Non-Markovian theory for the waiting time distributions of single electron transfers
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A ug Non-Markovian theory for the waiting time distributions of single electron transfers
Sven Welack and YiJing Yan
Department of Chemistry, Hong Kong University of Science and Technology, Kowloon, Hong Kong
We derive a non-Markovian theory for waiting time distributions of consecutive single electrontransfer events. The presented microscopic Pauli rate equation formalism couples the open electrodesto the many-body system, allowing to take finite bias and temperature into consideration. Numericalresults reveal transient oscillations of distinct system frequencies due to memory in the waiting timedistributions. Memory effects can be approximated by an expansion in non-Markovian corrections.This method is employed to calculate memory landscapes displaying preservation of memory overmultiple consecutive electron transfers.
I. INTRODUCTION
Detection of single electron transfers through quan-tum systems such as quantum dots has become exper-imentally feasible.
Theoretical investigations on theunderlying statistics were mostly obtained in terms ofhigher cumulants, e.g. noise and skewness, by using gen-erating function techniques.
Expan-sion of the higher cumulants in non-Markovian correc-tions has revealed significant memory effects in quan-tum dots when strong Coulomb interaction, phononbath or initial correlations are present.Statistics based on waiting time distribution (WTD)provides additional information on the system. Highercumulants can be derived from the WTD , but notvice versa. Waiting times were recently utilized toanalyze single electron transfers in the Markovianregime, for example, in double quantum dots, singlemolecules, single particle transport and Aharonov-Bohm interferometers. Non-Markovian treatment ofWTD has shown significant features in photon countingstatistics. Non-Markovian effects are induced by a small bias volt-age or a finite bandwidth of the system-electrode cou-pling. While the former can be eliminated easily, thelatter scenario is given by the setup of experiment. Inorder to explore both regimes, a non-Markovian Paulirate equation based on a microscopic description of theelectrode-system coupling using a Lorentzian spectraldensity is derived. It can be utilized for a variety ofsystem, such as single molecule and quantum dots.A formal connection of the WTD with the shot noisespectrum of electron transports through quantum junc-tions has been established in the Markovian regime. The non-Markovian shot noise spectrum provides amore accurate description of the signal and its relationto the physics of the junction than a Markovian version,since it reveals several distinct intrinsic system frequen-cies.In this paper we derive a non-Markovian theory forthe WTD of single particle transfer trajectories basedon the derivation of a non-Markovian microscopic Paulirate equation. It provides a general framework tostudy non-Markovian electron transport through many-body systems and allows us to distinguish between non-
QD1
ΓΓ ∆
QD2EE r i gh t e l ec t r od e l e f t e l ec t r od e QPC
FIG. 1: Illustrated set-up of the DQD in series and notationof the important parameters. The charge state of the DQDis measured by the quantum point contact (QPC) which pro-vides an electron transfer trajectory.
Markovian effects due to intrinsic properties of the sys-tem, finite electrode-system coupling band-width andsmall bias voltage. The WTD is evaluated in time do-main by perturbation theory leading to non-Markoviancorrections. We shall analyze the effect of memory onconsecutive electron transfers through double quantumjunctions (DQD), see Fig. 1, and demonstrate the influ-ence of many-body Coulomb coupling on memory land-scapes displaying the memory that is preserved in the sys-tem for several consecutive electron transfers. The non-Markovian spectrum is obtained from a Laplace trans-formation. The results reveal that the non-Markovianspectrum of the WTD provides similar information con-tent which qualifies it as an alternative method to thenon-Markovian shot noise spectrum.The paper is organized as follows. In section II, wepresent the derivation of the non-Markovian rate equa-tion. The expressions for the non-Markovian WTD areshown in section III. The formalism is applied to theDQD system and the results are given in section IV. Weconclude with a summary and outlook.
II. NON-MARKOVIAN RATE THEORY OFQUANTUM TRANSPORTA. Hamiltonian
Consider a junction consisting of a DQD in series asthe system, two electron reservoirs, and the respectivesystem–reservoir coupling, as shown in Fig. 1. The totalHamiltonian assumes H T = H S + H R + H SR . The systempart describes the DQD which is modeled by H S = X s =1 E s ˆ n s + U ˆ n ˆ n − ∆( c † c + c † c ) . (1)Here, ˆ n s = c † s c s is the electron number operator of quan-tum dot s = 1 or 2 with orbital energy E s , and U specifiesthe Coulomb interaction between two dots. The reser-voirs of left and right ( α = l and r ) electrodes are de-scribed by H R = X α = l,r H R α = X α = l,r X q ǫ αq c † αq c αq . (2)The system–reservoirs coupling responsible for electrontransfer between the system and the electrodes is H SR = X α = l,r X q h T ( l )1 q c † c lq + T ( r )2 q c † c rq + H . c . i . (3)The electron creation (annihilation) operators c † s ( c s )and c † αq ( c αq ) involved in Eqs. (1)–(3) satisfy the anti-commutator relations. In this system, single electrontransfer trajectories can be obtained from the chargestate of the DQD that is constantly measured by a quan-tum point contact (QPC). Such a configuration was em-ployed in experiment operated at a small bias voltage. B. Generalized non-Markovian rate equation
We now turn to the non–Markovian rate equation. Let ρ ( t ) ≡ tr R ρ T ( t ) be the reduced system density opera-tor. The total density operator is assumed to be ini-tially factorisable into a system and a reservoir part, ρ T ( t ) = ρ ( t ) ρ R ( t ), and the system–electrode cou-plings are assumed to be weak. Using the standard ap-proach, one can readily derive a non-Markovian quantummaster equation. For the present study, we adopt T ( α ) sq = T ( α ) s T ( α ) q for simplification. A rotating wave ap-proximation to the cross coupling terms between the sys-tem orbitals is not required here since each electrode iscoupled to one orbital site only. We denote the sys-tem Liouville operator L S · ≡ [ H S , · ] and set ~ = 1. Theresulting quantum master equation in the time-nonlocalform reads ˙ ρ ( t ) = − i L S ( t ) ρ ( t ) − X αs Z t d τ | T ( α ) s | × n C (+) l ( t − τ ) (cid:2) c s , e − iH S ( t − τ ) c † s ρ ( τ ) e iH S ( t − τ ) (cid:3) − C ( − ) l ( t − τ ) (cid:2) c s , e − iH S ( t − τ ) ρ ( τ ) c † s e iH S ( t − τ ) (cid:3) + H . c . o . (4)The reservoir correlation functions, C (+) α ( t ) = X q | T ( α ) q | h c † αq ( t ) c αq (0) i R α , (5) and C ( − ) α ( t ) = X q | T ( α ) q | h c αq (0) c † αq ( t ) i R α , (6)contain the properties of the electrodes. Here, c † αq ( t ) ≡ e iH Rα t c † αq e − iH Rα t and h O i R α ≡ tr R α { Oρ R α } , with ρ R α being the density operator of the bare electrode α undera constant chemical potential µ α . Physically, C (+) α ( t )describes the process of electron transfer from the α –electrode to the system, while C ( − ) α ( t ) describes thereverse process. These two correlation functions arenot independent; they are related via the fluctuation–dissipation theorem.Electron counting experiments are operated either inthe large–bias limit in order to achieve a directional tra-jectory of single transfer events or at small bias in or-der to realize transfer against the direction of the bias. Non-Markovian effects are either due to small bias orfinite band-width. In order to study both regimes, wederive a rate equation by projecting the master equa-tion (4) into the Fock space of system and by consideringonly the population part p m ≡ ρ mm . Some simple alge-bra leads from Eq. (4) to the non-Markovian Pauli rateequation˙ p m ( t ) = X αn Z t d τ (cid:2) Γ ( α ) mn C (+) α ( t − τ ) e − iω mn ( t − τ ) p n ( τ )+ Γ ( α ) nm C ( − ) α ( t − τ ) e − iω nm ( t − τ ) p n ( τ ) − Γ ( α ) nm C (+) α ( t − τ ) e − iω nm ( t − τ ) p m ( τ ) − Γ ( α ) mn C ( − ) α ( t − τ ) e − iω mn ( t − τ ) p m ( τ ) (cid:3) + c . c . ≡ X n Z t d τ K mn ( t − τ ) p n ( τ ) . (7)Here, ω mn ≡ E m − E n is the transition frequency betweentwo Fock states;Γ ( α ) mn = | T ( α ) s | |h m | c † s | n i| , (8)with s = 1 or 2 for α = l or r , respectively, is thestate–dependent non-Markovian system–reservoir cou-pling strength. As inferred from Eq. (8), Γ ( α ) mn = 0 onlyif | m i has one more electron than | n i . We can thereforeidentify the rate kernel elements involved in Eq. (7) withthree physically distinct contributions K ( t ) ≡ X α [ K ( α +) ( t ) + K ( α − ) ( t )] + K ( t ) . (9) K ( α +) ( t ) and K ( α − ) ( t ) realize an electron transfer in andout of the system through the α –electrode, respectively.They summarize the off–diagonal matrix elements of thetransfer rate kernel K ( t ) in Eq. (7), K ( α +) mn ( t ) = Γ ( α ) mn C (+) α ( t − τ ) e − iω mn t + c . c ., (10) K ( α − ) mn ( t ) = Γ ( α ) nm C ( − ) α ( t − τ ) e − iω nm t + c . c . (11) K ( t ) summarizes the diagonal matrix elements of K ( t )and leaves the number of electrons in system unchanged.These diagonal elements satisfy( K ) nn = − X α,m (cid:2) K ( α +) mn ( t ) + K ( α − ) mn ( t ) (cid:3) . (12)For the Lorentzian spectral density model [Eq. (A.2)],where the reservoir spectral density assumes the form J α ( ω ) = γ α / [( ω − Ω α ) + γ α ], we obtain for the off–diagonal rate kernel elements the following expressions, K ( α ± ) mn ( t ) = 2Γ ( α ) mn n e − γ α t [ a ± α cos(Ω αmn t ) − b ± α sin(Ω αmn t )]+ ∞ X k =1 e − ̟ k t [ c ± αk cos( µ αmn t ) − d ± αk sin( µ αmn t )] o . (13)Here, ̟ k = (2 k − π/β is the fermionic Matsubara fre-quency, while Ω αmn ≡ Ω α − ω mn and µ αmn ≡ µ α − ω mn .The coefficients a ± α , b ± α , c ± α , and d ± α are all real, givenexplicitly in Appendix by Eq. (A.10). The first term inthe curly brackets of Eq. (13) reflects the spectral prop-erties of the electrode-system coupling, while the secondterm arises from the decomposition into Matsubara fre-quencies which induces memory effects due to small biasvoltages. From the expressions one can infer that large γ α , wide bands, and large ̟ k , high bias, cause a fast de-cay of the transfer rates in time. The decay is responsiblefor the memory loss in the system.The non-Markovian Pauli rate equation (7), in termsof the population vector p ( t ) = { p m ( t ) } and the involvedtransfer matrices, is˙ p ( t ) = Z tt d τ K ( t − τ ) p ( τ ) . (14)It reads in Laplace frequency domain s ˜ p ( s ) − p = ˜ K ( s ) ˜ p ( s ) . (15)The corresponding electron transfer rates are˜ K ( α ± ) mn ( s ) = 2Γ ( α ) mn n a ± α ( s + γ α ) − b ± α Ω αmn ( s + γ α ) + (Ω αmn ) + ∞ X k =1 c ± αk ( s + ̟ k ) − d ± αk µ αmn ( s + ̟ k ) + ( µ αmn ) o . (16)The derived non-Markovian rate equation formalism isbased on a microscopic description of the electrode-system coupling, and is valid for arbitrary bias and tem-perature. Compared to the quantum master equationin the same regime , the exclusion of the coherencemakes it numerically feasible to calculate multilevel sys-tems such as large molecules. This allows to includenon–Markovian effects in large many–body systems, e.g.quantum-chemistry calculations, since the properties ofthe molecular-junction enter only through the couplings Γ ( α ) nm and the fitting parameters of the Lorentzian spec-trum.To rate equation (14), the Born-Markov approximationcan be applied by separating the integration variablesand extending the upper limit to infinity in Eq. (14). Theresulting integration over time, W ( α ± ) mn = Z ∞ d t K ( α ± ) mn ( t ) = ˜ K ( α ± ) mn ( s ) | s =0 , (17)gives the Markovian electron transfer rates. The secondidentity is via the Laplace domain rate equation (15), bywhich the Born-Markov approximation amounts to thezero frequency contribution. III. NON-MARKOVIAN WAITING TIMEDISTRIBUTIONA. Statistics analysis
