We calculate the system-size-over-wave-length ($M$) dependence of
sample-to-sample conductance fluctuations, using the open kicked rotator to
model chaotic scattering in a ballistic quantum dot coupled by two $N$-mode
point contacts to electron reservoirs. Both a fully quantum mechanical and a
semiclassical calculation are presented, and found to be in good agreement. The
mean squared conductance fluctuations reach the universal quantum limit of
random-matrix-theory for small systems. For large systems they increase
$\propto M^2$ at fixed mean dwell time $\tau_D \propto M/N$. The universal
quantum fluctuations dominate over the nonuniversal classical fluctuations if
$N < \sqrt{M}$. When expressed as a ratio of time scales, the
quantum-to-classical crossover is governed by the ratio of Ehrenfest time and
ergodic time.Comment: 5 pages, 5 figures: one figure added, references update

A. Kormanyos

A. Lodder

A. Ossipov

A. Tajic

B.L. Altshuler

C. W. J. Beenakker

E.B. Bogomolny

F. Borgonovi

H.U. Baranger

I. Adagideli

I.L. Aleiner

J. Tworzydło

M.C. Goorden

M.G. Vavilov

O. Agam

P.A. Lee

P.G. Silvestrov

Ph. Jacquod

R.A. Jalabert

R.E. Prange

Y.V. Fyodorov

English

arXiv

Wetensch. publicatieFaculteit der Wiskunde en Natuurwetenschappe

Beenakker, C.W.J.

Tworzydlo, J.

Tajic, A.

