Noise of a Quantum-Dot System in the Cotunneling Regime Eugene V. Sukhorukov, Guido Burkard, and Daniel Loss Department of Physics and Astronomy, University of Basel, Klingelbergstrasse 82, CH–4056 Basel, Switzerland 1 Westudythenoiseofthecotunnelingcurrentthroughoneorseveraltunnel-coupledquantumdots 0 in the Coulomb blockade regime. The various regimes of weak and strong, elastic and inelastic 0 cotunneling are analyzed for quantum-dot systems (QDS) with few-level, nearly-degenerate, and 2 continuous electronic spectra. We find that in contrast to sequential tunneling where the noise is n either Poissonian (due to uncorrelated tunneling events) or sub-Poissonian (suppressed by charge a conservationontheQDS),thenoiseininelasticcotunnelingcanbesuper-Poissonianduetoswitching J between QDS states carrying currents of different strengths. In the case of weak cotunneling we 2 prove a non-equilibrium fluctuation-dissipation theorem which leads to a universal expression for 2 the noise-to-current ratio (Fano factor). In order to investigate strong cotunneling we develop a microscopictheoryofcotunnelingbasedonthedensity-operatorformalism andusingtheprojection ] operator technique. The master equation for the QDS and the expressions for current and noise l l in cotunneling in terms of the stationary state of the QDS are derived and applied to QDS with a a h nearly degenerate and continuousspectrum. - s PACS numbers: 73.23.-b, 73.23.Hk, 72.70.+m, 73.63.Kv, 73.63.-b e m . t a I. INTRODUCTION which spoils the precision of single-electron devices due m toleakage.27 However,itis nowwellunderstoodthatco- - In recent years, there has been great interest in tunneling is interesting in itself, since it is responsible d transport properties of strongly interacting mesoscopic for strongly correlated effects such as the Kondo effect n o systems.1 As a rule, the electron interaction effects be- in quantum dots,28,29 or can be used as a probe of two- c comestrongerwiththereductionofthesystemsize,since electron entanglement and nonlocality,21 etc. [ the interacting electrons have a smaller chance to avoid In this paper we present a thorough analysis of the each other. Thus it is not surprising that an ultrasmall shot noise in the cotunneling regime. Since the single- 2 v quantumdotconnectedtoleadsinthe transportregime, electron “orthodox” theory cannot be applied to this 8 being under additional control by metallic gates, pro- case,wefirstdevelopamicroscopictheoryofcotunneling 5 vides a unique possibility to study strong correlation ef- suitable for the calculation of the shot noise in Secs. III 4 fects both in the leads and in the dot itself.2 This has and IV. [For an earlier microscopic theory of transport 0 led to a large number of publications on quantum dots, throughquantum dots see Refs. 30–32.] We consider the 1 which investigate situations where the current acts as a transport through a quantum-dot system (QDS) in the 0 probe of correlation effects. Historically, the nonequilib- Coulomb blockade (CB) regime, in which the quantiza- 0 / rium current fluctuations (shot noise) were initially con- tion of charge on the QDS leads to a suppression of the at sidered as a serious problem for device applications of sequential tunneling current except under certain reso- m quantum dots3–5 rather than as a fundamental physical nant conditions. We consider the transport away from - phenomenon. Later it became clearthat shot noise is an these resonances and study the next-order contribution d interestingphenomenoninitself,6 becauseitcontainsad- to the current, the so-called cotunneling current.25,26 In n ditionalinformationaboutcorrelations,whichisnotcon- general,the QDS can contain severaldots, which can be o tained, e.g., in the linear response conductance and can coupledbytunneljunctions,thedoubledot(DD)beinga :c beusedasafurtherapproachtostudytransportinquan- particular example.21 The QDS is assumedto be weakly v tumdots,boththeoretically4,5,7–22andexperimentally.23 coupledtoexternalmetallicleadswhicharekeptatequi- Xi Similarly, the majority ofpapers onthe noise ofquan- librium with their associated reservoirs at the chemical r tumdotsconsiderthesequential(single-electron)tunnel- potentials µl, l =1,2, where the currents Il can be mea- a ing regime, where a classical description (the so-called sured and the average current I through the QDS is de- “orthodox” theory) is applicable.24 We are not aware fined by Eq. (2.7). of any discussion in the literature of the shot noise in- Before proceeding with our analysis we briefly review duced by a cotunneling (two-electron, or second-order) the results available in the literature on noise of sequen- current,25,26 except Ref. 21, where the particular case tial tunneling. For doing this, we introduce right from of weak cotunneling (see below) through a double-dot the beginning all relevant physical parameters, namely (DD)systemisconsidered. Again,thismightbebecause the bath temperature T, bias ∆µ = µ1 µ2, charg- − until very recently cotunneling has been regarded as a ing energy EC, average level spacing δE, and the level minor contribution to the sequential tunneling current, widthΓ=Γ1+Γ2 oftheQDS,wherethetunnelingrates 1 tance G = G G /(G +G ), where G = πe2νν T 2 is T1 T2 the conducta1nce2 of t1he tu2nnel junctiolns to leadD|l,l|and ν is the density of dot states. Then the Fano fac- D tor is given by F = (G2 + G2)/(G + G )2, as it has 1 2 1 2 δE been found in Refs. 4,5,7. Thus, the shot noise is sup- . . . pressed, F < 1, and reaches its minimum value for the E2 symmetric QDS, G1 = G2, where F = 1/2. (2) The E1 low bias regime, δE ∆µ E . The first inequal- C ≪ ≪ ∆ (1,Ν) ∆ (2,Ν) ity δE ∆µ allows to assume a continuous spectrum + + ≪ on of the QDS and guarantees that the single-electron “orthodox” theory based on a classical master equation µ 1 ∆µ can be applied. The second inequality ∆µ E means C ≪ µ thatthe QDSisintheCB regime,wheretheenergycost 2 ∆ (l,N)=E(N 1) E(N) µ for the electron tun- lead 1 QDS lead 2 ± ± − ∓ l neling from the Fermi level of the lead l to the QDS (+) andviceversa( )oscillatesasafunctionofgatevoltage − FIG.1. Schematicrepresentationofthequantumdotsys- between its minimum value ∆± < 0 (where the energy tem(QDS)coupledtotwoexternalleads1and2(lightgrey) deficit turns into a gain, ∆± ∆µ) and its maximum | | ∼ via tunneling barriers (dark grey), where the energy scale is value ∆ E . Here, E(N) denotes the ground-state ± C ∼ drawn vertically. The tunneling between the QDS and the energy of the N-electron QDS. Thus the current I as leads l = 1,2 is parametrized by the tunneling amplitudes a function of the gate voltage consists of the CB peaks Tl, where the lead and QDS quantum numbers k and p have which are at the degeneracy points ∆± < 0, where the beendroppedforsimplicity,seeEq.(2.3). Theleadsareatthe number of electrons on the QDS fluctuates between N chemical potentials µ1,2, with an applied bias ∆µ=µ1−µ2. and N +1 due to single-electron tunneling. The peaks The (many-particle) eigenstates of the QDS with one added areseparatedbyplateaus,where the single-electrontun- electron(N+1electronsintotal)areindicatedbytheirener- nelingisblockedbecauseofthefiniteenergycost∆ >0 ± gies E , E , etc., with average level-spacing δE. The energy 1 2 and thus the sequential tunneling current vanishes. At costforaddingaparticlefromtheFermilevelofleadltothe the peaks the current is given by I = eγ γ /(γ +γ ), 1 2 1 2 N-electron QDS is denoted by ∆+(l,N) > 0 and is strictly whiletheFanofactorhasbeenreported5,7–10 tobeequal positive in the CB regime. Note that the energies ∆−(l,N) to F = (γ2 + γ2)/(γ + γ )2, 1/2 < F < 1, where for removing particles from the QDS containing N electrons 1 2 1 2 γ = e−2G ∆ (1,N) and γ = e−2G ∆ (2,N +1) arepositiveaswell,andarenotdrawnhere. Thecotunneling 1 1| + | 2 2| − | arethe tunneling rates tothe QDSfromlead1 andfrom process is visualized by two arrows, leading from the initial the QDS to lead 2, respectively. Within the “orthodox” state in, say, lead 1 (full circle), via a virtual state on the theory tunneling is still possible between the peaks at QDS (open circle), to thefinal state in lead 2 (full circle). finite temperature due to thermal activation processes, andthentheFanofactorapproachesthePoissonianvalue Γ =πν T 2totheleadsl=1,2areexpressedintermsof F = 1 from below. (3) Finally, the limit Γ ∆µ δE l | l| ≪ ≪ tunneling amplitudes T andthe density of states ν eval- is similar to the previous case, with the only difference l uatedattheFermienergyoftheleads. InFig.1themost thatthedotspectrumisdiscrete. Thesequentialtunnel- important parameters are shown schematically. This va- ing picture can still be applied; the result for the Fano riety of parameters shows that many different regimes factor at the current peak is F = (Γ21+Γ22)/(Γ1+Γ2)2, of the CB are possible. In the linear response regime, so that again 1/2<F <1.16 ∆µ k T, the thermal noise33 is given by the equilib- We would like to emphasize the striking similarity of B rium≪fluctuation-dissipationtheorem(FDT).34 Although the Fano factors in all three regimes, where they also the cross-over from the thermal to nonequilibrium noise resemble the Fano factor of the noninteracting double- is of our interest (see Sec. III), in this section we discuss barriersystem.6 The Fanofactorsinthe firstandsecond the shot noise alone and set T = 0. Then the noise at regimes become even equal if the ground-state level of zero frequency ω = 0, when δI = δI , can be charac- the QDS lies exactly in the middle between the Fermi 2 1 − terized by one single parameter, the dimensionless Fano levels of lead 1 and 2, ∆+ = ∆− . We believe that factor F = S(0)/eI , where the spectral density of the this “ubiquitous”7 doub|le-ba|rrier| cha|racter of the Fano | | noiseS(0) S (0)isdefinedbyEq.(2.7). TheFanofac- factor canbe interpretedas being the result ofthe natu- 22 ≡ tor acquires the value F =1 for uncorrelated Poissonian ral correlations imposed by charge conservation rather noise. than by interaction effects. Indeed, in the transport Next we discuss the different CB regimes. (1) In the throughadouble-barriertunneljunctioneachbarriercan limit of large bias ∆µ E , when the CB is sup- be thought of as an independent source of Poissonian C ≫ pressed, the QDS can be viewed as being composed of noise. And although in the second regime the CB is ex- two tunnel junctions in series, with the total conduc- plicitly taken into account, the stronger requirement of chargeconservationatzero frequency, δI +δI =0, has 1 2 2 to be satisfied,which leads to additionalcorrelationsbe- leadingtotheadditionalnoise∆S . Thusthetotalnoise h tweenthetwosourcesofnoiseandtoasuppressionofthe can be written as S = eI +∆S +∆S. For other cases h noisebelowthePoissonianvalue. Atfinitefrequency(but exhibiting super-Poissonian noise (in the strongly non- still in the classical range defined as ω ∆µ,E ) tem- linear bias regime) see Ref. 6. C ≪ porary charge accumulation on the QDS is allowed, and According to this picture we consider the following for frequencies larger than the tunneling rate, ω γ , different regimes of the inelastic cotunneling. We first 1,2 ≫ the conservation of charge does not need to be satisfied, discuss the weak cotunneling regime w w , where in whilethenoisepowerS approachesitsPoissonianvalue w Γ Γ ∆µ/∆2 is the average rate of th≪e