Abstract
Full counting statistics of electron transport is a powerful diagnostic tool for probing the nature of quantum transport beyond what is obtainable from the average current or conductance measurement alone. In particular, the nonMarkovian dynamics of quantum dot molecule plays an important role in the nonequilibrium electron tunneling processes. It is thus necessary to understand the nonMarkovian full counting statistics in a quantum dot molecule. Here we study the nonMarkovian full counting statistics in two typical quantum dot molecules, namely, serially coupled and sidecoupled double quantum dots with high quantum coherence in a certain parameter regime. We demonstrate that the nonMarkovian effect manifests itself through the quantum coherence of the quantum dot molecule system and has a significant impact on the full counting statistics in the high quantumcoherent quantum dot molecule system, which depends on the coupling of the quantum dot molecule system with the source and drain electrodes. The results indicated that the influence of the nonMarkovian effect on the full counting statistics of electron transport, which should be considered in a high quantumcoherent quantum dot molecule system, can provide a better understanding of electron transport through quantum dot molecules.
Introduction
Full counting statistics^{1} (FCS) of electron transport through mesoscopic system has attracted considerable attention both experimentally and theoretically because it can provide a deeper insight into the nature of electron transport mechanisms, which cannot be obtained from the average current^{2,3,4,5,6,7,8,9,10}. For instance, the shot noise measurements can be used to probe the dynamical in an open double quantum dots (QDs)^{11}, the coherent coupling between serially coupled QDs^{12}, the evolution of the Kondo effect in a QD^{13} and the conduction channels of quantum conductors^{14}. In particular, shot noise characteristics can provide information about the feature of the pseudospin Kondo effect in a laterally coupled double QDs^{15}, the spin accumulations in a electron reservoir^{16} and the charge fractionalization in the ν = 2 quantum Hall edge^{17}. In addition, the degree of entanglement of two electrons in the double QDs^{18}, the dephasing rate in a closed QD^{19}, the internal level structure of single molecule magnet^{20,21} can be characterized by the superPoissonian shot noise.
On the other hand, the quantum coherence in coupled QD system, which is characterized by the offdiagonal elements of the reduced density matrix of the QD system within the framework of the density matrix theory^{22}, plays an important role in the electron tunneling processes and has a significant influence on electron transport^{23,24,25,26,27,28,29,30,31,32,33}. In particular, theoretical studies have demonstrated that the highorder cumulants, e.g., the shot noise, the skewness, are more sensitive to the quantum coherence than the average current in the different types of QD systems^{12,34,35,36,37,38} and the quantum coherence information in a sidecoupled double QD system can be extracted from the highorder current cumulants^{35}. In fact, the nonMarkovian dynamics of the QD system also plays an important role in the nonequilibrium electron tunneling processes. However, the above studies on current noise or FCS were mainly based on the different types of Markovian master equations. Although the influence of nonMarkovian effect on the longtime limit of the FCS in the QD systems has received some attention^{33,39,40,41,42,43,44,45,46}, how the nonMarkovian effect affects the FCS is still an open issue, especially the influence of the interplay between the quantum coherence and nonMarkovian effect on the longtime limit of the FCS has not yet been revealed.
The aim of this report is thus to derive a nonMarkovian FCS formalism based on the exact timeconvolutionless (TCL) master equation and study the influences of the quantum coherence and nonMarkovian effect on the FCS in QD molecule systems. It is demonstrated that the nonMarkovian effect manifests itself through the quantum coherence of the considered QD molecule system and has a significant impact on the FCS in the high quantumcoherent QD molecule system, which depends on the coupling of the considered QD molecule system with the incident and outgoing electrodes. Consequently, it is necessary to consider the influence of the nonMarkovian effect on the full counting statistics of electron transport in a high quantumcoherent singlemolecule system.
Results
We now study the influences of the quantum coherence and nonMarkovian effect on the FCS of electronic transport through the QD molecule system. In order to facilitate discussions effectively, we consider three typical QD systems, namely, single QD without quantum coherence, serially coupled double QDs and sidecoupled double QDs with high quantum coherence in a certain parameter regime (see Fig. 1). In addition, we assume the bias voltage (μ_{L} = −μ_{R} = V_{b}/2) is symmetrically entirely dropped at the QDelectrode tunnel junctions, which implies that the levels of the QDs are independent of the applied bias voltage even if the couplings are not symmetric and choose meV as the unit of energy which corresponds to a typical experimental situation^{47}.
