a r X i v :0807.2033v 1 [q u a n t -p h ] 13 J u l 2008Mean parity of single quantum excitation of some optical fields in thermal channelShang-Bin LiShanghai research center of Amertron-global,Zhangjiang High-Tech Park,299Lane,Bisheng Road,No.3,Suite 202,Shanghai,201204,P.R.ChinaThe mean parity (the Wigner function at the origin)of excited binomial states,excited coherent states and excited thermal states in thermal channel is investigated in details.It is found that the single-photon excited binomial state and the single-photon excited coherent state exhibit certain similarity in the aspect of their mean parity in the thermal channel.We show the negative mean parity can be regarded as an indicator of nonclassicality of single-photon excitation of optical fields with a little coherence,especially for the single-photon excited thermal states.PACS numbers:42.50.Dv,03.65.Yz,05.40.CaAmong various kinds of indicators of nonclassicality of optical fields,the partial negativity of the Wigner func-tion (PNWF)indicates the highly nonclassical character of the optical fields.Quantum excitation of general op-tical fields can always exhibit PNWF which will be de-stroyed and eventually completely disappear at the same decay time γt c =ln 2+nπTr[(ˆO e −ˆO o )ˆρ],(1)where ˆO e ≡ ∞k =0|2k 2k |and ˆO o ≡ ∞k =0|2k +1 2k +1|are the even and odd parity operators respectively.Therefore,the value of Wigner function at the origin equals the constant 2dt=γ(n +1)2(2a †ρa −aa †ρ−ρaa †),(2)where γrepresents dissipative coefficient and n denotes the mean thermal photon number of the thermal channel.In the thermal channel described by the master equation (2),the time evolution Wigner function satisfies the fol-lowing Fokker-Planck equation [6]∂2(∂∂pp )W (q,p,t )+γ(2n +1)∂q 2+∂21+c 2]1+n,(6)andγt c 1=ln[2|α|2(1+n )+2n2n =0.5,k =1,M =2,η=0qγt −−−W H q,0Ln=0.5,k =1,M =2,η=0.5qγt W H q,0L n =0.5,k =1,M =2,η=1qγt −−−W H q,0LFIG.1:The cross section W (q,0)of Wigner functions of the EBSs |1,0,2 ≡|1 ,|1,0.5,2 ,and |1,1,2 ≡|3 in thermal channel with n =0.5are plotted as the function of decay time γt .field can be generated by repeated application of the pho-ton creation operator on binomial states [10].The bino-mial states of optical fields can be generated in some non-linear processes [11,12,13].Then,by a scheme similar to that preparing the ECS [7],in which the excited atoms pass through a cavity and provide the field in the cavity is initially in a binomial state,one can produce the EBS.If a traveling optical field in the binomial state has been produced,one can also adopt the experimental scheme of Zavatta et al.[14]to generate the single-photon-excited binomial state.Let us briefly recall the definition of the EBS [10].Then =0.5,k =1,M =3ηγt −−−−0W H 0,0L FIG.2:The value W (0,0)of Wigner function at the originof the EBS with k =1,and M =3in thermal channel with n =0.5is plotted as the function of decay time γt and η.EBS is defined by|k,η,M =N (k,η,M )a †k |η,M ,(8),where|η,M =M l =0(C l M )1/2ηl(1−η2)(M −l )/2|l(9)is the binomial state [15].Here k ,l and M are integers.ηis real number with 0≤η≤1.|l is Fock state.a †(a )is the creation operator (annihilation operator)of the optical mode.N (k,η,M )is normalization constant of EBS,which is given by N (k,η,M )=[η2M (M +k )!η2)]−13n =0.5,k =1,M =4ηγt −−−−0W H 0,0L FIG.3:The value W (0,0)of Wigner function at the originof the EBS with k =1,and M =4in thermal channel with n =0.5is plotted as the function of decay time γt and η.ηγt −−−−0W H 0,0L FIG.4:The value W (0,0)of Wigner function at the origin ofthe EBS with k =2,and M =2in photon loss channel with n =0is plotted as the function of decay time γt and η.the threshold decay time γt c .For η(3)1≤η<1,W (0,0)is initially non-negative and then becomes negative until the threshold decay time γt c .As η=1,W (0,0)is al-ways non-negative.In Fig.3,W (0,0)of Wigner function at the origin of the single-photon EBS with M =4in thermal channel with n =0.5is plotted as the function of decay time γt and η.In this case,for small valuesof η<η(4)1≃0.4,W (0,0)is always negative before thethreshold decay time γt c .For η(4)1≤η<η(4)2≃√πξ3e γt ,ξ=2(¯n −n )+(1+2n )e γt ,κ=−8(¯n −n )(1+n )+2(1+2n )2e 2γt+4[¯n (1+2n )−(1+2n )2]e γt ,(11)where ¯n denotes the mean photon number of the ini-4 tial thermal state.Its mean parity is always negativebefore the threshold decay timeγt c=ln2+n。
