Semiclassical approach to the ac-conductance of chaotic cavities
Cyril Petitjean, Daniel Waltner, Jack Kuipers, Inanc Adagideli, Klaus Richter
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Semi lassi al approa h to the a - ondu tan e of haoti avitiesCyril Petitjean , Daniel Waltner , Ja k Kuipers , (cid:157)nanç Adagideli , and Klaus Ri hter Institut für Theoretis he Physik, Universität Regensburg, 93040 Regensburg, Germany. Fa ulty of Engineering and Natural S ien es, Saban i University, 34956 Tuzla Istanbul, Turkey.(Dated: September 16, 2018)We address frequen y-dependent quantum transport through mesos opi ondu tors in the semi- lassi al limit. By generalizing the traje tory-based semi lassi al theory of d quantum transport tothe a ase, we derive the average s reened ondu tan e as well as a weak-lo alization orre tionsfor haoti ondu tors. Thereby we on(cid:28)rm respe tive random matrix results and generalize themby a ounting for Ehrenfest time e(cid:27)e ts. We onsider the ase of a avity onne ted through manyleads to a ma ros opi ir uit whi h ontains a -sour es. In addition to the reservoir the avityitself is apa itively oupled to a gate. By in orporating tunnel barriers between avity and leadswe obtain results for arbitrary tunnel rates. Finally, based on our (cid:28)ndings we investigate the e(cid:27)e t ofdephasing on the harge relaxation resistan e of a mesos opi apa itor in the linear low-frequen yregime.PACS numbers: 05.45.Mt,74.40.+k,73.23.-b,03.65.YzI. INTRODUCTIONIn ontrast to d -transport experiments, the appliedexternal frequen y ω of an a -driven mesos opi stru -ture provides a new energy s ale ~ ω that permits one toa ess further properties of these systems, in luding theirintrinsi harge distribution and dynami s.The interest in the a -reponse of mesos opi ondu -tors goes ba k to the work of Pieper and Pri e1 on thedynami ondu tan e of a mesos opi Aharonov-Bohmring. This pioneering work was followed by several exper-iments ranging from photon-assisted transport to quan-tum shot noise2,3,4,5,6,7. More re ently, the a -regime hasbeen experimentally reinvestigated a hieving the mea-surement of the in and out of phase parts of the a - ondu tan e8 and the realization of a high-frequen y sin-gle ele tron sour e9. Moreover, the re ent rise of interestin the full ounting statisti s of harge transfer has ledto a reexamination of the frequen y noise spe tra10,11,12.This experimental progress has sin e triggered renewedtheoreti al interest in time dependent mesos opi trans-port13,14,15,16,17.One way to ta kle the a -transport problem is to startfrom linear response theory for a given potential distri-bution of the sample18,19,20. This involves the di(cid:30) ultythat, in prin iple, the potential distribution and morepre isely its link to the s reening is unknown. Anotherapproa h onsists of deriving the a -response to an ex-ternal perturbation that only enters into quantities de-s ribing the reservoirs. Su h approa hs were initiatedby Pastawski21 within a non-equilibruium Green fun -tion based generalized Landauer-Büttiker formalism, andthen the s attering matrix formalism of a time-dependentsystem was developed by Büttiker et al.22,23. Sin e theenergy is in general no longer onserved for an a -bias,the formalism is based on the on ept of a s attering ma-trix that depends on two energy arguments24 or equiv-alently on two times25. Fortunately, when the inversefrequen y is small ompared to the time to es ape the avity, the a -transport an be expressed in terms of thederivative of the s attering matrix with respe t to en-ergy26. In this arti le we start from the time dependents attering matrix formalism and