Non-Markovian full counting statistics of cotunneling assisted sequential tunneling in open quantum systems

TL;DR

This paper introduces a non-Markovian full counting statistics formalism for electron tunneling in open quantum systems, accounting for quantum coherence and cotunneling, revealing their impact on shot noise and higher-order cumulants.

Contribution

It develops a novel non-Markovian formalism combining particle number resolved master equations and perturbation theory, highlighting the interplay of coherence and cotunneling in quantum transport.

Findings

Quantum coherence and cotunneling influence shot noise, skewness, and kurtosis.

Sign transitions in skewness and kurtosis depend on quantum coherence strength.

The formalism provides deeper insights into electron transport in quantum systems.

Abstract

We develop a non-Markovian full counting statistics formalism taking into account both the sequential tunneling and cotunneling based on the exact particle number resolved time-convolutionless master equation and the Rayleigh-Schr\"{o}dinger perturbation theory. Then, we study the influences of the quantum coherence and the cotunneling assisted sequential tunneling processes on the non-Markovian full counting statistics of electron tunneling through an open quantum system, which consists of a side-coupled double-quantum-dot molecule weakly coupled to two electron reservoirs. We demonstrate that in the sequential tunneling regime the competition or interplay between the quantum coherence and the cotunneling processes determines whether the super-Poissonian of the shot noise, the skewness and the kurtosis take place and whether the sign transitions of the values of the skewness and the kurtosis occur in the strong quantum-coherent side-coupled double-quantum-dot molecule. These results suggest that it is necessary to consider the effects of the quantum coherence and the cotunneling processes on the full counting statistics of electron tunneling through open strong quantum-coherent quantum systems in the sequential tunneling regime, which provide a deeper insight into understanding of electron tunneling through open quantum systems.

Equations

$$H=H_{\text{electrodes}}+H_{\text{OQS}}+H_{\text{hyb}}.$$

$$\frac{\partial }{\partial t}{\rho }_I\left(t\right)=-i\left[H_{\text{hyb}}^I\left(t\right),{\rho }^I\left(t\right)\right]\equiv \mathcal{L}\left(t\right){\rho }_I\left(t\right),$$

$$H_{\text{hyb}}^I\left(t\right)=\sum _{\alpha ,\mu }\left[f_{\alpha \mu }^{†}\left(t\right)d_{\mu }\left(t\right)+f_{\alpha \mu }\left(t\right)d_{\mu }^{†}\left(t\right)\right]$$

$$f_{\alpha \mu }^{†}\left(t\right)=\sum _k{t}_{\alpha \mu k}^{\ast }\exp \left(i{H}_{\text{electrodes}}t\right)a_{\alpha \mu k}^{†}\exp \left(-i{H}_{\text{electrodes}}t\right)$$

$$d_{\mu }\left(t\right)=\exp \left(i{H}_{\text{OQS}}t\right)d_{\mu }\exp \left(-i{H}_{\text{OQS}}t\right)$$

$$\mathcal{P}\rho ={\text{tr}}_B\left[\rho \right]\otimes {\rho }_B={\rho }_S\otimes {\rho }_B,$$

$$\mathcal{Q}\rho =\rho -\mathcal{P}\rho .$$

$${\frac{\partial }{\partial t}\mathcal{P}\rho \left(t\right)|}_{\text{sceond-order}}={\mathcal{K}}_2\left(t\right)\mathcal{P}\rho \left(t\right)={\int }_{-\infty }^{t}d{t}_1\mathcal{P}\mathcal{L}\left(t\right)\mathcal{L}\left(t_1\right)\mathcal{P}\rho \left(t\right)$$

$${\frac{\partial }{\partial t}\mathcal{P}\rho \left(t\right)|}_{\text{fourth-order}}$$

$${\frac{\partial }{\partial t}{\rho }_{I,S}\left(t\right)|}_{\text{sceond-order}}$$

$${\frac{\partial }{\partial t}{\rho }_{I,S}^{\left(n\right)}\left(t\right)|}_{\text{sceond-order}}$$

$${\frac{\partial }{\partial t}{\rho }_S^{\left(n\right)}\left(t\right)|}_{\text{sceond-order}}$$

$$C_{\alpha ij}^{\left(+\right)}\left(t-t_1\right)={\text{tr}}_R\left[f_{\alpha i}^{†}\left(t\right)f_{\alpha j}\left(t_1\right){\rho }_B\right]=⟨f_{\alpha i}^{†}\left(t\right)f_{\alpha j}\left(t_1\right)⟩,$$

$$C_{\alpha ij}^{\left(-\right)}\left(t-t_1\right)={\text{tr}}_R\left[f_{\alpha i}\left(t\right)f_{\alpha j}^{†}\left(t_1\right){\rho }_B\right]=⟨f_{\alpha i}\left(t\right)f_{\alpha j}^{†}\left(t_1\right)⟩.$$

