Quantum noise theory for quantum transport through nanostructures

We develop a quantum noise approach to study quantum transport through nanostructures. The nanostructures, such as quantum dots, are regarded as artificial atoms, subject to quasi-equilibrium fermionic reservoirs of electrons in biased leads. Noise operators characterizing the quantum fluctuation in the reservoirs are related to the damping and fluctuation of the artificial atoms through the quantum Langevin equation. The average current and current noise are derived in terms of the reservoir noise correlations. In the white-noise limit, we show that the current and current noise can be exactly calculated by the quantum noise approach, even in the presence of interaction such as Coulomb blockade. As a typical application, the average current and current noise through a single quantum dot are studied.


I. INTRODUCTION
Quantum transport through nanostructures is of importance in nano-science and nano-technology. Many electronic devices based on nanostructures, such as single-electron transistors, have been studied in the past decades for their potentials in various applications. Recently, in the efforts aiming at the coherent control of single electrons or electron spins, the quantum transport methods have been used to detect the quantized motion of electrons in nanostructures. 1,2 Besides the average current, the current noises also contain useful information about the quantum dynamics in nanostructures. 3,4,5 Various theoretical approaches have been developed to treat the quantum transport problem. The Landauer-Bütikker formula has established the basic relationship between scattering amplitudes and currents through nanostructures. 6,7 The nonequilibrium Green's function method provides a perturbation scheme to deal with the many-body interaction effects in quantum transport. 8,9 In the past few years, approaches based on notions in quantum optics were developed to study timedependent quantum transport processes in solid-state structures. Most of these quantum optics approaches adopt the density matrix formalism, with master equations or rate equations in the Schrödinger picture. 10,11,12,13,14,15,16,17,18 Very recently, quantum Langevin equation in the Heisenberg picture was also used to establish the quantum rate equations to study the transport problem. 19 In this paper, instead of the master equations or the rate equations, we will develop a quantum noise approach based on the quantum Langevin equation to the quantum transport problem. Essentially, we recognize that a general quantum transport problem can be regarded as a system-plus-reservoir problem. In this sense, the total system is divided into several sub-systems (see Fig 1). The central system (system for short) is a nanostructure, such as a quantum dot or coupled quantum dots. This subsystem contains several discrete electronic energy levels, resembling an artificial atom. The electrons in the leads, which have a continuous energy spectrum and are kept in quasi-equilibrium, constitute the fermionic reservoirs. The electrons in the reservoirs can be treated as free quasi-particles with the screened Coulomb interaction taken into account as a renormalization of electron effective mass. The central system and the reservoirs are coupled together to each other through hopping across the barriers. With this observation, it is natural to treat the quantum transport problem in the framework of the quantum open system method, the quantum Langevin equation, a standard approach in quantum optics to study cavity photon decay and atom damping.
As compared to the application in quantum optics, the quantum Langevin approach in the quantum transport problem has two features to be singled out: (i) The reservoirs consist of electrons, which are fermions while the baths in quantum optics are bosonic, and (ii) when finite biases are applied between different leads, the electronic reservoirs in different leads are in quasi-equilibrium with different chemical potentials but do not stay in equilibrium with each other. Our investigation in this paper will refine these features. As illustrative applications of our approach, the resonant transport through a single quantum dot is investigated for both the single-level case and the Coulomb blockade case.
The quantum Langevin approach is a natural formalism to study the noise spectroscopy of quantum dynamics in nanostructures, 5 which is particularly interesting for small quantum systems where the signals are often much weaker than the shot noises. When the coupling between the leads and the nanostructures can be described in the Markovian approximation, which is justified in large bias cases, the quan- 1: (color online) Schematic illustration of transport through a quantum dot. The whole system is divided into three parts, the central system, and the left and right reservoirs. The central system is characterized by eigen-states |i with discrete energies. Quantum noise operators L(t) and R(t) are introduced to describe the reservoirs.
tum Langevin approach provides an exact treatment of the interaction within the nanostructure. Furthermore, the quantum Langevin equation establishes a fundamental relationship and analogy between photon emission and electron tunneling processes, providing new understanding of quantum transport phenomena with notions and methods from quantum optics. The paper is organized as follow. In Sec. II, we introduce the basic concepts and the general formalism of the quantum noise approach to treat the quantum transport problem. In Sec. III and IV, we apply the quantum noise approach to transport through a single quantum dot containing a single level and double energy levels, respectively. In Sec. V, we show the relations between our approach and other quantum transport theories. We conclude and give an outlook of our approach in Sec.VI.

