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8 0 0 2 Non-Markoviansignaturesinthecurrentnoiseofachargequbit n a AlessandroBraggioa,1,ChristianFlindtb,andTomáˇsNovotnyć J 4 aLAMIA-INFM-CNR,DipartimentodiFisica,UniversitàdiGenova,ViaDodecaneso33,I-16146, Genova,Italy 2 bMIC–DepartmentofMicroandNanotechnology, NanoDTU,TechnicalUniversityofDenmark,Building345East, DK-2800KongensLyngby,Denmark ] cDepartmentofCondensedMatterPhysics,FacultyofMathematicsandPhysics,CharlesUniversity,KeKarlovu5,121 l 16Prague,CzechRepublic l a h - s e Abstract m Weinvestigatethecurrentnoiseofachargequbitcoupledtoaphononbathindifferentparameterregimes.Wefind,using . t the theory of Full Counting Statistics of non-Markovian systems, that the current fluctuations are strongly influenced by a m memoryeffects generatedfromtheinterplaybetweenquantum coherenceandthedynamicsofthephononbath. - d Keywords: Noise,non-Markoviandynamics,qubit n PACS:72.70.+m,73.23.-b,73.63.Kv,05.40.-a o c [ 1 The recent experimental realizations of charge that allows us to treat not only the Markovian, but v qubits in double quantum dots clearly illustrate the also thenon-Markoviandynamicsofthechargetrans- 3 importance of controlling quantum coherence within portprocess.Weshowthattheobserved,asymmetric 2 a system [1]. Detecting quantum coherence via indi- super-poissonian behavior is determined bytheinter- 7 rect andnon-invasivemeasurementsisthusbecoming playbetweenmemory effectsinducedbythecoherent 3 . a crucial task. Currents fluctuations, in particular dynamics and the coupling to the dissipative phonon 1 the current noise, have been suggested as a useful bath.Thisindicatesthatitmaybedisputabletoasso- 0 8 investigationtool[2,3].Inarecentexperiment,fluctu- ciatesuper-poissoniannoisepurelytocoherenteffects. 0 ations of the current through two coherently coupled We conclude that current fluctuations in general can : quantum dots were measured [4], and the observed, bestronglymodifiedbymemoryeffectsthatmaybeof v i unexpected super-poissonian noise was subsequently bothintrinsicand/orextrinsicorigintothesystem[7]. X explainedasapureeffectofquantumcoherence[5].In Weconsiderthemodeloftransportthroughacharge r the present work, we consider the same system, how- qubit in a dissipative environment previously studied a ever, using an approach based on the theory of Full inRefs.[2,3].Themodelconsistsofadoublequantum Counting Statistics for systems with Non-Markovian dot operated in the strong Coulomb blockade regime Dynamics (FCSNMD) [6,7]. The main advantage of andtunedtoadegeneracypoint,whereelectrontrans- thistechniqueliesinthepossibilitytocalculateallor- portproceedsonlyviathethreechargestates 0 (un- | i dersofthecurrentcumulantsusingaunifiedapproach occupieddots), + / (left/rightdotoccupied).The | i |−i Hamiltonianofthesystemreads 1 Correspondingauthor.E-mail:braggio@fisica.unige.it PreprintsubmittedtoPhysicaE February5,2008 ε Hˆ = σˆ +T σˆ +V σˆ +H +Hˆ +Hˆ , (1) theory of the FCSNMD to calculate the current and 2 z c x B z B T res thenoise[7].Thestartingpointisthenon-Markovian wherewedenotethetunnelcouplingbetween|+iand equationofmotion for P(χ,t)whichinLaplacespace |−i by Tc and the energy difference (detuning) of the translates to [z W(χ,z)]P(χ,z)=P(χ,t=0) with twostates byε,havingintroducedthestandard spin- P(χ,t = 0) bein−g the initial condition at t = 0.3 For boson pseudo-spin operators σˆ + + z ≡ | ih | − |−ih−| theproblemathand,thememorykernelreads and σˆ + + +. Relaxation and dephasing x ≡ | ih−| |−ih | are assumed to occur due to the coupling VBσˆz = Γ+ 0 Γ−eiχ gσˆ (aˆ†+aˆ )/√2toasurroundingdissipativeheat 0− 1 z j j j W(χ,z)= Γ Γ(+) Γ(−) (2) bathPdescribedasareservoirofnon-interactingbosons, B + − z z C B C kHnBow=nPspji~nω-bjoaˆs†joaˆnj.pWroebalreemt.huFsindaelalyli,nwgewiitnhtrtohdeuwceella- BB@ 0 Γ(z+) −Γ−−Γ(z−)CCA left/right(α=+, )leaddescribedasnon-interacting where Γ± = 2π̺± t± 2 are the lead tunneling − | | fermions, i.e., Hˆ = ǫ cˆ† cˆ coupled rates for energy independent densities of states res k,α=± k,α k,α k,α to the spin-boson systemPvia the tunnel-Hamiltonian ̺±. The phonon assisted hopping rates are Γ(z±) = HˆT = k,α=±(tαcˆ†kα|0ihα|+h.c.). Tc2[G(±)(z+)+G(∓)(z−)], where z± = z+Γ−/2±iε TodPescribechargetransportthroughthesystemwe and the bath correlation functions in Laplace follow theapproachdevelopedbyGurvitzandPrager space are defined as (±)(z) == ∞dte−zt (±)(t). G 0 G [8]. Assuming a constant tunneling density of states In time domain these functions rRead (±)(t) = and a large bias across the double-dot, we can derive TrB{e−iHB(+)tρ(β±)eiHB(−)t} with the equilibGrium den- aρˆn(n)e,qdueafitnioendoinf mthoetiHonilbfoerrttshpearceedoufctehdedeelencstitryonmicastyrisx- sity matrix ρ(β±) = e−βHB(±)/TrB{e−βHB(±)} of the displaced phonon bath with Hamiltonians H(±) = tem(0 , + , )andthephononbath,resolvedwith B | i | i |−i H V andinversetemperatureβ =1/k T.Assum- respect to the number of electrons n which havetun- B± B B ing an Ohmic spectral density, J(ω) = 2g2ωe−ω/ωc, neledfromtheleftbarrier[8].Tocalculatethecurrent we obtain to order g2 in the bath coupling and to cumulants it is convenient to introduce the counting leadingorderin1/βω ,theexpression field χ and consider the Fourier transform ρˆ(χ) = c einχρˆ(n) [9]. Current moments can be obtained 1 βz z n (±)(z)= 1 2g2 H +V(±) Pfrom thedynamicsof thesum of theχ-dependentoc- G z  − » „2π« „ωc«–ff cupation probabilities contained in the vector P(χ,t) (3) with elements [P(χ,t)] = iTr ρˆ(χ)(t) i , i = where H(x) = ln(x) Ψ(x) 1/2x with Ψ(x) be- 0,+, . Here, Tr denoites ahp|artBia{l trace}|ovier the ing the digamma func−tion and−V(±)(x) = [sin(x) bath−degrees of fBreedom and it is easy to verify the icos(x)][π/2 si(x) ici(x)],wheresi(x)= xdtsin(t)∓/t − ∓ 0 normalization for χ = 0. We proceed by assuming a andci(x)= ∞dtcos(t)/t.WenotethaRtevenwith- − x local Born approximation for the diagonal parts of outthephononRbath,correspondingtog =0,we still the density matrix: at any time the bath is assumed have a contribution of the form (±)(z) = 1/z oc- G quickly to reach local equilibrium corresponding to curring due to the coherent dynamics, and only if we the given charge state. Within this approximation take thez 0 limit we recover the standard Fermi’s → we may solve the problem for small values of T .2 goldenruleratesforincoherenttunnelingbetweenthe c The rate equation for thecounting probability vector dots,Γ(z±→)0=Tc2Γ−/[(Γ−/2)2+ǫ2].ThecurrentI and is P˙(χ,t)= tdτW(χ,t τ)P(χ,τ) with the mem- thenoiseFanofactorF =S(0)/eI forthesystemcan ory kernel WR0(χ,t) contai−ning the rates for tunneling beobtainedusingthetheoryofFCSNMD[6,7].4 The to and from the leads and phonon-assisted rates for theory shows that noise and higher order cumulants, tunneling between the two dots. In the following we but not the mean current, may contain signatures of apply some recently developed results for the general 3 Forthe zero-frequencyfluctuations consideredhere, the 2 Asweshallseeourtheoryagreeswellwithaperturbative initialconditionsdonotplayarole. approachdeveloped inthehybridizedbasiswhichisexact 4 Thetheoryalsoallowsustocalculatehighercumulants, toallordersinTc [2]. whichwe,however,willnotconsiderhere. 