limit our investigationsto open, lassi ally haoti ballisti ondu tors in the low-frequen y regime27.For a -transport we al ulate the average orrelator ofs attering matri es S ( E ) at di(cid:27)erent energies E . For thiswe need to know the joint distribution of the matrix el-ements S αβ ; ij at di(cid:27)erent values of the energy or otherparameters. (We label the reservoirs onne ted to the ondu tor by a greek index and the mode number by alatin index.) To our knowledge a general solution to thisproblem does not yet exist for haoti systems. How-ever, in the limit of a large number of hannels, the (cid:28)rstmoments of the distribution S αβ ; ij ( E ) S † αβ ; ij ( E ′ ) were de-rived using both semi lassi al methods28,29 and variousrandom matrix theory (RMT) based methods25,30,31,32.Although the a -transport properties of ballisti haoti systems seem to be well des ribed by the RMT of trans-port32, we develop a semi lassi al approa h for three rea-sons: First, this allows us to on(cid:28)rm the random matrixpredi tion by using a omplementary traje tory-basedsemi lassi al method. Se ond, the energy dependen ein the random matrix formalism was introdu ed by re-sorting to arti(cid:28) ial models su h as the "stub model"25.While being powerful, this treatment is far from mi ro-s opi or natural. The third and strongest reason is to gobeyond the RMT treatment and investigate the rossoverto the lassi al limit. Similarly as for the stati aseRMT is not appli able in this regime. As (cid:28)rst noti ed byAleiner and Larkin33, ballisti transport is hara terizedby a new time s ale, known as the Ehrenfest time τ E L and the lead width W . We an thus de(cid:28)ne an Ehrenfest time asso iated with ea hone36,37, the losed- avity Ehrenfest time, τ clE = λ − ln[ L/λ F ] , (1)and the open- avity Ehrenfest time, τ opE = λ − ln[ W /λ F L ] , (2)where λ is the lassi al Lyapunov exponent of the avity.Although the su ess of the semi lassi al method(beyond the so- alled diagonal approximation, see be-low) to des ribe quantitatively universal and nonuniversal d -transport properties is now learly es-tablished38,39,40,41,42,43,44,45,46,47,48,49, the orrespond-ing semi lassi al understanding of frequen y dependenttransport is far less developed. Based on an earliersemi lassi al evaluation of matrix element sum rules byWilkinson50 and a semi lassi al theory of linear responsefun tions51, a semi lassi al approa h to the frequen y-dependent ondu tivity within the Kubo-formalism ledto an expression of the a -(magneto-) ondu tivity σ ( ω ) in terms of a tra e formula for lassi al periodi orbits52.Closely related to this evaluation of σ ( ω ) is the problemof frequen y-dependent (infrared-) absorption in ballisti mesos opi avities whi h has been treated semi lassi- ally in Ref. [51℄. Peaks in the absorption ould be as-signed to resonan e e(cid:27)e ts when the external frequen y ω orresponds to the inverse periods of fundamental pe-riodi orbits in the avity. Ref. [33℄ ontains a (cid:28)rst, σ -model based approa h to weak lo alization e(cid:27)e ts inthe a -Kubo ondu tivity, where the (cid:28)ndings were inter-preted in a quasi lassi al traje tory pi ture (beyond thediagonal approximation). We note also that the semi- lassi al treatment of the produ t of s attering matri es S ( E ) at di(cid:27)erent energies, has been investigated in dif-ferent ontext su h as the Eri son (cid:29)u tuations41 and thetime delay48, however without onsidering the Ehrenfesttime dependen e.The outline of this arti le is as follows: In Se tion IIwe introdu e our model to treat the system of interestnamely a quantum dot under a bias, and re all some ba-si results about