inelastic co- 22 ∼ 1 2 ± frombelow,andthe crosscorrelationsvanish,S =0.35 tunneling transitions on the QDS [see Eqs. (4.23-4.26)], 12 Basedonthisobservationweexpectthatthedirectmea- and w is the intrinsic relaxation rate of the QDS to its in surement of interaction effects in noise is only possible equilibriumstateduetothecouplingtotheenvironment. either in the quantum (coherent) CB regime16 ∆µ Γ Inthisregimethecotunnelinghappenssorarelythatthe or in the Kondo regime,17–19 where both charge con∼ser- QDS always relaxes to its equilibrium state before the vationandmany-electroneffects leadto asuppressionof next electron passes through it. Thus we expect no cor- the noise. Another example is the noise in the quantum relations between cotunneling events in this regime, and regime,∆µ ω E ,whereitcontainssingularitiesas- the zero-frequency noise is going to take on its Poisso- C ≤ ∼ sociated with the “photo-assisted transitions” above the nian value with Fano factor F = 1, as first obtained for Coulomb gap ∆ .20,21,36 a special case in Ref. 21. This result is generalized in ± To conclude our brief review we would like to em- Sec.III, wherewefindauniversalrelationbetweennoise phasize again that while the zero-frequency shot noise and current of single-barrier tunnel junctions and, more in the sequential tunneling regime is always suppressed generally, of the QDS in the first nonvanishing order in below its full Poissonian value as a result of charge con- the tunneling perturbation V. Because of the universal servation (interactions suppressing it further), we find character of the results Eqs. (3.10) and (3.21) we call that, in the present work the shot noise in the cotun- them the nonequilibriumFDT in analogywith linear re- neling regime37 is either Poissonian F = 1 (elastic or sponse theory. weak inelastic cotunneling) or, rather surprisingly, non- Next, we consider strong cotunneling, i.e. w w . in ≫ Poissonian F = 1 (strong inelastic cotunneling). There- The microscopic theory of the transport and noise in 6 fore the non-Poissonian noise in QDS can be considered this regime based on a projector operator technique is as being a fingerprint of inelastic cotunneling. This dif- developed in Sec. IV. In the case of a few-level QDS, ference of course stems from the different physical origin δE E ,38 noise turns out to be non-Poissonian, as we C ∼ of the noise in the cotunneling regime, which we dis- havediscussedabove,andthiseffectcanbeestimatedas cuss next. Away from the sequential tunneling peaks, follows. The QDS is switching between states with the ∆ > 0, single-electron tunneling is blocked, and the different currents I ew, and we find δI ew. The ± ∼ ∼ only elementary tunneling process which is compatible QDS stays in each state for the time τ w−1. There- ∼ with energy conservation is the simultaneous tunneling fore,forthepositivecorrectiontothenoisepowerweget of two electrons called cotunneling25,26. In this process ∆S δI2τ e2w,andtheestimateforthecorrectionto ∼ ∼ one electron tunnels, say, from lead 1 into the QDS, and the Fano factor follows as ∆S/eI 1. A similar result ∼ theotherelectrontunnelsfromtheQDSintolead2with isexpectedforthe noiseinducedbyheating,∆S ,which h a time delay on the order of ∆−1 (see Ref. 21). This canroughlybeestimatedbyassuminganequilibriumdis- ± means that in the range of frequencies, ω ∆ , (which tribution on the QDS with the temperature k T ∆µ ± B weassumeinourpaper)thechargeonthe≪QDSdoesnot and considering the additional noise as being therm∼al,33 fluctuate,andthusincontrasttothesequentialtunneling ∆S Gk T (eI/∆µ)k T eI. The characteris- h B B ∼ ∼ ∼ thecorrelationimposedbychargeconservationisnotrel- tic frequency of the noise correction ∆S is ω w, with ∼ evantforcotunneling. Furthermore,inthecaseofelastic ∆S vanishing for ω w (but still in the classicalrange, ≫ cotunneling (∆µ < δE), where the state of the QDS re- ω ∆µ). In contrast to this, the additional noise due ≪ mainsunchanged,theQDScanbeeffectivelyregardedas to heating, ∆S , does not depend on the frequency. h a single barrier. Therefore, subsequent elastic cotunnel- In Sec. V we consider the particular case of nearly de- ing events are uncorrelated, and the noise is Poissonian generate dot states, in which only few levels with an en- with F =1. On the other hand, this is not so for inelas- ergy distance smaller than δE participate in transport, tic cotunneling(∆µ > δE), where the internal state of and thus heating on the QDS can be neglected. Specif- the QDSis changed,thereby changingthe conditions for ically, for a two-level QDS we predict giant (divergent) the subsequent cotunneling event. Thus, in this case the super-Poissoniannoiseiftheoff-diagonaltransitionrates QDS switches between different current states, and this vanish. The QDS goes into an unstable mode where it createsacorrectiontonoise∆S,sothatthetotalnoiseis switchesbetweenstates1and2with(generally)different non-Poissonian, and can become super-Poissonian. The currents. We consider the transport through a double- other mechanism underlying super-Poissonian noise is dot (DD) system as an example to illustrate this effect the excitationofhighenergylevels(heating)ofthe QDS [see Eq. (5.12) and Fig. 3]. caused by multiple inelastic cotunneling transitions and Finally, we discuss the case of a multi-level QDS, 3 δE E . In this case the correlations in the cotun- To describe the transport through the QDS we apply C nelin≪g current described above do not play an essential standardmethods40 andadiabaticallyswitchonthe per- role. In the regime of low bias, ∆µ (δEE )1/2, elas- turbation V in the distant past, t = t . The C 0 tic cotunneling dominates transport≪,25,39 and thus