$$A_{\alpha i}^{\left(+\right)}\left(t\right)=\sum _j{\int }_{-\infty }^{t}d{t}_1{C}_{\alpha ji}^{\left(+\right)}\left(t_1-t\right)e^{-i{H}_S\left(t-t_1\right)}{d}_j{e}^{i{H}_S\left(t-t_1\right)},$$

$$A_{\alpha i}^{\left(-\right)}\left(t\right)=\sum _j{\int }_{-\infty }^{t}d{t}_1{C}_{\alpha ij}^{\left(-\right)}\left(t-t_1\right)e^{-i{H}_S\left(t-t_1\right)}{d}_j{e}^{i{H}_S\left(t-t_1\right)},$$

$${\frac{\partial }{\partial t}{\rho }_S^{\left(n\right)}\left(t\right)|}_{\text{sceond-order}}$$

$${\frac{\partial }{\partial t}{\rho }_{I,S}\left(t\right)|}_{\text{fourth-order}}={\int }_{-\infty }^{t}d{t}_1{\int }_{-\infty }^{t_1}d{t}_2{\int }_{-\infty }^{t_2}d{t}_3\sum _{ijkl}$$

$$\times \{{\text{tr}}_B\left[H_I\left(t\right),\left[H_I\left(t_1\right),\left[H_I\left(t_2\right),\left[H_I\left(t_3\right),{\rho }_S\otimes {\rho }_B\right]\right]\right]\right]$$

$$-{\text{tr}}_B\left[H_I\left(t\right),\left[H_I\left(t_1\right),{\text{tr}}_B\left[H_I\left(t_2\right),\left[H_I\left(t_3\right),{\rho }_S\otimes {\rho }_B\right]\right]\otimes {\rho }_B\right]\right]$$

$$-{\text{tr}}_B\left[H_I\left(t\right),\left[H_I\left(t_2\right),{\text{tr}}_B\left[H_I\left(t_1\right),\left[H_I\left(t_3\right),{\rho }_S\otimes {\rho }_B\right]\right]\otimes {\rho }_B\right]\right]$$

$$-{\text{tr}}_B\left[H_I\left(t\right),\left[H_I\left(t_3\right),{\text{tr}}_B\left[H_I\left(t_1\right),\left[H_I\left(t_2\right),{\rho }_S\otimes {\rho }_B\right]\right]\otimes {\rho }_B\right]\right]$$

$$H_{\text{hyb}}^I\left(t\right)=\sum _{\alpha ,\mu }{F}_{\alpha \mu }\left(t\right)D_{\mu }\left(t\right),$$

$$F_{\alpha \mu }\left(t\right)=\sum _k{e}^{i{H}_{\text{electrodes}}t}{F}_{\alpha \mu k}{e}^{-i{H}_{\text{electrodes}}t},$$

$$D_{\mu }\left(t\right)=e^{i{H}_{\text{dot}}t}{D}_{\mu }{e}^{-i{H}_{\text{dot}}t},$$

$${\frac{\partial }{\partial t}{\rho }_{I,S}\left(t\right)|}_{\text{fourth-order}}$$

$$C_{02}=\sum _{\alpha ik}{\text{tr}}_B\left[F_{\alpha i}\left(t\right)F_{\alpha k}\left(t_2\right)\right]\text{, }{C}_{03}=\sum _{\alpha il}{\text{tr}}_B\left[F_{\alpha i}\left(t\right)F_{\alpha l}\left(t_3\right)\right]\text{,}$$

$$C_{12}=\sum _{\alpha jk}{\text{tr}}_B\left[F_{\alpha j}\left(t_1\right)F_{\alpha k}\left(t_2\right)\right]\text{, }{C}_{21}=\sum _{\alpha kj}{\text{tr}}_B\left[F_{\alpha k}\left(t_2\right)F_{\alpha j}\left(t_1\right)\right]\text{,}$$

$$C_{13}=\sum _{\alpha jl}{\text{tr}}_B\left[F_{\alpha j}\left(t_1\right)F_{\alpha l}\left(t_3\right)\right]\text{, }{C}_{31}=\sum _{\alpha lj}{\text{tr}}_B\left[F_{\alpha l}\left(t_3\right)F_{\alpha j}\left(t_1\right)\right]\text{,}$$

$$\hat{0}=D_i\left(t\right)\text{, }\hat{1}=D_j\left(t_1\right)\text{, }\hat{2}=D_k\left(t_2\right)\text{, }\hat{3}=D_l\left(t_3\right).$$