Single quantum dot without quantum coherence
In this subsection, we consider a single QD weakly coupled to two ferromagnetic electrodes. The Hamiltonian of the considered system is described by the H_{total} = H_{dot} + H_{leads} + H_{T}. The QD Hamiltonian H_{dot} is given by
where creates (annihilates) an electron with spin σ and onsite energy ε_{σ} (which can be tuned by a gate voltage V_{g}) in this QD system. U is the intradot Coulomb interaction between two electrons in the QD system.
The relaxation in the two ferromagnetic electrodes is assumed to be sufficiently fast, so that their electron distributions can be described by equilibrium Fermi functions. The two electrodes are thus modeled as noninteracting Fermi gases and the corresponding Hamiltonians can be expressed as
where creates (annihilates) an electron with energy ε_{α}_{k}, spin s and momentum k in α (α = L, R) electrode and s = + (−) denotes the majority (minority) spin states with the density of states g_{α}_{,s}. The polarization vectors p_{L} (left lead) and p_{R} (right lead) are parallel to each other and their magnitudes are characterized by p_{α} = p_{α} = (g_{α}_{,+} − g_{α}_{,−})/(g_{α}_{,+} + g_{α}_{,−}). The tunneling between the QD and the electrodes is described by
where spinup ↑ and spindown ↓ are defined to be the majority spin and minority spin of the ferromagnet, respectively.
The QDelectrode coupling is assumed to be sufficiently weak, thus, the sequential tunneling is dominant and can be well described by the quantum master equation of reduced density matrix spanned by the eigenstates of the QD. The particlenumberresolved TCL quantum master equation for the reduced density matrix of the considered single QD is given by
For more details, see Methods section. Here, the complete basis {0, 0〉, ↑, 0〉, ↓, 0〉, ↑, ↓〉} is chosen to describe the electronic states of this single QD system and the single QD system parameters are chosen as , U = 5, p = 0.9 and k_{B}T = 0.04.
Figure 2 shows the first four current cumulants as a function of the bias voltage for different ratios Γ_{L}/Γ_{R} describing the leftright asymmetry of the QDelectrode coupling. We found that the nonMarkovian effect has no influence on the current noise behaviors of the single QD considered here, see Fig. 2. Scrutinizing Eq. (4), it is found that for the nonMarkovian case the elements of the reduced density matrix are equivalent to that for the Markovian case because there are not the offdiagonal elements of the reduced density matrix. Thus, the equations of motion of the four elements of the reduced density matrix can be expressed as
Here, f_{α}_{,+} is the Fermi function of the electrode α and f_{α}_{,−} = 1−f_{α}_{,+}. The detailed procedure for calculation of the equation of motion of a reduced density matrix, see Methods section. Within the framework of the density matrix theory, the offdiagonal elements of the reduced density matrix characterize the quantum coherence of the considered QD system. Thus, the influence of the nonMarkovian effect on the FCS may be associated with the quantum coherence of the considered QD system. In order to confirm this conclusion, we take serially coupled and sidecoupled double QDs for illustration in the following two subsection.