Leiden University Scholary Publications

PHYS1CAL REVIEW B 69 165318 (2004)Quantum-to-classical crossover of mesoscopic conductance fluctuationsJ Twoizydlo ' " A Tajic ' and C W J Beenakkei'{lnstili/itt Loicnt Uiu\ tisileil Leiden PO Bo\ 9^06 2 WO RA Leiden The Nethei Ιαικ/ΊInstitute of l heoictical Plr\sic$ Wen scni Lfnucisin Ho a 69 00 681 Wen sau Poland(Receivcd 12 No\embci 2003 i c \ ised m i n u s u i p l iccened 1 5 J i n u i i > 2004 pubhshed 27 A p u l 2004)Wc cnlcuhte ihc s>stcm si/c o \ e i \\ ive length (M) dependence of sample to sample conduclince fiuciuaüons usmg the open kicked l ü t i t o i to modcl chiotic scutcung in a balhstic quantum dot coupled hy twoyV modc point contacls to elecuon leseivons Both i l u l l y quantum mechamcal and a scmiclassical cilculationaic picscnted ind iound to be in good igiecment The mean squaied conductince fluctuations leich theuniveisal quantum hmit of undom matux theoiy foi simll Systems Foi laige Systems they mciease ^M7 atfixed mean dwell tunc ro-rMIN The univeisal quantum fluctuations dommate ovei the nonumveisal classicüfluctuations if N< \[M When expiessed äs a laüo of Urne scales the quantum to classical ciossovei is go\emed by the latio öl Ehienfest time and ergodic UrneDOI 101103/PhysRevB 69 165318 PACS numbei(s) 73 23 -b 73 63 K\ 05 45 Ml 05 45 PqI INTRODUCIIONSample to sample fluctuations of the conductance of disoideied Systems have a univeisal legime in which they aieindependent of the mean conductance The lequnement foithese unneisal conductance fluctuations1 Ί is Chat the samplesize should be small compaied to the locahzation length Themean conductance is then much laigei than the conductancequantum e~"/hThe same condition applies to the univeisahty of conductance fluctuations in balhstic chaotic quantum dots 1 4 although theie is no locahzation in these Systems Randommatnx theoiy (RMT) has the univeisa l hmitllim vaiG= — (11)foi the vanance of the conductance G m units of e~*/h HeieN is the numbei of modes tiansmitted thiough each of thetwo balhstic pomt contacts that connect the quantum dot toelection leseivons Smce the mean conductance ( G ) —N/2the condition foi univeisahty lemams that the mean conductance should be laige compaied to the conductance quantumIn the piesent papei we wi l l show that theie is actually anuppei hmit on N beyond which RMT bieaks down in aquantum dot and the unneisahty of the conductance ftuctuations is lost Smce the width W of a pomt contact should besmall compaied to the size L of the quantum dot in oidei tohave chaotic scattenng a timal lequnement is N^Mwheie M is the numbei of tiansveise modes in a cioss sectionof the quantum dot (In two dimensions, N— W/\r and M— LI\Y with λ ( the Feimi wavelength ) By considenng thequantum to classical ciossovei we anive at the moie sinngent lequnement(12)with λ the L)apuno\ exponcnt and rei„ the eigodic Urne ölihe classical chaotic dynamics The lequnemenl is moiestnngent than N-^M becau.se t>pical ly λ ' and rc „ aieboth equal to the time of flight TO acioss the System so theexponential factoi in Eq (l 2) is not fai f iom unityExpiessed in teims of time scales the uppei hmit in Eq(l 2) says that rei„ should be laigei than the Ehienfest time56TF=maxΟ λ ,VThe condition τίιί>τΕ which we find foi the univeisahty ofconductance fluctuations is much moie stnngent than thecondition TD>TF foi the vahdity of RMT founci in otheicontexts 3~'' Heie το!=(ΜΙΝ)τ0 is the mean dwell time mthe quantum dot which is ^"rui in any chaotic SystemThe outline of this papei is äs tollows In See II wedescnbe the quantum-mechanical model that we use to calculate vai G numencally which is the same stioboscopicmodel used m pievious mvestigations of the Ehienfestt ime9 1 1 14 The data aie mteipieted semiclassically in SeeIII leading to the ciossovei cntenon (l 2) We conclude inSee IVII STROBOSCOPIC MODELThe physical System we have in mind is a balhstic (clean)quantum dot m a two dimensional election gas connected bytwo balhstic leads to election leseivons While the phasespace of this System is foui dimensional it can be leduced totwo dimensions on a Pomcaie suiface of section14 ^0 17-19 .!