Dynamic of Scaled Atomic Phase Entropy of a Single Two-Level Atom Interaction with SU(1,1) Quantum System

Transcription

1 Appl Mah Inf Sci 8, No 3, Applied Mahemaics & Informaion Sciences An Inernaional Journal hp://dxdoiorg/11785/amis/8319 Dynamic of Scaled Aomic Phase Enropy of a Single Two-Level Aom Ineracion wih SU1,1 Quanum Sysem M M A Ahmed, M Qohamey and S Abdel-Khalek Mahemaics Deparmen, Faculy of Science, Taif Universiy, Taif, Saudi Arabia Received: 1 May 13, Revised: 5 Sep 13, Acceped: 6 Sep 13 Published online: 1 May 1 Absrac: In his paper, we sudy he dynamics of he aomic inversion, scaled aomic Wehrl enropy and marginal aomic Q-funcion for a single wo-level aom ineracing wih SU1,1 quanum sysem We obain he wave funcion and sysem densiy marix using specific iniial condiions We examine he effecs of differen parameers on he scaled aomic Wehrl enropy, aomic Q-funcion and heir marginal disribuion We observe an ineresing monoonic relaion beween he differen physical quaniies for differen values of he iniial aomic posiion and deuning parameer Keywords: Scaled aomic Wehrl enropy, aomic Q-funcion, aomic inversion 1 Inroducion The mos imporan problems in quanum opics are he sudies of differen sysems ineracion such as field-aom, aom-aom and he field-field ineracion These problems have considered he subjec of grea deal of research works during he las decades In his way, here are numerous papers on hese problems For example he aom-field ineracion has been considered in[1]-[11], bu field-field ineracion [1]-[], while aom-aom ineracion [5]-[37] These ineracions has been classified from he poin of view of Lie algebra depending on he naure of he ineracion For example, he Hamilonian which represens he ineracion beween wo fields is described in he form of he parameric frequency converer is of SU Lie algebra ype While he Hamilonian which represens he non-degenerae parameric amplifier is of SU1,1 Lie algebra ype On he oher hand, he degenerae parameric amplifier, which conains in is ineracion erm he second harmonic generaion, is of SU1,1 Lie algebra ype In his conex a sysem which describes he ineracion beween SU and SU1,1 Lie algebra has been considered [38], in which a Hamilonian of he following from was reaed { H = h ωk z + ω σ z+ λk σ + + k + σ, Corresponding auhor sayedquanum@yahoocouk ω is he frequency of he sysem, hω is he energy difference beween he aomic levels and λ is a coupling consan Recenly, much aenion has been focused on informaion enropies as a measure or quanifier he enanglemen in quanum informaion [39] In his way he von Neumann enropy [], linear enropy, and Shannon informaion enropy [1] have been frequenly used in enanglemen-discussions concerning a variey of quanum sysems Some problem appear wih some of hese measures such as he SE involves only he diagonal elemens of he densiy marix so in can gives informaion similar o ha obained from he NE On he oher hand, here is an addiional enropic quaniy, namely, he semiclassical, aomic phase-space aomic Wehrl enropy AWE [] This measure has been successfully applied as enanglemen quanifier in he JCM For example, AWE of he modes are iniially prepared in a finie dimensional rio-coheren sae FTCS has discussed [3] Also, he dynamical properies of he AWE for a single wo-level rapped ion ineracing wih a laser field has been invesigaed [] I is shown ha he AWE gives quaniaive qualiaive informaion on he enanglemen of he biparie sysem c 1 NSP Naural Sciences Publishing Cor

