Entanglement Dynamics of Atoms in Double Jaynes Cummings Models with Kerr Medium

Transcription

1 Commun. Theor. Phys. (Bejng, Chna) 54 (010) pp c Chnese Physcal Socety and IOP Publshng Ltd Vol. 54, No. 5, November 15, 010 Entanglement Dynamcs of Atoms n Double Jaynes Cummngs Models wth Kerr Medum XIE Qn ( ) and FANG Mao-Fa (à Ù) College of Physcs and Informaton Scence, Hunan Normal Unversty, Changsha 41001, Chna (Receved February 4, 010) Abstract Entanglement dynamcs of the atoms n the double Jaynes Cummngs models wth the Kerr medum s studed, and the effect of the Kerr medum on that s examned. The result shows that, the Kerr medum can control the entanglement dynamcs of the atoms and repress entanglement sudden death. We can obtan the maxmum entanglement between the atoms by strengthenng the nonlnear nteracton of the Kerr medum. PACS numbers: Mn, Bg Key words: entanglement, Kerr medum, concurrence 1 Introducton Entanglement s now regarded to be the man resource of quantum computaton and quantum nformaton processng (QIP). Thus the ssue of creatng, enhancng, and controllng entanglement n varous composte systems has great practcal mportance n actual quantum nformaton processng and has attracted much attenton n theory and experment technology. [1 11] In the last couple of years a good deal of work has been devoted to lookng for ways of storng and controllng entangled qubts [1 13] of atom and feld, where the atoms only nteract wth a sngle mode feld n a cavty. [14 15] Recently, Isabel Sanz et al. [16] studed entanglement nvarant for the double Jaynes Cummngs model. They found a natural entanglement nvarant under evoluton, and demonstrated that entanglement spreads out over all of the system s degrees of freedom that become entangled through the nteracton. They also provded an analyss of why certan ntal states lose all ther entanglement n a fnte tme, although ther exctaton and coherence vansh only asymptotcally wth tme. Meanwhle, they studed the entanglement evoluton of two remote atoms nteractng ndependently wth a cavty feld, as n the double Jaynes Cummngs (J-C) model, and proved that perodc and drected transfer of entanglement nto a specfc qubt par s possble, entanglement transfer and entanglement sudden death (ESD) can be prevented usng off-resonant nteractons. However ther work does not consder the case of the cavtes flled wth the Kerr medum. And much work has studed the nfluence of the nonlnear nteracton of the Kerr medum (NIKM) but only on one J-C modes. As we know, the entanglement dynamcs of the double J-C models wth the Kerr medum has not been nvestgated. In ths paper, we study the entanglement dynamc of the double J-C models wth the Kerr medum, va an entanglement measure based on the concurrence. We focus on the nfluences of the NIKM on the entanglement between the atoms. For two dfferent cases, the results reveal that we can get maxmal entanglement through renforcng the strength of the NIKM. For a certan ntal atom entanglement state, there s an entanglement sudden death effect between the two atoms. The NIKM can effctvely prevent the undesrable entanglement sudden death from occurrng. The paper s organzed as follows. In Sec., we ntroduce the model under consderaton. In Sec. 3, the effect of the NIKM n sngle exctaton state s dscussed. In Sec. 4, the effect of the NIKM n two-exctaton state s dscussed. In Sec. 5, a summary s gven. Model Frst, we ntroduce a system composed of two separated sngle mode cavtes, each contanng a sngle twolevel atom, wth the Kerr