Fractional Langevin Equation in Quantum Systems with Memory Effect

Transcription

1 Applie Mahemaics, 4, 5, Publishe Olie Jue 4 i SciRes. hp:// hp://x.oi.org/.436/am Fracioal Lagevi Equaio i Quaum Sysems wih Memory Effec Jig-Nuo Wu, Hsi-Chie Huag, Szu-Cheg Cheg *, We-Feg Hsieh * Deparme of Opoelecric Physics, Chiese Culure Uiversiy, Taiwa Deparme of Phooics a Isiue of Elecro-Opical Egieerig, Taiwa jiguowu@gmail.com, hchuag@faculy.pccu.eu.w, * sccheg@faculy.pccu.eu.w, * wfhsieh@mail.cu.eu.w Receive 8 April 4; revise May 4; accepe 9 May 4 Copyrigh 4 by auhors a Scieific Research Publishig Ic. This wor is licese uer he Creaive Commos Aribuio Ieraioal Licese (CC BY. hp://creaivecommos.org/liceses/by/4./ Absrac I his paper, we irouce he fracioal geeralize Lagevi equaio (FGLE i quaum sysems wih memory effec. For a paricular form of memory erel ha characerizes he quaum sysem, we obai he aalyical soluio of he FGLE i erms of he wo-parameer Miag-Leffler fucio. Base o his soluio, we suy he ime evoluio of his sysem icluig he qubi excie-sae eergy, polarizaio a vo Neuma eropy. Memory effec of his sysem is observe irecly hrough he rappig saes of hese yamics. Keywors Fracioal Geeralize Lagevi Equaio, Memory Effec, Miag-Leffler Fucio, Memory Kerel, Trappig Saes, Polarizaio, vo Neuma Eropy. Iroucio Applicaio of he fracioal calculus o physics has arace a icreasig aeio his ecae icluig physical ieics [], aomalous raspor heory i soli-sae physics [], a oliear yamics [3]. These applicaios aim o explore he olocal quaum pheomea fou for eiher log-rage ieracios or ime-epee processes wih log-ime memory effec. Fracioal space a ime erivaives are use o escribe hese sysems wih olocal yamics, e.g. aomalous iffusio or fracioal Browia moio [4] [5]. Recely, he fracioal ime erivaive is give a physical ierpreaio by Iomi which escribes a effecive ieracio of a quaum sysem wih is evirome [6]. Here we will irouce he pracical quaum sysem wih log-ime memory effec [7]. * Correspoig auhors. How o cie his paper: Wu, J.-N., Huag, H.-C., Cheg, S.-C. a Hsieh, W.-F. (4 Fracioal Lagevi Equaio i Quaum Sysems wih Memory Effec. Applie Mahemaics, 5, hp://x.oi.org/.436/am.4.567