·> 16 Theopen kicked lotatoi is a stioboscopic model witha two dimensional phase space We summanze how thismodel is constmcted followmg Ref 11One staits fiom the closed System (without the leads) Thekicked lotatoi descubes a paiticle mo\mg along a cnclekicked peiiodically at time inteivals TO We set to unity ihestioboscopic time TO and the Plank constant ti The stioboscopic time evolution of a wa\e funct ion is gi\en by theFloquet opeiatoi T wh ich can be lepiesented by an MΧ Λ / t imtaiy symmetnc m a t i i x The e \ e n integei M dehnesthe eHectne Planck constant \i^\\— UM In the disciete cooidmate lepiesentation (v =inlM ;/i = 0 1 M—i) thematux elements of J- die given by01631829/2004/69( 16)1165318(5)/$22 50 69 165318 l ©2004 T h e Amencan Physical SocietyJ TWORZYDLO A TAJIC AND C W J BEENAKKER PHYSICAL RDVIEW B 69 165318 (2004)F m = M U2e '^e'iTMS^ χ > ^ (2 l)wheie S is the map geneiating tunction,(22)and K is the kickmg stiengthThe eigenvalues exp( — ιέ,,,) of-Fdefine the quasieneigiesε „ e (0 2rr) The mean spacing 2rr/M of the quasieneigiesplays the lole of the mean level spacing δ in the quantumdotTo model a pan of N mode balhstic leads, we imposeopen boundaiy conditions m a subspace of Hilbeit spacelepiesented by the mdices m^ The subscupt n= 1,2, N labels the modes and the supeiscnpt a= 1,2 labels the leads A 2NXM piojection matiix P descubes thecouplmg to the balhstic leads Its elements aie1 rI !l if m = ne{m(na>}0 otheiwise(23)The mean dwell time is το = Μ/2Ν (in units of TO)The matnces P and ^togethei deteimme the quasieneigydependent scatteiing matuxThe symmetiy of J^ensuies that S is also symmetnc, äs itshould be in the piesence of time-ieveisal symmetiy Bygiouping togethei the N mdices belonging to the same leadthe 2NX2N matux S can be decomposed mto foui subblocks contaming the NXN tiansmission and leflection matnces,(25)The conductance G (in units of e2lh) follows fiom the Landauei foimula? = Ti «r (26)The open quantum kicked lotatoi has a ciassical limitdescnbed by a map on the toius {A p|modulol} The ciassi-cal phase space, includmg the leads, is shown in Fig l Themap lelates x,p at time k to \',p' at time k+lp' = S(x',x) p= S(x' x) (27)äx' dxThe ciassical mechanics becomes fully chaotic foi K 3^7,with Lyapunov exponent λ^\η(Κ/2) Foi smallei K thephase space is mixed contaming both legions of chaotic andof legulai motion We wi l l lestiict ouiselves to the f u l l y cha-otic legime in this papeiIII NUMI RICAI RESULfSTo calculate the conductance (26) we need to mveit theMX M matux between squaie biackets in Eq (24) We dothis numencally using an iteiative pioceduie " The iteialiontv«\& ίFIG l Classical plnse space of the open kicked rotatoi Thedashed Imes indicate the two leads (shown foi the case TD 5)Inside each lead we plot the init ia l and final cooidtnates of tiajectones which aie innsmitted fiom the lelt to the nght lead after atmosl thiee iteiations (with K = 1 5) The pomts clustei along nariow tnnsmission bandscan be done efficiently using the fast Founei-tiansfoim algonthm to calculale the application of J-to a vectoi The timelequned to calculate 5 scales äs M 2 lnM which foi laige Mis quickei than the M1 scalmg of a dnect mveision Thememoiy leqimements scale äs M, because we need not stoiethe füll scatteiing matiix to obtam the conductanceWe distmguish two types of mesoscopic ftuctuations inthe conductance The fiist type appeais upon vaiymg thequasieneigy ε Ιοί a given scatteiing matux 5(ε) Smce thesefluctuations have no ciassical analog [the classicai map (2 7)bemg ε mdependent], we lefei to them äs quantum fluctuations The second type appeais upon vaiymg the position ofthe leads so these mvolve vanation of the scatteiing matiixat fixed ε We i efei to them äs sample to sample fluctuations They have both a quantum mechamcal component anda ciassical analog One could intioduce a thnd type of fluctuations mvolvmg both vanation of ε and of the lead positions We have found (äs expected) that these aie statisticallyequivalent to the sample to-sample fluctuations at fixed ε sowe need not distmguish between fluctuations of type 2 and 3We have calculated the vanance vai G = (G~) — (G)" ofthe conductance eithei by vaiymg ε at fixed lead positions(quantum fluctuations) 01 by vaiymg both ε and lead positions (sample to sample fluctuations) Smce the quantum mteiteience pattein is completely ditfeient only toi eneigyvauations oi oidei of the Thouless eneigy l/ro we choose anumbei TD of equal ly spated values of ε in the i n t e i v a l(02π) We take ten d i f l e i e n t lead positions, landomly located at the \ axis in Fig l To investigate the quantum tociassical ciossovei we change /i t i r =l/M while keepmg the165318 2QUANTUM-TO-CLASSICAL CROSSOVER OF . . .0.2PHYSICAL REVIEW B 69, 165318 (2004)0 150 1005K = 75τη = 10100 1000 10000MFIG. 2. Variance of the conductance flucluations obtamed nu-mencally by varying ε with fixed iead positions. Error bars indicatethe scatter of values oblained for different Iead positions. Resultsare shown äs a function of l//;eH = M, for two values of the dwellUrne TD = M/2N. The dashed line is the RMT prediction var Gdwell time rD = MI2N constant. The results are plotted inFiss. 