the perturbed state of the system is described→by−t∞he time- noise is Poissonian. In the opposite case of large bias, dependent density matrix ρ(t) = e−iH(t−t0)ρ0eiH(t−t0), ∆ ∆µ (δEE )1/2, the transport is governed by which can be written as ± C ≫ ≫ inelastic cotunneling, and in Sec. VI we study heating effects which are relevant in this regime. For this we use ρ(t)=e−iL(t−t0)ρ0, LA [H,A] , A, (2.4) ≡ ∀ the results of Sec. IV and derive a kinetic equation for the distribution function f(ε). We find three universal with the help of the Liouville operator L = L0+LV.41 regimes where I ∆µ3, and the Fano factor does not Here ρ0 is the grand canonical density matrix of the un- depend on bias th∼e ∆µ. The first is the regime of weak perturbed system, cotunneling, τ τ , where τ and τ are time scales in ≪ c in c ρ =Z−1e−K/kBT, (2.5) characterizing the single-particle dynamics of the QDS. 0 The energy relaxation time τ describes the strength of in where we set K =H µN . the coupling to the environment while τc ∼ eνD∆µ/I Because of tunneli0ng−theltoltall number of electrons in is the cotunneling transition time. Then we obtain for eachleadN = c† c Pis no longer conserved. For the the distribution f(ε) = θ( ε), reproducing the result of l k lk lk Ref. 25. We also find that−F =1, in agreement with the outgoing currenPts Iˆl =eN˙l we have FDT proveninSec. III. The other tworegimesof strong Iˆ =ei[V,N ]=ei(D† D ). (2.6) cotunneling τin τc are determined by the electron- l l l − l ≫ electronscatteringtimeτ . Forthecold-electronregime, ee The observables of interest are the average current I τc τee,wefindthedistributionfunctionbysolvingthe ≡ ≪ I = I through the QDS, and the spectral density of integral equations (6.11) and (6.12), while for hot elec- 2 − 1 trons, τ τ , f is given by the Fermi distribution the noise Sll′(ω)= dtSll′(t)exp(iωt), c ee ≫ feunnecrgtiyonbawlaitnhceaneqeuleacttioronn(6te.1m5p).erWateurueseobft(aεi)nteodcfraolcmultahtee Il =Trρ(0)Iˆl, SlRl′(t)=ReTrρ(0)δIl(t)δIl′(0), (2.7) theFanofactor,whichturnsouttobeverycloseto1. On whereδI =Iˆ I . Belowwewillusetheinteractionrep- the other hand, the current depends not only on G1G2 l l− l resentationwhereEq.(2.7)canberewrittenbyreplacing but also on the ratio, G /G , depending on the cotun- neling regime [see Fig. 4]1. De2tails of the calculations are ρ(0) ρ0 and Iˆl(t) U†(t)Iˆl(t)U(t), with → → deferred to four appendices. t U(t)=Texp i dt′V(t′) . (2.8) − (cid:20) Z−∞ (cid:21) II. MODEL SYSTEM In this representation, the time dependence of all opera- tors is governedby the unperturbed Hamiltonian H . Thequantum-dotsystem(QDS)understudyisweakly 0 coupled to two external metallic leads which are kept in equilibrium with their associatedreservoirsatthe chem- III. NON-EQUILIBRIUM ical potentials µ, l = 1,2, where the currents I can l l FLUCTUATION-DISSIPATION THEOREM FOR be measured. Using a standard tunneling Hamiltonian TUNNEL JUNCTIONS approach,40 we write H =H0+V , H0 =HL+HS +Hint, (2.1) Inthissectionweprovetheuniversalityofnoiseoftun- HL = εkc†lkclk, HS = εpd†pdp, (2.2) nel junctions in the weak cotunneling regime w ≪ win keeping the first nonvanishing order in the tunneling l=1,2 k p X X X HamiltonianV. SinceourfinalresultsEqs.(3.10),(3.12), V = (Dl+Dl†), Dl = Tlkpc†lkdp, (2.3) (3.13),and(3.21)canbeappliedtoquitegeneralsystems l=1,2 k,p out-of-equilibriumwecallthisresultthenon-equilibrium X X wherethetermsH andH describetheleadsandQDS, fluctuation-dissipationtheorem(FDT).Inparticular,the L S respectively (with k and p from a complete set of quan- geometry of the QDS and the interaction Hint are com- tum numbers), and tunneling between leads and QDS is pletelyarbitraryforthediscussionofthenon-equilibrium described by the perturbation V. The interaction term FDT in this section. Such a non-equilibrium FDT was H is specified below. The N-electron QDS is in the derived for single barrier junctions long ago.42 We will int cotunneling regime where there is a finite energy cost need to briefly review this case which allows us then to ∆ (l,N) > 0 for the electron tunneling from the Fermi generalize the FDT to QDS considered here in the most ± level of the lead l to the QDS (+) and vice versa ( ), so direct way. − that only processes of second order in V are allowed. 4 A. Single-barrier junction nρ n = Z−1exp[ E /k T]. Next we introduce the 0 n B h | | i − spectral function, The total Hamiltonian of the junction [given by (ω)=2π (ρ +ρ ) mAn 2 Eqs. (2.1)-(2.3)] and the currents Eq. (2.6) have to be A n m |h | | i| replaced by H =HL+Hint+V, where nX,m δ(ω+E E ), (3.7) n m × − V =A+A†, A= Tkk′c†2kc1k′, (3.1) andrewriteEqs.(3.5)and(3.6)inthematrixforminthe Xk,k′ basis n takingintoaccountthatthe operatorAcreates Iˆ = Iˆ =ei[V,N ]=ei A† A . (3.2) (annih|iliates) an electronin the lead 2 (1) [see Eq.(3.1)]. 2 1 2 − − We obtain following expressions For the sake of generality, we do(cid:0)not spec(cid:1)ify the interac- ∆µ tionH inthissection,northeelectronspectruminthe int I(∆µ)=etanh (∆µ), (3.8) leads, and the geometry of our system. (cid:20)2kBT(cid:21)A Applying the standard interaction representation e2 technique,40 weexpandthe expression(2.8)forU(t)and S(ω,∆µ)= (∆µ ω), (3.9) 2 A ± keep only first non-vanishing contributions in V, obtain- ± X ing where ∆µ = µ µ . From these equations our main 1 2 − result follows t I(t)=i dt′ V(t′),Iˆ(t) , (3.3) e ∆µ ω 2 S(ω,∆µ)= coth ± I(∆µ ω), (3.10) h i −Z∞ h i 2 ± (cid:20) 2kBT (cid:21) ± X where we use the notation ... = Trρ (...). Analo- where we have neglected contributions of order 0 gously, we find that the firsthnoni-vanishing contribution ∆µ/εF,ω/εF 1. We call the relation (3.10) non- ≪ to the noise power S(ω) S (ω) is given by equilibrium fluctuation-dissipation theorem because of 22 ≡ its generalvalidity (we recallthat no assumptionsonge- ∞ ometry or interactions were made). 