2 J.-N. Wu e al. Quaum iformaio sysems are currely aracig eormous ieres for heir fuameal aure a poeial applicaios o compuaio a secure commuicaio. May ovel mehos have bee propose o geerae corollable qubi saes hrough he evirome of he qubis [8]-[]. Whe a qubi mae of a wolevel aom is embee isie a srucure reservoir wih memory effec, he correlaio bewee he qubi a evirome will affec he yamics of he qubi [7] [] []. I his case, he effecive ieracio of he qubi wih he phoo moes of he evirome is expresse as a memory erel. I his paper, we ae a paricular memory erel of a aisoropic phooic crysal (PhC as a example o illusrae he meho a suiabiliy of applyig fracioal calculus o his qubi sysem. We fi ha he ieic equaio ca be expresse as a fracioal geeralize Lagevi equaio (FGLE whe he fracioal ime erivaive is use o express memory erel erm. The expressio of FGLE opes a ew roue for he applicaio of fracioal calculus o quaum iformaio sysems. This paper is orgaize as follows. I Secio, we prese a FGLE for he quaum sysem wih memory effec. The geeral soluio of his FGLE is expresse i erms of wo-parameer Miag-Leffler fucio hrough he mehoology of Laplace rasform. I Secio 3, a paricular memory erel characerizig he quaum sysem of a qubi i a aisoropic PhC is use o illusrae he solvig proceures of he FGLE. Base o he aalyical soluio of his FGLE, we suy he yamics of he qubi eergy, polarizaio a vo Neuma eropy. Fially, we summarize our resuls i Secio 4.. Fracioal Lagevi Equaio Whe he quaum sysem wih memory effec is cosiere, he geeral form of he ieic equaio ca be erive from he ime epee Schrőiger equaio as A A = K τ τ τ (. where A eoes he ime evoluio of he quaum sysem a K( τ o he memory erel of he reservoir. This ieic equaio reas ha he fuure of he sysem is eermie by he memory of he reservoir i is previous sae. To solve Equaio (., we irouce he Laplace rasform. Whe he Laplace image of he memory erel K is cosiere, we ee o cosruc a Caor se wih ifiie umber of ivisios of ime ierval because o characerisic ime scale exiss i his memory erel. As he Caor se is cosruce, we coul choose he eire ime ierval as T wih uiy heigh. Whe he ceral par of he ime ierval is remove, he ierval leaves wo iervals wih legh ξ T ( ξ <. I orer o eep he iegral memory, he heighs of he wo iervals mus be icrease from uiy o he value T ( ξ. I he ex sage, each remaiig ierval wih legh ξ T is subjece o he same ivisio process. As he ivisio process is performe imes, he memory erel i Laplace image will be represee by a se of iervals wih legh ξ T a heigh T ( ξ. The iegral of he memory erel i Laplace image is approache as he ivisio ime is ae o be ifiiy. This approach leas o he Laplace rasform of he memory erel has he form of e s K s K ( st ν wih ν beig he fracal imesio of ime blocs paricipaig i he cosruc of he Caor se. As he iverse Laplace rasform is performe o he erel K ( s s ν, we obai he memory erel approximaely as ν ( τ ( τ ( ν K Γ (. wih Gamma fucio Γ ( x. Subsiuig his memory erel io Equaio (., we have he ieic equaio as A ν = ( τ A( τ τ. Γ ( ν (.3 Comparig he righ-ha-sie erm of his equaio wih Riema-Liouville fracioal erivaive ν f ν = ( τ f ν ( τ τ, Γ (.4 ( ν 74

3 J.-N. Wu e al. we coul express he ieic equaio as a iffereial equaio wih fracioal orer, i.e., A ν A + =. (.5 ν I orer o o lose he iiial coiios of he quaum sysems, we apply he iegral operaor ( o his equaio a obai ν A A A + =. (.6 ν This iffereial equaio wih egaive fracioal orer coul be furher processe hrough maipulaig he fracioal iffereial operaor. This maipulaio yiels A + A = ν + ν + ν A. Γ Here we have expresse he ieic equaio of his quaum sysem wih memory effec as a FGLE hrough usig he fracioal ime erivaive for he memory erel. Whe we procee o solve he FGLE hrough Laplace rasform, we ee he formula of Laplace rasform i fracioal orer ( ν (.7 µ µ s ν µ L f e f s L{ f } s D f µ µ = (.8 = = wih Laplace variable s. Here he operaor of fracioal calculus a wih [ ] a D ν ν ν, Re[ ν ] > =, Re[ ν ] =, Re < ν ( τ [ ν ] a D ν is efie as ν ν Re ν beig he real par of he orer ν a he fracioal erivaive beig efie hrough he Riema-Liouvile form i Equaio (.4. A he Laplace rasform of expoeial orer ν coul be obaie from elemeary calculus as Γ( ν ( ν L = L =Γ s Γ = + ν ν+ ν ( ν if ν + >. Wih hese wo formula, he Laplace rasform of he FGLE has he algebraic form as, (.9 (. m m+ ν + m A ν+ ν ( s s + s A ( ( ν s. m+ ν + =Γ (. = = This algebraic form coul be expresse as a sum of parial fracios as wih expasio coefficies a a (. A s = a ν ( s X µ X beig he roos of he iicial Equaio (.. Whe he iverse Laplace rasform is performe o hese fracioal expasios of A ( s formula wih posiive ieger q ν of he FGLE is obaie as, we ee he q j q L X E ( jν, X ν = (.3 ( s X j= = a he wo-parameer Miag-Leffler fucio E ( α, a. The aalyical soluio 743