$${\frac{\partial {\rho }_S\left(t\right)}{\partial t}|}_{\text{fourth-order}}$$

Full text

Non-Markovian; full counting statistics; cotunneling; quantum dot molecules

††preprint: HEP/123-qed

I INTRODUCTION

In an open quantum system, for an intermediate coupling strength between the open quantum system and electron reservoir, the high-order tunneling processes, i.e., the cotunneling processes can influence the electron tunneling processes and bring out novel physical properties. Therefore, the electron cotunneling in open quantum systems, especially quantum dot (QD) systems, which is artificial molecules made from coupled QDs, and single molecules have been extensively studied both experimentally FranceschiEx ; LiuEx ; SchleserEx ; SigristEx ; JoEx ; RochEx and theoretically HartmannTDia ; GolovachTCAS ; WeymannTDia ; ElsteTDia ; MisiornyTDia ; BegemannTDia . Particularly, the shot noise OnacSEx ; GustavssonSEx ; ZhangSEx ; ZarchinSEx ; GustavssonSEx08 ; OkazakiSEx ; SukhorukovSToff ; ThielmannST ; WeymannST ; WeymannCAS ; AghassiCAS ; WeymannST10 ; WeymannST11 ; CarmiSTCAS ; KaasbjergST ; WrzesniewskiST and the full counting statistics (FCS) KaasbjergST ; BraggioFCS ; UtsumiFCS ; EmaryFCS of cotunneling in QD systems have attracted considerable attention due to they can allow one to identify the intrinsic properties of the QD systems and access the information of electron correlation that cannot be obtained through the average current measurements. For example, in the Coulomb blockade regime, in which the transfer of electrons is dominated by cotunneling processes, it has been demonstrated experimentally OnacSEx ; GustavssonSEx ; ZhangSEx ; ZarchinSEx ; GustavssonSEx08 ; OkazakiSEx and theoretically SukhorukovSToff ; ThielmannST ; WeymannST ; WeymannCAS ; AghassiCAS ; WeymannST10 ; WeymannST11 ; CarmiSTCAS ; KaasbjergST ; WrzesniewskiST that the transport current displays super-Poissonian shot noise, which indicates that the super-Poissonian distribution of transferred-electron number and has a width being broader than its mean.

On the other hand, the quantum coherence, characterized by the off-diagonal elements of reduced density matrix of the considered system, also plays an important role in the electron tunneling through the strong quantum-coherent systems SukhorukovSToff ; Kieblichoff ; Lindebaumoff ; Urbanoff ; Xueoff1 ; Xueoff2 ; Xueoff3 ; Xueoff4 . In particular, theoretical studies have demonstrated that the non-Markovian effect of a strong quantum-coherent system plays an important role in the non-equilibrium electron tunneling processes DMarcos11 , and manifests itself through the quantum coherence Xueoff3 . Consequently, in the intermediate coupling strength case, the electron tunneling through an open quantum system are mainly governed by the competitions or interplays between the cotunneling, sequential tunneling and quantum coherence. In the Coulomb blockade regime, it has been demonstrated that the electron cotunneling processes play a crucial role; whereas that, in the sequential tunneling regime where the transfer of electrons being dominated by sequential tunneling, has a slightly influence on the conduction and shot noise SchleserEx ; WrzesniewskiST ; DMarcos11 ; WeymannCAS0 . Outside the Coulomb blockade regime, including the transition region from Coulomb blockade to sequential tunneling and the sequential tunneling regime, theoretical studies have demonstrated that the cotunneling assisted sequential tunneling processes have an important influence on the conduction GolovachTCAS ; AghassiCAS and shot noise CarmiSTCAS ; WeymannCAS . However, in the sequential tunneling regime, the influences of quantum coherence and the cotunneling assisted sequential tunneling processes on the non-Markovian FCS has not yet been revealed.

In this work, we derive a non-Markovian FCS formalism taking into account both the sequential tunneling and cotunneling based on the exact particle number resolved time-convolutionless (TCL) master equation and the Rayleigh-Schrödinger perturbation theory developed in references Flindt01 ; Flindt02 ; Flindt03 . Then, in the sequential tunneling regime, we study the influences of the quantum coherence and the cotunneling processes on the non-Markovian FCS of electron tunneling through open quantum systems. For the sake of discussion, the considered open quantum system consists of a side-coupled double quantum-dot system weakly coupled to two electron electrodes (reservoirs). Here, the corresponding quantum coherence can be tuned by modulating the hopping strength between the two QDs relative to the coupling of this QD molecule with the source and drain electrodes. It is numerically demonstrated that, for the strong quantum-coherent side-coupled double QD system, the competition or interplay between the cotunneling processes and the quantum coherence, in the sequential tunneling regime, can take place in a range of the QD-electrode coupling strength and has a remarkable influence on the FCS. These characteristics depend on the temperature and the left-right asymmetry of the QD-electrode coupling. Therefore, in the open strong quantum-coherent quantum systems, it should be considered the effects of the quantum coherence and the cotunneling processes on the FCS, even through the electron tunneling is mainly dominated by the sequential tunneling processes.