2 19 M M A Ahmed e al : Dynamic of Scaled Aomic Phase Enropy of a Single In his aricle, we consider he exension of he problem by considering which is called he scaled aomic Wehrl enropy associaed wih he reduced aomic densiy operaor as an enanglemen quanifier beween SU1,1 and SU quanum sysem We focus on he effec of he exciaion number, iniial aomic sae and deuning parameer on he evoluion of he aomic inversion, scaled aomic Wehrl enropy and marginal aomic Q-funcion The paper is organized as follows: In Sec, he sysem Hamilonian of he ineracion beween SU1,1 and SU is inroduced, followed by a discussion of he mehod o calculae he scaled aomic Wehrl enropy and marginal aomic Q-funcion in Sec 3 Numerical resuls of he calculaed scaled aomic Wehrl enropy are presened and compared wih he marginal aomic Q-funcion in Sec, We conclude in Sec 5, wih a summary and an oulook The Sysem Hamilonian The Hamilonian which describe he ineracion beween a single wo-level aom and SU1,1 quanum sysem ake he following form H = ωk z + Ω 1 S 11 + Ω S + λk S 1 + k + S 1, 1 ω is he frequency of he sysem, Ω i is he energy and S i j are elemens of he SU1,1 group obeying he following commuaion relaion [S i j,s kl ]=S il δ k j S k j δ il, while k ± and k z saisfy he following commuaion relaion [k z,k ± ]=±k ±, [k,k + ]=k z and [S i j,k ±,z ]= 3 The Heisenberg equaion of moion for any operaor O is given by i do d =[O,H], h=1, hus, he equaions of moion for S i j and k z are given by i dk z d = [k z,h] = λk S 1 k + S 1, 7 i ds 11 i ds d d i dk z d + i dk z d = λk S 1 k + S 1 = 1 S 11 S +k z = consan of moion 8 from he above equaion, we can see ha N = 1 S 11 S +k z is consan of moion, herefore, he Hamilonian akes he following form H = ωn+c, 9 C = S 11 S + λk σ + + k + σ wih = Ω 1 Ω We noe ha [N,C] =, herefore [N,H] = [H,C] =, ie N and C are he consans of moion, he ime evoluion operaor is defined as hus U=exp ih, 1 U=exp iωnexp ic, 11 [ exp iω kz + exp iωn=[ 1 ] ] exp [ iω k z 1 ], 1 [ ] [ ] λk C=, C µ = 1 λk + µ, 13 µ j = +ν j, j= 1,, ν 1 = λ k k + and ν = λ k + k 1 we noe ha k µ = µ 1 k 15 also k + µ 1 = µ k + 16 i ds 11 d = [S 11, H] = λk S 1 k + S 1, 5 C 3 = [ µ 1 µ 1 λk ] [ ] µ λk +, C µ = 1 µ µ 17 i ds d = [S,H] = λk S 1 k + S 1, 6 exp ic = I+ ic 1! = I ic i C + ic!! + i C3 3 3! + ic3 3! c 1 NSP Naural Sciences Publishing Cor

3 Appl Mah Inf Sci 8, No 3, / wwwnauralspublishingcom/journalsasp 195 hen, one can wrie he ime evoluion operaor as [ ] F11 F U= 1, 19 F 1 F [ F 11 = exp iω k z + 1 ] cos µ 1 i [ F 1 = iλ exp iω k z + 1 ] sin µ1 k, µ 1 [ F 1 = iλ exp iω k z 1 ] sin µ k +, µ [ F = exp iω k z 1 ] cos µ + i sin µ 1, µ 1 sin µ µ The ime evoluion for he expecaion value of any operaor can be calculaed hrough he following relaion O = Ψ O Ψ = Ψ U + OU Ψ 1 Le us assume he iniial sae of he sysem can be wrien as Ψ = Ψ SU Ψ S1,1 = cos θ e +sin θ g m,k, k z m,k = m+k m,k, k + m,k = m+1m+k m+1,k, k m,k = mm+k 1 m 1,k 3 Ψ = U Ψ [ ][ ] F11 F = 1 cos θ F 1 F sin θ m,k = F 11 cos θ + F 1 sin θ m,k e +F 1 cos θ + F sin θ m,k g Subsiuing from Eqs in Eq, hen he final form of he wave funcion can be wrien as { Ψ = e iωk z+ 1 cos µ 1 i sin µ 1 cos θ µ 1 iλ sin µ 1 m,k,e k sin θ µ 1 { cos µ + i +e iωk z 1 iλ sin µ µ k + cos θ sin µ sin θ µ m,k,g 5 Then, he wave funcion can be wrien in he form Ψ =A m,k e +B m,k g, 6 and consequenly he densiy marix ρ= Ψ Ψ becomes ρ = A m,k e e k,m A +B m,k g g k,m B +B m,k g e k,m A +A m,k e g k,m B, 7 { A = e iωk z+ 1 cos µ 1 i iλ sin µ 1 k sin θ µ 1 { B = e iωk z 1 iλ sin µ µ k + cos θ one can easily cheek ha cos µ + i sin µ 1 cos θ µ 1, 8 sin µ sin θ µ, 9 A + B = 1 3 Thus, he expecaion value for any operaor can be calculaed hrough he following equaion O = Ψ O Ψ = Ψ O Ψ, 31 Ψ and Ψ are defined by Eqs and 6 Therefore, he expecaion values of he aomic operaors σ x and σ y can be obained as follows { σ x = 1 cosµ 1 cosµ µ 1 µ cosω sinµ 1 cosµ µ + cosµ 1sinµ { 1 µ σ y = 1 sinω+ cosµ 1 cosµ sinµ 1 sinµ sinµ 1 cosµ µ 1 + cosµ 1sinµ µ sinω sinθ, sinµ 1 sinµ µ 1 µ cosω sinθ, Now, we close his secion by presening he concep of he aomic populaion inversion which is he simples imporan quaniy o be calculaed I is relaed o he difference beween he probabiliies of finding he aom in he upper and lower sae = cosθ { + λ mm+k 1 sin µ m+1m+k sin µ 1 µ 1 µ cos θ 3 sin θ, 33 c 1 NSP Naural Sciences Publishing Cor

4 196 M M A Ahmed e al : Dynamic of Scaled Aomic Phase Enropy of a Single µ 1 = + λ m+1m+k, µ = + λ mm+k 1 3 Now, we are in a posiion o use he resuls obained in his secion o discuss he dynamical behavior of he aomic inversion, marginal aomic Q-funcion and scaled AWE in he following secions a b c d Scaled aomic Wehrl enropy, marginal disribuion and enanglemen quanifiers In his secion: we invesigae he marginal aomic Q-funcion and aomic Wehrl enropy AWE We sar our invesigaion by defining he aomic Q-funcion as [1] Q A Θ,Φ,= 1 π Θ,Φ ˆρ 11 Θ,Φ, 35 ˆρ is he densiy marix which is given in equaion 7 and Θ,Φ is he aomic coheren sae expressed as Fig 1: Time evoluion of he aomic inversion, for =,λ = 5, k = 1 and wih differen values of he exciaion number m and iniial aomic posiion θ and relaive phase φ = θ : Fig a m,θ=1,, Fig b m,θ= 1, π, Fig c m,θ=, and Fig d m,θ=, π Θ,Φ =cosθ/ e +sinθ/e iφ g, 36 5 a b 5 Θ π, Φ π he definiion 35 means ha wo differen spin coheren saes overlap unless hey direced ino wo anipodal poins on he sphere [1] The scaled aomic Wehrl enropy can be wrien in erms of he aomic Q- funcion as [1]: = ln 1 ln {lnπ e + π π Q AΘ,Φ,lnQ A Θ,Φ,sinΘdΘdΦ One can easily check ha he Q A 37 is normalized By c d inegraing he aomic Q-funcion Q A over he aomic variable Φ, we obain he marginal aomic Q-funcion as follows π Q Φ = Q A sinθdθ 38 Fig : Time evoluion of he aomic inversion, for =,λ = 5, k = 1 and wih differen values of he exciaion number m and iniial aomic posiion θ and relaive phase φ = θ : Fig a m,θ=1,, Fig b m,θ= 1, π, Fig c m,θ=, and Fig d m,θ=, π 3 Numerical resuls The populaion inversion of he aom is one of he imporan aomic dynamic variables of he sysem This in fac would give us informaion abou he behavior of he aom sae during ineracion ime In figure 1, we have ploed he dynamical behavior for differen values of he involved parameers We concenrae on he variaion of he iniial aomic posiion θ from he excied sae ie θ = o he superposiion sae ie θ = π/ as well as on he exciaion number m, which is in analogy wih he usual Jaynes Cummings model, corresponding o he number of phoons Firsly, we consider ha he sysem is iniially in he excied sae θ = and he absence of he deuning parameer = I is observed ha he aomic populaion inversion has a regular and periodic oscillaion c 1 NSP Naural Sciences Publishing Cor