4 J.-N. Wu e al. q j q = ( ν, (.4 A a X E j X j= which is a liear combiaio of he wo-parameer Miag-Leffler fucios. These wo-parameer Miag- Leffler fucios are efie as wih he erivaive formula α a E ( α, a = e = α α = They are relae o he oe-parameer Miag-Leffler fucio E α q q ( α E a = a E a wih posiive ieger q = α [4]. α, = 3. A Paricular Memory Kerel ( a ( α (.5 Γ + + µ E ( α, a = E ( α µ, a. (.6 µ α = hrough = Γ + ( α I his secio, we cosier a quaum sysem of a qubi i a aisoropic phooic crysal (PhC wih a paricular memory erel as show i Figure. The Hamiloia of his quaum sysem is Here he aomic operaors (,, H= ω σ + ω aa + i g aσ σ a ( σ ij = i j i j = obey he commuaio relaio of σij, σ lm = δ jlσlm δimσlj wih he Kroecer ela fucio δ ij. A he phoo operaors a a a + follow he commuaio rules of a, a =, a +, a + + = a a, a = δ. The frequecy ω sas for he aomic rasiio frequecy from excie sae o grou sae a ω for he phoo moe frequecy of he reservoir wih wave- vecor. The couplig sregh bewee he aom a he phoo (elecromageic fiel is characerize by ω = eˆ uˆ εω V g wih he fixe qubi ipole mome = ˆ u, sample volume V, ielecric cosa a polarizaio ui vecor eˆ of he phoo moe wih frequecy ω. Figure. (Color olie (a A qubi wih excie sae a grou sae. The rasiio frequecy ω is early resoa wih he frequecy rage of he PhC reservoir; (b Direcioal epee ispersio relaios ear ba ege expresse by he effecive-mass approximaio wih he ege frequecy ω by soli a ashe curves; (c Phoo DOS c ρ( ω of he aisoropic PhC reservoir exhibiig forbie phoo moe below he ege frequecy ω c. 744

5 J.-N. Wu e al. As we assume oly oe phoo is creae or aihilae for oe aomic rasiio (sigle-phoo secor, he wave fucio of he sysem has he form ψ iω iω = A e, { } + D, { } + B e, { } θ θ iφ wih he iiial coiio A = e cos, D = si a he ampliues A, D( a B ψ equaio H i a (.8 B =. The equaios of moio for ca be obaie whe we projec he ime-epee Schrőiger ψ = o he sigle-phoo secor of he Hilber space as Ω i A = gb e, (.9 i e, B ga Ω = (. D = (. wih euig frequecy Ω = ω ω. The wo Equaios of (.9 a (. ca be combie as A = K( A τ τ τ (. wih he memory erel Ω i τ K τ g = e. This evoluio equaio relaes he excie ampliue A of he qubi o he reservoir memory hrough he memory erel K( τ his memory erel has a paricular form of ( τ K = π β ( τ wih he couplig cosa β = ( ω ( 3 ω c 6π c e i 3π 4 ( τ 3. For he aisoropic PhC reservoir, (.3 a he euig frequecy = ω ωc of he qubi rasiio frequecy ω from he ba ege frequecy ω c [7]. Wih his special form of memory A becomes erel, he ieic equaio of he excie ampliue if he rasformaio e i A C C β + = i C e iπ 4 π C ( τ ( τ τ (.4 3 = is performe. Here we express his memory effec as a fracioal ime erivaive which escribes he effecive ieracio of he qubi wih he eviromeal PhC reservoir. Tha is, he righ-ha-sie erm of he ieic Equaio (.4 is wrie as a Riema-Louville fracioal ime erivaive wih orer ν = a = such ha ( τ ( τ Γ C C τ. (.5 3 = This expressio leas o he fracioal form of he ieic equaio as C + i C + e C =. (.6 iπ 4 β π We efie his fracioal iffereial equaio as a fracioal geeralize Lagevi equaio (FGLE of his quaum sysem. We solve his FGLE by applyig Laplace rasform a obai C ( s =. (.7 π 4 s + i + β e i s 745