1 S(ω)= dteiωt Iˆ(t),Iˆ(0) , (3.4) The fact that the spectral function Eq. (3.7) depends 2 2 2 h{ }i Z only on one parameter can be used to obtain further −∞ useful relations. Suppose that in addition to the bias where ... stands for anticommutator, and I2 = 0 in ∆µ a small perturbation of the form δµe−iωt is applied leading{ord}er. 2 to the junction. This perturbation generates an ac cur- We notice that in Eqs. (3.3)and (3.4)the terms AA rent δI(ω,∆µ)e−iωt through the barrier, which depends and A†A† areresponsibleforCooperpairtunnelinghandi on both parameters, ω and ∆µ. The quantity of in- vanishh in tihe case of normal (interacting) leads. Taking terest is the linear response conductance G(ω,∆µ) = thisintoaccountandusingEqs. (3.1)and(3.2)weobtain eδI(ω,∆µ)/δµ. The perturbation δµ can be taken into account in a standard way by multiplying the tunnel- ∞ ing amplitude A(t) by a phase factor e−iφ(t), where φ˙ = I =e dt A†(t),A(0) , (3.5) δµe−iωt. Substituting the new amplitude into Eq. (3.3) h i Z and expanding the current with respect to δµ, we arrive −∞ (cid:2) (cid:3) at the following result, ∞ S(ω)=e2 dt cos(ωt) A†(t),A(0) , (3.6) ∞ h{ }i ie2 Z ReG(ω,∆µ)= dtsin(ωt) [A†(t),A(0)] . (3.11) −∞ ω h i Z −∞ where we also used A†(t)A(0) = A†(0)A( t) . h i h − i Finally,applyingthespectraldecompositiontothisequa- Next we apply the spectral decomposition to the cor- tion we obtain relators Eqs. (3.5) and (3.6), a similar procedure to that which also leads to the equilibrium fluctuation- (2/e)ωReG(ω,∆µ)=I(∆µ+ω) I(∆µ ω), (3.12) dissipation theorem. The crucial observation is that − − [H ,N ] = 0, l = 1,2 (we stress that it is only whichholds fora generalnonlinearI vs ∆µ dependence. 0 l the tunneling Hamiltonian V which does not commute FromthisequationandfromEq.(3.10)itfollowsthatthe with N , while all interactions do not change the num- noise power at zero frequency can be expressed through l ber of electrons in the leads). Therefore, we are al- the conductance at finite frequency as follows lowed to use for our spectral decomposition the ba- sis n = E ,N ,N of eigenstates of the operator S(0,∆µ)+S(0, ∆µ)= n 1 2 − | i | i K = H µ N , which also diagonalizes the grand- ω canonica0l−dePnslityl mlatrix ρ0 [given by Eq. (2.5)], ρn = 2ωcoth(cid:20)2kBT(cid:21)ReG(ω,0)|ω→∆µ. (3.13) 5 Andforthe noisepoweratzerobiasweobtainS(ω,0)= more general, +∞dtD (t)e±iωt =0 (note that we have −∞ l lωibcroituhm(ω/F2DkTBT.34)REeGq.(ω(,30.1),0)whriecphroidsutcheessttahnedarerdsuletquoi-f aanssdumtheedcyecalrilcieRprrothpaetrtωyo≪fth∆e±tr)a.ceUwsiengobtthaeinsetheqeufoallliotiwes- Ref. 42. The current is not necessary linear in ∆µ (the ingresult(fordetailsofthederivation,seeAppendix A), case of tunneling into a Luttinger liquid43 is an obvi- ous example), and in the limit T,ω 0 we find the ∞ Poissonian noise, S = eI. In the limit→T,∆µ 0, the I =e dt B†(t),B(0) , (3.14) → h i quantum noise becomes S(ω) = e[I(ω) I( ω)]/2. If Z − − −∞ (cid:2) (cid:3) I( ∆µ)= I(∆µ),wegetS(ω)=eI(ω),andthusS(ω) − − B =D D¯†+D†D¯ . (3.15) can be obtained from I(∆µ ω). 2 1 1 2 → Applyingasimilarprocedure(seeAppendixA),wearrive at the following expression for the noise power S = S , B. Quantum dot system 22 see Eq. (2.7), We consider now tunneling through a QDS. In this ∞ case the problem is more complicated: In general, the S(ω)=e2 dt cos(ωt) B†(t),B(0) . (3.16) twocurrentsIˆ arenotindependent, because [Iˆ,Iˆ]=0, h{ }i l 1 2 Z 6 −∞ and thus all correlatorsSll′ are nontrivial. In particular, it has been proven in Ref. 21 that the cross-correlations where we have dropped a small contribution of order ImS12(ω) aresharplypeakedatthe frequencies ω =∆±, ω/∆±. which is caused by a virtual charge-imbalance on the Thus, we have arrivedat Eqs. (3.14) and (3.16) which QDS during the cotunneling process. The charge accu- areformallyequivalenttoEqs.(3.5)and(3.6). Similarly mulationontheQDSforatime oforder∆−1 leadstoan to A in the single-barrier case, the operator B plays the ± additionalcontributiontothenoiseatfinitefrequencyω. role of the effective tunneling amplitude, which annihi- Thus,weexpectthatforω ∆± thecorrelatorsSll′ can- latesanelectroninlead1andcreatesitinlead2. Similar ∼ notbe expressedthroughthe steady-statecurrentI only to Eqs.(3.7), (3.8), and (3.9) we canexpress the current and thus I has to be complemented by some other dis- and the noise power sipative counterparts, such as differential conductances Gll′ (see Sec. IIIA). I(∆µ)=etanh ∆µ (∆µ), (3.17) Ontheotherhand,atlowenoughfrequency,ω ≪∆±, (cid:20)2kBT(cid:21)B the charge conservation on the QDS requires δIs = e2 (δI +δI )/2 0. Below we concentrate on the limit S(ω,∆µ)= (∆µ ω), (3.18) 2 1 ≈ 2 B ± of low frequency and neglect contributions of order of ± X ω/∆ to the noise power. In Appendix A we prove that ± in terms of the spectral function S (ω/∆ )2, and this allows us to redefine the cur- ss ± ∼ rent and the noise power as I I = (I I )/2 and S(ω) Sdd(ω).44 In addition w≡e redquire t2h−at t1he QDS B(ω)=2π (ρn+ρm)|hm|B|ni|2 is in t≡he cotunneling regime, i.e. the temperature is low nX,m enough, kBT ≪ ∆±, although the bias ∆µ is arbitrary ×δ(ω+En−Em). (3.19) (i.e. it can be of the order of the energy cost) as soon as The difference, however, becomes obvious if we notice thesequentialtunnelingtothedotisforbidden,∆ >0. ± thatincontrasttotheoperatorA[seeEq.