We consider two consecutive electron transfers con-tained in a time series as illustrated in Fig. 1. An electronentered the system from the left electrode at an earliertime t is detected at time t leaving the system throughthe right electrode. No other electron transfers are de-tected in between. The joint-probability for the consec-utive electron transfer events is P ( t ) = hh W ( r − ) G ( t, t ) W ( l +) p ( t ) ii . (18)Here, hh· · ·ii denotes the sum over the final system states.We assume that the transfer events are instantaneouscompared to the time-scale of the system propagationin between as shown in Fig. 1. Therefore we have usedthe Markovian forms of rate matrices, for the ascribedtwo consecutive events. This assumption is reasonablein accordance with electron counting experiments, wheretypical waiting times are long compared to the fast trans-fer events. The memory of the system is containedin G ( t, t ), the non-Markovian propagator of the systemfrom t to t in absence of transfers. It is therefore as-sociated with the diagonal rate matrix K of Eq. (12),satisfying ddt G ( t, t ) = Z tt d τ ′ K ( t − τ ′ ) G ( τ ′ , t ) . (19)For the given two–event case, the joint–probability isequivalent to a waiting time distribution. Now consider the event of an electron transferredinto the system and the subsequent waiting time be-fore any other transfer takes place, also referred to assurvival probability. In the present notation it is givenby hh G ( t, t ) W ( α ± ) p ( t ) ii . While the joint probability issubject to the nature of the second transfer, the spe-cific form of the second event is irrelevant to the survivalprobability. Consequently, we introduce the survival timeoperator Z ( α ± ) ( t, t ) = G ( t, t ) W ( α ± ) . (20)If memory is absent, the survival probability is indifferentfrom the previous waiting times. To study the memory ofa previous survival time that carries on into the follow-ing survival time, we introduce two–time joint survivalprobabilities of the form Q ( τ , τ ) = hh Z ( r − ) ( τ , τ ) Z ( l +) ( τ , t ) p ( t ) ii . (21) B. Non-Markovian corrections
The formal solution to the propagator in Laplace do-main is given by ˜ G ( s ) = 1 s − ˜ K ( s ) . (22)The complex Laplace frequency s = γ + iω is associ-ated with the system residing in its state. The bilateralLaplace transformation reduces to a Fourier transforma-tion by setting γ = 0. Since ˜ K ( s ) is strictly diagonal inthe many-body eigenspace of the system, the matrix in-version required in Eq. (22) can be efficiently carried outfor large systems.The technique of expanding the propagation into non-Markovian corrections has been applied to electron trans-port recently. Here we apply it to the WTD. Letus first express Eq. (22) by its series˜ G ( s ) = ∞ X n =0 [ ˜ K ( s )] n s n +1 . (23)Assuming the derivative ∂ ms [ ˜ K ( s )] exists for all m ,the kernel can then be expanded into a Taylor series[ ˜ K ( s )] n = P ∞ m =0 ∂ ms [ ˜ K ( s )] n | s =0 s m m ! . Thus,˜ G ( s ) = ∞ X n =0 ∞ X m =0 ∂ ms [ ˜ K ( s )] n | s =0 m ! s n +1 − m . (24)Now we apply the inverse Laplace transform x ( t ) = πi R γ + i ∞ γ − i ∞ d s e st ˜ x ( s ) to switch back into time domain.One can simplify the poles by using m = n which ne-glects the transient terms. We obtain G ( t ) = ∞ X n =0 n ! (cid:20) ∂ n ∂s n (cid:16) [ ˜ K ( s )] n e ˜ K ( s ) t (cid:17)(cid:21) s =0 ≡ ∞ X n =0 G ( n ) ( t ) , (25)with G ( n ) ( t ) denoting the individual term involved,where G (0) ( t ) = e ˜ K ( s ) t | s =0 describes the Markovian dy-namics. The first identity of expression (25) is asymp-totically exact since the dynamics is reduced to the poles m = n . The WTD can also be expressed in terms of P ( t ) = P n P ( n ) ( t ), with P (0) ( t ) denoting the Markoviancontribution; so can the survival probabilities. F ( ω ) ∆=1∆=5∆=8 V=1V=2V=3 ω F ( ω ) U=1U=2U=4 0 2 4 6 8 10 ω ∆Ε=0.1∆Ε=1∆Ε=2 a) (x1.5) b)c) d)V=1 ∆=5 U=0
FIG. 2: The relative non-Markovian spectrum F ( ω ) of theWTD. The parameters used for the four panels are given asfollows. Upper left panel (a): U = 0, ∆ E = 0, V = 1 . . , . , .