II MODEL AND FORMALISM

II.1 Hamiltonian of open quantum system and TCL master equations

We consider an open quantum system (OQS) weakly coupled to the two electrodes (reservoirs), which is described by the following Hamiltonian

[TABLE]

Here, the first term $H_{\text{electrodes}}=\sum_{\alpha,k,\sigma}\varepsilon_{\alpha k}a_{\alpha k\sigma}^{{\dagger}}a_{\alpha k\sigma}$, characterized by the two noninteracting reservoirs, stands for the Hamiltonian of the two electrodes, with $\varepsilon_{\alpha k}$ being the energy dispersion, and $a_{\alpha k\sigma}^{{\dagger}}$ ($a_{\alpha k\sigma}$) the creation (annihilation) operators in the $\alpha$ electrode. The second term $H_{\text{OQS}}=H_{S}\left(d_{\mu}^{{\dagger}},d_{\mu}\right)$, which may contain vibrational or spin degrees of freedom and different types of many-body interactions, represents the OQS Hamiltonian, where $d_{\mu}^{{\dagger}}$ ($d_{\mu}$) is the creation (annihilation) operator of electron in a quantum state denoted by $\mu$. The third term $H_{\text{hyb}}=\sum_{\alpha,\mu,k}\left(t_{\alpha\mu k}d_{\mu}^{{\dagger}}a_{\alpha\mu k}+t_{\alpha\mu k}^{\ast}a_{\alpha\mu k}^{{\dagger}}d_{\mu}\right)$, which is assumed to be a sum of bilinear terms that each create an electron in the OQS and annihilate one in the electrodes or vice versa, describes the tunneling coupling between the OQS and the two electrodes.

Due to the OQS-electrode coupling being sufficiently weak, thus, $H_{\text{hyb}}$ can be treated perturbatively. In the interaction representation, the equation of motion for the total density matrix reads

[TABLE]

with

[TABLE]

where

[TABLE]

[TABLE]

To derive an exact equation of motion for the reduced density matrix $\rho_{S}$ of the OQS, it is convenient to define a super-operator $\mathcal{P}$ according to

[TABLE]

where $\rho_{B}$ is some fixed states of the two electrodes. Accordingly, a complementary super-operator $\mathcal{Q},$

[TABLE]

For a factorizing initial condition $\rho\left(t_{0}\right)=\rho_{S}\left(t_{0}\right)\otimes\rho_{B}$, $\mathcal{P}\rho\left(t_{0}\right)=\rho\left(t_{0}\right)$, and then $\mathcal{Q}\rho\left(t_{0}\right)=0$. Using the above TCL projection operator method, one can obtain the second-order and the fourth-order TCL master equations book

[TABLE]

[TABLE]

Both Eqs. (7) and (8) are the starting point of deriving the particle number resolved quantum master equation.

II.2 The second-order particle number resolved TCL master equation

In this subsection, we derive the second-order particle number resolved quantum master equation based on Eq. (7). Using both Eqs. (2) and (5), Eq. (7) can be rewritten as

[TABLE]

To fully describe the electron transport processes, the electron numbers, which emitted from the source electrode and passed through the OQS and arrived at the drain electrode, should be recorded. Following Refs. Lixq ; WangSK , the Hilbert subspace $B^{\left(n\right)}$ $\left(n=1,2,...\right)$, which corresponds to $n$ electrons arriving at the drain electrode and spanned by the product of all many-particle states of the two electrodes, is introduced and formally denoted as $B^{\left(n\right)}\equiv$ span$\left\{\left|\Psi_{L}\right\rangle^{\left(n\right)}\otimes\left|\Psi_{R}\right\rangle^{\left(n\right)}\right\}$. Consequently, the entire Hilbert space of the two electrodes can be expressed as $B=\oplus_{n}B^{\left(n\right)}$. With this classification of the states of the two electrodes, the average over states in the entire Hilbert space $B$ in Eq. (9) should be replaced with the states in the subspace $B^{\left(n\right)}$. Then, Eq. (9) can be expressed as a conditional TCL master equation

[TABLE]

Before proceeding, two physical considerations are implemented. (i) Instead of the conventional Born approximation for the entire density matrix $\rho_{T}\left(t\right)\simeq\rho\left(t\right)\otimes\rho_{B}$, the ansatz $\rho^{I}\left(t\right)\simeq\rho^{\left(n\right)}\left(t\right)\otimes\rho_{B}^{\left(n\right)}$ is proposed, where $\rho_{B}^{\left(n\right)}$ being the density operator of two electrodes associated with $n$ electrons arriving at the drain electrode. With this ansatz for the entire density operator (i.e., tracing over the subspace $B^{\left(n\right)}$), Eq. (10) can be reexpressed as

[TABLE]

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. Additionally, 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. After the procedure done in Eq. (11), the density matrices of two electrodes $\rho_{B}^{\left(n\right)}$ and $\rho_{B}^{\left(n\pm 1\right)}$ should be replaced by $\rho_{B}^{\left(0\right)}$. In the Schrödinger representation, Eq. (11) can be written as

[TABLE]

where the correlation functions are defined as

[TABLE]

[TABLE]

Introducing the following super-operators

[TABLE]

[TABLE]

then, Eq. (12) can be rewritten as a compact form

[TABLE]

where $A_{i}^{\left(\pm\right)}\left(t\right)=\sum_{\alpha}A_{\alpha i}^{\left(\pm\right)}\left(t\right)$. The equation (17) is the starting point of the non-Markovian FCS calculation taking into account the sequential tunneling processes only.