(3.1)]whichis Inthis limitthe currentthroughaQDSarisesduetothe aproductoftwofermionicSchr¨odingeroperatorswithan direct hopping of an electron from one lead to another equilibrium spectrum, the operator B contains an addi- (through a virtual state on the dot) with an amplitude tional time integration with the time evolution governed which depends on the energy cost ∆ of a virtual state. ± by H = K + µN . Applying a further spectral de- Although this process can change the state of the QDS, 0 l l l composition to the operator B [given by Eq. (3.15)] we thefastenergyrelaxationintheweakcotunnelingregime, P arrive at the expression w w ,immediately returnsittothe equilibriumstate in ≪ (fortheoppositecase,seeSecs.IV-VI). Thisallowsusto mD n′ n′ D† n apply a perturbation expansion with respect to tunnel- i mB n = h | 2| ih | 1| i ing V and to keep only first nonvanishing contributions, h | | i n′ En′ −En−µ1 X which we do next. It is convenient to introduce the notation D¯l(t) + hm|D1†|n′′ihn′′|D2|ni, (3.20) −t∞dt′Dl(t′). We notice that all relevant matrix el≡e- Xn′′ En′′ −En+µ2 emRi∆en−tts,,ahrNe f|aDslt(to)s|cNill+ati1nig∼funec−tii∆on+st,ofhNtim−e.1|TDhlu(ts),|Nunide∼r wherethetwosumsovern′andn′′onthelhsaredifferent bytheorderoftunnelingsequenceinthecotunnelingpro- the above conditions we can write D¯ ( )= 0, and even l ∞ cess. Thus we see that the current and the noise power 6 depend on both chemical potentials µ separately (in where Z = Tr exp[ K /k T], K = H µ N , 1,2 L − L B L L − l l l contrast to the one-parameter dependence for a single- andµ isthechemicalpotentialofleadl. Notethatboth l P barrierjunction,seeSec.IIIA),andthereforetheshiftof leadsarekeptatthesametemperatureT. Physically,the ∆µinEq.(3.18)by ω willalsoshifttheenergydenom- product form of ρ in Eq. (4.1) describes the absence of 0 ± inators of the matrix elements on the lhs of Eq. (3.20). correlationsbetweentheQDSandtheleadsintheinitial However,sincethe energydenominatorsareoforder∆ stateatt . Furthermore,weassumethattheinitialstate ± 0 the lasteffect canbe neglectedandwe arriveatthe final ρ isdiagonalintheeigenbasisofH ,i.e. thattheinitial 0 0 result state is an incoherent mixture of eigenstates of the free Hamiltonian. e ∆µ ω S(ω,∆µ)= coth ± I(∆µ ω) In systems which can be divided into a (small) system 2 ± (cid:20) 2kBT (cid:21) ± (like the QDS) and a (possibly large) external“bath” at X thermalequilibrium(here,theleadscoupledtotheQDS) +O(ω/∆ ). (3.21) ± itturnsouttobeveryusefultomakeuseofthesuperop- ThisequationrepresentsournonequilibriumFDTforthe erator formalism,41,45,46 and of projectors PT = ρLTrL, transportthroughaQDSintheweakcotunnelingregime. which project on the “relevant” part of the density ma- A special case with T,ω = 0, giving S = eI, has been trix. We obtain PTρ by taking the partialtrace TrL of ρ derived in Ref. 21. To conclude this section we would with respect to the leads and taking the tensor product like to list again the conditions used in the derivation. of the resulting reduced density matrix with the equi- The universality of noise to current relation Eq. (3.21) librium state ρL. Here, we will consider the projection provenhere is validin the regime in which it is sufficient operators to keep the first nonvanishing order in the tunneling V P =(P P 1 )P , Q=1 P, (4.2) which contributes to transport and noise. This means D N ⊗ L T − that the QDS is in the weak cotunneling regime with satisfying P2 = P, Q2 = Q, PQ = QP = 0, where P is ω,k T ∆ , and w w. B ≪ ± in ≫ composed of PT and two other projectors46 PD and PN, where P projects on operators diagonalin the eigenba- D sis n of H , i.e. nP Am = δ nAm , and P IV. MICROSCOPIC THEORY OF STRONG {| i} S h | D | i nmh | | i N projects on the subspace with N particles in the QDS. COTUNNELING The particle number N is definedby having minimalen- ergy in equilibrium (with no applied bias); all other par- A. Formalism ticle numbers have energies larger by at least the energy deficit37 ∆. Above assumptions about the initial state In this section, we give a systematic microscopic Eq. (4.1) of the system at t can now be rewrit- 0 derivation of the master equation, Eq. (4.22), the av- ten as → −∞ erage current, Eq. (4.37), and the current correlators, Eqs. (4.52)-(4.54) for the QDS coupled to leads, as in- Pρ =ρ . (4.3) 0 0 troduced in Eqs. (2.1)-(2.3), in the strong cotunneling regime, w w. Under this assumption the intrinsic For the purpose of deriving the master equation we in ≪ relaxation in the QDS is very slow and will in fact be taketheLaplacetransformofthetime-dependentdensity neglected. Thermalequilibrationcanonly take place via matrix Eq. (2.4), with the result coupling to the leads, see Sec. IVB. Due to this slow re- laxationintheQDSwefindthattherearenon-Poissonian ρ(z)=R(z)ρ0. (4.4) correlations∆S inthe currentthroughthe QDSbecause Here, R(z) is the resolvent of the Liouville operator L, theQDShasa“memory”;thestateoftheQDSafterthe i.e. the Laplace transform of the propagator exp( itL), transmission of one electron influences the transmission − ofthenextelectron. Abasicassumptionforthefollowing ∞ i procedure is that the system and bath are coupled only R(z)= dteit(z−L) =i(z L)−1 , (4.5) − ≡ z L weakly and only via the perturbation V, Eq. (2.3). The Z0 − interactionpartH oftheunperturbedHamiltonianH , int 0 where z = ω+iη. We choose η > 0 in order to ensure Eq. (2.1), must therefore be separable into a QDS and a convergence (L has real eigenvalues) and at the end of lead part, H = Hint +Hint. Moreover, H conserves int S L 0 the calculation take the limit η 0. We can split the the number of electrons in the leads, [H ,N ]=0, where → 0 l resolventinto fourparts bymultiplying itwith the unity Nl = kc†lkclk. operator P +Q from the left and the right, We assume that in the distant past, t , the 0 P → −∞ system is in an equilibrium state R=PRP +QRQ+PRQ+QRP. (4.6) 1 ρ =ρ ρ , ρ = e−KL/kBT, (4.1) Inserting the identity operator i(z L)R(z) = i(z 0 S ⊗ L L ZL L)(P +Q)R(z) between the two−facto−rs on the lef−t han−d 7 side of QP = 0, PQ = 0, Q2 = Q, and P2 = P, we N). Inthecotunnelingregime37,thesequentialtunneling obtain contribution(secondorderinL )toEq.