5 Appl Mah Inf Sci 8, No 3, / wwwnauralspublishingcom/journalsasp a c b d Fig 3: Time evoluion of he scaled aomic Wehrl enropy, for =,λ = 5, k = 1 and for differen values of he exciaion number m and iniial aomic posiion θ and relaive phase φ = θ : Fig a m,θ=1,, Fig b m,θ= 1, π, Fig cm,θ=, and Fig dm,θ=, π Fig 5: The surface plo of he marginal aomic Q-funcion Q Φ versus he ime and he phase space parameer Φ for =,λ = 5, k = 1 and wih differen values of he exciaion number m and iniial aomic posiion θ and relaive phase φ = θ : Fig a m,θ = 1,, Fig b m,θ = 1, π, Fig c m,θ=, and Fig dm,θ=, π a c b d Fig : Time evoluion of he scaled aomic Wehrl enropy, for =,λ = 5, k = 1 and for differen values of he exciaion number m and iniial aomic posiion θ and relaive phase φ = θ : Fig a m,θ=1,, Fig b m,θ= 1, π, Fig cm,θ=, and Fig dm,θ=, π he ampliude of oscillaion is decrease by increasing he number of phoon exciaion The srucure of he aomic inversion oscillaions is changed when he aom is iniially in he superposiion sae see Fig 1b,d The number of oscillaion is increased when he effec of he aomic inversion is aken ino accoun see Fig The scaled aomic phase space enropy as a quanifier of he enanglemen beween wo-level aom and SU1,1 quanum field is ploed in Fig 3, As seen from Fig 3 has a periodic behavior and regular oscillaion The sysem reurns o is separable sae = a s = 5m m =,1,, On he oher hand he sysem is maximally enangled sae = ln a he middle of he ime inerval < < s Fig 3 d, depics ha he enanglemen is gradually decreases by increases he number of phoon exciaion when he aom is iniially in he superposiion sae Now, we are going o answer he quesion Wha is he impac of he deuning parameer on he aom-su1,1 field enanglemen for differen values of he number of phoon exciaion and iniial aomic posiion? As presened in Fig, he was ploed as a funcion of he ime when he aom is iniially in he excied and superposiion sae I is ineresing o noe he high amoun of he quanum enanglemen can be obained in he presence of he deuning parameer during he ime evoluion In Figure 5 depics he evoluion of Q Φ as a funcion of he ime and aomic phase space parameer Φ for differen modes of exciaions I is ineresing o menion here ha he behavior of Q Φ for differen values of he non-flucuaing componens of Rabi frequency I is observed ha Q Φ oscillaes beween minimum and maximum peaks The disribuion of he marginal aomic Q-funcion peaks in depending he iniial sae seing of he wo-level aom On he oher hand he number of peaks in increased by increasing he aomic Q-funcion peaks c 1 NSP Naural Sciences Publishing Cor

6 198 M M A Ahmed e al : Dynamic of Scaled Aomic Phase Enropy of a Single Conclusion I is well known ha he sudy of he physical properies of he aom field ineracion is a cenral and imporan problem in he field of quanum informaion In his paper, we have discussed he problem of he ineracion beween wo-level aom and SU1,1 quanum sysem The model was considered when he wo-level aom is iniially in superposiion sae and he wave funcion is obained analyically Using he scaled aomic phase space enropy he sysem enanglemen have been invesigaed The analysis herein has been carried ou a wo disinc consideraions of he deuning parameer and number of phoon exciaion Firsly, when he deeuning parameer was negleced, he scaled aomic phase space enropy has a regular oscillaions beween and ln during he ime