Casimir effect for quantum graphs

Transcription

1 Casmr eect or quantum graphs D.U. Matrasulov, J.R.Yusupov, P.K.Khaullaev, A.A.Saov Heat Physcs Department o the Uzek Acaemy o Scences, 8 Katartal St., 75 Tashkent, Uzekstan Astract The Casmr eects or one-mensonal ractal networks, so-calle quantum graphs s stue. Base on the Green uncton approach or quantum graphs zero-pont energy or some smplest topologes s wrtten explctly. The Casmr eect s a manestaton o the zero-pont oscllatons o the vacuum o a quantum (electromagnetc, scalar, etc.) el [-6]. In the smplest case t s the nteracton o two neutral, parallel perectly conuctng plates. Ths zero-pont energy can e calculate va solvng quantum el theoretcal equatons wth gven ounary contons. Despte the act that almost sxty years have passe snce rom H.B.G. Casmr pulshe hs amous paper, Casmr eect s stll remanng as a hot topc n contemporary physcs. Especally, Casmr eect has gane new evelopment ue to recent progress mae n nanoscale physcs, such as nanomechancs an mesoscopc physcs. In ths paper, we treat Casmr eect on quantum graphs. Namely, usng the Green uncton approach or quantum graphs we explore posslty o computatons o zero-pont energy n such systems. Quantum graphs are an mportant class o moels, whch have proven to e valuale or unerstanng quantum phenomena n mesoscopc an sorere systems as well as prolems n quantum normaton an molecular electroncs [7-9]. These are one-mensonal networks, whch were ntrouce earler n quantum chemstry to moel ree electron moton n organc molecules. Durng the last ecae they oun applcaton as moels n sol state physcs, acoustcs, mesoscopc physcs an quantum normaton theory as well as or organc molecules eng the key ulng locks or a possle new generaton o electronc evces on molecular scales. Quantum graphs have een shown to serve as accurate moels or the stuy o quantum transport an spectral statstcs [6,9] n nanosze systems. Recently quantum graphs are stue expermentally y usng mcrowave networks consstng o coaxal wavegues []. Zero pont energy n quantum graphs may play mportant role n varous systems (e.g., polymers, molecular networks, mcrowave networks an other supramolecular structures), whose ynamcs can e moelle y quantum graphs, as well as n nanomechancs. We gve rst re enton o quantum graph. Graph conssts o V vertces connecte y B ons. The valency o a vertex s the numer o ons meetng at that vertex. When the vertces an are connecte, we enote the connectng on y = (,). Partcle ynamcs n a quantum graph s escre y the ollowng onemensonal Schrönger [6] (n unts h = m = ):

2 A Ψ ( x) = k Ψ ( x), = (, ), () x where on each on, the component Ψ o the total waveuncton Ψ s a soluton o the eq.(). Here a magnetc vector potental s ntrouce to reak tme-reversal symmetry. The waveuncton, satses ounary contons at the vertces, whch ensure contnuty an curent conservaton [6]. The ounary contons are gven as ollows [6]: or every =,..., V Ψ A Contnuty, Ψ, ( x) = ϕ Ψ =, x, ( x) Current conservaton, C, A, Ψ, ( x) < x L L,, = ϕ, > C, A or all < an C, x Ψ, ( x), = λ ϕ () The parameters λ are ree parameters whch etermne the type o ounary contons. The specal case o zero λ 's correspons to Neumann ounary contons. Drchlet ounary contons correspon to the case when all the λ =. The egenunctons o the graph are completely etermne y the set o V unctons { ϕ }, values o the waveuncton at the vertces. At any on = (, ) = the component Ψ can e wrtten n terms o ts values on the vertces an as Ψ, A x, e = sn kl, A, L, ( ϕ sn[ k( L x)]) ϕ e sn kx) C, <. (), In the case o Drchlet ounary contons the egenuncton has smple structure: A x e Ψ = nπx sn L L an the egenvalues are gven y ( ) nπ kn = or all n >. L Several approaches are use to treat o such propertes o quantum graphs as spectral statstcs, transport an other propertes. In partcular, peroc ort theory s use to escre spectral statstcs n semclasscal approach [6]. Scatterng approach allows to treat partcle transport n quantum graphs [6]. Recently the Green uncton approach s evelope or explorng general quantum graphs [8]. Green uncton o a quantum graph, s ene y the equatons,

3 x x ) k ) k x m G ( x, ) = δ ( x ), h m G ( x, ) = δ ( x l h ) k ), < x, < l < x, < l ) =, {,ˆ} ˆ = l Note that x' s xe to elong to the on. The Green uncton satses the same ounary conton as the wave uncton. Recently the explct orm o the Green uncton or some graphs was erve an a prescrpton or nng the exact Green uncton or general (open or close) graphs was evelope [8]. Fgure. Fgure. For the graph whch s gven n Fg. the exact Green uncton s gven y G [ k x x ] S (k) [ k(x x )] = { δnl exp nl exp }. k nl where (k) S nl s the scatterng matrx ene rom the equaton ψ ( x; k) = δ exp[ kx] S ( k) exp[kx]. n n Ths scatterng matrx satses the ollowng contons n SS = S S =, S ( k) = S( k), an can e relate to the transmsson an relecton coecents, an R as ollows: S = R S = T nn n n n T n, For the graph n Fg. whch has the orm o tetraheron, the Green uncton can e wrtten as n

4 G = kg { ( I ( I exp ( [ kl] ) exp[ k( x x )] R exp[ k( x x )] R R I I ) exp[ kl] ) exp[ k( l x x )] R exp[ k(l x x )]}. () R = ( γ ( N )k) /( Nk γ ) ( T = k /( Nk γ )), I I = T ( T = I R = I exp[kl]{( R T )( R exp[ kl]) T R T R = R = R )exp[kl] ( R = k[ R exp[ kl] ( R T ) R T R T exp[kl] (T where exp[ kl]}/, RT RT T R N = R = { R R T R )exp[kl]}/ exp[ kl] )exp[kl]]. In the Green uncton ase approach the Casmr energy s calculate as [,8] L EC = lm k Γ( x, x, τ )x (5) τ τ where Γ ( x r, x r, τ ) s the nhomogeneous part o the Green uncton whch s gven y r r r r G x,, x x ) = G ( x ) Γ( x, x, x ) ( x The uncton Γ ( x r, x r, τ ) s ene y gven ounary contons, geometry or topology o the gven conguraton. The parameter L enes the sze characterstcs o the gven geometry (n the case o parallel plates L s the stance etween the plates). In the case o quantum graph L s the length o a on. Insertng the Green uncton n eq. () nto the eq. (5) an calculatng the otane ntegral numercally we get the Casmr energy or the graph. The ntegral over x can e taken easly. In partcular, or the graph gven y Fg. we have kl kl [( ( R I ) e ) kl ( e ) R] τ E = lm e k (6) τ k g The epenence o the Casmr energy o the lentgh o the on (here or the smplcty we assume that all the on have the same length, whch s equal to L ) can e otane y ntegratng numercally the eq.(6). Thus, we have evelope a prescrpton that allows to calculate zero-pont energy or quantum graphs. It shoul e note that wthn ths prescrpton we cannot speak aout Casmr orce snce the graphs consere wthn our approach are one-mensonal networks. Ths work s supporte n part y the grants o Founaton o Support o Funamental Research o the UzAS (Re.Nr.7-6) an y the grant o the CCSTD (F---). kτ

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