onservation laws in presen e of a timedependent (cid:28)eld. In Se t. III we present the method usedto treat s reening, whi h is based on a self- onsistent ap-proa h developed by Büttiker et al.23. The admittan e,i.e. the a - ondu tan e, is then al ulated semi lassi allyfor the parti ular ase of strong oupling to the leads(transparent onta t) in Se t. IV, where we illustrate ourresult by treating the time dependen e of a pulsed avity.We generalize the method to ope with arbitrary tunnelrates in Se t. V, and (cid:28)nally we use our general resultsto investigate dephasing e(cid:27)e ts on the harge relaxationresistan e of a mesos opi apa itor in Se t. VI.II. THE MODELWe onsider a ballisti quantum dot, i.e. a two-dimensional haoti avity oupled to M ele tron reser-voirs via M leads. Ea h lead α has a width W α and Figure 1: Two dimensional haoti avity with M leads andone gate . Ea h lead α has a width W α and is oupled toa reservoir at potential U α ( ω ) and urrent I α ( ω ) . Ea h tun-nel barrier is hara terized by the set of transmission prob-abilities Γ α = { Γ α, , · · · , Γ α,N α } . The gate and the sam-ple are apa itively oupled, whi h leads to a gate urrent I ( ω ) = − i ωC [ U ( ω ) − U ( ω )] .is oupled to the avity through a tunnel barrier (seeFig. 1). In addition to the treatment of Ref. [45℄ we as-sign a parti ular tunnel probability to ea h lead mode.The tunnel barrier is thus hara terized by a set of trans-mission probabilities, Γ α = { Γ α, , · · · , Γ α,N α } , with N α the maximum mode number of lead α . The haoti dot isadditionally apa itively oupled to a gate onne ted toa reservoir at voltage U ( ω ) , from whi h a urrent I ( ω ) (cid:29)ows. This apa itive oupling with the gate is takeninto a ount via a geometri al apa itan e C L , but still semi lassi allylarge, ≪ N α ≪ L /λ F . This requirement ensures thatthe parti le spend enough time inside the avity to expe-rien e the haoti dynami s.As usual for su h mesos opi stru tures we need to dis-tinguish between quantum and lassi al time s ales. Onthe quantum side we have already introdu ed the Ehren-fest times ( τ opE , τ clE ) in Eqs. (1,2), while another time s aleis the Heisenberg time τ H , the time to resolve the meanlevel spa ing of the system. On the lassi al side the timeof (cid:29)ight τ f between two onse utive boun es at the sys-tem avity wall is relevant. In most ballisti systems orbilliards we have τ f ≃ λ − . Another relevant time s aleis the ballisti ergodi time τ erg whi h determines howlong it takes for an ele tron to visit most of the availablephase spa e. However, as we deal with transport proper-ties, a further important time s ale is the dwell time τ D ,the average time spent in the avity before rea hing the onta t, we have τ D /τ erg ≫ . The related es ape ratetherefore satis(cid:28)es τ − = τ − M X α =1 N α X i =1 Γ α,i . (3)For small openings whi h we onsider here, we have λ τ D ≫ .The a -transport properties of su h a mesos opi sys-tem are hara terized by the dimensionless admittan e g αβ ( ω ) = G αβ ( ω ) /G = G − ∂I α ( ω ) /∂U β ( ω ) , (4)with G = d s e /h , where d s = 1 or in the absen eor presen e of spin degenera y. In this study we limitourselves to the oe(cid:30) ients g αβ ( ω ) with α, β = 1 , · · · , M where the oe(cid:30) ients denoting the gate are determinedby urrent onservation and the freedom to hoose thezero point of energy22, M X α =0 g αβ ( ω ) = M X β =0 g αβ ( ω ) = 0 . (5)We note that Eq. (5) is a straightforward onsequen eof the underlying gauge invarian e. Owing to the on-servation of harge, the total ele tri urrent ful(cid:28)lls the ontinuity equation ∇∇∇ · j p + ∂ρ∂t = 0 , (6)where ρ is the harge