II.3 The fourth-order particle number resolved TCL master equation

In this subsection, we derive the fourth-order particle number resolved quantum master equation based on Eq. (8). Using both Eqs. (2) and (5), Eq. (8) can be expressed as

[TABLE]

To further facilitate this derivation, the tunneling coupling between the OQS and the two electrodes $H_{\text{hyb}}^{I}\left(t\right)$ is rewritten as the following equation

[TABLE]

with

[TABLE]

[TABLE]

where $F_{\alpha\mu k}=t_{\alpha\mu k}a_{\alpha\mu k}+t_{\alpha\mu k}^{\ast}a_{\alpha\mu k}^{{\dagger}}$, $D_{\mu}=d_{\mu}+d_{\mu}^{{\dagger}}$. Inserting Eq. (19) into Eq. (18), one can obtain book

[TABLE]

with

[TABLE]

[TABLE]

[TABLE]

[TABLE]

In the Schrödinger representation, Eq. (20) can be expressed as

[TABLE]

with

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Here, we define the super-operator $\mathcal{L}_{S}$ as $e^{-iH_{S}t}Oe^{iH_{S}t}\equiv e^{-i\mathcal{L}_{S}t}O$. Now, we derive the fourth-order particle number resolved quantum master equation based on Eq. (21). Without loss of generality, we consider the case of Eq. (22). Considering the Hamiltonian $H_{\text{hyb}}^{I}\left(t\right)$ $=\sum_{\alpha,\mu}\left[f_{\alpha\mu}^{{\dagger}}\left(t\right)d_{\mu}\left(t\right)+f_{\alpha\mu}\left(t\right)d_{\mu}^{{\dagger}}\left(t\right)\right]$, the $C_{02},C_{13}$ and $D_{i,j,k,l}$ have the following forms

[TABLE]

[TABLE]

[TABLE]

[TABLE]

respectively. Therefore, the particle number resolved formation of Eq. (22) can be expressed as follows

[TABLE]

with

[TABLE]

[TABLE]

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

According to the procedure described above, one can obtain the particle-number-resolved density matrices corresponding to Eq. (21), which is the starting point of the non-Markovian FCS calculation taking the cotunneling processes into account. Therefore, the particle number resolved quantum master equation taking into account both the sequential tunneling and cotunneling can be written as

[TABLE]

II.4 FULL COUNTING STATISTICS

The FCS formalism based on Eq. (38) can be obtained from the cumulant generating function (CGF) $F\left(\chi\right)$ Bagrets

[TABLE]

where $\chi$ is the counting field, and $P\left(n,t\right)=$Tr$\left[\rho_{S}^{\left(n\right)}\left(t\right)\right]$. Thus, one has $e^{-F\left(\chi\right)}=$Tr$\left[S\left(\chi,t\right)\right]$ by defining $S\left(\chi,t\right)=\sum_{n}\rho_{S}^{\left(n\right)}\left(t\right)e^{in\chi}$, where the trace is over the eigenstates of the OQS. Since Eq. (38) has the following form

[TABLE]

then $S\left(\chi,t\right)$ satisfies

[TABLE]

where $S$ is a column matrix, and $A$, $C_{1}$, $D_{1}$, $C_{2}$ and $D_{2}$ are five square matrices. Here, for the second-order case $C_{2}=D_{2}=0$, and the specific form of $L_{\chi}$ can be obtained by performing a discrete Fourier transformation to the matrix elements of Eq. (38).

In the low frequency limit, the low order cumulants of transferred-electron number $C_{k}$ can be calculated based on Eq. (41) and the Rayleigh–Schrödinger perturbation theory developed in Refs. Xueoff3 ; Flindt01 ; Flindt02 ; Flindt03 ; WangSK ; Bagrets ; Kielich ; Groth . Here, the first four cumulants are directly related to the peak position (i.e., the average current $\left\langle I\right\rangle=eC_{1}/t$), the peak-width (i.e., shot noise characterized by Fano factor $C_{2}/C_{1}$), the skewness ($C_{3}/C_{1}$) and the kurtosis ($C_{4}/C_{1}$) of the distribution of transferred-electron number. In general, the shot noise, skewness and kurtosis are represented by the Fano factors $F_{2}=C_{2}/C_{1}$, $F_{3}=C_{3}/C_{1}$ and $F_{4}=C_{4}/C_{1}$, respectively.