0. Upper right panel (b): U = 0, ∆ E = 0, ,∆ = 5 . V = 1 . , . , .
0. Bottom left panel (c): ∆ E = 0, V = 1 .
0, ∆ = 5 . U = 1 . , . , .
0. Bottom right panel (d): V = 1 .
0, ∆ = 5 . U = 0 .
0, ∆ E = 0 . , . , . IV. DEMONSTRATION AND DISCUSSION
We employ a non-Markovian rate equation to calcu-late the two–electron system as illustrated in Fig. 1. Thissystem resembles the counting experiment conducted inRef. 4. Here, the DQD provides a total number of foureigenstates: the unoccupied ( | i ) two single–occupied( | i and | i ), and one double-occupied ( | i ), with the en-ergies of ǫ = 0, ǫ / = ( E + E ) ∓ q ( E − E ) + ∆ ,and ǫ = E + E + U , respectively. The equilib-rium of the chemical potential of the electrodes is set to µ eq = ( E + E ) /
2. For numerical demonstrations, we usethe numbers in accordance with recent electron countingexperiments of electron transfers through quantum dotsystems at small temperatures. A coupling strength ofΓ = 10 Hz serves as the unit for all values. This is equiv-alent to an energy unit of [ E ] = 10 h = 6 . × − J,and a time unit of [ t ] = 0 . We also use a low temperature of T = 2 × [ E ] = 10 mK. If mentioned, we set a smallenergy detuning of ∆ E = E − E in order to deduce spe-cific frequencies of the systems. The bandwidth γ is setsufficiently large in order to neglect the finite bandwidtheffects; thus the non–Markovian effect is studied in thewide band region. In addition, the Lorentzian spectraldensities are aligned to the orbitals of the system. A. Transients and Fourier spectrum of WTD
Figure 2 shows the relative non-Markovian spectrumof the WTD represented by F ( ω ) = 1Γ | P ( ω ) − P (0) ( ω ) | P (0) ( ω ) . (26)It is noteworthy that F ( ω ) is independent of the systemreservoir coupling strength parameter Γ. The WDT spec-trum reveals several frequencies that are present in thetransient oscillations. These depend only on the internaltransfer rate ∆, Coulomb coupling U , and bias voltage V . The specific values of the parameters are given in thecaption of the figure. Figure 2(a) shows the main charac-teristics of F ( ω ), consisting of two overlapping sub-peakscentered around the value of ∆. Changing the value of∆ leads to the shift of both sub-peaks equally by ∆.In Fig. 2(b), ∆ is kept constant and the bias voltage isvaried. We find that the splitting of the two sub-peaksis determined by the applied voltage. Labeling the twopeaks with ± , respectively, we can deduce the followingrelation for the corresponding characteristic frequencies. ω ± = | ∆ ± V / | . In the presence of Coulomb interac-tion, we observe an additional double–peaks feature at | U − ∆ ± V / | , as demonstrated in Fig. 2(c). This is sim-ilar to a non-Markovian shot noise spectrum, wherea finite Coulomb interaction U induces also additionalpeaks due to the energy gap between the two-particle oc-cupation state and lower states. On the other hand, theorbital detuning does not induce additional peaks in thedouble quantum dot in series as shown in Fig. 2(d).Oscillations of Rabi frequency, which were observed inparallel DQD systems, are however absent in thepresent series DQD system. In the parallel cases, thetransport proceeds via two channels, and the Rabi oscil-lations in the WDT can be observed as the consequence ofquantum mechanical interferences. It is also notedthat the information contained in the spectrum of thenon-Markovian WDT is mostly equivalent to a measure-ment of the non-Markovian shot noise spectrum. For thispurpose, the WTD can be considered as an alternativeapproach to the shot noise spectrum measurement.