density and j p the parti le urrentdensity. For d -transport, the harge density is time in-dependent and so we have ∇∇∇ · j p = 0 . Thus the sum of all urrents that enter into the dot is always zero. Moreoverthe urrent properties must remain un hanged under a si-multaneous global shift of the voltages of the reservoirs.These onditions imply the well know unitarity of thes attering matrix54, X α,i S † αβ ; ij ( E ) S αγ ; ik ( E ) = δ βγ ; jk . (7)For a -transport, the produ t of s attering matri esat di(cid:27)erent energies no longer obey a similar prop-erty54,55,56,57 i.e. X α,i S † αβ ; ij ( E ) S αγ ; ik ( E ′ ) = δ βγ ; jk , (8)indeed this inequality expresses the fa t that, due to thepossible temporary pile up of harge in the avity, theparti le urrent density no longer satis(cid:28)es ∇∇∇ · j p = 0 .However one an instead use the Poisson equation ∇∇∇ · D = ρ, (9)where D = − ǫ ∇∇∇ ϕ with ϕ the ele tri potential, to de(cid:28)nethe total ele tri urrent density whi h satis(cid:28)es ∇∇∇ · j = 0 ,as a sum of a parti le and a displa ement urrent: j = j p + ∂ D ∂t . (10)In order to (cid:28)nd j one needs to know the ele tri al (cid:28)eld D . In general its al ulation is not a trivial task be ausethe intrinsi many-body aspe t of the problem makes thetreatment of the Poisson equation (9) tri ky, espe ially if it is ne essary to treat the parti le and displa ement urrent on the same footing.In this work we shall adopt the approa h of Ref. [23℄ tosimplify the problem. In this approa h the environmentis redu ed to a single gate, the Coulomb intera tion isdes ribed by a geometri al apa itan e C , and the two urrents are treated on di(cid:27)erent footing; the parti le ur-rent is al ulated quantum me hani ally via the s atter-ing approa h, while the displa ement urrent is treated lassi ally via the ele trostati law (Eqs. (6,9)). This sim-pli(cid:28) ation will permit us below to re-express the Poissonequation (9) to obtain the simplest gauge invariant the-ory that takes are of the s reening. We emphasize thateven though our model ould be thought of as oversim-pli(cid:28)ed it has the advantage of being able to probe thee(cid:27)e ts due to the long range Coulomb intera tion. In-deed, for non-intera ting parti les it is possible to treatthe dot and the gate via two sets of un orrelated onti-nuity equations. The Coulomb intera tion removes thispossibility, and we need to onsider the gate and dot asa whole system.III. EXPRESSION FOR THE ADMITTANCEThe method to ompute the admittan e pro eeds intwo steps55: First the dire t response (parti le urrent)to the hange of the external potential is al ulated un-der the assumption that the internal potential U ( ω ) ofthe sample is (cid:28)xed. This leads to the de(cid:28)nition of theuns reened admittan e g uαβ ( ω ) . Se ond, a self- onsistentpro edure based on the gauge invarian e ( urrent on-servation and freedom to hoose the zero of voltages) isused to obtain the s reened admittan e g αβ ( ω ) .The uns reened admittan e reads22 g uαβ ( ω ) = Z d E f ( E − ~ ω ) − f ( E + ~ ω )) ~ ω (11) × Tr (cid:20) δ αβ α − S αβ (cid:18) E + ~ ω (cid:19) S † αβ (cid:18) E − ~ ω (cid:19)(cid:21) , where f ( E ) stands for the Fermi distribution, S αβ is the N α × N β s attering matrix from lead β to lead α , and α is an N α × N α identity matrix. Under the assumptionthat U ( ω ) is spatially uniform, the s reened admittan e g αβ ( ω ) is straightforward to obtain22. For sake of om-pleteness we present here only the outline of the methodand refer to Ref. [26℄ for more details.On the one hand the urrent reponse at onta t α is I α ( ω ) = G M X β =1 g uαβ ( ω ) U β ( ω ) + g iα ( ω ) U ( ω ) , (12)where