2 F temperature (dark and light grey dashed lines). On 1.2 the other hand, including non-Markovian terms we seetheasymmetric super-poissonian behavior for low 1.0 temperatures(darkgreysolid lines)which disappears withincreasing temperature(light greysolid lines) as 0.8 alsoseenintheexperiment[4].We,moreover,findan optimal agreement with the results of the standard perturbationtheoryinthehybridizedbasis(darkand 0.6 −800 −400 ε 0 400 800 light grey dotted lines), allowing us to exclude possi- ble spurious effect of the perturbation in T . We see c Figure1.Fanofactorvs.leveldetuningεfordifferentphys- that the noise becomes super-poissonian due to the ical parameter regimes and two temperatures T = 1.4 K non-Markovian corrections. By direct inspection of (light grey) and T =12 K (dark grey). The other param- the expression for the Fano factor, we see that the eters correspond to those used in Fig. 2b of Ref. [5]. Co- conditionforhavingsuper-poissoniannoiseisgivenby herentregime(solidblackline),g=0sequentialtunneling (blackdashedline),phonon-assistedincoherent tunnelling theinequalityΓ+(∂zΓ(0+)(Γ−+Γ(0−))−Γ(0+)∂zΓ(0−))> (dashed grey lines), phonon-assisted tunneling (solid grey Γ(0+)(Γp+Γ−+Γ+).Thisshowsthatapurecoherent lines) and, for comparison, perturbative approach in the contributioncannoteasily bedisentangledfrom other hybridizedbasis(dotted lines). classical orquantumbathmemoryeffects,sincethese all appear mixed in the memory kernel. Our analy- non-Markovian dynamics,signaled bythepresenceof sis supports the suspicion, previously arisen, that an z-derivativesoftheratesΓ(±).Thecurrentisgivenas z anomalous behavior of the current cumulants may I = Γ Γ Γ(+)/Γ2 with Γ2 = Γ (Γ +Γ(+))+Γ Γ − + 0 t t − + 0 + p be due to classical memory effects [7]. Such effects, andΓ =(Γ(−)+Γ(+)).TheFanofactorF reads p 0 0 independently of their origin, can destroy the signa- 2Γ Γ2[(Γ +Γ(−))∂ Γ(+) ∂ Γ(−)Γ(+)] turesofpurequantumeffects,andshouldbecarefully F =Fm+ − + − 0 Γ4z 0 − z 0 0 , taken into account in any interpretation of current t fluctuations. 2Γ Γ2Γ(−)+Γ2Γ2+Γ2[Γ2 +(Γ(+))2] F = − + 0 + p − + 0 , SupportbytheEuropeanScienceFoundation(ESF) m Γ4 t Research Networking Programme entitled “Arrays (4) of Quantum Dots and Josephson Junctions” and by where F denotes the Markovian contribution to the EU via MCRTN-CT2003-504574 network and by the m Fano factor (taking only the z = 0 contribution) and grant 202/07/J051 of the Czech Science Foundation wherewehavedefined∂ Γ(±) ∂ [Γ(±)] . aregratefullyacknowledged. z 0 ≡ z z z=0 In Fig. 1 we show the Fano factor calculated in different parameter regimes. Without coupling to the phonons(g=0),wefindinthelimitz 0thesequen- → tialtunneling regimedescribedonlybytheMarkovian References terms[6]. TheFanofactor is symmetricaround ǫ=0 [1]T. Fujisawa et al. Science 282 (1998) 932; and sub-poissonian (black dashed curve). Taking the T.H.Oosterkampetal.Nature395(1998)873. full z-dependence of the rates, non-Markovian contributions are included, and we find the so-called [2]R.Aguadoetal.,Phys.Rev.Lett.92(2004)206601. coherent regime [8,10]. The Fano factor is again sym- [3]G.Kießlichetal.Phys.Rev.B73(2006)033312. metric, but now super-poissonian (solid black curve) [4]P.Bartholdetal.,Phys.Rev.Lett.96(2006)246804. [5]. Including coupling to the phonon bath (g > 0), [5]G.Kießlichetal.,condmat/0706.1737 (2007). we are in the phonon-assisted regime, where electron tunneling between the dots is assisted by phonon [6]A.Braggioetal.,Phys.Rev.Lett.96(2006)026805. absorption and/or emission. In the incoherent ap- [7]C.Flindtetal.,condmat/0706.2925;C.Flindtetal.,in proximation without non-Markovian contributions, preparation(2007). we find sub-poissonian noise independently of the [8]S.A.Gurvitzetal.,Phys.Rev.B53(1996)15932. 3 [9]L.S.Levitovetal.,J.Math.Phys.37(1996)4845. [10] T.H.Stoofetal.,Phys.Rev.B53(1996)15932. 4