epl draft Can dephasing generate non-local spin correlations? 2 1 0 Colin Benjamin1 2 n 1 National Institute of Science Education and Research, IOP Campus, Sachivalaya Marg, a Bhubaneswar 751005, India J 7 ] l PACS 74.50.+r– l a PACS 74.78.Na– h PACS 85.25.-j– - s Abstract –By examining the full counting statistics of a non adiabatic pure spin pump with e m particular emphasis on the second and third moments, it is shown that incoherent or sequential transport,incontrasttocoherenttransport,canchangenon-localspinshotnoisecross-correlations . t frombeinganti-correlatedtobeingcompletelycorrelated,atrulycounterintuitiveresult. Thethird a moment on the other hand is shown to be much more resilient and its nature remains unaltered m in incoherent transport regime. However, phenomenologically including dephasing modifies this - picture as both Shot noise and more so the third moment are non-trivially affected. In fact non- d local spin correlations are completely positive for maximal dephasing. n o c [ 2 v Introduction. – Non-local shot noise correlations, the second moment, in solid state 5 3 devices have been studied for a long time. Some of these studies include normal metal- 2 superconductor hybrid structures [1], coulomb blockaded quantum dots [2], exploiting the 5 Rashba scattering[3], etc. However,anexperimentaldemonstrationhas thus far been lack- . 0 ing. This is mainly due to the difficulty in controlling environmental effects like dephasing 1 or decoherence. The origin of these environmental effects can be traced to magnetic im- 1 purities in the experimental system which affects electronic spin and to temperature which 1 leads to electron-phonon interaction and dephasing. It begs the question how to deal with : v decoherence and reduce it. In this work a novel scheme is proposed in which the dephas- i X ing present in such systems can be used as a resource. We particularly concentrate on the electronic spin. The reason for dwelling on the spin instead of charge is because there r a have been many works on the charge counting statistics howeverworks on the full counting statistics for spin are less visible. However, they have been attempted in different context to that which is the topic of this letter. For example, in [4], the FCS of spin currents was first attempted, the FCS of spin transfer through ultra small quantum dots in context of Kondo effect was attempted in [5] while in [6] a study of FCS in interacting quantum dots attached to ferromagnetic leads revealed super-poissonian transport. Many works revolve around the non-local spin shot-noise correlations. Among the notable works on non-local spin shot-noise correlations mention may be made of: spin current shot noise of (i) a single quantumdotcoupledtoanopticalmicro-cavityandaquantizedcavityfield[7],(ii)arealis- ticsuperconductor-quantumdotentangler[8],and(iii)aspintransistor[9,10]. Inthisletter the properties of the third moment are also calculated. The reason for looking at the third moment is because, for charge in contrast to spin transport, the third moment is predicted p-1 Colin Benjamin1 to be much more resilient to decoherence [11]. In our work we see that this statement does not strictly hold for third moment non-local spin correlations especially when decoherence is large. Charge or spin transport is a statistical process involving electrons carrying definite amounts of spin or charge, since charge or spin current fluctuates in time. Therefore, in addition to knowing the mean charge or spin current