g iα ( ω ) is the unknown internal reponse of themesos opi ondu tor generated by the (cid:29)u tuating po-tential U ( ω ) . On the other hand the urrent indu ed atthe gate is I ( ω ) = − i ωC [ U ( ω ) − U ( ω )] . (13)Gauge invarian e permits a shift of − U ( ω ) and providesan expression for the unknown internal response, g iα ( ω ) = − M X β =1 g uαβ ( ω ) . (14)Then urrent onservation, P Mα =1 I α ( ω ) + I ( ω ) = 0 ,yields the result of the s reened admittan e22, g αβ ( ω ) = g uαβ ( ω ) + P Mδ =1 g uαδ ( ω ) P Mδ ′ =1 g uδ ′ β ( ω ) i ωC/G − P Mδ =1 P Mδ ′ =1 g uδδ ′ ( ω ) . (15)In the self- onsistent approa h used to obtain Eq. (15),the only ele tron-ele tron intera tion term that has been onsidered is the apa itive harging energy of the avity.This implies that we should onsider a su(cid:30) iently largequantum dot58. We note that, using a /N -expansion,the self- onsistent approa h above was re ently formally on(cid:28)rmed in Ref. [59℄. Moreover, Eq. (15) an be gener-alized to non-equilibrium problems, using Keldysh non-equilibrium Green fun tions60.In the next se tions we present the semi lassi al evalu-ation of Eq. (11) in the zero temperature limit (in luding(cid:28)nite temperature is straightforward). For reasons of pre-sentation we (cid:28)rst give the semi lassi al derivation for thetransparent ase in Se t. IV, and then we explore thegeneral ase in Se t. V. In Se t. VI we present an appli- ation of the s reened result for tunnel oupling, when we ompute the relaxation resistan e of a mesos opi haoti apa itor.IV. SEMICLASSICAL THEORY FOR THEADMITTANCEA. Semi lassi al approximationWe (cid:28)rst onsider the multi-terminal ase assumingtransparent barriers, i.e. Γ α,i = 1 , ∀ ( α, i ) . In the limit k B T → the uns reened admittan e, Eq. (11), redu esto g uαβ ( ω ) = N α δ αβ − Tr (cid:20) S αβ ( E F + ~ ω S † αβ ( E F − ~ ω (cid:21) . (16)Semi lassi ally, the matrix elements for s attering pro- esses from mode i in lead β to mode j in lead α read29,61 S αβ ; ji ( E F ± ~ ω (17) − Z β d x Z α d x h j | x ih x | i i (2 π i ~ ) / X γ A γ e i ~ S γ ( x,x ; E F ± ~ ω ) , where | i i is the transverse wave fun tion of the i -th mode.Here the x (or x ) integral is over the ross se tion of the β th (or α th) lead. At this point S αβ is given by a sumover lassi al traje tories, labelled by γ . The lassi al paths γ onne t X = ( x , p x ) (on a ross se tion oflead β ) to X = ( x, p x ) (on a ross se tion of lead α ).Ea h path gives a ontribution os illating with a tion S γ (in luding Maslov indi es) evaluated at the energy E F ± ~ ω/ and weighted by the the omplex amplitude A γ . This redu es to the square root of an inverse elementof the stability matrix62, i.e. A γ = | (d p x / d x ) γ | .We insert Eq. (17) into Eq. (16) and obtain doublesums over paths γ , γ ′ and lead modes | i i , | j i . The sumover the hannel indi es is then performed with the semi- lassi al approximation45, P N β i =1 h x | i ih i | x ′ i ≈ δ ( x ′ − x ) ,and yields g uαβ ( ω ) − N α δ αβ = − Z β d x Z α d x X γ,γ ′ A γ A ∗ γ ′ π ~ e i ~ δS ( E F ,ω ) . (18)Here, δS ( E F , ω ) = S γ ( x , x ; E F + ~ ω − S γ ′ ( x , x ; E F − ~ ω . (19)As we are interested in the limit ~ ω ≪ E F , we an expand δS ( E F , ω ) around E F . The dimensionless a - ondu tan eis then given by g uαβ ( ω ) − N α δ αβ = − Z β d x Z α d x X γ,γ ′ A γ A ∗ γ ′ π ~ (20) × exp (cid:20) i ~ δS ( E F ) + i ω t γ + t γ ′ ) (cid:21) , where δS ( E F ) = S γ ( x , x ; E F ) − S γ ′ ( x , x ; E F ) and t γ ( t γ ′ ) is the total duration of the path γ ( γ ′ ). Eq. (20) isthe starting point of our further investigations.B. Drude Admittan eWe are interested in quantities