6 J.-N. Wu e al. Here C ( s is he Laplace rasform of C( wih he iiial coiio C A = =. I orer o fi he soluio of his excie ampliue, saar proceures of expressig his algebraic equaio as a sum of parial fracios a performig iverse Laplace rasform o hese parial fracios are ae. I he firs sep, iπ 4 we ee o fi he roos of he iicial equaio Y + β e Y + i =, where he variable s has bee covere io Y. Two is of roos exis i his iicial equaio: oe wih iffere roos Y Y a he oher wih egeerae roo Y = Y. For he case of iffere roos, C ( s is expresse as C ( s = (.8 ( s Y ( Y Y s Y wih a iπ 4 e Y = β + β (.9 iπ 4 Y e ( β β. π 4 For he egeerae case, we have β = which leas o he iicial equaio as ( Y β i parial fracios of C( s = (.3 + e =. The is hus wrie as C s =. (.3 ( s + β e iπ 4 As he iverse Laplace rasform is applie o hese parial-fracioal forms of C( s of he variables s, we use he formulas of a L = E, a + ae, a ( s a L ae,,, = a + ae a + + a E a ( s a wih he fracioal expoeial fucio E (, a wih fracioal powers (.3 (.33 α, whose efiiio a properies are lise i Equaios (.5 a (.6. This proceure leas o he soluio of he fracioal ieic equaio beig expresse as he liear combiaio of he fracioal expoeial fucios such ha e C E Y E Y YE Y Y E Y β iπ 4 =,, + (, (, for he iffere-roo case β ; a iπ 4 3/ i3π 4 C = ( + iβ E (, iβ β e E, iβ β e E, iβ (.34 (.35 for he egeerae case β =. Base o his aalyic soluio, we ca obai he reuce esiy marix of he qubi from he wave fucio i Equaio (.8 hrough racig over he reservoir egrees of freeom as ρ iφ cos e si ( θ ρ ˆ q =. ρ ρ iφ θ A e si ( θ A cos ρ θ A A (

7 J.-N. Wu e al. The elemes i his marix are associae wih he iformaio of he qubi eergy a coherece. I he followig, we will suy he yamics of he excie-sae eergy, polarizaio a vo Neuma eropy of he qubi. 3.. Excie-Sae Probabiliy We show he ime evoluio of he qubi excie-sae eergy hrough he probabiliy P ρ A = = i Figure. The eergy yamics exhibis oscillaory behavior a oes o ecay wih ime as he qubi frequecy lies isie he phooic bag gap (PBG regio ( β <. The memory effec of his quaum sysem is observe irecly hrough hese rappig saes of he qubi eergy. 3.. Polarizaio Dyamics As a qubi ieracs wih he evirome, i will raomize he polarizaio of he qubi. The quaum phase iformaio of a qubi carrie by he qubi polarizaio will hus escape from he qubi io he evirome hrough his raomizaio of polarizaio a lea o he quaum ecoherece. Here we show he polarizaio yamics i Figure 3 hrough he expressio of qubi polarizaio PZ = ρ + ρ = Re Up (if φ =, θ = π. The polarizaio yamics of he qubi wih frequecy lyig isie he PBG regio ( β < exhibis o-ecayig oscillaio. The qubi loses parial of is polarizaio i he very begiig perio of ime a he preserves he remaiig polarizaio hrough he seay oscillaio. This rappig sae of he qubi polarizaio reveals he memory effec of he sysem which leas o he preservaio of he qubi phase iformaio Dyamics of vo Neuma Eropy Eropy, a measureme of iformaio amou sore i a qubi, will be chage as he qubi is correlae o he evirome. The correlaio bewee he evirome a he sae will rasform he iiially pure sae of he qubi io a fially mixe sae where he amou of iformaio of he qubi is chage. For a qubi sae wih esiy marix ˆ ρ, vo Neuma eropy is efie as S = Tr ˆ ρ log ˆ ρ = λi log λ i i wih λ i beig he eigevalues of he marix ˆ ρ. for he esiy marix i Equaio (.36, we show he vo Neuma eropy i Figure 4 for he iiially excie qubi ( θ =. The eropy has is miimal value zero a = a reaches is maximal value log =.693 a he very begiig of ime. Afer a perio of ime o he orer of he ecay imescale, he eropy becomes seay wih ozero value for he qubi frequecy i PBG regio ( β <. This resul shows ha he iiially pure sysem becomes maximally mixe i he very begiig perio of ime. As he qubi equilibraig wih he PhC reservoir, he sysem becomes seay wih less mixe sae. The rappig sae of he vo Neuma eropy reveals ha he amou of iformaio sore i he qubi Figure. (Color olie Dyamics of he qubi exciesae probabiliy wih iffere euig frequecies δ β = ( ω β from he ba ege frequecy ω c. 747