passing through a normal conductor one needs to know the noise as well as the other transport moments in order to fully char- acterize charge or spin electron motion. To do this one takes recourse to the full counting statistics(FCS), which gives us the complete knowledge about all the moments of the dis- tribution of the number of transferred charges or spins. The full counting statistics for a non-adiabatic pure spin pump is analyzed in both the completely coherent and incoherent transport regimes. In this letter we find that in the coherent transport regime the current and non-local spin shot-noise correlations are similar to that in Ref. [12]. In the sequential or incoherent transportregimesthe oddmoments arealmostunchanged. In contrastthe secondmoment, i.e., spin shot noise becomes completely positive. An extremely counter-intuitive result. In absence of any dephasing the third moment spin auto or non-local correlations do not change much from the coherent and incoherent transport regimes. To connect the coherent and incoherent transport regimes we introduce a phenomenological model of dephasing. In this work it is shown that for maximal rate of dephasing the coherent and sequential transport regimes match exactly. Themainbodyofthisletter startswithanexplanationofthe model. Thecoherentden- sity matrix equationis then analyzedseparate from the incoherent density matrix equation to bring out the differences. Lastly we bring out a perspective on future endeavors. 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A single quantum dot is connected to two leads. A large coulomb repulsion is assumed for the quantum dot leading to prevention of double occupation. A large oscillatingmagneticfieldwithstrengthdenotedbyRrf =g⊥µBBrf/2establishespurespincurrents in theleads bypumpingspins from lower to higher energy level. Model. – The model of Ref. [12] is the starting point. It is depicted in Fig.1. It is a single quantum dot connected to two leads. The single electron levels in the dot are split by an external magnetic field B. Thus, ǫ −ǫ =g µ B =∆(Zeeman energy), where g is ↑ ↓ z B z effectiveelectrong-factorinz-directionandµ isBohrmagneton. Nobiasvoltageisapplied B acrosstheleads. AnadditionaloscillatingmagneticfieldB (t)=(B cos(ωt),B sin(ωt)) rf rf rf applied perpendicularly to constant field B with frequency ω nearly equal to ∆ can pump the electronto higher levelwhere its spinis flipped, then the spin downelectroncantunnel outofthe leads. Coulombinteractioninthe quantumdotisconsideredtobe strongenough to prohibit double occupation. No extra electrons can enter the quantum dot before the spin-downelectronexits. TheHamiltonianofESRinducedspinbatteryunderconsideration is written as: H = ǫ c† c + ǫ c† c +Un n η,k,σ η,k,σ η,k,σ σ dσ dσ d↑ d↓ X X η,k,σ σ p-2 Can dephasing generate non-local spin correlations? + (V c† c +h.c.)+H (t) (1) η η,k,σ dσ rf X η,k,σ Intheaboveequation,c† (c )andc† (c )arethecreationandannihilationoperators η,k,σ η,k,σ dσ dσ for electrons with momentum k, spin σ and energy ǫ in lead η(= L,R) and for spin σ η,k,σ electrononthequantumdot. Thethirdtermdescribescoulombinteractionamongelectrons on the quantum dot. The fourth term describes tunnel coupling between quantum dot and reservoirs. H (t) describes the coupling between the spin states due to the rotating field rf B (t) and can be written in rotating wave approximation as: rf H (t)=R (c† c eiωt+c† c e−iωt) (2) rf rf d↑ d↓ d↓ d↑ with,ESRrabifrequencyR =g µ B /2,withg-factorg andamplitudeofrffieldB . rf ⊥ B rf ⊥ rf The quantum rate equations for the density matrix can be easily derivedas in Ref. [12]. ρ and ρ describe occupationprobabilityin QD being respectivelyunoccupied and spin- 00 σσ σ states and off-diagonal term ρ denotes coherent superposition of two coupled spin ↑↓(↓↑) statesinquantumdot. Thedoublyoccupiedisprohibitedduetoinfinitecoulombinteraction U →∞. To derive the density matrix, we proceed as follows. The time dependence can be removed from Eqs. [1-2], by using the following unitary transformation [13]: U =e−i2ωt[ k,η(c†η,k,↓cη,k,↓−c†η,k,↑cη,k,↑)+(c†d,↓cd,↓−c†d,↑cd,↑)] (3) P The Hamiltonian is then redefined in the rotating reference form as follows: dU−1 H = U−1HU +i U RF dt = ¯ǫ c† c + ǫ¯ c† c +Un n η,k,σ η,k,σ η,k,σ σ dσ dσ d↑ d↓ X X η,k,σ σ + (V c† c +h.c.)+R (c† c +c† c ) η η,k,σ dσ rf d↑ d↓ d↓ d↑ X η,k,σ (4) In the above equation, ǫ¯ = ǫ − ∆ + w, and ǫ¯ =ǫ + ∆ − w, while ǫ¯ = ǫ + w and ↑ D 2 2 ↓ D 2 2 ηk↑ ηk 2 ǫ¯ =ǫ − w. ηk↓ ηk↓ 2 To get the density matrix from the above Hamiltonian, the following procedure is used. An electron operator affecting only the electron on the dot can be written in terms of |p >< p|,p = 0,↑,↓. Writing, for the annihilation operator of the dot c = |0 >< σ|, dσ and for the creation operator for the dot c† = |σ >< 0|, the Hamiltonian is rewritten in dσ terms of the three states: |0 >,| ↑>,| ↓>, corresponding to empty state, a single electron with spin-up and single electron with spin-down. The doubly occupied state in the dot is prohibitedbythefactthatUistakentobeextremelylarge. Thisistheinfinite-UAnderson model. We invoke this approximation to avoid double occupancy of the dot. This entails that the probability for an electron entering the dot depends on whether dot is already occupied or not and not on the mean occupation of the dot. Thus the Hamiltonian reduces to: H = ǫ¯ c† c + ¯ǫ |σ ><σ|+Un n η,k,σ η,k,σ η,k,σ σ d↑ d↓ X X η,k,σ σ + (V c† |0><σ|+h.c.)+R (|↑><↓| η η,k,σ rf X η,k,σ + |↓><↑|) (5) Theelementsofthedensitymatrixρ indotspinbasisareexpectationvaluesofoperators mn |n >< m|, with n,m = 0,↑,↓, so we can write-ρ =< |0 >< 0| >,ρ =< |σ >< σ| > 00 σσ p-3 Colin Benjamin1 ,ρ =<|σ¯ >< σ| >. The time evolution of the density matrix elements can be expressed σσ¯ in terms of expectation values for new operators [17]. For instance, ∂ iρ˙ = i <|0><0|>=<[|0><0|,H]> 00 ∂t ρ˙ = i[H|0><0|−|0><0|H], 00 = i[V∗|σ ><0|c −V |0><σ|c† ], η ηkσ η ηkσ = [V∗G< (t,t)−V G< (t,t)] (6) η ηkσ η 0σ,ηkσ To derive the Greens functions for the dot-lead system we assume that the dot-lead system is weakly coupled. This means the dot Greens function in presence of tunneling can be written is the same form as the decoupled dot Greens function. This is also called the Markov approximation in which the probability of a tunneling event at a given time dependsonlyontheoccupationatthatparticulartime,i.e. thereisnomemorystructurein the system. Invoking Markov approximation is reasonable as tunneling happens rarely and the system is in same state at each tunneling event, i.e., when system is weakly coupled. The approximated current Green’s functions are (using Ref. [17]) as a guide we have: G< (t,t′) = dt [GR (t,t )V∗ g< (t ,t′) 0σ,ηkσ′ Z 1 0σσ′ 1 ηkσ′ ηkσ′ 1 + G< (t,t )V∗ gA (t ,t′)], 0σσ′ 1 ηkσ′ ηkσ′ 1 G< (t,t′) = dt [gR (t,t )V G< (t ,t′) ηkσ′,0σ Z 1 ηkσ′ 1 ηkσ′ 0σ′σ 1 + g< (t ,t′)V GA (t ,t′)] (7) ηkσ′ 1 ηkσ′ 0σ′σ 1 The G ’s are the green functions for the dot, while g is the Green’s function for the 0σσ′ ηkσ η-lead in absence of tunneling. From