(4.19)vanishes. V Theleadingcontribution[usingEqs.(4.11)and(4.16)]is 1 QR(z)P =Q QL PR(z)P, (4.7) of fourth order in L , V V z QLQ − PR(z)Q=−iPR0(z)PLVQR(z)Q, (4.8) Wnm =Trpn(LVQR0)3LVpmρL. (4.20) i QR(z)Q=Q Q, (4.9) Note that since we study the regime of small frequen- z QLQ+iQL PR (z)PL Q V 0 V − cies Rez = ω L Q E E , where m = n, i ≪ || 0 || ≈ | n − m| 6 PR(z)P =P P, (4.10) we can take the limit ω 0 here. In addition to this, z Σ(z) → we have assumed fast relaxation in the leads and have − taken the Markovian limit z = iη 0, i.e. we have re- where we define the self-energy superoperator → placed W (z) in Eq. (4.19) by W lim W (z) nm nm z→0 nm ≡ 1 in Eq. (4.20). The trace of ρ is preserved under the Σ(z)=PL Q QL P, (4.11) V z QLQ V time evolution Eq. (4.18) since nWnm has the form − TrP L A=Tr[V,A] TrQ [V,A]wherethe firstterm N V N and the free resolvent R (z) = i(z L )−1. Here, we vanishes exactly and −the secondPterm invloving Q = 0 − 0 N have used the identities 1 P is O(κ). After some calculation, we find that N − W is of the form Tr (c ρ )=Tr (c† ρ )=0, (4.12) nm L lk L L lk L PTLVPT =PTIˆlPT =0, (4.13) Wnm =wnm−δnm wm′n, (4.21) [P,L0]=[Q,L0]=0, (4.14) Xm′ L P =PL =0. (4.15) with w > 0 for all n and m. Substituting this equa- 0 0 nm tion into Eq. (4.18) we can rewrite the master equa- Equation(4.13)followsfromEq.(4.12),while Eq.(4.14) tion in the manifestly trace-preserving form ρ˙ (z) = n hnoolrddsobeesciatucsheaHng0entehitehedriamgoixneasl tehleemQenDtSs owritthhethpearletaicdles m[wnmρm(z)−wmnρn(z)], or in real time, numberofastate. Finally,Eq.(4.15)canbe shownwith P ρ˙ (t)= [w ρ (t) w ρ (t)]. (4.22) n nm m mn n Eq. (4.14) and using that P contains P . For an expan- − D m sioninthesmallperturbationL inEqs.(4.7),(4.9)and X V (4.11) we use the von Neumann series This “classical” master equation describes the dynamics of the QDS, i.e. it describes the rates with which the 1 1 Q= Q probabilities ρn for the QDS being in state n change. z QLQ z L0 QLVQ After some algebra (retaining only47 O(κ0), c|f.iApp. B), − − − ∞ we find = iR (z)Q [ iL R (z)Q]n. (4.16) 0 V 0 − − nX=0 wnm =wn+m+wn−m+wn0m, (4.23) where (in the cotunneling regime) B. Master Equation w+ =w (2,1), w− =w (1,2), (4.24) nm nm nm nm w0 = w (l,l), (4.25) Using Eqs.(4.3),(4.4),and(4.10)the diagonalpartof nm nm the reduced density matrix ρS(z) = PDPNTrLρ(z) can lX=1,2 now be written as with the “golden rule” rate from lead l to lead l′, i ρS(z)=TrLPR(z)Pρ0 = z−Σ(z)ρS. (4.17) wnm(l′,l)=2π |hn|(Dl†,Dl′)|mi|2 m¯,n¯ X This equation leads to ρ˙ (z) = izρ (z) ρ = S − S − S δ(Em En ∆µll′)ρL,m¯. (4.26) iΣ(z)ρ (z). The probability ρ (z) = nρ (z)n for × − − S n S − h | | i the QDS being in state |ni then obeys the equation Ipnottehnitsiaelxpdrreospsiobne,tw∆eµelnl′l=eaµdll−aµnld′ dleeandotle′s, tahnedchρemica=l L,m¯ ρ˙n(z)= Wnm(z)ρm(z), (4.18) m¯ ρL m¯ . We have defined the second order hopping h | | i m operator X W (z)= iTr p Σ(z)p = iΣ (z), (4.19) nm S n m nn|mm − − (Dl†,Dl′)=Dl†R0Dl′ +Dl′R0Dl† with p = n n, which is a closed equation for the densitynmatr|ixihin|the subspace defined by P (with fixed =−(Dl†D¯l′ +Dl′D¯l†), (4.27) 8 where D is given in Eq. (2.3), D¯ = 0 D (t)dt. wherewehaveinsertedP+Q=1andusedEqs.(4.3)and l l −∞ l tNhoetel,eathdatl(tDol†,tDhel′)leiasdthle′a(minplpitaurdtiecuolfacro,tuwRneneclainngwfrroitme i(s4h.4es).foTrhρe(zse).coAncdcoterrdmingcatnobEeq.re(4w.r1i3t)tetnheusfiirnsgtEteqrsm.(v4a.7n)- B = (D†,D ), see Eq. (3.15)). The combined index and (4.17), with the result − 1 2 m = (m,m¯) contains both the QDS index m and the 1 lead index m¯. Correspondingly, the basis states used Il(z)=TrIˆlQ QLVρS(z)ρL z QLQ above are m = m m¯ with energy Em = Em +Em¯, − where|mi|is aineig|enis|taiteofHS+HSint withenergyEm, =TrSWI(z)ρS(z)= WnIm(z)ρm(z). (4.33) and m¯ is an eigenstate of H +Hint µ N with nm energ|y iE . The terms w± aLccountLfo−r thel clhanlge of X m¯ nm P Using the projector method, we have thus managed to state in the QDS due to a current going from lead 1 to express the expectation value of the current (acting on 2 (2 to 1). In contrast to this, the cotunneling rate w0 nm both the QDS and the leads) in terms of the linear su- involveseitherlead1orlead2and,thus,itdoesnotcon- peroperator WI which acts on the reduced QDS density tribute directly to transport. However, w0 contributes nm matrix ρ only. Taking z 0 in Eq. (4.33), the average to thermalequilibrationof the QDS via particle-hole ex- S → current in lead l in the stationary limit becomes citations in the leads and/or QDS (see Secs. VIA and VIB). 1 I = limTrIˆQ QL ρ¯ ρ . (4.34) l l V S L z→0 z QLQ − C. Stationary State Up to now this is exact, but next we use again the perturbation expansion Eq. (4.16). In the cotunneling In order to make use of the standard Laplace trans- regime37,47, i.e. away from resonances, the second-order form for finding the stationary state ρ¯of the system, we tunneling current shift the initial state to t =0 and define the stationary 0 state as ρ¯=limt→∞ρ(t)=limt→∞e−iLtρ0. This can be Il(2) =−iTrIˆlR0LVρ¯SρL (4.35) expressed in terms of the resolvent, is negligible [O(κ)], and the leading contribution is the ρ¯= ilimzR(z)ρ , (4.28) cotunneling current 0 − z→0 using the propertylimt→∞f(t)= ilimz→0zf(z)ofthe Il(4) =iTrIˆl(QR0LV)3ρ¯SρL. (4.36) − Laplace transform. The stationary state ρ¯ of the QDS S After further calculation we find in leading order (cf. can be obtained in the same way from Eq. (4.17), App. B) z ρ¯ = lim ρ . (4.29) S z→0z−Σ(z) S I2 =−I1 =e wnImρ¯m, (4.37) mn Multiplyingbothsideswithz Σ(z)andtakingthelimit X − wI =w+ w− , (4.38) z 0, we obtain the condition nm nm− nm → where w± are defined in Eq. (4.24). Note again that Σ ρ¯ =0, (4.30) nm 0 S w0 in Eq. (4.25) does not contribute to the current nm whereΣ =lim Σ(z). UsingEq.