medum, as show n Fg. 1. Fg. 1 Two dstant and non-nteractng subsystem 1 and each wth a sngle-mode cavty contanng a two-level atom and the Kerr medum. In rotatng-wave and the electrc-dpole approxmaton, the Hamltonan of the system can be wrtten as H = H feld + H atom + H nt, (1) where H feld s the Hamltonan of the cavty modes whch Supported by the Natonal Natural Scence Foundaton of Chna under Grant No , the Natural Scence Foundaton of Hunan Provnce under Grant No. 07JJ3013, and the Educaton Mnstry of Hunan Provnce under Grant No. 06A038 Correspondng author, E-mal: mffang@hunnu.edu.cn

2 No. 5 Entanglement Dynamcs of Atoms n Double Jaynes Cummngs Models wth Kerr Medum 841 can be modeled as ( H feld = ω 1 a a + 1 ) ( + ω b b + 1 ). () Here ω 1 and ω are the frequences of the two cavtes, respectvely. The parameters a (a) and b (b) respectvely are the creaton (annhlaton) operators for the modes of the cavtes labeled as a and b, and H atom s the Hamltonan of the atoms, whch can be wrtten as H atom = ω 0 S z A + ω 0S z B, (3) where ω 0 s the transton frequency of the atoms, and S z A,B s the atomc nverson operator. Whle H nt s the nteracton Hamltonan between the atoms and the cavty modes, whch can be reduced to H nt = g 1 (a S A + as+ A ) + g (b S B + bs+ B ) + χ 1 a aa a + χ b bb b. (4) The g 1 and g are the strengths of the couplng between the atom and the cavty modes, whch should be real always, for the sake of smplcty we only consder the case of g 1 = g = g, and S + and S are the rasng and lowerng operators of the -th atom respectvely. The χ 1 and χ are the nonlnear ampltudes of nteracton between the Kerr medum and cavty a and b modes respectvely, the values of χ represent the strengths of the nonlnear nteracton. 3 Entanglement Dynamcs of Atoms n Sngle Exctaton States In ths secton, we nvestgate the ntal state ψ where there s only a sngle exctaton present, so that d 1 (t) = exp [( χ { 1/)t] Ω 1 d 1 (0)cos(Ω 1 t) Ω 1 d (t) = exp [( χ { /)t] Ω d (0)cos(Ω t) Ω d 3 (t) = exp [( χ { 1/)t] Ω 1 d 3 (0)cos(Ω 1 t) + Ω 1 d 4 (t) = exp [( χ { /)t] Ω d 4 (0)cos(Ω t) + Ω the space of the system s spanned by four state vectors, defned as follows: [17] ψ 1 = 00, ψ = 00, ψ 3 = 10, ψ 4 = 01, (5) then the ψ can be wrtten as ψ(t) = d 1 (t) 00 + d (t) 00 + d 3 (t) 10 + d 4 (t) 01, (6) whch s a lnear combnaton of the avalable product states of the atoms and the cavty modes gven by Eq. (5) at tme t. The coeffcent d (t) determnes the probablty ampltude of the -th state at the tme t. By solvng the Schrödnger equaton, we can get the coeffcent d (t), and fnd they satsfy the dfferental equatons d 1 = gd 3, d = gd 4, d 3 = 1 d 3 gd 1 χ 1 d 3, d 4 = d 4 gd χ d 4, (7) where j = (ω 0 ω j ) (j = 1, ) s the detunnng of j-th cavty frequency from the atomc transton frequency, as for the smple that we defne 1 = =. By usng the Laplace transform technque, a smple soluton of the equatons, vald for the ntal state gven by Eq. (6). χ 1 χ χ 1 χ d 1 (0) + gd 3 (0) sn(ω 1 t), d (0) + gd 4 (0) sn(ω t), d 3 (0) gd 1 (0) sn(ω 1 t), d 4 (0) gd (0) sn(ω t), (8) where ( Ω 1 = g + χ ) ( 1, Ω = g + χ ) are the detuned Rab frequency of cavty mode a and b, respectvely. Our nterest s centered prncpally on the evoluton of entanglement between the dfferent parts of the two twoqubt subsystems 1 and. By denotng the two atoms as A and B, and the correspondng cavty modes as a and b, we may dstngush sx pars of subsystems AB, ab, Aa, Ab, Ba, Bb. It s convenent to quantfy the degree of any bpartte entanglement by the concurrence, and