Table Of Content

Frequency-dependent counting statistics in interacting nanoscale conductors C. Emary,1 D. Marcos,2 R. Aguado,2 and T. Brandes1 1 Institut fu¨r Theoretische Physik, Hardenbergstr. 36, TU Berlin, D-10623 Berlin, Germany 2 Departamento de Teor´ıa de la Materia Condensada, Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco 28049, Madrid, Spain 8 (Dated: February 5, 2008) 0 0 Wepresentaformalismtocalculatefinite-frequencycurrentcorrelationsininteractingnanoscopic 2 conductors. We work within the n-resolved density matrix approach and obtain a multi-time cu- n mulant generating function that provides the fluctuation statistics solely from the spectral decom- a position of the Liouvillian. We apply the method to the frequency-dependent third cumulant of J thecurrent through a single resonant level and through a double quantum dot. Ourresults, which 3 show that deviations from Poissonian behaviour strongly depend on frequency, demonstrate the 2 importance of finite-frequency higher-ordercumulants in fully characterizing transport. ] PACSnumbers: 73.23.Hk,72.70.+m,02.50.-r,03.65.Yz l l a h Following the considerable success of shot-noise in the n-resolvedDM can be written quite generally as - s understanding of transport through mesoscopic systems e [1], attention is now turning towards the higher-order ρ˙(n)(t)=L0ρ(n)(t)+LJρ(n−1)(t), (1) m statisticsofelectroncurrent. Theso-calledFullCounting where the vector ρ(n)(t) contains the nonzero elements at. Statistics (FCS)ofelectrontransportyields allmoments of the DM, written in a suitable many-body basis. The (or cumulants) of the probability distribution P(n,t) of m LiouvillianL describesthe ‘continuous’evolutionofthe the number of transferred electrons during time t. De- 0 system, whereas L describes the quantum jumps of the - spite their difficulty, measurements of the third moment J d transferofanelectrontothecollector. Wemaketheinfi- n of voltage fluctuations have been made [2, 3], and recent nite bias voltage approximationsuch that the transfer is o developmentsinsingleelectrondetection[4,5,6]promise unidirectional. Byconstruction,thismethodisverypow- c to open new horizons on the experimental side. [ erfulforstudyinginteractingmesoscopicsystemsthatare 3 The theory of FCS is now well established in the zero- weakly coupled to the reservoirs, such as coupled quan- v frequency limit [7, 8, 9]. However, this is by no means tum dots (QDs) in the Coulomb Blockade (CB) regime 1 thefullpicture,sincethehigher-ordercurrentcorrelators [9, 15, 16] or Cooper-pair boxes [17]. Within this frame- 8 atfinitefrequenciescontainmuchmoreinformationthan work, our theory of frequency-dependent FCS is of com- 7 their zero-frequencycounterparts. Already at second or- plete generality and therefore of wide applicability. 