the convolution theorem for Fourier transforms, dw dt A(t−t )B(t −t)= duA(u)B(−u)= A(w)B(w). (8) Z 1 1 1 Z Z 2π Inserting the approximated current Green’s functions from Eqs.7 into Eq.6 and Fourier transforming one gets: ρ˙ =|V |2[G< (w)(gR (w)−gA (w))+g< (w)(GA (w)−GR (w))] (9) 00 η 0σσ ηkσ ηkσ ηkσ 0σσ 0σσ The general property for Green’s functions G>−G< ≡GR−GA, is then used- ρ˙ =|V |2[G< (w)(g> (w)−g< (w))+g< (w)(G< (w)−G> (w))] (10) 00 η 0σσ ηkσ ηkσ ηkσ 0σσ 0σσ The lesser Green’s function then becomes- gη=ie−iǫηkσt =ie−iǫηkσtfη(ǫηkσ), (11) where, f(ǫ) is the Fermi function. Performing a fourier transformation yields g< (w)=2πif (ǫ )δ(w−ǫ ),and similarly g> (w)=−2πi[1−f (ǫ )]δ(w−ǫ )(12) ηkσ η ηkσ ηkσ ηkσ η ηkσ ηkσ Substituting the above expressions in Eqs.6, and using the coupling parameter Γη(ǫ) = σ 2π |V |2δ(ǫ−ǫ ) gives- k η ηkσ P −i ρ˙ = dw {Γη(1−f (w))G< (w)+Γηf (w)G> (w)} (13) 00 2π Z σ η 0σσ σ η 0σσ X ησ p-4 Can dephasing generate non-local spin correlations? ThelesserandgreaterGreensfunctionsforthedotcanbederivedusingthesameformalism as inRef. [17]. Thus, G< (w)=2πiρ δ(w−ǫ ), andG> (w)=−2πiρ δ(w−ǫ ). After 0σσ σσ σ 0σσ 00 σ substituting these expressions in Eq.13, and integrating gives- ρ˙ = Γη[(1−f (ǫ ))ρ −f (ǫ )ρ ] (14) 00 σ η σ σσ η σ 00 X ση Now in Ref.[12] the Fermi functions for the left andrightleads with respectto the electron spin f (ǫ ) = f (ǫ ) = 1 and f (ǫ ) = f (ǫ ) = 0. Further the coupling parameters Γ’s L ↑ R ↑ L ↓ R ↓ are independent of energy which implies that the density of states and tunneling matrix elements are constant. This approximationis called the wide band limit. It is also assumed that occupations are constant in time as we are only interested in steady state result where this approximation is valid. The weak coupling assumption as invoked above also implies that no broadening of energy level occurs in the dot due to tunneling and this means that coupling is much smaller than temperature. Thus, ρ˙ =−(ΓL+ΓR)ρ +(ΓL+ΓR)ρ (15) 00 ↑ ↑ 00 ↓ ↓ ↓↓ Proceeding in exactly the same way, and using the Ref. [17] as a guide one can derive the other rate equations as written below. To model incoherence we turn to Ref. [15] and use that as a model. Results. – We introduce density matrices ρ (t) meaning quantum dot is on the elec- ab tronic state |a > (a = b = 0,↑,↓) or on a quantum superposition state (a 6= b) at time t. We introduce counting fields [14], χ ,η = L/R and σ =↑/ ↓ to describe transitions from η,σ the dot to leads. Coherenttransportregime:. Wefirstdealwiththecoherentregime: ρ˙(t)=(ρ˙ ,ρ˙ ,ρ˙ ,ℜ(ρ˙ ),ℑ(ρ˙ ))= 00 ↑↑ ↓↓ ↑↓ ↑↓ Mρ(t), with −(ΓL↑+ΓR↑) 0 (ΓL↓eiχL↓ +ΓR↓eiχR↓) 0 0  (ΓL↑e−iχL↑ +ΓR↑e−iχR↑) 0 0 0 −2Rrf  M= 0 0 −(Γ +Γ ) 0 2R ,  L↓ R↓ rf   0 0 0 −(Γ +Γ ) −δ   L↓ R↓ ESR   0 R −R δ −(Γ +Γ )   rf rf ESR L↓ R↓  (16) andδ =∆−ω. Thenormalizationrelationρ + ρ =1holdsfortheconservation ESR 00 σσ σσ and Γ = 2π |V |2δ(w−ǫ ). We assume the Pspin relaxation time of an excited spin ησ k η ηkσ state into the tPhermal equilibrium to be very large. We calculate the eigenvalues of Eq.16. The minimal of these eigenvalues defines the full counting statistics (as, χ → 0,η = L,R;σ =↑,↓). After finding this eigenvalue Ev , and ησ 0 then by using the approachpioneeredin Ref. [14], We calculate the first, second andhigher cumulants. Note that the approach of Ref. [14] has been generalized in Refs. [15,16] to include both coherent and incoherent transport regimes. The first cumulant is defined as the current, we calculate the individual spin polarized