arising from averagingover variations in the energy or avity shapes. For mostsets of paths, the phase given by the linearized a tion dif-feren e δS ( E F ) will os illate widely with these variations,so their ontributions will average out. In the semi las-si al limit, the dominant ontribution to Eq. (20) is thediagonal one, γ = γ ′ , whi h leads to t γ = t γ ′ , δS ( E F ) = 0 and gives g u, D αβ ( ω ) = N α δ αβ − Z β d x Z α d x X γ | A γ | π ~ e i ωt γ . (21)In the following we pro eed along the lines of Ref. [42℄.The key point is the repla ement of the semi lassi al am-plitudes by their orresponding lassi al probabilities. Tothis end we use a lassi al sum rule valid under ergodi assumptions63, X γ | A γ | e i ωt γ [ · · · ] γ = (22) Z ∞ d t Z π/ − π/ d θ d θ e i ωt p F cos( θ ) P ( X , X ; t )[ · · · ] X . Figure 2: A semi lassi al ontribution to weak lo alizationfor a system with strong (transparent) oupling to the leads.The two paths follow ea h other losely everywhere ex eptat the en ounter, where one path (dashed line) rosses itselfat an angle ǫ , while the other one (full line) does not (goingthe opposite way around the loop). The ross-hat hed areadenotes the region where two segments of the solid paths arepaired (within W α ≃ W β ≃ W of ea h other)In Eq. (22), p F cos( θ ) is the initial momentum along theinje tion lead and P ( X , X ; t ) the lassi al probabilitydensity to go from an initial phase spa e point X =( x , θ ) at the boundary between the system and the leadto the orresponding point X = ( x, θ ) . The average of P over an ensemble or over energy gives a smooth fun tionthat reads h P ( X , X ; t ) i = cos( θ )2 τ D P Mα =1 W α e − t/τ D , (23)with the es ape rate τ − given in Eq. (3).Using Eqs. (21), (22) and (23), we re over the Drudeadmittan e g u, D αβ ( ω ) = N α δ αβ − N α N β N (cid:18) − i ωτ D (cid:19) , (24)where N = P Mα =1 N α .C. Weak lo alization for transmission, re(cid:29)e tionand oherent ba ks attering1. Weak lo alizationThe leading-order weak-lo alization orre tion to the ondu tan e was identi(cid:28)ed in Refs. [33,39℄ as those aris-ing from traje tories that are exponentially lose almosteverywhere ex ept in the vi inity of an en ounter. Anexample of su h a traje tory pair for haoti ballisti systems is shown in Fig. 2. At the en ounter, separat-ing the `loop' from the `legs', one of the traje tories ( γ ′ )interse ts itself, while the other one ( γ ) avoids the ross-ing. Thus, they travel along the loop they form in op-posite dire tions. In the semi lassi al limit, only pairsof traje tories with a small rossing angle ǫ ontributesigni(cid:28) antly to weak lo alization. In this ase, ea h tra-je tory remains orrelated for some time on both sides of the en ounter. In other words, the smallness of ǫ re-quires two minimal times: T L ( ǫ ) to form a loop, and T W ( ǫ ) in order for the legs to separate before es apinginto di(cid:27)erent leads. The en ounter introdu es a typi allength s ale δr ⊥ that orresponds to the perpendi ulardistan e between the two paths in the vi inity of theen ounter. In the ase of hyperboli dynami s, we get δr ⊥ = v F ǫ/ (2 λ ) ∼ Lǫ . Hen e, the typi al minimal timeis given by T ℓ ( ǫ ) = λ − ln[( ℓ/δr ⊥ ) ] , with ℓ = { L , W } that we an approximate as T L ( ǫ ) ≃ λ − ln[ ǫ − ] , (25a) T W ( ǫ ) ≃ λ − ln[ ǫ − ( W/L ) ] . (25b)The presen e of the external driving does not hangethis pi ture. Ea h weak-lo alization ontribution a u-mulates a phase di(cid:27)eren e given by the linearized a tion δS ( E F ) ≃ δS RS = E F ǫ /λ