Serially coupled double quantum dots with high quantum coherence
We now consider two serially coupled double QDs weakly connected to two metallic electrodes, see Fig. 1(a). For the sake of simplicity, the spin degree of freedom has not been considered. The doubleQD is described by a spinless Hamiltonian
where creates (annihilates) an electron with energy ε_{i} (which can be tuned by a gate voltage V_{g}) in ith QD. U is the interdot Coulomb repulsion between two electrons in the double QD system, where we consider the intradot Coulomb interaction U → ∞, so that the doubleelectron occupation in the same QD is prohibited. The last term of H_{dot} describes the hopping coupling between the two dots with J being the hopping parameter. The two metallic electrodes are modeled as noninteracting Fermi gases and the corresponding Hamiltonians are given by
where creates (annihilates) an electron with energy ε_{α}_{k} and momentum k in α (α = L, R) electrode. The tunneling between the double QDs and the two electrodes is described by
For the case of the weak QDelectrode coupling, the particlenumberresolved TCL quantum master equation for the reduced density matrix of the considered serially doubleQD system reads
Here, we can diagonalize the serially coupled double QDs Hamiltonian H_{dot,2} in the basis represented by the electron occupation numbers in the QD1 and QD2 denoted respectively by N_{L} and N_{R}, namely, {0, 0〉, 1, 0〉, 0, 1〉, 1, 1〉} and obtain the corresponding four eigenstates of the considered serially coupled double QDs system^{48}
with
and
Here, we focus on the regime , where the hopping coupling between the two QDs strongly modifies the internal dynamics and the offdiagonal elements of the reduced density matrix play an essential role in the electron tunneling processes^{23,49,50,51}. In the following numerical calculations, thus, the parameters of the serially coupled double QDs system are chosen as , J = 0.001, U = 4 and k_{B}T = 0.05.
When the coupling of the QD2 with the right (drain) electrode is stronger than that of the QD1 with the left (source) electrode, namely, Γ_{L}/Γ_{R} < 1, we plot the first four current cumulants as a function of the bias voltage for different values of the QD2electrode coupling Γ_{R} at Γ_{L}/Γ_{R} = 0.1 in Figs. 3(a)–3(d). We found that the nonMarkovian effect has a very weak influence on the FCS. Interestingly, the highorder current cumulants the skewness and the kurtosis can still show the tiny differences, see Figs. 3(c) and 3(d). Whereas for the Γ_{L}/Γ_{R} ≥ 1 case, the nonMarkovian effect has a significant impact on the FCS, see Fig. 4. Especially, for a relatively large value of the ratio Γ_{L}/Γ_{R} = 10 and the coupling of the QD1 with the left electrode being stronger than the hoping coupling, namely, Γ_{L}/J > 1, the nonMarkovian effect can induce a strong negative differential conductance (NDC) and superPoissonian noise, see Figs. 4(e) and 4(f). In addition, in the case of Γ_{L}/Γ_{R} ≥ 1 and Γ_{L}/J > 1, the transitions of the skewness and the kurtosis from positive (negative) to negative (positive) values are observed, see the dotted line in Fig. 4(c), the dotted and dashdotdotted lines in Fig. 4(d) and the dashdotdotted line in Fig. 4(h). It is well known that the skewness and the kurtosis (both its magnitude and sign) characterize, respectively, the asymmetry of and the peakedness of the distribution around the average transferredelectron number during a time interval t, thus that provides further information for the counting statistics beyond the shot noise.
To discuss the underlying mechanisms of the current noise clearly, for the system parameters considered here, the two singlyoccupied eigenstates and eigenvalues can be expressed as
Here we have utilized the equations and . In this situation, the equations of motion of the six elements of the reduced density matrix are given by
where , (Ψ is the digamma function) and . Compared with the Markovian case, it is obvious that the nonMarkovian effect manifests itself through the offdiagonal elements of the reduced density matrix, namely, the quantum coherence of the considered QDs system. In Fig. 5(a), we plot the functions Φ_{L} − 0.1Φ_{R} (Γ_{R} = 0.1Γ_{L}), Φ_{L} − Φ_{R} (Γ_{R} = Γ_{L}) and 0.1Φ_{L} − Φ_{R} (Γ_{L} = 0.1Γ_{R}) as a function of bias voltage. It is clearly evident that the values of the functions Φ_{L} − 0.1Φ_{R} and Φ_{L} − Φ_{R} show significant variations with increasing bias voltage, especially in the vicinity of the bias voltages V_{b} = 2 and V_{b} = 10 because the new transport channels begin to participate in quantum transport; while 0.1Φ_{L} − Φ_{R} has a gentle variation. Consequently, the nonMarkovian effects in the Γ_{L}/Γ_{R} ≥ 1 case have a remarkable impact on the FCS, see Fig. 4. Moreover, for Γ_{L}/Γ_{R} = 10 case, the nonMarkovian effect has a more significant on the FCS than the Γ_{L}/Γ_{R} = 1 case, which originates from the QD2electrode coupling Γ_{R} is weaker than the hoping coupling J, where the electron tunneling from QD1 can not tunnel out QD2 very quickly and still influence the internal dynamics.