currents as follows: I = ∂Ev0| . The spin current is thus Is = I −I , while the ησ ∂χησ χησ→0 η η↑ η↓ chargecurrentisIc =I +I . Thesecondcumulantdefinestheshot-noise. Theshotnoise η η↑ η↓ local and non-local correlations can be calculated as follows. The spin shot noise local and non-localcorrelationis whatwe concentrateon. Ss =S↑↑ +S↓↓ −S↑↓ −S↓↑ andSs = LL LL LL LL LL LR S↑↑ +S↓↓ −S↑↓ −S↓↑ wherein,Sσσ′ = ∂2Ev0 | . Similarlythethirdmoment LR LR LR LR ηη′ ∂χησ∂χη′σ′ χησ,χη′σ′→0 spin correlations are calculated as follows: Cs = C↑↑↑ +C↑↓↓ +C↓↑↓ +C↓↓↑ − ηη′η′′ ηη′η′′ ηη′η′′ ηη′η′′ ηη′η′′ (C↑↑↓ +C↑↓↑ +C↓↑↑ +C↓↓↓ ), wherein Cσσ′σ′′ = ∂3Ev0 | . ηη′η′′ ηη′η′′ ηη′η′′ ηη′η′′ ηη′η′′ ∂χησ∂χη′σ′∂χη′′σ′′ χησ,χη′σ′,χη′′σ′′→0 Thesecondandthirdspincumulantsaresumoftheindividualcorrelationsandaregiven as Cs = Ss +Ss +2Ss and Cs = Cs +Cs +3Cs +3Cs . The spin Fano 2 LL RR LR 3 LLL RRR LLR LRR p-5 Colin Benjamin1 0.4 0.05 0 CInochoehreernetnt Cs Cs LLR LLL 0.04 0.3 -0.1 0.03 0 s I0.2 0.02 -0.2 10 0.01 0.1 -0.3 100 0 0.0 -0.4 0.0 0.5 R1.0/Γ 1.5 2.0 0.0 0.5 R1.0/Γ 1.5 2.0 0.0 0.5 R1.0/Γ 1.5 2.0 rf rf rf Fig. 2: (Color online) A comparison of coherent and incoherent transport regimes. The odd moments- spin currents(left) and third moment cross and auto-correlations are plotted as function of strength of rotating field for various dephasing rates Γ /Γ = 0(black),10(red),100(blue). φ Solid lines are for Coherent regime and dashed lines are for sequential transport regime. The parameters are Γ=1,δESR=0. factor andnormalizedskewnessalsocanprovidemoreinformationasto howspin transport is affected by dephasing these are defined as- spin Fano factor Cs/Is and normalized spin 2 skewness as Cs/Is. 3 Sequential or Incoherent transport regime:. To go into the incoherent or sequential transport regime as exemplified in Refs. [15], we use the complete coherent matrix, Eq.16, The coefficient matrix for incoherent transport can be obtained from Eq.16, via setting ℜ(ρ˙ )=0 andℑ(ρ˙ )=0 and then solvingthe two simultaneous equations for ℜ(ρ ) and ↑↓ ↑↓ ↑↓ ℑ(ρ ) as in Refs. [15,16]. This leads to a 3X3 matrix: ρ˙(t)=(ρ˙ ,ρ˙ ,ρ˙ )=Mρ(t) with ↑↓ 00 ↑↑ ↓↓ −(ΓL↑+ΓR↑) 0 ΓL↓eiχL↓ +ΓR↓eiχR↓ M= ΓL↑e−iχL↑ +ΓR↑e−iχR↑ −z z , (17) 0 z −z−(Γ +Γ )  L↓ R↓  and,z = R2rf(ΓL↓+ΓR↓) . Theminimaleigenvalueofthisequationisagainwhatwerequire. δE2SR+(ΓL↓+ΓR↓)2 Model for Decoherence. In order to understand the coherent and sequential transport regimes better and how could the transition between these two regimes be connected we introduce a phenomenological model of decoherence via a charge detector. In this model the off-diagonalelementsofLiouvilleequationareconsideredtoexponentiallydecaytozero with the rateΓ or1/T , whereT is the spin decoherencetime, i.e., in the lasttwo rowsof φ 2 2 thecoefficientmatrix(16),ΓisreplacedbyΓ+Γ . Thismethodofintroducingdecoherence φ canbe substantiatedby the insertionofa quantumpointcontactcloseto the quantum dot. Whenever an electron enters the quantum dot the transmission through it changes. This chargedetectionleadstoexponentialdampingoftheoff-diagonalsasderivedinRefs.[18,19]. Similarly, in the sequential transport regime the decoherence factor can be introduced via the replacementΓ→Γ+Γ in z of the coefficient matrix for sequential tunneling, Eq.(17). φ Intheincoherentregimetoothespincurrent,spinshot-noiseautoandcrosscorrelations are calculated and finally the third moment auto and cross-correlations. In Fig. 2, the oddmomentsareplotted-purespincurrentIs andthethirdmomentautoCs ,andcross- LLL correlationsCs . InFig. 