evoluion There is some monoonic correlaion beween he behavior of he aomic inversion, scaled aomic phase space enropy and he marginal aomic Q-funcion Quanum enanglemen is a key resource which disinguishes quanum informaion heory from classical one I plays a cenral role in quanum informaion and compuaion and communicaion In he presen work, our resuls show ha he SU1,1 quanum field aom ineracion considering he effec of he number of phoons exciaion and deuning parameer have much richer srucure References [1] S Abdel-Khalek, Applied Mahemaics & Informaion Sciences, 1, 53 7 [] E T Jayness, F W Cummings, Proc IEEE, 51, [3] M Tavis, F W Cummings, Phys Rev, 17, [] M Tavis, F W Cummings, Phys Rev, 188, [5] E M Khalil, M S Abdalla, A S-F Obada, J Peřina, J Op Soc Am B, 37, 66 1 [6] M Abdel-Ay, S Abdel-Khalek, A S-F Obada, Op Rev, 7, 99 ; M Abdel-Ay, S Abdel-Khalek, A S-F Obada, Chaos, Solions & Fracals, 1, 15 1 [7] T M El-Shaha, S Abdel-Khalek, M Abdel-Ay, A S-F Obada, J of Mod Op, 5, 13 3 [8] T M El-Shaha, S Abdel-Khalek, M Abdel-Ay, A S-F Obada, Chaos, Solions & Fracals, 89, 18 3; T M El- Shaha, S Abdel-Khalek, A S-F Obada, Chaos, Solions & Fracals, 6, [9] M S Abdalla, A S-F Obada, M Abdel-Ay, Ann Phys, 318, 66 5 [1] W H Louisell, AYariv, A E Siegman, Phys Rev, 1, [11] W H Louisell, Coupled Mode and Parameric Elecronics, Wiley, New York, 196 [1] R J Glauber, Phys Rev, 131, [13] B R Mollow, R J Glauber, Phys Rev, 16, [1] E AMishkin, D FWalls, Phys Rev, 185, [15] D FWalls, R Baraka, Phys Rev, A, 1, [16] J Tucker, D F Walls, Ann Phys, N Y, 5, [17] J Tucker, D F Walls, Phys Rev, 178, [18] G P Agrawal, C L Meha, J Phys A: Mah Gen, 7, [19] M S Abdalla, Phys Rev A, 35, [] M S Abdalla, S S Hassan, A-S F Obada, Phys Rev A, 3, [1] S S Hassan, M S Abdalla, A-S F Obada, H A Baarfi, J Mod Op,, [] M S Abdalla, J Phys A: Mah Gen, 9, [3] M S Abdalla, M M Nassar, Ann Phys, 3, [] M S Abdalla, Phys A, 179, [5] M S Abdalla, M M A Ahmed, S Al-Homidan, J Phys A: Mah Gen, 31, [6] M S Abdalla, In J Mod Phys B, 16, 837 [7] D Loss, D P DiVincenzo, Phys Rev A, 57, [8] J Levy, Phys Rev A, 6, [9] C H Benne, D P Di Vincenzo, Naure,, 7 [3] G Burkard, D Loss, Phys Rev Le, 88, 7938 [31] A Imamoglu, D D Awschalom, G Burkard, D P DiVincenzo, D Loss, M Sherwin, ASmall, Phys Rev Le, 83, 1999 [3] G-F Zhang, S-S Li, Eur Phys J D, 37, 13 6 [33] Z-N Hu, S H Youn, K Kang, C S Kim, J Phys A: Mah Gen, 39, [3] M S Abdalla, E Lashin, G Sadiek, J Phys B: A Mol Op Phys, 1, [35] G Sadiek, E Lashin, M S Abdalla, Phys B,, [36] M S Abdalla, M M A Ahmed, A-S F Obada, Phys A, 16, [37] M S Abdalla, M M A Ahmed, A-SF Obada, Phys A, 17, [38] M S Abdalla, M M A Ahmed, Opics Communicaion, 8, [39] M A Nielsen and I L Chuang, Quanum Compuaion and Quanum Informaion, Cambridge, England, ; A S-F Obada, S Abdel-Khalek, Physica A, 389, 891 1; A S- F Obada, S Abdel-Khalek, A Plasino, Physica A, 39, [] von Neumann J Mahemaical Foundaions of Quanum Mechanics Princeon Universiy Press, Princeon, NJ, 1955 [1] C E Shannon and W Weaver The Mahemaical Theory of Communicaion Urbana Universiy Press, Chicago, 199 [] A-S F Obada and S Abdel-Khalek, J Phys A: Mah Gen, 37, 6573 ; R Dermez, S Abdel-Khalek, J of Russ Laser Research, 3, [3] S Abdel-Khalek, E M Khalil, and S I Ali, Laser Physics, 18, 1 8 [] S Abdel-Khalek, H F Abdel-Hameed and M Abdel-Ay, In J Quanum Informaion, 9, c 1 NSP Naural Sciences Publishing Cor