B. Memory landscape of consecutive waiting times
The expansion in non-Markovian corrections, Eq. (25),can be readily employed to calculate the two propagatorsinvolved in the two-times joint probabilities defined byEq. (21). Denote Q ( n ) ( τ , τ ) = n X k =0 n X j =0 hh G ( k ) ( τ − τ ) W ( r − ) × G ( j ) ( τ − t ) W ( l +) p ( t ) ii . (27)A memory landscape of the system can be calculatedby the difference between non-Markovian and Markovian two-times joint probabilities L ( n ) ( τ , τ ) = Q ( n ) ( τ , τ ) − Q (0) ( τ , τ ) Q (0) ( τ , τ ) . (28)The order n of the perturbative expansion in non-Markovian corrections has to be chosen in accordanceto the parameters in order to assure satisfactory conver-gence. We find that the summation to the fourth non-Markovian contribution already converges sufficiently forthe given parameters. As the memory in Eq. (28) de-cays, the relative non-Markovian landscape L ( n ) ( τ , τ )converges to zero. Figure 3 shows the memory landscapeof two survival times related to two consecutive electrontransfers through the left electrode. It visualizes howmemory of the waiting time τ after the first transfer iscarried over into the waiting time τ following the secondtransfer.We find that the non-Markovian effects are small forthe given parameters in case the DQD is coupled sym-metrically to the electrodes. This is due to the relativelyweak coupling of the DQD to the electrodes which is re-quired in present counting experiment in order to resolvesingle electron transfers on the measurable timescales.It is observed that a stronger coupling to only one elec-trode induces significantly larger non-Markovian effectsas shown in the left panels of Fig. 3.In general, fast electron transfers are necessary in orderto observe significant non-Markovian effects. The devi-ations for τ , τ approaching zero from the Markovianvalue are due to the truncation of the transients in thederivation of the expansion. The expansion follows thegeneral trend of a numerically exact solution. Both solu-tions overlap after the transients have decayed. Howeverthis causes relatively large inaccuracies for τ and τ closeto zero.There is an interesting dependency of the non-Markovian effects in the memory landscape on theCoulomb repulsion U . By comparing the upper panels ofFig. 3 where Coulomb repulsion is absent, with the bot-tom one, where a large U induces a Coulomb blockaderegime, we observe that memory decays faster with τ inthe Coulomb blockade regime. This can be explained asfollows. In the second regime, only a single electron canoccupy the DQD and the double occupancy state doesnot provide memory for the second survival time lead-ing to an overall smaller non-Markovian contribution. Inthis case, only one possible trajectory in the left to rightdirection is possible. An electron enters the unoccupiedDQD at time τ = 0 and leaves it at time τ = 0.The memory is preserved during τ by the single elec-tron inside the DQD. However, after the electron has lefta junction, the memory of its trajectory is lost rapidlysince no other electron can serve as a messenger insidethe DQD thus leading to comparatively short survivaltimes τ where memory is present. In the regime whereCoulomb repulsion is neglectible, a second electron canoccupy the junction along the described trajectory, which Τ @ ms D Τ @ m s D L H L H Τ , Τ L % Τ @ ms D Τ @ m s D L H L H Τ , Τ L % Τ @ ms D Τ @ m s D L H L H Τ , Τ L % Τ @ ms D Τ @ m s D L H L H Τ , Τ L % FIG. 3: L (4) l,r memory landscape of consecutive