III TRANSPORT THROUGH SIDE-COUPLED DOUBLE QD SYSTEM

III.1 Hamiltonian of the side-coupled double QD system

In order to facilitate discussions effectively, we consider a side-coupled double QD system weakly connected to two metallic electrodes, see Fig. 1. For the sake of simplicity, we neglect electron-spin. The Hamiltonian of the side-coupled double-QD system is described by

[TABLE]

where $d_{i}^{{\dagger}}$ ($d_{i}$) is the creation (annihilation) operator of an electron with energy $\varepsilon_{i}$ in $i$th QD, and $U_{12}$ the interdot Coulomb repulsion between two electrons in different QDs. Here, we assume that the intradot Coulomb interaction $U\rightarrow\infty$, thus, the double-electron occupation in different QDs is permitted only. The last term of $H_{dot}$ describes the hopping between the two QDs with $J$ being the hopping parameter. To facilitate the following calculation, the eigenstates of $H_{\text{dot}}$ are used to describe the electronic states of the side-coupled double QD system. Here, The Hamiltonian $H_{\text{dot}}$ can be diagonalized in the basis represented by the electron occupation numbers of the QD-1 and the QD-2, i.e., $\left|0\right\rangle_{1}\left|0\right\rangle_{2}$, $\left|1\right\rangle_{1}\left|0\right\rangle_{2}$, $\left|0\right\rangle_{1}\left|1\right\rangle_{2}$, $\left|1\right\rangle_{1}\left|1\right\rangle_{2}$. Consequently, the four eigenvalues of and the corresponding four eigenstates of the side-coupled double QD system are given by Xueoff3

[TABLE]

[TABLE]

[TABLE]

with

[TABLE]

[TABLE]

and

[TABLE]

[TABLE]

The electron distributions of the two metallic electrodes, in which the relaxation is assumed to be sufficiently fast, are described by the equilibrium Fermi functions and the corresponding Hamiltonian reads

[TABLE]

where $a_{\alpha\mathbf{k}}^{{\dagger}}$ ($a_{\alpha\mathbf{k}}$) is the $\alpha$-electrode electron creation (annihilation) operator with energy $\varepsilon_{\alpha\mathbf{k}}$ and momentum $\mathbf{k}$.

The tunneling between the QD-1 and the two electrodes is described by

[TABLE]

where the tunneling amplitudes $t_{\alpha}$ and the density of states $g_{\alpha}$ are assumed to be independent of wave vector and energy, thus, the electronic tunneling rate can be characterized by $\Gamma_{\alpha}=2\pi|t_{\alpha}|^{2}g_{\alpha}$.

In the side-coupled double QD system, the quantum coherence can be tuned by modulating the magnitude of the hopping parameter $J$ relative to the tunneling coupling strength between the QD-1 and the two electrodes. In the case of $J\ll\Gamma$ ($\Gamma=\Gamma_{L}+\Gamma_{R}$), the hopping strength between the two QDs strongly modifies the internal dynamics, and the off-diagonal elements of the reduced density matrix play an essential role in the electron tunneling processes Xueoff1 ; Xueoff3 ; LuoJPCM ; while in the regime $J\gg\Gamma$, the off-diagonal elements of the reduced density matrix have very little influence on the electron tunneling processes Xueoff1 . In the following calculation, the parameters of the side-coupled double QD system are taken as $\varepsilon_{1}=\varepsilon_{2}=2.35$, $U_{12}=4$ and $k_{B}T=0.1$ (if not explicitly stated otherwise), where the unit of energy is chosen as meV Elzerman .

III.2 The side-coupled double QD system with strong quantum coherence

We first study the influences of the off-diagonal elements of the reduced density matrix, namely, the quantum coherence, and the electron cotunneling processes on the FCS of electron transport through this QD system with strong quantum coherence. For the side-coupled double QD system, the quantum coherence has an important influence on the electron sequential tunneling processes in the bias voltage range in which the transitions between the singly-occupied and empty-occupied eigenstates take place Xueoff1 ; Xueoff3 . Consequently, the following discussions focus on this bias voltage region, the applied bias voltage is here chosen as $V_{b}=4.5$ based on the parameters of the QD system. To determine the dependence of the FCS on the quantum coherence and the electron cotunneling processes, we consider the average current, shot noise, skewness and kurtosis as a function of the tunneling rate $\Gamma_{\alpha}$ for the four different cases, (1) considering the diagonal elements of the reduced density matrix in the sequential tunneling processes only, (2) considering the diagonal and off-diagonal elements of the reduced density matrix in the sequential tunneling processes, (3) considering the diagonal elements of the reduced density matrix in the cotunneling assisted sequential tunneling processes only, (4) considering the diagonal and off-diagonal elements of the reduced density matrix in the cotunneling assisted sequential tunneling processes.