In order to illustrate whether the nonMarkovian effect has a weak influence on the FCS in a relatively small quantumcoherent QD system, we consider the regime (J = 1), where the offdiagonal elements of the reduced density matrix have little influence on the electron tunneling processes. We find that for the J = 1 case the diagonal elements of the reduced density matrix play a major role in the electron tunneling processes and the nonMarkovian effect in this case indeed has little impact on the FCS, see Figs. 3(e)–3(h). Consequently, the influence of the nonMarkovian effect on the FCS depends on the quantum coherence of the considered QD system. To prove whether this conclusion is universal or not, we take sidecoupled double QDs for further illustration in the following subsection.
Sidecoupled double quantum dots with high quantum coherence
We consider here a sidecoupled double QDs system. In this case, the QD1 is only weakly coupled to the two electrodes, see Fig. 1(b). The QDelectrode tunneling is thus described by
In the case of the QDelectrode weak coupling, the particlenumberresolved TCL quantum master equation for the sidecoupled double QDs can be expressed as
Here, the eigenstates and eigenvalues of the sidecoupled double QDs system are the same as the serially coupled double QDs system. In the following numerical calculations, the parameters of the sidecoupled QDs system are chosen as , J = 0.001, U = 5 and k_{B}T = 0.1.
For the present sidecoupled QDs system with high quantum coherence, we find that for Γ_{L}/Γ_{R} ≥ 1 case the nonMarkovian effect has a more remarkable impact on the FCS than that in the serially coupled double QDs system, but the NDC does not appear, see Figs. 4 and 6. For instance, in the case of Γ_{L}/J > 1 and Γ_{L}/Γ_{R} = 1, the nonMarkovian effect can further enhance the superPoissonian shot noise, see the dotted and dashdotdotted lines in Fig. 6(b); and the transitions of the skewness and the kurtosis from a relatively small positive to a large negative values take place, especially for a relatively large value Γ_{L}/J the kurtosis can be further decreased to a very large negative value, see the dotted and dashdotdotted lines in Figs. 6(c) and 6(d). While for the Γ_{L}/J > 1 and Γ_{L}/Γ_{R} = 10 case the nonMarkovian effect can enhance the shot noise to a superPoissonian value, see the dotted and dashdotdotted lines in Fig. 6(f) and the transition of the kurtosis from small positive to large negative values only takes place, see the dotted and dashdotdotted lines in Fig. 6(h). For the system parameters considered here, namely, in the limit of , the equations of motion of the six elements of the reduced density matrix read
From the above four equations, we find that these characteristics also originate from the quantum coherence of the sidecoupled double QDs and can also be understood in terms of the functions Φ_{L} + 0.1Φ_{R} and Φ_{L} + Φ_{R}, which have considerable variations in the vicinity of the bias voltages V_{b} = 2 and V_{b} = 12 because the new transport channels begin to enter the bias voltage window, see the solid and dashed lines in Fig. 5(b). As for the Γ_{L}/Γ_{R} < 1 case the nonMarkovian effect has a slightly influence on the FCS because the function 0.1Φ_{L} + Φ_{R} has a gentle variation with increasing the bias voltage, see the dotted line in Fig. 5(b), which is the same as the serially coupled double QDs system, see Figs. 3(a)–3(d) and 7.
In addition, it should be pointed out that for Γ_{L}/Γ_{R} = 1 the nonMarkovian effect has a stronger impact on the FCS than that for Γ_{L}/Γ_{R} > 1 case, which is contrary to the case of the serially coupled double QDs system. For the the sidecoupled double QDs system, the quantum coherence originates from the quantum interference between the direct electron tunneling process, namely, the conductionelectron tunneling into the QD1 and then directly tunneling out of the QD1 onto the drain electrode and the indirect tunneling process, namely, the conductionelectron from the source electrode first tunneling from the QD1 to the QD2, then tunneling back into the QD1 and at last tunneling out of the QD1. Thus, the fast direct tunneling process in the Γ_{L} = 10Γ_{R} case can be suppressed compared with the Γ_{L} = Γ_{R} case, which leads to the nonMarkovian effect has a relatively strong impact on the FCS in the Γ_{L}/Γ_{R} = 1 case.