39. Following the same linesas for the derivation of the Drude ontribution, thoughthe sum over paths is now restri ted to paths with anen ounter, the sum rule (22) still applies, provided theprobability P ( X , X ; t ) is restri ted to paths whi h rossthemselves. To ensure this we write P ( X , X ; t ) = Z C d R d R P ( X , R ; t − t ) × P ( R , R ; t − t ) P ( R , X ; t ) , (26)where the integration is performed over the energy sur-fa e C . Here, we use R i = ( r i , φ i ) , φ i ∈ [ − π, π ] for phasespa e points inside the avity, while X lies on the leadsurfa e as before.We then restri t the probabilities inside the integralto traje tories whi h ross themselves at phase spa epositions R , with the (cid:28)rst (or se ond) visit of the rossing o urring at time t (or t ). We an write d R = v sin ǫ d t d t d ǫ and set R = ( r , φ ± ǫ ) . Thenthe weak-lo alization orre tion is given by g u, wl αβ ( ω ) = 1 π ~ Z β d X Z d ǫ ℜ e h e i δS RS / ~ i h F ( X , ǫ, ω ) i , (27)with, F ( X , ǫ, ω ) = (28) v sin ǫ Z ∞ T L + T W d t Z t − T W / T L + T W / d t Z t − T L T W / d t × p F cos θ Z R d Y Z C d R P ( X , R ; t − t ) × P ( R , R ; t − t ) P ( R , X ; t ) e i ωt . Under our approximation t γ ′ ≃ t γ = t , the intro-du tion of the driving frequen y leads to performing aFourier transform of the survival probability, and we ob-tain h F ( X , ǫ, ω ) i = ( v F τ D ) p F sin ǫ cos θ π Ω N α N (29) × exp [ − T L /τ D ] exp [ i ω ( T L + T W )](1 − i ωτ D ) , with Ω the avity area. Inserting Eq. (29) into Eq. (27),the ǫ integral is dominated by small angle ( ǫ ≪ ) ontributions, allowing for the approximation sin ǫ ≃ ǫ and pushing the upper limit to in(cid:28)nity. This yieldsan Euler Gamma fun tion times an exponential term e − τ clE /τ D e i ω ( τ clE + τ opE ) (with τ opE and τ clE given by Eqs. (1,2)that reads, to leading order in ( λ τ D ) − , Z ∞ d ǫ ℜ e (cid:20) exp (cid:20) i E F ǫ λ ~ (cid:21)(cid:21) ǫ λτ D (1 − i ωτ D ) (cid:18) WL (cid:19) i ωλ ≃ − π ~ mv τ D e − τ clE τ D + i ω ( τ clE + τ opE ) (1 − i ωτ D )+ O (cid:20) λτ D (cid:21) . (30)Performing the X integral and using N β = ( π ~ ) − p F W β and N = ( ~ τ D ) − m Ω , the weak-lo alization orre tion tothe uns reened admittan e is g u, wl αβ ( ω ) = N α N β N e − τ clE /τ D (1 − i ωτ D ) e i ω ( τ clE + τ opE ) (1 − i ωτ D ) . (31)We note that due to the absen e of unitarity of the un-s reened admittan e we need to expli itly evaluate all theelements of g uαβ ( ω ) . The weak-lo alization ontributionto re(cid:29)e tion r u, wl αα ( ω ) is derived in the same manner as g u, wl αβ ( ω ) , repla ing however the fa tor N β /N by N α /N .We then obtain r u, wl αα ( ω ) = (cid:18) N α N (cid:19) e − τ clE /τ D (1 − i ωτ D ) e i ω ( τ clE + τ opE ) (1 − i ωτ D ) . (32)However as in the d - ase there is another leading-order ontribution to the re(cid:29)e tion, the so- alled oherentba ks attering. This di(cid:27)ers from weak lo alization asthe path segments that hit the lead are orrelated. Thisme hanism should be treated separately when omputingthe Ehrenfest time dependen e, whi h is the obje t of thenext paragraph.2. Coherent ba ks atteringThough the orrelation between two paths does notin(cid:29)uen e the treatment of the external frequen y, it in-du es an a tion di(cid:27)eren e δS ( E F ) = δS cbs = − ( p ⊥ + mλr ⊥ ) r ⊥ where the perpendi ular di(cid:27)eren e in po-sition and momentum are r ⊥ = ( x − x ) cos θ and p ⊥ = − p F ( θ − θ ) . As for weak lo alization, we an identify two times ales, T ′ L , T ′ W , asso iated with thetime for paths to spread to L, W , respe tively. Howeverunlike for weak lo alization we de(cid:28)ne these times alesas times measured from the lead rather than from theen ounter. Thus we have T ′ ℓ ( r ⊥ , p ⊥ ) ≃ λ ln [( mλℓ ) / | p ⊥ + mλr ⊥ | ] , (33)with ℓ = { L, W }