Figures 2, 3 and 4 show the influence of the temperature of the QD system on the first four current cumulants with the different values of the left-right asymmetry of the QD-electrode coupling $\Gamma_{L}/\Gamma_{R}$. In the case of the coupling of the QD-1 with the source-electrode is stronger than that of the QD-1 with the drain-electrode, i.e., $\Gamma_{L}/\Gamma_{R}>1$, we in Fig. 2 plot the first four current cumulants as a function of the tunneling rate $\Gamma_{L}$ with different temperatures $k_{B}T$ at $\Gamma_{L}/\Gamma_{R}=10$. We found that, in the $\Gamma/J<1$ case, the electron cotunneling processes play a essential role in determining the values of the shot noise and high-order current cumulants; whereas in the $\Gamma/J\gg 1$ case the quantum coherence plays a crucial role in determining whether the Fano factors of the shot noise, the skewness and the kurtosis are larger than one or not, see Fig. 2. In the case of the intermediate value of $\Gamma/J$, the competition between the electron cotunneling processes and the quantum coherence takes place. This leads to the formation of a crossover region, but the range of which depends on the temperature $k_{B}T$, see Fig. 2.

The underlying physics of the cotunneling effect can be understood in terms of the cotunneling-induced redistribution of the occupation probabilities of the QD’s different eigenstates. In the $\Gamma_{L}/\Gamma_{R}=10$ case, the occupation probabilities of the two singly-occupied eigenstates are much larger than that of empty-occupied eigenstate, leading to a relatively long dwell time of the conduction electron before tunneling out the QD system. When $\Gamma/J\ll 1$, the conduction electrons can tunnel back and forth between the two singly-occupied eigenstates very rapidly, and then enhance the cotuneling processes induced by the transitions between the doubly-occupied $\left|2\right\rangle$ and singly-occupied $\left|1\right\rangle^{\pm}$ eigenstates. Consequently, the cotunneling processes can dramatically decrease and increase the occupation probabilities of the singly-occupied and empty-occupied eigenstates with decreasing ratio of $\Gamma$ to $J$, respectively, see Figs. 5(a1)-5(a3). This indicates that the sequential-induced blocking of electron tunneling can be removed by the cotunneling processes, which leads to the shot noise being decreased, see Figs. 2(b1)-2(b3). Whereas in the $\Gamma/J\gg 1$ case the conduction electrons can tunnel back and forth between the two singly-occupied eigenstates very slowly, and then suppress the cotuneling processes. Thus, the quantum coherence has a very significant influence on the electron tunneling processes and the cotunneling-induced probability distributions for the singly-occupied and empty-occupied eigenstates have a slight variation, see Figs. 5(a1)-5(a3). In addition, the cotunneling-induced non-equilibrium electron distribution depends on the temperature $k_{B}T$, see Figs. 5(a1)-5(a3), which are responsible for the slight influence of the cotunneling effect on the FCS with decreasing the temperature $k_{B}T$.

Compared with the $\Gamma_{L}/\Gamma_{R}>1$ case, in the case of $\Gamma_{L}/\Gamma_{R}\leq 1$, the range of the crossover region is very small, which also depends on the temperature $k_{B}T$, see Figs. 3 and 4. Particularly, in the cases of $\Gamma_{L}/\Gamma_{R}\leq 1$ and $\Gamma/J\gg 1$, the interplay between the electron cotunneling processes and the quantum coherence determine the FCS properties of transferred-electron number, such as, whether the super-Poissonian distributions of the shot noise, the skewness and the kurtosis ($F_{i}>1$) occur or not, and whether the signs of the values of the skewness and the kurtosis become negative from a positive value or not, see Figs. 3 and 4. In particular, the magnitudes and signs of the skewness and kurtosis characterize the asymmetry of and the combined weight of the tails relative to the rest of the probability distribution of transferred-electron number, respectively. Thus, they can provide much more information for the counting statistics that the shot noise. Moreover, in the cases of $\Gamma_{L}/\Gamma_{R}=1$ and $\Gamma/J\gg 1$, the behavior of the shot noise is mainly governed by the quantum coherence, see Figs. 4(b1), 4(b2) and 4(b3). However, these characteristics also depend on the temperature $k_{B}T$, i.e., the quantum coherence will play an essential role in the electron tunneling processes with decreasing temperature, see Figs. 3(a3)-3(d3) and 4(a3)-4(d3).

These properties of the $\Gamma_{L}/\Gamma_{R}\leq 1$ case can also be explained through the cotunneling-induced redistribution of the occupation probabilities. In the $\Gamma_{L}/\Gamma_{R}=0.1$ case, the occupation probabilities of the two singly-occupied eigenstates are much smaller than that of empty-occupied eigenstate, thus, the conduction electrons have a very short dwell time, which is contrary to the $\Gamma_{L}/\Gamma_{R}=10$ case. In the $\Gamma/J\gg 1$ case, the two electron tunneling can occur through the cotunneling processes induced by the transitions between the doubly-occupied $\left|2\right\rangle$ and singly-occupied $\left|1\right\rangle^{\pm}$ eigenstates and the succeed sequential processes induced by the transitions between the singly-occupied $\left|1\right\rangle^{\pm}$ and empty-occupied $\left|0\right\rangle$ eigenstates. Thus, the cotunneling assisted sequential tunneling processes in the $\Gamma/J\gg 1$ case still play a important role in determining the FCS. Additionally, in the $\Gamma/J\ll 1$ case, the cotunneling processes can further increase the occupation probability of the empty-occupied eigenstate with decreasing ratio of $\Gamma$ to $J$, see Figs. 5(b1)-5(b3). This effect can further block the electron tunneling, then leading to the shot noise being relatively enhanced, see Figs. 3(b1)-3(b3).