3 0 der (shot-noise), one can extract valuable information In this n-resolved picture, electrons are transferred to the leads via quantum jumps and there exists no quan- 7 about transport time scales and correlations. When the 0 conductorhasvariousintrinsictimescaleslike,forexam- tumcoherencebetweenstatesinthesystemandthosein t/ ple,thechargerelaxationtimeandthedwellingtimeofa theleads. Thus,althoughthesystemitselfmaybequan- a tum, the measuredcurrentmay be considereda classical chaotic cavity [10], one needs to go beyond second-order m in order to fully characterizeelectronic transport. Apart stochastic variable and therefore amenable to classical - from this example, and some other notable exceptions counting [18]. This observation allows us to derive vari- d n [11, 12, 13], the behaviour of finite-frequency correlators ousgeneralizationsofclassicalstochasticresults,andob- o beyond shot-noise is still largely unexplored. tain a multi-time cumulant generating function in terms c of local propagators. We illustrate our method by calcu- v: In this Rapid Communication, we develop a theory lating the frequency-resolved third cumulant (skewness) i of frequency-dependent current correlators of arbitrary for two paradigms of CB mesoscopic transport: the sin- X order in the context of the n-resolved density matrix gle resonant level (SRL) and the double quantum dot r (DM) approach, — a Quantum Optics technique [14] (DQD). a that has recently found application in mesoscopic trans- Equation(1) canbe solvedby Fourier transformation. port [15]. Within this approach, the DM of the sys- Defining ρ(χ,t) = ρ(n)(t)einχ, we obtain ρ˙(χ,t) = n tem, ρ(t), is unravelledinto components ρ(n)(t) in which M(χ)ρ(χ,t), with M(χ) ≡ L +eiχL . Let N be the 0 J v P n=n(t)=0,1,...electronshavebeentransferredtothe dimension of M(χ), and λ (χ); i = 1,...,N , its eigen- i v collector. Considering a generic mesoscopic system with values. Intheχ→0limit,oneoftheseeigenvalues,λ (χ) 1 Hamiltonian H = H +H +H , where H and H say,tendstozeroandthecorrespondingeigenvectorgives S L T S L refer to the system and leads respectively, and provided the stationary DM for the system. This single eigen- thattheBorn-Markovapproximationwithrespecttothe value is sufficient to determine the zero-frequency FCS tunnelling termH isfulfilled, the time-evolutionofthis [9]. In contrast, here we need all N eigenvalues. Using T v 2 the spectral decomposition, M(χ) = V(χ)Λ(χ)V−1(χ), The above procedure can be easily generalized to obtain with Λ(χ) the diagonal matrix of eigenvalues and V(χ) the N-time CGF, which reads: the corresponding matrix of eigenvectors,the DM of the N system at an arbitrary time t is given by e−F(χ;t) = T Tr Ω(σ ;τ )ρ(t ) , (5) k N−k 0 ρ(χ,t)=Ω(χ,t−t0)ρ(χ,t0), (2) (kY=1 ) whereΩ(χ;t)≡eM(χ)t =V(χ)eΛ(χ)t)V−1(χ)istheprop- where σ ≡ N χ , χ ≡ (χ ,...,χ ), t ≡ k i=N+1−k i 1 N agator in χ-space, and ρ(χ,t0) is the (normalized) state (t1,...,tn)andτk ≡tk+1−tk [22]. The multi-time CGF of the system at t0, at which time we assume no elec- in Eq. (5) contPains a product of local-time propagators, trons have passed so that ρ(n)(t0) = δn,0ρ(t0) and thus and expresses the Markovian character of the problem. ρ(χ,t0) ≡ ρ(t0). The propagator in n-space, G(n,t) ≡ It allows one to obtain all the frequency-dependent cu- d2πχe−inχΩ(χ,t), such that ρ(n)(t) = G(n,t−t0)ρ(t0), mulants from the spectral decomposition of M(χ). The for t > t0, fulfils the property: G(n − n0,t − t0) = N-time current-cumulant (e=1) is calculated using: R G(n−n′,t−t′)G(n′ −n ,t′ −t ), for t > t′ > t , n′ 0 0 0 (n′ ≡ n(t′), n0 ≡ n(t0)). This is an operator version of S(N)(t1,...,tN)≡hδI(t1)...