we solve for all the two qubts concurrence, ther smple expressons can be gven as C AB (t) = d 1 (t) d (t) C ab (t) = d 3 (t) d 4 (t), C Aa (t) = d 1 (t) d 3 (t), C ab (t) = d (t) d 3 (t), C Ab (t) = d 1 (t) d 4 (t), C Bb (t) = d (t) d 4 (t). (9) Now we consder an avalable ntal energy state, wth a unform populaton dstrbuton, and we wrte ψ 0 = 1 [( ψ 1 + e θ ψ ) ± ( ψ 3 e φ ψ 4 )], (10) there θ and φ are arbtrary phase factors. From the state, n the case of zero detunngs, we fnd that the ntal entanglement s maxmally shared between the sx subsystem pars, and C AB (0) = C Ab (0) = C Aa (0) = C ab (0) = C ab (0) = C Bb (0) = 1/. Here, we stress that the entanglement of two atoms can be manpulated by changng the strength of the NIKM easly. In Fg., we show that the nfluences of the NIKM on the entanglement evoluton of the atom-atom concur-

3 84 XIE Qn and FANG Mao-Fa Vol. 54 rence C AB (t) wth the equal NIKM, namely χ 1 = χ = χ and the rato /g = 0. It should note that the curve of entanglement evolves n the form of sn and cos functon, and the perods of C AB decrease wth ncrease the NIKM. The nterestng thngs s that from ths ntal state, gven by Eq. (10), when χ/g = 0 the value of C AB = 0.5 stably exts. It stems from the fact that n ths case from the Eq. (8) the probablty ampltude d (t) = d (0)exp(±jgt) ( = 1;, j s the magnary number), whch means that the tme evoluton only changes the phase of the probablty ampltude d (t), whle the ampltude s unaltered. Hence, C AB (t) = d 1 (t) d (t) = C AB (0) = 1/. Wth ncrease of the NIKM the fluctuaton of the concurrence becomes quas-perodc and the oscllatng frequency of concurrence C AB ncrease wth ncreasng the strength of the NIKM. Therefore, the Kerr medum can stablze the entanglement between the two atoms and cause hgh frequency surge, whch may be useful n quantum nformaton process based on the entanglement. Fgure 3 whch for the detunnng 0 has a dfferent parameter settng and dsplays the smlar property to that n Fg.. Next, we nvestgate the evoluton of the atomc concurrence for the followng Bell ntal state ψ 0 = 1 ( ψ 1 ± ψ ) = 1 ( 00 ± 00 ). (11) Ths correspondng to the case where d 1 (0) = 1/, d (0) = ±1/, d 3 (0) = 0, d 4 (0) = 0 n Eq. (6) and from Eq. (9), we dscover the ntal entanglement restrcted to just the two atoms, namely C AB = 1. Fg. 4 The tme evoluton of C AB, and χ 1 = χ = χ, when /g = 0, where χ/g = 0 (sold curve), χ/g = 0.5 (dashed curve), χ/g = (dash-dotted curve), χ/g = 10 (dotted curve). Fg. The tme evoluton of the concurrence C AB(t), and χ 1 = χ = χ, when /g = 0, where χ/g = 0 (sold curve), χ/g = 0.5 (dashed curve), χ/g = (dash-dotted curve), χ/g = 10 (dotted curve). [18] Fg. 3 The tme evoluton of the concurrence C AB(t), and χ 1 = χ = χ, when /g = 1, where χ/g = 0 (sold curve), χ/g = 0.5 (dashed curve), χ/g = (dash-dotted curve), χ/g = 10 (dotted curve). Fg. 5 The tme evoluton of C AB, and χ 1 = χ = χ, when /g = 1, where χ/g = 0 (sold curve), χ/g = (dashed curve), χ/g = 5 (dash-dotted curve), χ/g = 10 (dotted curve). Then we dscuss the case of /g = 0, n Fg. 4 wth χ 1 = χ = χ. The fgure reveals that, as χ ncreases the perods of the atomc entanglement C AB s shortened. We notce that the NIKM wll be stronger f the value of C AB more approaches to 1. Therefore, we can adjust the large strength of the NIKM n order to have the maxmum atomc entanglement to the greatest extent. We now turn to another case /g = 1, n Fg. 5. We fnd that t s qute dfferent about the case we dscussed before, when χ/g, the larger value of χ/g s, the longer s the perod of entanglement evoluton, and the mnmum value decreases as the value of χ/g ncreases and t s opposte when χ/g >. We can obtan the smlar result of the case above when the NIKM s strong enough. It s worth notng that when the atom-atom entanglement s maxmum, at C AB = 1, other bpartte entanglement