Discussion
We have developed a nonMarkovian FCS formalism based on the exact TCL master equation and studied the influence of the interplay between the quantum coherence and nonMarkovian effect on the longtime limit of the FCS in three QD systems, namely, single QD, serially coupled double QDs and sidecoupled double QDs. It is demonstrated that the nonMarkovian effect manifests itself through the quantum coherence of the considered QD molecule system and especially has a significant impact on the FCS in the high quantumcoherent QD molecule system, which depends on the coupling of the considered QD molecule system with the source and drain electrodes. For the single QD system without quantum coherence, the nonMarkovian effect has no influence on the current noise properties; whereas for the serially coupled and sidecoupled double QDs systems with high quantum coherence, that has a remarkable impact on the FCS when the coupling of the considered QD molecule with the incident electrode is equal to or stronger than that with the outgoing electrode. For instance, for the high quantumcoherent serially coupled double QDs system, the nonMarkovian effect can induce a strong NDC and change the shot noise from the subPoissonian to superPoissonian distribution in the case of and Γ_{L} > J; while for the high quantumcoherent sidecoupled double QDs system, that can remarkably enhance the superPoissonian noise or the subPoissonian noise for the Γ_{L}/Γ_{R} ≥ 1 case. Moreover, the nonMarkovian effect can also lead to the occurrences of the skewness and kurtosis from small positive to large negative values. These results indicated that the influence of the nonMarkovian effect on the longtime limit of the FCS should be considered in a highly quantumcoherent singlemolecule system.
Methods
Particlenumberresolved timeconvolutionless quantum master equation
We consider a general transport setup consisting of a singlelevel QD molecule weakly coupled to the two electrodes, see Fig. 1, which is described by the following Hamiltonian
Here, the first term stands for the Hamiltonians of the two electrodes, with ε_{αk} being the energy dispersion and the annihilation (creation) operators in the α electrode. The second term , which may contain vibrational or spin degrees of freedom and different types of manybody interaction, represents the QD molecule Hamiltonian, where is the creation (annihilation) operator of electrons in a quantum state denoted by μ. The third term describes the tunneling coupling between the QD molecule and the two electrodes, which is assumed to be a sum of bilinear terms that each create an electron in the QD molecule and annihilate one in the electrodes or vice versa.
The QDelectrode coupling is assumed to be sufficiently weak, so that H_{hyb} can be treated perturbatively. In the interaction representation, the equation of motion for the total density matrix reads
with
where and . In order to derive an exact equation of motion for the reduced density matrix ρ_{S} of the QD molecule system, it is convenient to define a superoperator according to
with ρ_{B} being some fixed state of the electron electrode. Accordingly, a complementary superoperator reads
For a factorizing initial condition , and . Using the TCL projection operator method^{52}, one can obtain the secondorder TCL master equation
The Eq. (31) is the starting point of deriving the particlenumberresolved quantum master equation. Using Eqs. (28) and (29), after some algebraic calculations we can rewrite Eq. (31) as
In order to fully describe the electron transport problem, we should record the number of electrons arriving at the drain electrode, which emitted from the source electrode and passing through the QD molecule. We follow Li and coauthors^{53,54} and introduce the Hilbert subspace B^{(n)} (n = 1, 2, …) corresponding to n electrons arriving at the drain electrode, which is spanned by the product of all manyparticle states of the two isolated electrodes and formally denoted as . Then, the entire Hilbert space of the two electrodes can be expressed as . With this classification of the electrode states, the average over states in the entire Hilbert space B in Eq. (32) should be replaced with the states in the subspace B^{(n)} and leading to a conditional TCL master equation
To proceed, two physical considerations are further implemented. (i) Instead of the conventional Born approximation for the entire density matrix , the ansatz is proposed, where being the electrode density operator associated with n electrons arriving at the drain electrode. With this ansatz for the entire density operator, tracing over the subspace B^{(n)}, the Eq. (33) can be rewritten as
Here we have used the orthogonality between the states in different subspaces. (ii) The extra electrons arriving at the drain electrode will flow back into the source electrode via the external closed transport circuit. Moreover, the rapid relaxation processes in the electrodes will bring the electrodes to the local thermal equilibrium states quickly, which are determined by the chemical potentials. Consequently, after the procedure done in Eq. (34), the electrode density matrices and should be replaced by . In the Schrödinger representation, the Eq. (34) can be expressed as
where the correlation function are defined as
Introducing the following superoperators
then, the Eq. (35) can be rewritten as a compact form
where . The above equation is the starting point of the nonMarkovian FCS calculation.