(4.19),thiscondition directly,butindirectlyviathemasterequationEq.(4.31) 0 z→0 for the stationarystate canalsobe expressedin termsof which determines ρ¯m (note that ρ¯m is non-perturbative W , in V). We finally remark that for Eqs. (4.34)-(4.37) we nm do not invoke the Markovianapproximation. W ρ¯ = (w ρ¯ w ρ¯ )=0, (4.31) nm m nm m mn n − m m X X E. Current Correlators whichisobviouslythestationarityconditionforthemas- ter equation, Eq. (4.22). Nowwestudythecurrentcorrelatorsinthestationary limit. We let t , therefore ρ(t = 0) ρ¯. The 0 → −∞ → symmetrized current correlator [cf. Eq. (2.7)], D. Average Current Sll′(t)=ReTrδIl(t)δIl′ρ¯, (4.39) The expectation value I (t) = TrIˆρ(t) of the current l l Iˆ in lead l [Eq. (2.7)] can be obtained via its Laplace where δI = Iˆ I , can be rewritten using the cyclic l l l − l transform property of the trace as Il(z)=TrIˆlρ(z)=TrIˆl(P +Q)R(z)Pρ0, (4.32) Sll′(t)=ReTrδIle−itLδIl′ρ¯, (4.40) 9 where e−itL acts on everything to its right. Taking the We find SQ =SQ = SQ = SQ SQ, where 11 22 − 12 − 21 ≡ LaplacetransformandusingEq.(4.28)forthestationary state ρ¯, we obtain SQ =e2 (w+ +w− )ρ¯ . (4.51) nm nm m mn Sll′(z)= lim Re( iz′)TrδIlR(z)δIl′R(z′)Pρ0, (4.41) X z′→0 − Finally, we can combine Eqs. (4.47) and (4.51), using Eq. (4.42) and we obtain the final result for the current where z = ω +iη and η 0+. We insert P +Q = 1 → correlators, twice and use Eq. (4.12) with the result S (ω)=S (ω)= S (ω)= S (ω) S(ω), (4.52) Sll′(z)=SlPl′(z)+SlQl′ −(i/z)IlIl′, (4.42) 1S1(ω)=e222 (w+−+12w− )ρ¯−+2∆1 S(ω≡), (4.53) nm nm m where SQ = SQQ+SQP. We further evaluate the con- mn ll′ ll′ ll′ X tributions to Sll′(z) using Eqs. (4.7) and (4.29), and we ∆S(ω)=e2 wnImδρmn′(ω)wnI′m′ρ¯m′, (4.54) obtain n,m,n′,m′ X SlPl′(z)=ReTrIˆlRQLVPR(z)PIˆl′RQLVρ¯, (4.43) where δρnm(ω) = ρnm(ω) − 2πδ(ω)ρ¯n. Here, ρnm(ω) is the Fourier-transformed conditional density matrix, where RQ =limz→0(z−QLQ)−1, and whichis obtainedfromthe symmetrizedsolutionρn(t)= ρ ( t) of the master equation Eq. (4.22) with the ini- n SlQl′Q =−ReTrIˆlR0LVQR0Iˆl′R0LVρ¯ tial−condition ρn(0) = δnm. Note that ρnm(ω) is re- −ReTrIˆlR0Iˆl′QR0LVR0LVρ¯, (4.44) lρated(ωt)o=thρeLLTa(pωla)c+etρrLaTn(sfoωrm).Eq.(4.49)viatherelation SlQl′P =−ReTrIˆlR0LVQR0LVR0Iˆl′ρ¯. (4.45) nFmor a fewn-lmevel QDSnm, δ−E ∼ EC, with inelastic cotun- neling the noise will be non-Poissonian, since the QDS While SP(z)asgiveninEq.(4.43)isa non-perturbative ll′ is switching between states with different currents. An result, we have used Eq. (4.16) to find the leading con- explicit result for the noise in this case can be obtained tribution in the tunneling amplitude Tlkp for SlQl′Q and by making further assumptions about the QDS and the SlQl′P in Eqs. (4.44) and (4.45). Also note that QR(z)Q couplingtotheleads,andthenevaluatingEq.(4.54),see was replaced by QR Q in Eqs. (4.44) and (4.45), since the following sections. For the general case, we only es- 0 ω E E for n = m and therefore SQQ and SQP timate ∆S. The current is of the order I ew, with w ≪ | n− m| 6 ll′ ll′ ∼ do not depend on z, i.e. they do not depend on the fre- sometypicalvalueofthecotunnelingratewnm,andthus quency ω. δI ew. Thetimebetweenswitchingfromonedot-state In order to analyze Eq. (4.43) further, we insert the toa∼notherduetocotunnelingisapproximatelyτ w−1. ∼ resolutionof unity p =1 next to the P operators The correction ∆S to the Poissonian noise can be esti- m m S in Eq. (4.43) with the result SP = SP = SP = SP mated as ∆S δI2τ e2w, which is of the same order where P 11 22 − 12 − 21 as the Poisson∼ian cont∼ribution eI e2w. Thus the cor- ∼ rectiontotheFanofactorisoforderunity. Incontrastto SP =∆S+(i/z)I2, (4.46) this, we find that for elastic cotunneling the off-diagonal 11 1 rates vanish, w δ , and therefore δρ = 0 and nm nm nn with the non-Poissonianpart ∆S = 0. Moreover∝, at zero temperature, either w+ or nn w− mustbezero(dependingonthesignofthebias∆µ). nn ∆S(z)=e2 wnImδρmn′(z)wnI′m′ρ¯m′. (4.47) As a consequence,for elastic cotunneling we find Poisso- n,m,n′,m′ nian noise, F =S(0)/eI =1. X | | In summary, we have derived the master equation, The conditional density matrix is defined as Eq. (4.22), the stationary state Eq. (4.29) of the QDS, the average current, Eq. (4.37), and the current correla- δρ (z)=ρ (z) (i/z)ρ¯ , (4.48) nm nm − n tors, Eqs. (4.52)- (4.54) for the QDS system coupled to ρnm(z)=TrpnR(z)pmρL. (4.49) leads in the cotunneling regime under the following as- sumptions. (1) Strong cotunneling regime, w w, i.e. in Eq. (4.17) shows that ρ (z) must be a solution of ≪ nm negligible intrinsic relaxation in the QDS compared to the master equation Eq. (4.22) for the initial condition the cotunneling rate; (2) the weak perturbation V is the ρ (0)=p ,orρ (0)=δ . Wenowturntotheremain- S m n nm only coupling between the QDS and the leads, in partic- itfnrroagmncsofiontrstmrLibaSuplFtlli′Taoc(neωS)tlrQola′fnttsohfoeSrlnml′o(izSs)elLliT′snp(zeEc)qtbr.uy(m4s.4yc2ma)nm. TbetehreiozbFintoaguinrtiheeder u(annlaedritHhHeLiirnnttboe=ntwtHeheSeinntlet+ahdeHsQLoinDntl,Syw;a(hn3ed)renthoHeqSilnuetaandatscutnsmoorcnowrtirhtehelianQtiotDhnSes latter, QDS or the leads) in the initial state, ρ = Pρ ; (4) no 0 0 degeneracy in the QDS, E = E for n = m; (5) small SlFl′T(ω)=SlLlT′ (ω)+SlL′Tl (−ω). (4.50) frequencies, ω Em Enn6. Fomr the m6aster equation ≪ | − | 10