4 No. 5 Entanglement Dynamcs of Atoms n Double Jaynes Cummngs Models wth Kerr Medum 843 C ab = C ab = C Ab = 0, t means the entanglement s localzed at atom-atom. 4 Entanglement Dynamcs of Atoms n Two- Exctaton States In ths secton, our am s to nvestgate the dynamcs of entanglement of the atoms when two exctaton state can be presented for ntal state of the system. We defne the wavefuncton of two exctaton state as [17] ψ = d 1 (t) 00 + d (t) 01 + d 3 (t) 10 + d 4 (t) 11 + d (1) There the ground state 0 = 00, whch has no exctaton, s not the egenstate of the Hamltonan gven by Eq. (1) and we know the probablty ampltude d 0 does not evolve n tme, e.g. d 0 (t) = d 0. Usng the Schrödnger equaton, we can obtan the evoluton of the set nvolves for equatons Fg. 6 The tme evoluton of concurrence C AB(t) for symmetrc state wth χ 1 = χ = χ, when /g = 0, where χ/g = 0 (sold curve), χ/g = 1 (dashed curve), χ/g = (dash-dotted curve), χ/g = 10 (dotted curve). d 1 = d 1 g( d + d 3 ), d = χ d g( d 1 + d 4 ), d 3 = χ 1 d3 g( d 1 + d 4 ), d 4 = ( χ 1 χ ) d 4 g( d + d 3 ). (13) In order to remove the fast oscllatng terms, we ntroduce a rotatng frame through the relaton d = e (ω1+ω)t d. (14) Then we can have the four parwse concurrence between the subsystems Fg. 7 The tme evoluton of the concurrence C AB(t) for antsymmetrc state wth χ 1 = χ = χ, when /g = 0, where χ/g = 0 (sold curve), χ/g = 1 (dashed curve), χ/g = (dash-dotted curve), χ/g = 5 (dotted curve). C AB (t) = max{0, ( d 1 (t) d 0 (t) d (t) d 3 (t) )}, C ab (t) = max{0, ( d 4 (t) d 0 (t) d (t) d 3 (t) )}, C ab (t) = max{0, ( d (t) d 0 (t) d 1 (t) d 4 (t) )}, C Ab (t) = max{0, ( d 3 (t) d 0 (t) d 1 (t) d 4 (t) )}. (15) and the concurrence C Aa and C Bb have no relatonshp wth d 0, C Aa (t) = d 1 (t) d 3 (t) + d (t) d 4 (t), C Bb (t) = d 1 (t) d (t) + d 3 (t) d 4 (t). (16) To further nvestgate, we consder two dfferent ntal states, one s a lnear superposton of the zero-exctaton state wth a two-quanta symmetrc state 1 ( 1 ( ) + 00 ), (17) another s the zero exctaton state wth a two-quanta antsymmetrc state 1 ( 1 ) ( ) (18) Fg. 8 The tme evoluton of the concurrence C AB(t) for symmetrc state wth χ 1 = χ = χ, when /g = 1, where χ/g = 0 (sold curve), χ/g = 0.5 (dashed curve), χ/g = 1 (dash-dotted curve), χ/g = 10 (dotted curve). We know the detunng results n the ESD havng been dscussed for the two ntal state wrtten above n double Jayness Cummng models. In ths secton, we are nterested n the nfluence of the NIKM, wth the defnte detunng. At frst we assume /g = 0. Fgure 6 shows

5 844 XIE Qn and FANG Mao-Fa Vol. 54 the tme evoluton of the concurrence C AB for the symmetrc state wth χ 1 = χ = χ. There s an ESD effect between two atoms n ths ntal state when χ/g = 0 and χ/g = 1. Gradually ncreasng the Kerr nonlnearty coeffcent, the ESD dsappears. When we ncrease χ/g, the mnmum value of the entanglement ncreases, but the maxmum value s fxed, near about 0.7, and the atomc entanglement s pretty close to 0.7 f we go on ncreasng the strength of the NIKM. In Fg. 7, t shows that the evoluton of C AB for the antsymmetrc state wth χ 1 = χ = χ. From the fgure, It s obvous that, the stable atomc entanglement exhbts at χ/g = 0, entanglement oscllaton appears at χ/g = 1 and the ESD exhbts when χ/g =. But on condton that we stll ncrease χ, we dscover that the entanglement would revval, so one can control the occurrence of ESD by usng the Kerr medum. Ths may be useful for the quantum nformaton process based on the entanglement. Fg. 9 The tme evoluton of the concurrence C AB(t) for antsymmetrc state wth χ 1 = χ = χ, when /g = 1, where χ/g = 0 (sold curve), χ/g = 0.5 (dashed curve), χ/g = 1 (dash-dotted curve), χ/g = 10 (dotted curve). Moreover we now focus on the case of /g = 1. It shows the atomc entanglement C AB wth χ 1 = χ = χ of the symmetrc state n Fg. 8. From Fg. 8, t s clear that as the value of χ/g ncreases the perod of C AB s extended. If we contnue to strengthen the NIKM, the ESD arses and when the NIKM s strong enough, the ESD dsappears and the value of C AB comes very closely to 0.7. Now we consder the effect of the NIKM for antsymmetrc state. Fgure 9 descrbes the evoluton of the concurrence C AB. Dependng on χ/g, dfferent dynamc behavor s clearly vsble. For χ/g = 0 and χ/g = 0.5, the ESD contnues and the perod of the ESD s postponed. For χ/g = 1, the ESD dsappears and the perod s prolonged. We also can obtan that the stronger of NIKM s the value of C AB more approaches to Summary and Dscusson In concluson, we have studed entanglement dynamcs of the atoms n the double Jaynes Cummngs models wth the Kerr medum, and examned the effects of the NIKM on the entanglement dynamcs of the atoms. Concretely, we have consdered two dfferent cases wth the equal NIKM: the frst one where there s only one exctaton present n the entre system, and we have proved that we can be near to the maxmum entanglement as much as possble va modulaton and control of the strength of the NIKM; The second one where there s two exctatons present, and we fnd that there occurs the entanglement sudden death effect only for a certan ntal atom state. By adjustng the Kerr nonlnear coeffcent, one can effectvely control the occurrence of entanglement sudden death when t s not desrable. Ths may be useful for quantum nformaton processng based on the entanglement, and the entanglement s more approxmate to 0.7 when the NIKM gets more stronger. References [1] M.B. Pleno and S.F. Huelga, Phys. Rev. Lett. 88 (00) [] S. Bose, K. Jacobs, and P.L. Knght, Phys. Rev. A 59 (1999) 304. [3] Abdel-Aty Mahmoud, M.R.B. Wahddn, and A.S.F. Obada, quant-ph/ (005). [4] R. Tanas and Z. Fcek, J. Opt. B: Quantum Semclass. Opt. 6 (004) S610. [5] Y. Wu, M.G. Payne, E.W. Hagley, and L. Deng, Phys. Rev. A 70 (004) [6] R. Tanaś and Z. Fcek, J. Opt. B: Quantum Semclass. Opt. 6 (004) S90. [7] V.S. Malnovsky and I.R. Sola, Phys. Rev. Lett. 93 (004) [8] S.J. Gu and H.Q. Ln, quant-ph/ (005). [9] H. Fu, X. Wang, and A.I. Solomon, Phys. Lett. A 91 (001) [10] Q. Wu, M.F. Fang, and Y.H. Hu, Chn. Phys. 16 (007) [11] Y.H. Hu, M.F. Fang, and X.P. Lao, Chn. Phys. 16 (007) [1] E. Schrnger, Proc. Cambrdge Phlos. Soc. 31 (1995) 555. [13] J.S. Bell, Physcs 1 (1964) 195. [14] E.T. Jaynes and F.W. Cummngs, Comparson of Quantum and Semclasscal Radaton Theores wth Applcaton to the Beam Maser Proc. IEEE (1963) [15] B.W. Shore and P.L. Knght, Topcal revew: the Jaynes- Cummngs Model J. Mod. Opt. 40 (1993) 1195 and references theren. [16] Isabel Sanz and Gunnar Björk, Phys. Rev. A 76 (007) [17] Stanley Chan, M.D. Red, and Z. Fcek, quant-ph/ (008). [18] Yan Xang-An, Wang L-qang, Jang Wen-Juan, Y Bao- Yn, Zheng Hua-Bng, and Song Jan-Png, Acta Photonca Snca 38 (009) 7.