δI(tN)i= P the Chapman-Kolmogorovequation [19]. =∂ ...∂ hhn(t )...n(t )ii= t1 tN 1 N The joint probability of obtaining n electrons after t1 and n2 electrons after t2, namely P1 >(n1,t1;n2,t2) =−(−i)N∂t1...∂tN∂χ1...∂χNF(χ;t) χ=0. (6) (the superscript ‘>’ implies t > t ), can be written in (cid:12) 2 1 (cid:12) termsofthesepropagatorsbyevolvingthelocalprobabil- The Fourier transform of S(N) with respect to(cid:12) the time ities P(n,t) = Tr{ρ(n)(t)} [20], and taking into account intervals τk, gives the Nth-order correlation functions P(n ,t )= P>(n ,t ;n ,t ), such that as functions of N − 1 frequencies. In particular, the 2 2 n1 1 1 2 2 frequency-dependent skewness is a function of two fre- P>(n ,tP;n ,t ) = Tr{G(n −n ,t −t ) 1 1 2 2 2 1 2 1 quencieswhich,asaconsequenceoftime-symmetrization ×G(n1,t1−t0)ρ(t0)}.(3) and the Markovian approximation, has the symmetries S(3)(ω,ω′) = S(3)(ω′,ω) = S(3)(ω,ω −ω′) = S(3)(ω′ − The total joint probability reads P(n ,t ;n ,t ) = 1 1 2 2 ω,ω′) = S(3)(−ω,−ω′), and is therefore real. The Nth- TP>(n ,t ;n ,t ) = P>(n ,t ;n ,t )θ(t − t ) + 1 1 2 2 1 1 2 2 2 1 order Fano-factor is defined as F(N) ≡S(N)/hIi. P<(n ,t ;n ,t )θ(t −t ) where T is the time-ordering 1 1 2 2 1 2 InthecasewherethejumpmatrixL containsasingle operator, θ(t) the unitstep function defined as θ(t) = 1 J element, (L ) =Γ δ δ , which is the case for a wide fort≥0,andzerootherwise,andwhereP<(n ,t ;n ,t ) J ij R iα jβ 1 1 2 2 class of models including our two examples below, all is the joint probability with t < t . It should be noted 2 1 thecorrelationfunctionscanbeexpressedsolelyinterms that,incontrasttothelocalprobabilityP(n,t),thejoint of the eigenvalues λ of M(0), and the N coefficients probability P(n ,t ;n ,t ) contains information about k v 1 1 2 2 c ≡(V−1L V) =Γ V V−1. Thesecond-orderFano the correlations at different times. k J kk R βk kα factor then has the simple, general form Result(3)maybealternativelyderivedusingtheBayes formula for the conditional density operator [21]: Nv c λ P>(n1,t1;n2,t2)=P(n1,t1)P>(n2,t2|n1,t1) F(2)(ω) = 1−2 ω2k+kλ2, (7) k=2 k X = Tr ρ(n1)(t ) Tr ρ(n2|n1)(t ) 1 2 whichhasalsobeenderivedinotherways[23]. Theskew- = Trnρ(n1)(t )oTrnG(n −n ,ot −t ) ρ(n1)(t1) . ness has the form F(3)(ω,ω′) = −2+ 3i=1F(2)(νi) + 1 2 1 2 1 F(3)(ω,ω′), with ν = ω, ν = ω′, and ν = ω − ω′. n o ( Tr ρ(n1)(t1) ) F(3)(ω,ω′) is an ir1reducible2contributioPn, 3the form of n o The normalization in the denominator accounts for the wehich is too cumbersome to be given here. The high- collapse n = n1 at t = t1 using Von Neumann’s pro- ferequency limit of the skewness is F(3)(ω,∞)=F(2)(ω). jection postulate [21]. Equation (3) is recovered when AsafirstexampleweconsideraSRL,describedbyρ= ρ(n1)(t1) is written as a time evolution from t0. (ρ ,ρ )T and M(χ) = −ΓL eiχΓR , in the basis Thetwo-timecumulantgeneratingfunction(CGF)as- 00 11 Γ −Γ L R (cid:18) (cid:19) sociated with these joint probabilities is of ‘empty’ and ‘populated’ states, {|0i,|1i}. Employing Eq. (5), we obtain the known results for the currentand e−F(χ1,χ2;t1,t2) ≡ P(n ,t ;n ,t )ein1χ1+in2χ2, 1 1 2 2 noise, and arrive at our result for the skewness: nX1,n2 which,using Eq.