3,thesecondmoment,shotnoiseautoSs andcross-correlations LLR LL Ss . Finally,inFig. 4the spincumulants(bothsecondandthird) andthe Fanofactorand LR normalized spin skewness are plotted. In the insets of the right hand panels of Fig. 4 the largeR isshown. Onethingwhichisquiteclearinallfiguresisthatdephasingwashesout rf p-6 Can dephasing generate non-local spin correlations? 0.02 10 100 0.01 Coherent 0 Incoherent 0.00 0.4 -0.01 SsLR 0.3 SsLL -0.02 0.2 0.1 -0.03 0.00.0 0.5 1.0R /Γ 1.5 2.0 -0.04 rf 0.0 0.5 1.0 1.5 2.0 R /Γ rf Fig. 3: The second moment: The non-local spin shot-noise correlations plotted as function of strength of rotating field for various dephasing rates Γ /Γ = 0(black),10(red),100(blue). Solid φ linesareforCoherentregimeanddashedlinesareforsequentialtransportregime. Intheinsetspin auto (or, local) correaltions are depicted. Parameters are Γ=1,δESR=0. all features in cumulants making them just increase monotonically with increasing strength of the rotating field. Both the spin cumulants are zero for small R with increasing R rf rf the second and third spin cumulants differ markedly, with the third showing much more feature including a pronounced dip at just above R =0.5Γ. The spin Fano factor plotted rf on top right hand corner of Fig 4 saturates in the large R limit at just above 2 spin flux rf quantumsthisisinagreementwiththenormalizedskewnesswhichalsosaturatesjustunder this value. In all of these figures the results for the coherent and incoherent transport regimes are contrastedas dephasingrateis increasedfromΓ /Γ=0 to 100. Inboth regimesthe charge φ current is absolutely zero. Thus there is a pure spin current. The physics behind the pure spin current can be outlined as follows. In the model (Fig. 1) coulomb interaction in the quantum dot is strong enough to prohibit the double occupation, no more electrons can enter the quantum dot before the spin-down electron exits. As a result, the number of electrons exiting from the quantum dot is equal to that of electrons entering the quantum dot;namely,thechargecurrentsexactlycancelouteachotherimplyingzerochargecurrent. Conclusions. – The pure spin current obtained in our set-up is shown to give rise to completely positive shot noise cross-correlations in the sequential transport regime. This is seen to be sustained when dephasing is included. In fact maximum dephasing gives rise to completely correlated non-local spin correlations. An analysis of the case when there is no dephasing is presented in Table 1. For maximal dephasing the coherent and sequential transport regimes merge. ThemainresultofourworkisdepictedinFig. 3,thisisperhapsthefirstworkwhereitis shownexplicitly that the shot noise cross-correlationsturn completely positive for maximal dephasing (blue lines in Fig. 3) in either regime. What are the reasons for the completely positive shot noise cross-correlations? One can see from the formula for the spin shot noise cross-correlations it is a difference between same spin and opposite spin correlations. For maximaldephasing one notices that the magnitude of the same spincorrelations,which are negative, is always less than that of the opposite spin case. In Fig. 2, the third moment auto and cross-correlations are also plotted. The odd moment doesn’t change markedly as compared to the second moment in case of sequential transport. Dephasing however has a dramatic impact. The third moment auto-correlationsare completely negative as expected since the possibility of detecting three electrons