandreduce to the TMBS.Besides,the conditionm=ncorresponds to the symmetrical non-Gaussian operation,whilenrefers to the asymmetrical case.Note that photon addition and subtraction operations can be realized by using the parametric downconverter and high transmittance optical beamsplitter under experimentally realistic conditions,[23?25]hence we confirm that,in physical realization,the statesandmay be produced at the beamsplitter or a parametric downconverter output ports when the TMBS is on the input port.

Much work has demonstrated that photon addition and subtraction can bring about some markedly different physical properties from the original input states,so we shall investigate how the operationsa?mbnandamb?ninfluence the PND,the negativity of the WF and the entanglement of the statesandin the following several sections.

3 Photon Number Distribution

As a key characteristic of any two-mode quantum state,its PNDrefers to the probability of findingnaandnbphotons in this field.[26?27]

Fig.1 (Color online) Photon number distribution Ps,a for the state in the Fock space (na,nb) for some different values of (q,m,n,?),where (2,0,0,0.1),(4,0,0,0.1),(4,3,0,0.1),(4,0,3,0.1),(4,1,1,0.1),(4,1,1,0.6) respectively refer to (a),(b),(c),(d),(e),(f).

So,we now calculate the PNDs of the statesandas

which show that there exist the constrained condition,i.e.,na?m=q ?n?nborna+m=q+n?nb,for the PNDs.In particular,whenm=n=0,we have

which is just the PND of the TMBS.

In Fig.1,we plot the distributionPs,ain the Fock space (na,nb) for some different values ofq,m,n,andζ.From Fig.1,we clearly see that the PND is constrained by the conditionna?m=q ?n ?nbsince the probabilities of finding only several (na,nb) numbers of photons in the field cannot be zero.As the parametersm,n,andqincrease,the maximum probability of finding (na,nb)photons appears in different number states and the maximum probabilities are very different.As the parameterζis increased,more photons are presented in larger number states in this optical field,which is because the coefficientcs,aprovides a higher weight to the initial higher-number photon component.In sum,the positions and values of the peaks depend on how many photons are added or subtracted,the two-mode number sumqand the value of the parameterζ.A similar result appears for the distributionPa,sof the state|ξ〉a,s,so no more details here.

4 Wigner-Function Negativity

In quantum statistics,the negativity of WF is indeed a good indicator of the highly nonclassicality of the state.[28?30]So,it is very essential for us to investigate the WFs for the statesandIn order to get the analytical expressions of the WFs for these two states,we first review the two-mode Wigner operator ?(σ,γ) in the entangled staterepresentation,[31?32]i.e.,

where=e?(1/2)|?|2+?a????b?+a?b?|00〉 is the entangled state,the parameter?=?1+i?2.Also,it can be easily proved that the states|?〉are the common eigenstates of the operators (Qa?Qb) and (Pa+Pb) with the eigenvaluesandQiandPi(i=a,b) are respectively the coordinate and momentum operators.Only whenγ=α+β?andσ=α?β?,there exists the identity?(σ,γ)≡?(α,α?)?(β,β?),where ?(i,i?) (i=α,β) is the single-mode Wigner operator.Using Eqs.(4)and(8),the WFs for the statesandare respectively obtained as

where we have used the WF for the number statei.e.,

Obviously,whenm=n=0,both the WFsWs,aandWa,scan reduce to the WFWqfor the TMBS,i.e.,

In Fig.2,we plot the WFWs,afor the stateas a function of Reαand Reβfor different values ofm,n,q,andζ.Obviously,whenm=n=0,the WFWqhas an upward main peak for evenqand a downward main peak for oddq,and the downward secondary peaks have regular distributions and their number is also related to the twomode number sumq.As the parametersm(n=0) andζincrease,the regular distribution can be destroyed and develop into an irregular complex structure.However,the effects of the parametersmandζon the WF distributions are different.In particular,the change of the parameterζcannot change the distributions of the main peak,but the direction of the main peak relies on the even or odd quality of two-mode number sumq+m.With the increasement ofn(m=0),the characteristics of the WFWs,afor the two-mode number sumq?nis consistent with that of the WFWq′forq′=q ?n,which is understandable by comparing the analytical expressions (3) and (4).For the the same number of operations (m=n),the WF distributionWs,adepends entirely on the parameterm,but the direction of the main peak is only related to the two-mode number sumq.In sum,the parametersmandnlead to the increasingly complicated WF distributions,different from the original distributionsWq.Noticing that the impacts of the parametersm,n,qandζon the WF distributionWa,sare very similar to the WFWa,s,so we cannot show it here.