(3), ande−F =e−TF> =Te−F>, gives F(3)(ω,ω′) = 1−2Γ Γ 2i=1 γi+ω2−ωω′+ω′2 , L R 3 Γ2+ν2 e−F(χ2,χ1;t2,t1) = T Tr(Ω(χ ,t −t ) Q (cid:0) j=1 j (cid:1) 2 2 1 ×Ω(χ +χ ,t −t )ρ(t )). (4) with γ = Γ2 +Γ2, γ = 3Γ2, aQnd Γ(cid:0)= Γ +Γ(cid:1) . The 1 2 1 0 0 1 L R 2 L R 3 CB regime [25, 26]. In the basis of ‘left’ and ‘right’ states |Li and |Ri, which denote states with one excess electron with respect to the many body ‘empty’ state |0i, the Hamiltonian reads H = ǫ(|LihL|−|RihR|)+ S T (|LihR|+|RihL|), with detuning ǫ and coupling c strengthT . The twolevels|Li, |Ri,arecoupledtotheir c respective leads with rates Γ and Γ . The DM vector L R is now ρ = (ρ ,ρ ,ρ ,Re(ρ ),Im(ρ ))T, and the 00 LL RR LR LR Liouvillian in this basis reads: −Γ 0 eiχΓ 0 0 L R Γ 0 0 0 −2T L c   M(χ)= 0 0 −Γ 0 2T . R c  0 0 0 −12ΓR 2ǫ   0 Tc −Tc −2ǫ −12ΓR    Comparisonof the quantum-mechanicallevel-splitting ∆ ≡ 2 T2+ǫ2 with the incoherent rates Γ divides c L,R the dynamicalbehaviourofthe systeminto tworegimes. p For ∆≪Γ , alleigenvalues of M(0) are real, and cor- L,R respondingly, the noise and skewness are slowly-varying functions of their frequency arguments. In this regime, dephasing induced by the leads suppresses interdot coherence, and the transport is largely incoherent. In the opposite regime, ∆≫Γ , two of the eigenvalues form L,R acomplexpair,λ ≈±i∆−Γ /2+O(Γ/∆)2,whichsig- FIG. 1: The third-order frequency-dependent Fano factor ± R F(3)(ω,ω′) for the single resonant level. (a) Contour plot nals the persistence of coherent oscillations in the dots. of F(3)(ω,ω′) as function of its arguments for ΓR = ΓL. The finite-frequency correlators then show resonant fea- (b) Sections F(3)(ω,0) and F(3)(ω, ω) show that theskew- tures at ∆, since these eigenvalues enter into the de- − ness is suppressed throughout frequency space both with nominators, as in Eq. (7), giving rise to poles such as respect to the Poissonian value of unity and to the noise ω∓∆−iΓ /2. Thestructureoftheskewnessissimilarto R F(2)(ω). In contrast to the shotnoise, the skewness has a the SRL forweak coupling(∆≪Γ ), butmuchricher L,R minimum at a finite frequency ωm, which exists in the cou- in the strong coupling regime (∆≫Γ ) (Fig. 2). Now pling range 3 √5 /2 ΓL/ΓR 3+√5 /2. In di- L,R rection ω =` −ω′, thís m≤inimum occ≤ur`s, for Γ´L = ΓR, at the skewness exhibits a series of rapid increases. From oωfmt1h/eΓRmin=impu−m√1s0h−ift2s/s√lig3htwlyhetroeaωsmf2o/rΓωR′==20/√th3e. (pco)sitTiohne tmhieniomriugminaotufitwnaitredfsreinqutehnecωy-aωn′dptlahnene,iwnfleeoxbiosnerpveoifinrtsstaat maximum suppression occurs at ΓL=ΓR. ω ∼ ΓR, |ω| = ∆, |ω′| = ∆ and |ω−ω′| = ∆. Fig. 2b shows sections in the ω-ω′ plane, and the resonant behaviour,in the form of Fano shapes, is most