NonMarkovian full counting statistics
In this subsection, we outline the procedure to calculate the nonMarkovian FCS based on Eq. (38). The FCS can be obtained from the cumulant generating function (CGF) F (χ) which related to the probability distribution P (n, t) by^{54,55} , where χ is the counting field. The CGF F (χ) connects with the particlenumberresolved density matrix ρ^{(n)} (t) by defining . Evidently, we have e^{−F(χ)} = Tr[S (χ, t)], where the trace is over the eigenstates of the QD molecule system. Since Eq. (38) has the following form , then, S (χ, t) satisfies , where S is a column matrix and A, C and D are three square matrices. The specific form of L_{χ} can be obtained by performing a discrete Fourier transformation to the matrix element of Eq. (38). In the low frequency limit, the counting time, namely, the time of measurement is much longer than the time of tunneling through the QD molecule system. In this case, F (χ) is given by^{34,40,43,55,56,57} F (χ) = −λ_{1} (χ) t, where λ_{1} (χ) is the eigenvalue of L_{χ} which goes to zero for χ → 0. According to the definition of the cumulants one can express λ_{1} (χ) as . The low order cumulants can be calculated by the Rayleigh–Schrödinger perturbation theory in the counting parameter χ. In order to calculate the first four current cumulants we expand L_{χ} to four order in χ
and define the two projectors^{40,43,56,58} and Q = Q^{2} = 1 − P, obeying the relations PL_{0} = L_{0}P = 0 and QL_{0} = L_{0}Q = L_{0}. Here, 0〉〉 is the right eigenvector of L_{0}, i.e., L_{0} 0〉〉 = 0 and is the corresponding left eigenvector. In view of L_{0} being singular, we also introduce the pseudoinverse according to , which is welldefined due to the inversion being performed only in the subspace spanned by Q. After a careful calculation, λ_{1} (χ) is given by
From Eq. (40) we can identify the first four current cumulants:
Here, it is important to emphasize that the first four cumulants C_{k} are directly related to the transport characteristics. For example, the firstorder cumulant (the peak position of the distribution of transferredelectron number) gives the average current 〈I〉 = eC_{1}/t. The zerofrequency shot noise is related to the secondorder cumulant (the peakwidth of the distribution) . The thirdorder cumulant and fourorder cumulant characterize, respectively, the skewness and kurtosis of the distribution. Here, . In general, the shot noise, skewness and kurtosis are represented by the Fano factor F_{2} = C_{2}/C_{1}, F_{3} = C_{3}/C_{1} and F_{4} = C_{4}/C_{1}, respectively.
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Acknowledgements
This work was supported by the NKBRSFC under grants Nos. 2011CB921502, 2012CB821305, NSFC under grants Nos. 11204203, 61405138, 11004124, 11275118, 61227902, 61378017, 11434015, SKLQOQOD under grants No. KF201403, SPRPCAS under grants No. XDB01020300.
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H.B.X. conceived the idea and designed the research and performed calculations. H.J.J., J.Q.L. and W.M.L. contributed to the analysis and interpretation of the results and prepared the manuscript.
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Xue, HB., Jiao, HJ., Liang, JQ. et al. NonMarkovian full counting statistics in quantum dot molecules. Sci Rep 5, 8978 (2015). https://doi.org/10.1038/srep08978
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