A. Quantum Langevin equations for quantum transport
In general, the quantum transport problem of nanostructures can be modeled by the following Hamiltonian: where H sys describes the nanostructure, such as a quantum dot, with multiple discrete energy levels. The leads, which play the role of reservoirs, are described by the Hamiltonian H lead . The electron tunneling between the leads and the nanostructure is included in H T . For the two-lead case, the leads Hamiltonian H lead and the tunneling Hamiltonian H T can be written as where b k and c j are the annihilation operators of the left and right leads with continuous spectra ω (L) k and ω (R) j , respectively. The tunneling is characterized by the coefficients ξ ik and ζ i j . Note that we have neglected the interaction in the leads as a common approximation for a Fermi sea with the Coulomb interaction effectively taken into the renormalized quasi-particle spectra. Now, we consider the Heisenberg equations of motion of the system and reservoir operators. For simplicity, we show equations of motion for the simplest single level case, i.e. a i (t) = a(t), ξ ik = ξ k , ζ i j = ζ j and H sys = ω 0 a † a. The multilevel case will be discussed later in this paper. Straightforward calculation giveṡ In the following, we try to eliminate the lead variables from the equation of motion (3a) of the system operator. To this end, the formal solution for b k (t) is written as With this formal solution, the following relation is obtained where the input noise operator L in (t) due to the left lead is defined as The damping term in Eq. (5) arises from the Markovian approximation 21 under the continuous limit where D(ω k ) is the density of the states of the lead. Here we have assumed that D(ω k )ξ 2 (ω k ) is flat around the frequency ω 0 and is widely used as the tunneling rate in nanostructure quantum transport problems. Similarly, for the right lead where the noise operator of the right lead R (t) is defined as and γ R is the tunneling rate to the right lead. Using Eqs. (5) and (9), we obtain the quantum Langevin equation for the system operator a(t): Similar to the cases in quantum optics, the two electronic leads, which play the role of fermionic reservoirs, induce the damping and the fluctuations through the noise operators.

B. Projection operator formalism for interacting systems
For an interacting system, the complexity of the quantum Langevin equations arises from the evolution induced by the system Hamiltonian H sys . To deal with such complexity, we introduce the projection operators of the interacting system in this subsection.
Though there may be interaction between the electrons in the system Hamiltonian H sys , the artificial atom can always be considered as consisting of a few discrete many-body energy levels. In other words, we can diagonalize the system Hamiltonian H sys as where σ i j = |i j| is the projection operator from | j to |i , with |i being the eigen-state of H sys of energy ω i . In general cases, determined by the system Hamiltonian H sys , the fermionic operators a i and a † i can be written in terms of the projection operators as where T (i) andT (i) are N × N matrices associated with the fermionic operators a i and a † i , andT (i) ≡ T (i) † . The commutative relation between the projection operators σ i j can be calculated with σ i j σ kl = σ il δ jk . In the following calculations we also need to identify the commutative relation between the projection operators σ i j and the reservoir operators, i.e. b k (b † k ) and c j (c † j ). Note that the eigen-state |i is also an eigen-state of the electron number operator in the quantum dotN = i a † i a i , i.e.N |i = N i |i . Thus, the projection operator σ i j corresponds to a definite electron number change N i − N j , which is either odd or even. Consequently, σ i j and b k (b † k ) have the following (anti-)commutative relation: where the factor The Heisenberg equation for the σ i j iṡ With the help of the definition of noise operators and the first Markovian approximation, we obtain the quantum Langevin equation for the projection operator σ i j where ∆ i j = ω j −ω i , and the coefficients D i j with In principle, the quantum Langevin equation for the system operators is equivalent to a quantum stochastic equation if we introduce the quantum Wiener process, 30 and the properties of the their solution can be discussed by defining the quantum stochastic integration. 30 Thus, we point out that the quantum transport problem provides an experimentally accessible proving ground for the quantum stochastic theory. Instead of further discussing the mathematical properties of Eq. (17), in this paper we will focus, through concrete models, on how to derive the observable quantities in quantum transport.