pronounced inF(3)(ω,−ω). Startingfromhighfrequencies,theonset zero-frequency limit F(3)(0,0) agrees with Ref. [24]. of antibunching occurs at ω =∆. At higher frequencies, This skewness is plotted in Fig. 1, from which the six- the system has no information about correlations and is fold symmetry of F(3) is readily apparent. The third- Poissonian. The overall behaviour is seen in Fig. 2(c) order Fano factor gives, in accordance with the noise, a where we plot F(3)(ω,−ω) as a function of T and ω. c sub-Poissonian behaviour for all frequencies. This can The line ω = ∆ delimits two regions: at high frequen- be easilyunderstoodasaCBsuppressionofthe longtail cies the skewness is Poissonian. At resonance, and after of the probability distribution of instantaneous current: a small super-Poissonian region at ω & ∆, the system duetotheinfinitebias,thedistributionisboundedonthe becomes sub-Poissonian (and even negative, for certain leftbyzero,but,inprinciple,isnotboundedontheright internalcouplings). In the limit ω →0, our results qual- (large, positive skewness). CB suppresses large current itatively agree with those of Ref. [27] for a noninteract- fluctuations which explains a sub-Poissonianskewness. ing DQD: as a function of T , the skewness presents two c Alongthesymmetrylinesinfrequencyspacewithω′ = minima and a maximum (where the noise is minimum, 0, ω′ = ω and ω = 0, the skewness is highly suppressed. not shown). In our case, however, the maximum occurs In contrastwiththe noise, the minimum in the skewness around∆=Γ /2—halfthatofthenoninteractingcase. R occurs at finite frequency (Fig. 1b) with the strongest Finally, we plot dF(3)(ω,−ω)/dω as a function of both ǫ suppression occuring at ΓL =ΓR (Fig. 1c). andω(Fig.2d)wheretheresonancesatω =∆,ω =∆/2 As a secondexample weconsidera DQDinthe strong and ω ∼Γ are clearly resolved. In contrast,the deriva- R 4 [2] B.Reulet1,J.Senzier,andD.E.Prober,Phys.Rev.Lett. 91, 196601 (2003). [3] Yu. Bomze, G. Gershon, D. Shovkun, L. S. Levitov and M. Reznikov,Phys.Rev.Lett. 95 176601 (2005). [4] T. Fujisawa, T. Hayashi, R. Tomita, and Y. Hirayama, Science 312, 1634 (2006). [5] S. Gustavsson, R. Leturcq, B. Simoviˇc, R. Schleser, T. Ihn, P. Studerus, K. Ensslin, D. C. Driscoll, and A.C.Gossard,Phys.Rev.Lett.96,076605(2006);Phys. Rev. B 75 075314 (2007). [6] E. V. Sukhorukov, A. N. Jordan, S. Gustavsson, R. Leturcq, T. Ihn, and K. Ensslin, Nature Physics 3, 243 (2007). [7] L. S.Levitov and G. B. Lesovik,Pis’ma Zh. E´ksp.Teor. Fiz. 58, 225 (1993); L. S. Levitov, H. -W. Lee, and G. B. Lesovik, J. Math. Phys. 37, 4845 (1996). [8] Yu. V. Nazarov, Ann. Phys. (Leipzig) 8, Special Issue, SI-193 (1999). [9] D. A. Bagrets and Yu. V. Nazarov, Phys. Rev. B 67, 085316 (2003). [10] K. E. Nagaev, S. Pilgram, and M. Bu¨ttiker Phys. Rev.Lett. 92, 176804 (2004). [11] S. Pilgram, K. E. Nagaev, and M. Bu¨ttiker, Phys. Rev. FIG. 2: Frequency-dependent Fano skewness for the dou- B 70, 045304 (2004). ble quantum dot in Coulomb blockade. (a) Contour plot in [12] A. V. Galaktionov, D. Gloubev, and A. D Zaikin, Phys. the strong coupling regime, Tc = 3ΓR, with ΓL = ΓR and Rev. B 68, 235333 (2003). ǫ = 0. (b) Sections F(3)(ω,0) and F(3)(ω, ω), and shot- [13] J. Salo, F. W. J. Hekking, and J. P. Pekola, Phys. Rev. − noise F(2)(ω) show a series of abrupt increases with increas- B 74, 125427 (2006). ingω. Both noiseand skewnessexhibitboth sub-and super- [14] R. J. Cook, Phys. Rev.A, 23, 1243 (1981). Poissonian behaviour (c) Varying the internal coupling Tc, [15] S. A. Gurvitz, Phys.Rev. B 57, 6602 (1998). the skewness shows rapid increases along the lines ω = ∆ [16] R. Aguado and T. Brandes, Phys. Rev. Let. 92, 206601 and ω = ∆/2. For ω > ∆ the system is Poissonian (slightly (2004). super-Poissonianforω&∆),whileforω<∆thetransportis [17] M.-S. Choi, F. Plastina, and R. Fazio Phys. Rev. Lett. always sub-Poissonian. The skewness is strongly suppressed 87, 116601 (2001). at low frequencies. (d) The derivative dF(3)(ω, ω)/dω as a [18] This is in contrast with quantum detection, where the function of frequency and detuning ǫ for Tc =−3ΓL = 3ΓR. non-commutativity of current at different times must be Resonances occur at ω=∆, ∆/2 and ΓR. explicitly considered [7]. ∼ [19] N.G.Van.Kampen,Stochastic Processes inPhysics and Chemistry (North-Holland P.L., 1992). [20] With ρ a vector, the trace operation corresponds to a tive dF(3)(ω,0)/dω (not shown), exhibits a minimum at sum overall thediagonal elements of theoriginal DM. w = ∆ for small ǫ, which transforms into a maximum [21] A. N.Korotkov, Phys. Rev.B 63, 115403 (2001). [22] One can also obtain the CGF in terms of a single time- for ǫ ≫ T . As expected, transport tends to be more c dependent counting field from the expression e−F[χ] := Poissonianas ǫ increases, signaling lost of coherence. Despite the simplicity of the models we have stud- G[χ] ≡ DeiRn(t)χ(t)dtET, where h...iT denotes a time- ied, our results demonstrate the importance of finite- ordered path integral with the joint probability as the weight function [9].Establishing thebiyectionbetween t frequencystudies. DeviationsfromPoissonianbehaviour and n, one recovers Eq. (5) in terms of products of evo- ofhigher-ordercumulantsarefrequency-dependent,such lutionoperatorsbydividingthetimeintervalintosubin- that a comprehensive analysis in the frequency domain tervalsτ :=t t ,k=0,...,N 1,withassociated k k+1 k is needed in order to fully characterize correlations and time-independent−countingfields χ −and variables n . k k statistics in electronic transport. [23] S. Hershfield,J. H.Davies, P.Hyldgaard, C.J. Stanton, Worksupportedby the WEHeraeusfoundation,DFG and J. W. Wilkins, Phys. Rev. B 47, 1967 (1993); C. grant BR 1528/5-1, and by the grants MAT2006-03741 Flindt, T. Novotny´,and A.-P.Jauho, Physica E 29, 411 (2005). andFPUAP2005-0720(MEC-Spain),20060I003(CSIC) [24] M. J. M. de Jong, Phys.Rev.B 54, 8144 (1996). and 200650M047(CAM). [25] T. H. Stoof and Yu.V. Nazarov, Phys. Rev. B 53, 1050 (1996). [26] T. Brandes, Phys. Rep. 408/5-6, 315, (2005). [27] G. Kiesslich, P. Samuelsson, A. Wacker, and E. Schöll, Phys. Rev.B 73, 033312 (2006). [1] Ya. M. Blanter and M. Bu¨ttiker, Phys. Rep. 336, 1 (2000).