2 and 3.IV. INTERPRETATIONWe Interpret the numerical data by assuming that the vari-ance of the conductance is the s um of two contributions: auniversal quantum-mechanical contribution VRMT given byrandom-matrix theory and a nonuniversal quasiclassical con-tribution Vd determined by sample-to-sample fluctuations inthe classical Iransmission probabilities.The RMT contribution equals3'4Vn ΜΗ— g , (4.1)in the presence of time-reversal symmetry. The classical con-tribution is calculated from the classical map (2.7), by deter-mining the probability f i _ 2 °f a particle injected randomlythrough Iead l to escape via Iead 2. Since the conductance isgiven semiclassically by GC\ = NP{_,2, we obtainV d =/V2 var (4.2)100100.1 <K = 75 O14 O40 D20.a'··.·· ,->ra..=, ··_:'·%··' .*'·-'m'"' i^V£*'-'--r*i'-'-'--·100 1000 10000MFIG 3 Same äs Fig. 2, but now for an ensemblc in which theIead positions and thc quasiencrgy are both vaned The dashed lincsare thc sum öl thc RMT value (4 . I ) and thc classical rcsult (4.2).Results are shown foi th icc values of thc k ick ing sticngth K OpcnSymbols are foi the dwell Urne TD= 10 and closcd ones foi TD= 20ΙΟ'10'ΙΟ'β·oTD = 20 D30 ·40 o60 ·15 2 25 3 35 4λFIG. 4. Variance of the classical fluctuations of the transmissionprobability P\ — i upon changcs of Iead positions, calculated nu-merically from the map (2.7). The data are shown for four values ofthe dwell time TD , äs a function of the Lyapunov exponent λ= ln(Ä72). The dotted hnes are the analytical prediction (4.3), withfit parameters c= 1.6 and Te,„ = 0.68 (the same for all data sets).We plot var G = VRMT+ Vci in Fig. 3 (dashed curves), forcomparison with the results of our füll quantum-mechanicalcalculation. The agreement is excellent.We now would like to investigate what ratio of timescales governs the crossover from quantum to classical fluc-tuations.To estimate the magnitude of the sample-to-sample fluc-tuations in the classical transmission probability, we use re-sults from Ref. 6. There it was found that the starting points(and end points) of transmitted trajectories are not homoge-neously distributed in phase space. Instead, they cluster to-gether in nearly parallel, narrow bands. These transmissionbands are clearly visible in Fig. 1. The largest band has anarea Anu^=A0e~XTae determined by the ergodic time reig.This is the time required for a trajectory to explore the wholeaccessible phase space. The values of relg and AQ depend onthe degree of collimation of the beam of trajectories injectedinto the System.6 For our model, without collimation, one hasTeig of order unity (one stroboscopic period) and A0— (N l M ) 2 . The typical transmission band has an areaA 0 e~X 7 / ) which is exponentially smaller than Amwi (sincerD=M/2N>rerg).As the position of the Iead is moved around, transmissionbands move into and out of the Iead. The resulting fluctua-tions in the transmission probability P{ _^2 are dominated bythe largest band. Since there is an exponentially large num-ber eKTr> of typical bands, their fluctuations average out. Thetotal area in phase space of the Iead is ΑίΐΆά=Ν/Μ, so weestimate the mean-squared fluctuations in /Ί_2 atvar (4.3)with c and Tel„ of order unity. We have tested this functionaldependence numerically for the map (2.7), and find a reason-able agreement (see Fig. 4). Both the exponential depen-dence on λ and the quadratic dependence on rD = MI2N areconsistenl with the data. We find Te,„ = 0.68 of order unity, äsexpected.Equations (4.2) and (4.3) implyvar G= (4.4)165318-3J TWORZYDLO, A TAJIC, AND C W J BEENAKKER PHYSICAL REVIEW B 69, 165318 (2004)100 r1000FIG 5 Same data äs in Fig 3 lescaled to show the approach toa single limitmg cutve m ihe laige dwcll Urne limit The solid hne iscalculated fiom Eq (44), with the same paiameteis c=\ 6, ταί,= 068 äs m Fig 4In Fig 5 we plot the same data äs in Fig 3, but now äs afunction of (N4/M2)e~2XTe"' We see that the functional de-pendence (44) is appioached toi laige dwell timesThe quantum fluctuations of RMT dommate ovei the clas-sical fluctuations if N2 vai P\^2(4 3), this amounts to the conditionUsmg the estimaterF (4 5)that the eigodic time exceeds the Ehienfest time Notice thatcondition (45) is always satisfied if N2<M= l//zeff Thisagiees with the findings ot Ref 6 that the bieakdown ofRMT staits when /Vä \ΪΜV. CONCLUSIONSIn summaiy, we have piesented both a fully quantum-mechanical and a semiclassical calculation of the quantum-to-classical ciossovei fiom umveisal to nonuniveisal con-ductance fluctuations The two calculations aie in veiy goodagieement, without any adjustable paiametei (compaie datapoints with cuives in Fig 3) We have also given an analyti-cal appioximation to the numeiical data, which allows us todeteimme the paiametnc dependence ot the ciossoveiWe have found that univeisahty ot the conductance fluc-tuations lequues the eigodic time reie to be laigei than theEhienfest time rc This condition is much moie stimgentthan the condition