is prohibited via Paulli exclusion while p-7 Colin Benjamin1 0.8 4.0 C2s C2s/Is 100 10 0 0.4 2.0 2.5 0 10 100 2.00.0 10.0 20.0 0.00.0 0.5 R1.0/Γ 1.5 2.00.00.0 0.5 R1.0/ΓRrf/Γ 1.5 2.0 Crofherent 8.0 rf 0.6 Cs3 Incoherent Cs3/ Is 2.0 0.4 1.00.0 10.0 20.0 4.0 100 10 0.2 0 10 100 0 0.00.0 0.5 R1.0/Γ 1.5 2.00.00.0 0.5 R1.0/Γ 1.5 2.0 rf rf Fig. 4: Spin cumulants: C2s(top left panel) and C3s (bottom left panel) alongwith their respective Normalized counterparts, Spin Fano factor (top right panel) and Normalized skewness (bottom right panel) plotted as function of strength of rotating field for various dephasing rates Γ /Γ = φ 0(black),10(red),100(blue). Solid lines are for Coherent regime and dashed lines are for sequential transport regime, in theinsets the large field limit is shown. Parameters are Γ=1,δESR=0. third moment cross-correlations turn negative for maximal dephasing. We have compared and contrasted the absolutely incoherent and absolutely coherent regimes. An effective parameter which shows the transition from completely coherent to completely incoherent canbeintroducedinthecoherentdensitymatrix,Eq. 16,tomodelthis. Phenomenologically introducing a spin relaxation time into the coherent density matrix does indeed show the transition between completely coherent and incoherent regimes attesting our results. In this article for the first time the dramatic nature of Non-local spin shot noise correlations as function of dephasing is shown. Future endeavors on effects of incoherence on different geometries especially including superconductors are contemplated. Table1: Comparingfirstthreemomentsincoherentandincoherent(orsequential)transportregimes for zero dephasing Moment Coherent Incoherent 1st Pure spin current Purespin current 2nd Non-local shot-noise Non-local shot-noise correlations positive for certain correlations range of parameters always positive. 3rd Third moment finite Third moment finite No qualitative change REFERENCES [1] C. Benjamin and J. K.Pachos, Phys. RevB 78, 235403 (2008). [2] A.Cottet, W. Belzig, C. Bruder, Phys.Rev.Lett. 92, 206801 (2004). [3] J. C. Egues, et. al., Phys. Rev B 72, 235326 (2005). p-8 Can dephasing generate non-local spin correlations? [4] A.D. Lorenzo and Y.V. Nazarov, Phys. Rev.Lett.93, 046601 (2004). [5] T. L. Schmidt,A. Komnik and A. O. Gogolin, Phys. Rev.B 76, 241307(R) (2007). [6] S.Lindebaum, D. Urbanand J. König, Phys.Rev.B 79, 245303 (2009). [7] I. Djuric, M. Zivkovic, C. P. Search, G. Recine, arxiv: 0807.2468; I. Djuric and C. P. Search, cond-mat/0611288. [8] O. Sauret,T. Martin and D. Feinberg, Phys.Rev. B 72, 024544 (2005). [9] Y.He, D. Hou and R. Han,J. Appl.Phys.101, 023710 (2007). [10] B. Wang, J. Wangand H. Guo, Phys. Rev.B 67, 092408 (2003). [11] C. W. J. Beenakker,Journal club for Condensed matterphysics October 2003. [12] Bing Dong, H. L. Cui and X. L. Lei, Phys. Rev.Lett.94, 066601 (2005). [13] P. Zhang, Q-K Xue and X. C. Xie, Phys. Rev. Lett. 91, 196602 (2003), Y. Kondo, et.al., quant-ph/0503067. [14] D.A. Bagrets and Y. V.Nazarov, Phys.Rev.B 67, 085316 (2003). [15] G. Kieβlich, P. Samuelsson, A. Wackerand E. Schöll, Phys.Rev.B 73, 033312 (2006). [16] H.Sprekeler, G. Kieβlich, A.Wackerand E. Schöll, Phys. Rev.B 69, 125328 (2004). [17] J. N.Pedersen, Cand.scient thesis, Orsted Laboratory, Niels Bohr InstitutefAPG, University of Copenhagen, June 2004. [18] S.A. Gurvitz, Phys.Rev.B 56, 15215 (1997). [19] G. Kiesslich, Ph. D thesis, TU Berlin, October 2005. p-9

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