survival times.The bandwidth is large and a finite bias of V = 0 . k b T isapplied. The left coupling strength is Γ ( l ) = 10 Hz . Theupper panels are calculated in absence of Coulomb coupling, U = 0, bottom panels display the Coulomb blockade regime, U = ∞ . Left panels are calculated for a symmetric system,Γ ( l ) = Γ ( r ) . In the right panels, a stronger coupling strengthis applied to the right electrodes Γ ( r ) = 10 Hz . is represented in the model by the presence of an occu-pied double occupancy state. The presence of the secondelectron preserves the memory during τ after the otherelectron has left the junction. V. CONCLUSION
We find that non-Markovian effects are small in theregimes of recent single electron counting experiments.The sampling rate of current experiments is slow, a re-quirement which is imposed by the detection process ofsingle electron transfers with currently available technol-ogy. This verifies the reason that the Markovian approx-imation of previous studies considering FCS or WTD isreasonable for the previously investigated systems.Non-Markovian effects in the electron transfer statis-tics have to be taken into consideration for strongerelectrode-DQD couplings, which then also requires fastersampling rates or a strongly asymmetric system. For ex-ample they affect the decay rates of the WTD which aredirectly related to the electronic structure of the systemin junction. Non-Markovian effects also induce severaloscillations with characteristic system frequencies.Note that the form of Pauli rate equation remains validitself in the strong coupling limit. In the present paperwe employ a perturbative approach to the rate equation and observe that the non-Markovian effects increase withthe coupling strength. This observation is expected to re-main true based on general Pauli rate equation dynamics.In other words, the non-Markovian effects are mainly vis-ible for stronger couplings.The employed microscopic non-Markovian rate equa-tion provides a general framework to study similar sys-tems and allows us to distinguish between non-Markovianeffects due to intrinsic properties of the system, finiteelectrode-system coupling band-width and small biasvoltages. It can be combined with quantum chemistrycalculations that can calculate the employed parametersfor molecules and their binding to the electronic bandsof the metal electrodes. The approaches derived for thenon-Markovian WTD are general and can be applied toa variety of processes in physics, chemistry and biologythat are described by rate equations.
Acknowledgments
Support from the RGC (604007 & 604508) of HongKong is acknowledged.
APPENDIX: RATE COEFFICIENTS
Introducing the coupling reservoir spectrum density J α ( ω ) = π P q | T ( α ) q | δ ( ω − ǫ αq ) and applying Fermistatistics to the reservoir modes, the correlation func-tions (5) and (6) can be written as C ( ± ) α ( t ) = Z ∞−∞ d ωπ J α ( ω ) f ( ± ) α ( ω ) e ∓ iωt . (A.1)Here, f (+) α ( ω ) = 1 − f ( − ) α ( ω ) = [1 + e β ( ω − µ α ) ] − is theFermi distribution function, with β = 1 /k b T being theinverse temperature and µ α the chemical potential of tothe α –electrode. Adopting a Lorentzian form of spectraldensity, J α ( ω ) = γ α / [( ω − Ω α ) + γ α ] , (A.2)the finite spectral width parameter γ α is used to char-acterize the non-Markovian nature of system–reservoircoupling. With the complex roots of the Fermi functionand of the Lorentzian spectral density, the integrals inEq. (A.1) can be determined by the residues of the