C. Boundary Relation and Causality
Besides the input noise operators L in (t) and R in (t), the output noise operators 29,30 can be defined as where t f is a time in the remote future. Similar to Eqs. (5) and (9), the first Markovian approximation gives the following relations According to Eqs. (5) and (21), the "boundary relation" between the noise operators and the system operator is 29 Similarly, for the right lead According to the quantum Langevin equation (11), the fermionic system operator d(t) ∈ {a(t), a † (t)} at time t only depends on the input noise operators at time t ′ < t. As a result, in the Markovian limit, the causality relation reads 30,32 For the similar reason, the system operator at t is independent of the output noise operators at time t ′ < t According to Eqs. (22∼25), the anti-commutators between noise and system operator are converted to those between system operators 30,32 where step function θ(t) is defined as For the multi-level case, this causality relation Eq. (26) can be generalized to the system projection operators σ i j , i.e.
The choice of the commutative and anti-commutative relation in Eq. (28) is determined by the parity of the electron number change, see Eq. (15).
In the following, to simplify the notation, we will omit the subscript "in" of the input noise operators, unless stated otherwise.

D. Current and current noise
For the quantum transport problem, we are interested in the average current and the current noise spectra. In this subsection, we will give the expressions of such quantities in terms of the noise operators.
We consider the current through the right lead for example. For simplicity, let us first study the single level case. The formula for the multi-level case with Coulomb blockade will be discussed later. The current operator can be defined as the changing rate of the electron number on the right lead, i.e.
The second line is obtained by noticing the relations in Eq. (5) and Eq. (9). We point out that the current operator can be divided into two parts: (i) the damping part γ R a † (t) a (t), which is proportional to the level occupation and the escaping rate γ R ; and (ii) the fluctuation part (the last two terms), which is induced by the noise operators R (t) and R † (t).
For the average current, we take the average of the current operatorÎ R over the thermal states of the leads And for the current noise, we first calculate the current-current correlation function At steady state, its Fourier transformation gives the current noise spectrum 20 To calculate the correlation Î R (t)Î R (t + τ) in g (2) (τ), by the definition ofÎ R in Eq. (29b), one need to calculate the two time correlation such as In Sect. III, we will show that the fluctuation part in the current operator does not contribute to the average current, so the average current Î R = γ R a † (t) a (t) is held. But the fluctuation terms will contribute to the current noise through the correlations in Eq. (33).

III. APPLICATION I: SINGLE LEVEL TRANSPORT
In this section, the general quantum Langevin formula is applied to the resonant transport through a quantum dot. As the first example, we consider a model in which only one single energy level in the quantum dot is relevant. The system Hamiltonian reads We consider the large bias condition and assume that the single-particle energy level with energy ω 0 is well within the bias window, i.e. µ L − ω 0 , ω 0 − µ R ≫ γ L , γ R , for µ L/R being the chemical potentials of the left/right leads. According to the discussion in Sec. II A, the quantum Langevin equation readsȧ are defined in the rotating reference frame to single out the slow-varying dynamics. In the white-noise limit, the correlation between the noise operators can be written as (see Appendix A) Using these relations, we calculate average current and current noise.