that the dwell time TD should be laigeithan TE, found pieviously foi univeisahty of the shot noisem a quantum dot 6 1 0 " The univeisahty of the excitation gapm a quantum dot connected to a supeiconductoi is also gov-eined by the latio TD/TE lathei than reig/TC,. ι , r . i i i i , rr _ , 1257-9 as is theuniveisahty of the weak-locahzation effect 1 2 1 3 These twopiopeities have in common that they lepiesent ensemble av-eiages, lathei than sample-to-sample fluctuationsWe piopose that what we have found heie foi the conduc-tance is genenc foi othei tianspoit piopeities That thebieakdown of RMT with incieasmg TC occuis when TL>TD foi ensemble aveiages and when TC>Tets foi the fluc-tuations This has immechate expenmental consequences, be-cause it is much easiei to violate the condition rL> Te,e thanthe condition TE>TDTo test this pioposal, an obvious next step would be todeteimme the latio ot time scales that govein the bieakdownof univeisahty of the fluctuations m the supeiconducting ex-citation gap The numeiical data m Refs 14 and 21 weiemteipieted in teims of the latio T L / T D , but an alternativedescnption m teims of the latio r£/r e i g was not consideiedOne final lemaik about the distmction between classicaland quantum fluctuations (explamed in See III) is as fol-lows It is possible to suppiess the classical fluctuations en-tnely, by vaiymg only the quasieneigy at fixed lead posi-tions In that case we would expect the bieakdown ofuniveisahty to be govemed by rD/rc mstead of Te i g/T£Oui numeiical data (Fig 2) clo not show any systematic de-viation fiom RMT, piobably because we coulcl not leachsufficiently laige Systems in oui Simulationl Note addedOui final lemaik above has been cnticized by Tacquodand Sukhoiukov 22 They aigue that the numeiical data of Fig2 (and similai data of then own) do not show any systematicdeviation fiom RMT because quantum fluctuations lemamumveisal if TF>TD Then aigument lehes on the assump-tion that the effective RMT ot Ref 6 holds not only toi theclassical fluctuations (as we assume heie), but also foi thequantum fluctuations The ettective RMT says that quantumfluctuations aie due to a numbei Ncii^Ne~T>Ιτ» of tiansmis-sion channels with a RMT distubution Univeisahty of thequantum fluctuations is then guaianteed even if N&K<N, aslong as /Veff is st i l l laige compaied to unityThis hne ot leasomng, if puisued tuithei, contiadicts theestabhshed theoiy1 2 n of the r£ dependence of weak local-ization RMT says that the weak-locahzation conection öG= - j is independent ot the numbei of channels ^ 4 Validityof the effective RMT at the quantum level would theiefoieimply that weak locahzation temains umveisal if rr> το , aslong as Ne~T''~n>i This contiadicts the lesult 8GT» of Rets 12 and 13ACKNOWLEDGMENTSThis woik was suppoited by the Dutch Science Founda-tion NWO/FOM J T acknowledges the financial suppoitpiovided thiough the Euiopean Commumty's Human Poten-tial Piogiam undei Contiact No I-TPRN-CT-2000-00144,Nanoscale Dynamics'B L A l l s h u l e i Pis ma Zh Eksp Tcoi Fiz 41 530 (1985) [JETPLeu 41 648 (1985)]2 P A Lee and A D Stone Phys Re\ Lett 55, 1622(1985)3 H U Baiangei and PA Mello Phys Rev Lctt 73, 142(1994)4 R A J a l a b c i l l - L Pichaid and C W l Bcenakkei CuiophysLeu 27 255 (1994)Ί Μ G Vavi lov and A l Laikm, Phys Rev B 67 115335 (2003)6 PG Silvestiov, M C Gooiden, and C W J Bcenakkei Phys Rev165318-4QUANTUM-TO-CLASSICAL CROSSOVER OFB 67, 241301 (2003)7 A Loddei and Yu V Nazarov, Phys Rev B 58, 5783 (1998)b PG Silvestiov MC Gooiden, and C W J Beenakkei, Phys RevLeu 90, 116801 (2003)9Ph Jacquod, H Schomeius, and C W J Beenakkei, Phys RevLeu 90. 207004 (2003)10O Agam, l Alemei, and A Laikm, Phys Rev Leu 85. 3153(2000)"j Twoizydlo, A Tajic, H Schomeius, and CWJ Beenakkei,Phys Rev B 68, 115313 (2003)121 L Alemei and A I Laikm, Phys Rev B 54, 14 423 (1996)nl Adagideh. Phys Rev B 68. 233308 (2003)1 4 MC Gooiden, Ph Jacquod, and C W J Beenakkei, Phys Rev B68, 220501 (2003)PHYSICAL REVIEW B 69. 165318 (2004)' ^ E B Bogomolny Nonhneanty 5 805(1992)1 6 R E Piange, Phys Re\ Lett 90,070401 (2003)I 7 F Boigonovi I Guainen, and D L Shepelyansky, Phys Rev A43, 4517 (1991)18F Boigonovi and I Guainen, J Phys A 25, 3239 (1992)I 9 Y V Fyodoiov and H - J Sommcis, Pis ma Zh Eksp Teoi Fiz72, 605 (2000) [JETP Lett 72, 422 (2000)]20A Ossipov T Kottos, and T Geisel, Euiophys Lett 62, 719(2003)2 1 A Koimanyos, Z Kaufmann, C J Lambeil, and J Cseiti, PhysRev B 67, 172506 (2003)22Ph Jacquod and E V Sukhoiukov, Phys Rev Lett 92, 116801(2004)L65318-5

Quantum-to-classical crossover of mesoscopic conductance fluctuations

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Quantum-to-classical crossover of mesoscopic conductance fluctuations

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