In order to clearly see how the negativity of the WFWs,aas an indicator for quantifying nonclassicality of the statechanges with the parametersm,nandζ,we adopt the integration formula for calculating the negative volume of the WF,[33]that is,

Inserting the WF(9)into the integration(12)and directly carrying out it,the negative volume ?s,aof the WFWs,acan be easily obtained.

Fig.2 (Color online) Wigner function Ws,a for the state of Re α and Re β for different values of (q,m,n,?),where(2,0,0,0.1),(5,0,0,0.1),(5,0,0,0.6),(5,2,0,0.1),(5,3,0,0.1),(5,0,2,0.1),(5,0,3,0.1),(5,3,3,0.1) respectively refer to (a),(b),(c),(d),(e),(f),(g),(h).

Fig.3 (Color online) Negative volume ?s,a of the WF Ws,a for the state for a give value of q=5 and different values of (m,n),where (0,0),(2,0),(3,0),(0,2),(0,3),(2,2),(3,3) respectively refer to the black solid,red dashed,green dotted,blue dash-dotted,cyan dash-dot-dotted,magenta shortdashed,yellow short-dotted lines.

In Fig.3,we present the negative volume ?s,aof the WFWs,afor the stateforq=5 and the different values ofmandn.Clearly,the negative volume ?s,adecreases at first and then increases as the parameterζincreases,which is independence on the photon numbersm,n.Form=0(n=0)and any oddn(m),the negative volume ?s,aofWs,ais always smaller than that ofWqin the whole range ofζ,which shows that the odd-number photon-addition or subtraction operations can weaken the nonclassicality of the original statewhen another mode is zero.For any evenm(n=0)or the same number of operations (i.e.,m=n),the negative volume ?s,aofWs,ais smaller than that ofWqin the regime of low values ofζ,but larger than that ofWqwhen the parameterζexceeds a certain threshold value.However,for any evenn(m=0),the negative volume ?s,aofWs,ais always larger than that ofWqfor all values ofζ.In a word,the nonclassicality of the statefor the cases ofm=nor any evenm(n=0) andn(m=0) can be enhanced for the initial stateBesides,the variations of the negative volume ?a,sof the WFWa,swith the parametersζ,m,andnare very similar to those of ?s,a.

5 Entanglement

In this section,we want to discuss the entanglement properties of the statesandvia calculating their Neumann entropy.For a pure entangled statewith the Schmidt expansion,i.e.,=are the mutual orthonormal states andclis some positive real number,its entanglement can be quantified by the partial von Neumann entropy of the reduced density operator,i.e.,whereρa=Based on the Schmidt expressions of the statesandin Eq.(4),and thus their entanglement can be easily obtained as

In the case ofm=n=0,Eq.(13) refers to the amount of the entanglement of the state|ξ〉q,i.e.,Eq=

Fig.4 (Color online) Entanglement entropy as a function of the parameter ? for the states and with a given value of q=5 and different values of(m,n),where the values of (m,n) are the same of those of Fig.3.

In Fig.4,we plot the entanglement entropy as a function of the parameterζfor the statesandwithq=5 and different values ofmandn.Obviously,for any states,andthe entanglement always increases at first and then decreases as the parameterζincreases,and their maximum values appear nearζ=1.For both the statesandone-mode photon addition enhance the entanglement of the initial statebut one-mode photon subtraction weakens it.Comparing with the original state,the entanglement of the state() can be enhanced for smaller (larger)ζwhen the same number of operations (m=n) is applied to both modes.With the increasement of the number of operations,the entanglement of the statesanddecreases and totally less than that offor enough large value ofm=n.

6 Conclusions

In summary,we have investigated the PNDs,the WF distributions and the entanglement of two new multiplephoton-added and -subtracted TMBSs.The analytical results indicate that the PND is constrained by the conditionna?m=q?n?nborna+m=q+n?nb,and the WF is represented as a finite dimensional summation over the WFs for the number states.Besides,the numerical results show that the different number of operations leads to different photon number distributions with the different positions and amplitudes of the maximum probabilities.For the initial state,both one-mode even-number photon operations and two-mode symmetrical operations can enhance its nonclassicality,and one-mode photon-addition and two-mode symmetrical operations can enhance its entanglement.