Figures 6 and 7 show the influence of the left-right asymmetry of the QD-electrode coupling $\Gamma_{L}/\Gamma_{R}$ on the first four current cumulants for a given temperature $k_{B}T=0.1$. In the cases of $\Gamma_{L}/\Gamma_{R}>1$ and $\Gamma>J$ (the intermediate value), the range of the crossover region, in which the electron cotunneling processes decrease the values of Fano factors while the quantum coherence increase that of Fano factors, increases with increasing the ratio of $\Gamma_{L}$ to $\Gamma_{R}$, see Fig. 6. Whereas in the cases of $\Gamma_{L}/\Gamma_{R}<1$ and $\Gamma/J\gg 1$, the interplay between the electron cotunneling processes and the quantum coherence has a relatively remarkable influence on the high-order current cumulants with decreasing the ratio of $\Gamma_{L}$ to $\Gamma_{R}$, and determines whether the super-Poissonian distributions of the shot noise and the kurtosis take place or not, and whether the transition of the skewness from positive to negative values occurs or not, see Fig. 7. These results can also be understood with the help of the redistribution of the occupation probability induced by the left-right asymmetry of the QD-electrode coupling, see Fig. 8.

III.3 The side-coupled double QD system with weak quantum coherence

We finally discuss the influences of the electron cotunneling processes on the first four current cumulants in the side-coupled double QD system with weak quantum coherence. Here, the hopping parameter is thus chosen as $J=1$. According to the parameters of the QD system, we choose the three fixed bias voltages, under which the different transitions between the QD eigenstates participate in the electron tunneling processes, namely, $V_{b}=2.5$ corresponding to the transitions between the singly-occupied $\left|1\right\rangle^{-}$ and empty-occupied eigenstates, $V_{b}=4.5$ corresponding to the transitions between the singly-occupied $\left|1\right\rangle^{\pm}$ and empty-occupied eigenstates, and $V_{b}=6.5$ corresponding to the transitions between the singly-occupied $\left|1\right\rangle^{\pm}$ and empty-occupied eigenstates and the transitions between the doubly-occupied $\left|2\right\rangle$ and singly-occupied $\left|1\right\rangle^{+}$ eigenstates. In this situation, the properties of the first four current cumulants are well determined by the electron cotunneling processes because the quantum coherence indeed has a very small influence on the values of the first four current cumulants, see Figs. 9 and 10. In the case of $\Gamma_{L}/\Gamma_{R}>1$, the electron cotunneling processes have a relatively obvious influence on the first four current cumulants, see Fig. 9; whereas in the case of $\Gamma_{L}/\Gamma_{R}<1$ that have a slight influence on the first four current cumulants, see Fig. 10. It is important to note that the electron cotunneling processes do not change the intrinsic statistical properties of current cumulants, namely, whether the super-Poissonian distribution of the current cumulants take place or not, and whether the sign transitions of the values of the skewness and the kurtosis occur or not, see Figs. 9 and 10.

IV CONCLUSIONS

We have developed an efficient non-Markovian FCS formalism taking into account both the sequential tunneling and cotunneling processes, and studied the influences of the quantum coherence and the cotunneling assisted sequential tunneling processes on the first four current cumulants in a side-coupled double QD system. In the strong quantum-coherent side-coupled double QD system, it is numerically demonstrated that, in the sequential tunneling regime, the competition or interplay between the cotunneling processes and the quantum coherence determines that whether the super-Poissonian distributions of the shot noise, the skewness and the kurtosis take place, and whether the sign transitions of the values of the skewness and the kurtosis occur. These characteristics depend on the temperature of the QD system, the left-right asymmetry of the QD-electrode coupling, and the magnitude of the coupling strengths. However, in the weak quantum-coherent side-coupled double QD system, the cotunneling processes has a relatively slight influence on the statistical properties of current cumulants, which also depends on the left-right asymmetry of the QD-electrode coupling and the corresponding coupling strengths. Consequently, the dependence of the FCS on the quantum coherence and the cotunneling processes is necessary to be considered in the open strong quantum-coherent quantum systems, even through the electron tunneling is mainly dominated by the sequential tunneling processes.

V ACKNOWLEDGMENTS

This work was supported by the Shanxi Natural Science Foundation of China under Grant No. 201601D011015, Program for the Outstanding Innovative Teams of Higher Learning Institutions of Shanxi, NKRDP under Grant No. 2016YFA0301500, NSFC under Grants Nos. 11204203, 11434015, 61227902, 61378017, KZ201610005011, SKLQOQOD under Grant No. KF201403, SPRPCAS under Grants Nos. XDB01020300, XDB21030300.