8 J.-N. Wu e al. Figure 3. (Color olie Dyamics of he qubi polariδ β < zaio for he qubi frequecy lyig isie a ousie ( δ β > he ba gap. Figure 4. (Color olie Dyamics of he vo Neuma eropy for iffere qubi frequecies. is preserve hrough he seay mixe sae. The memory effec of he sysem is observe irecly hrough he preservaio of he qubi iformaio. 4. Coclusio We have use he fracioal ime erivaive o express he ieic equaio of he quaum sysem wih memory effec as a FGLE. For a paricular memory erel, we obai he soluio of he FGLE i erms of he woparameer Miag-Leffler fucio. I he suy of he qubi yamics i he paricular memory erel of a aisoropic PhC, we observe he memory effec irecly hrough he rappig saes of he qubi eergy, polarizaio a vo Neuma eropy. Acowlegemes We woul lie o graefully acowlege parially fiacial suppor from he Naioal Sciece Coucil (NSC, Taiwa uer Corac Nos. NSC -8-M-34-, NSC --M-34--MY3 a NSC -- M-9-6-MY3. Refereces [] Zaslavsy, G.M. (5 Hamiloia Chaos a Fracioal Dyamics. Oxfor Uiversiy Press, Oxfor. 748

9 J.-N. Wu e al. [] Mezler, R. a Klafer, J. ( The Raom Wals Guie o Aomalous Diffusio: A Fracioal Dyamics Approach. Physics Repors, 339, -77. hp://x.oi.org/.6/s37-573(7-3 [3] Zaslavsy, G.M. ( Chaos, Fracioal Kieics, a Aomalous Traspor. Physics Repors, 37, hp://x.oi.org/.6/s37-573(33-9 [4] Miller, K.S. a Ross, B. (993 A Iroucio o he Fracioal Calculus a Fracioal Differeial Equaios. Joh Wiley a Sos, Ic., Hoboe. [5] Samo, S.G., Kilbas, A.A. a Marichev, O.I. (993 Fracioal Iegrals a Derivaives: Theory a Applicaios. Goro a Breach, Loo. [6] Iomi, A. (9 Fracioal-Time Quaum Dyamics. Physical Review E, 8, Aricle ID: 3. hp://x.oi.org/.3/physreve.8.3 [7] Wu, J.-N., Huag, C.-H., Cheg, S.-C. a Hsieh, W.-F. ( Spoaeous Emissio from a Two-Level Aom i Aisoropic Oe-Ba Phooic Crysals: A Fracioal Calculus Approach. Physical Review A, 8, Aricle ID: 387. hp://x.oi.org/.3/physreva [8] Mazzola, L., Maiscalco, S., Piilo, J., Suomie, K.-A. a Garraway, B.M. (9 Sue Deah a Sue Birh of Eagleme i Commo Srucure Reservoirs. Physical Review A, 79, Aricle ID: 43. hp://x.oi.org/.3/physreva [9] Maiscalco, S. a Fracica, F., Zaffio, R.L., Lo, Gullo, N. a Plasia, F. (8 Proecig Eagleme via he Quaum Zeo Effec. Physical Review Leers,, Aricle ID: 953. hp://x.oi.org/.3/physrevle..953 [] Bellomo, B., Lo Fraco, R., Maiscalco, S. a Compago, G. (8 Eagleme Trappig i Srucure Eviromes. Physical Review A, 78, Aricle ID: 63(R. [] Jorgese, M.R., Galusha, J.W. a Barl, M.H. ( Srogly Moifie Spoaeous Emissio Raes i Diamo-Srucure Phooic Crysals. Physical Review Leers, 7, [] Joh, S. a Quag, T. (994 Spoaeous Emissio ear he Ege of a Phooic Ba Gap. Physical Review A, 5, 764. hp://x.oi.org/.3/physreva

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