Ker-nel. The resulting infinite series are C (+) α ( t ) = γ α f (+) α (Λ α ) e i Λ t − iβ m X k =1 J α ( υ k ) e iυ k t (A.3)and C ( − ) α ( t ) = γ α f ( − ) α ( − Λ ∗ α ) e − i Λ ∗ t − iβ m X k =1 J α ( υ ∗ k ) e − iυ ∗ k t , (A.4)with the abbreviation Λ α = Ω α + iγ α and υ k = µ α + i̟ k ,where ̟ k ≡ (2 k − π/β are the Fermion Matsubarafrequencies. In order to completely separate real andimaginary parts of the correlation functions (5) and (6),we first separate its individual components. For the twocomplex Fermi functions we calculate f ( ± ) α ( ± Ω α + iγ α ) = 1 + e ± β (Ω α − µ α ) e − iβγ α X ± α , (A.5)where X ± α ≡ βγ α ) e ± β ( µ α − Ω α ) + e ± β ( µ α − Ω α ) . (A.6)The complex spectral densities can be separated into J α ( µ α ± i̟ k )= [ ̟ k + ( µ α − Ω α ) + γ α ] ∓ i̟ k ( µ α − Ω α ) Y α , (A.7)where Y α ≡ γ α [ ̟ k +( µ α − Ω α ) + γ α ] +4 ̟ k ( µ α − Ω α ) . (A.8)Based on the separation in real and imaginary contribu-tions, we can write the correlation functions as C ( ± ) α ( t ) = ( a ± α + ib ± α ) e ( ± i Ω α − γ α ) t + m X k =1 ( c ± αk + id ± αk ) e ( ± iµ α − ̟ k ) t . (A.9) The coefficients are all real: a ± α = γ α X ± α [1 + e ± β (Ω α − µ α ) cos( βγ α )] , (A.10a) b ± α = γ α X ± α [ e ± β (Ω α − µ α ) sin( βγ α )] , (A.10b) c ± αk = ∓ µ α − Ω α ) ̟ k βY α , (A.10c) d ± αk = ∓ ̟ k + ( µ α − Ω α ) + γ α ] βY α . (A.10d)Rigorously, the sum over the Matsubara values would beinfinite; i.e., k max = m → ∞ in Eqs. (A.3) and (A.4),but it can be truncated for practical purposes at a finitevalue that depends on the temperature of the system T and the spectral width. W. Lu, Z. Ji, L. Pfeiffer, K. W. West, and A. J. Rimberg,Nature , 422 (2003). T. Fujisawa, T. Hayashi, Y. Hirayama, H. D. Cheong, andY. H. Jeong, Appl. Phys. Lett. , 2343 (2004). S. Gustavsson, R. Leturcq, B. Simovic, R. Schleser, T. Ihn,P. Studerus, K. Ensslin, D. C. Driscoll, and A. C. Gossard,Phys. Rev. Lett. , 076605 (2006). T. Fujisawa, T. Hayashi, R. Tomita, and Y. Hirayama,Science , 1634 (2006). L. S. Levitov, H. W. Lee, and G. B. Lesovik, J. Math.Phys. , 4845 (1996). L. S. Levitov and M. Reznikov, Phys. Rev. B , 115305(2004). J. Wabnig, D. V. Khomitsky, J. Rammer, and A. L. She-lankov, Phys. Rev. B , 165347 (2005). J. Rammer, A. L. Shelankov, and J. Wabnig, Phys. Rev.B , 115327 (2004). A. L. Shelankov and J. Rammer, Europhys. Lett. , 485(2003). C. Flindt, T. Novotny, and A.-P. Jauho, Europhys. Lett. , 475 (2005). Y. Utsumi, D. S. Golubev, and G. Schoen, Phys. Rev. Lett. , 086803 (2006). J. N. Pedersen and A. Wacker, Phys. Rev. B , 195330(2005). A. Bachtold, P. Hadley, T. Nakanishi, and C. Dekker, Sci-ence , 1317 (2001). G. Kießlich, P. Samuelsson, A. Wacker, and E. Sch¨oll,Phys. Rev. B , 033312 (2006). H.-A. Engel and D. Loss, Phys. Rev. Lett. , 136602(2004). A. Braggio, J. K¨onig, and R. Fazio, Phys. Rev. Lett. ,026805 (2006). A. Braggio, C. Flindt, and T. Novotn´y, Physica E , 1745(2008). C. Flindt, T. Novotny, A. Braggio, M. Sassetti, and A.-P.Jauho, Phys. Rev. Lett. , 150601 (2008). T. Brandes, Ann. Phys. , 477 (2008). S. Welack, M. Esposito, U. Harbola, and S. Mukamel,Phys. Rev. B , 195315 (2008). J. Koch, M. Raikh, and F. v. Oppen, Phys. Rev. Lett. ,056801 (2005). S. Welack, J. B. Maddox, M. Esposito, U. Harbola, andS. Mukamel, Nano Lett. , 1137 (2008). S. Welack, S. Mukamel, and Y. J. Yan, EPL , 57008(2009). H. Zaidi, Phys. Rev. A , 061802 (2006). J. S. Jin, X. Q. Li, and Y. J. Yan, arXiv:0806.4759 (2008). U. Weiss,
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