A. Average current
From Eq. (35), the system operatorã (t) in terms of the noise operators is where Γ = γ L + γ R . Multiplying the noise operatorL † (t) on both sides of Eq. (37), we have Here we have assumed that at initial time t = 0, the system and the reservoir are independent, i.e., L and Thus, according to Eq. (29b), the fluctuation part of the current operator does not contribute to the average current, and the average current becomes In order to determine the mean occupation number ã †ã , we use the equation of motion The ensemble average leads to Thus, the averaged population in the quantum dot is As a result, the average current at steady state for t → +∞ is which is the well-known result for the resonant tunneling transport. 9,11

B. Current noise
To investigate the current noise, we calculate the currentcurrent correlation Î R (t)Î R (t + τ) . With the definition of the current operator in Eq. (29b), the noise contains typically twotime correlations like and We will discuss such correlations one by one.
Noticing that the electron number correlation function n (t)n (t + τ) contains only the system operators, we use the quantum regression theorem 24 and Eq. (43) and obtain d dτ n (t)n (t + τ) = −Γ n (t)n (t + τ) + γ L n (t) .
This equation, together with the initial condition with respect to τ, i.e. for τ = 0, n (t)n (t + τ) = n (t)n (t) = n (t) , determines the occupation number fluctuation in the quantum dot. The steady state correlation is lim t→+∞ n (t)n (t + τ) The other terms contain the correlations between the system and noise operators. Taking ã † (t)R (t)R † (t + τ)ã (t + τ) for example, according to Eq. (37), we have where the four-time noise correlation is defined as According to the independent noise assumption and the whitenoise approximation, Thus, we have Similarly, It can be checked that all the other terms in the current-current correlation function vanish. Consequently, the current-current correlation function is and its Fourier transformation gives the current noise spectra This result accords with the ones derived from other approaches, 17 and shows that the presence of the single level quantum dot suppresses the zero-frequency current noise to half of the Poisson value S P = 2e I R ss in the case γ L = γ R . It is worth to emphasize that, clearly shown in our quantum noise approach, although the fluctuation part [see Eq. (29b)] of the current operator does not contribute to the average current, it does the current noise. According to our approach, the current-current correlation originates from three different kinds of sources: (i) the on-site number-number correlation [Eq. (49)], which always contributes a positive correlation, (ii) the correlation between the fluctuation terms Eq. (53), which induces a white-noise correlation, and (iii) and the correlation between the on-site number and the fluctuation term Eq. (54), which always provides a negative correlation. This classification of current-current correlation is also valid in the interacting case, as will be discussed below.

A. Average current
Now we apply the general theory to the Coulomb blockade case. For simplicity, we assume that only one single orbital level in the quantum dot is relevant (i.e., within the energy range of interest). The system Hamiltonian reads where ω ↑,↓ are the single electron energy for spin-up and spin-down electrons in the quantum dot, and U is the Coulomb interaction strength between two electrons. In this paper, we consider the large U limit, i.e. ω ↑ + U, As has been discussed in Sec.II B, though there is interaction between the electrons in the system Hamiltonian, H sys is diagonalized as and the projection operators are related to the Fermion operators by where σ i j = |i j| for i, j = v, ↑, ↓ and d. The subscripts v, ↑, ↓ and d represent the vacuum state, spin-up, spin-down, and doubly occupied state, respectively (Fig. 2). Annihilating an electron with definite spin (say spin-up) from the quantum dot consists of two different projection processes depending on whether the spin-down level is occupied or not.
Here, we assume that the quantum dot is coupled to ferromagnetic leads. Thus, the electron with different spin can tunnel on and off the quantum dot with different rates. The quantum Langevin equations of the projection operators σ i j in this Coulomb blockade case follow the general formula in Sec. II B. The resultant equations for the diagonal elements are and those for the off-diagonal elements arė where The damping rate Γ s = γ Ls + γ Rs , with the spin dependent tunneling rates where ξ s and ζ s are the coupling amplitudes of the quantum dot to the left and right leads, and D (L) s (ω) and D (R) s (ω) are the spin-resolved density of states of left and right leads, respectively.
These Langevin equations of the system variables are analogous to the ones used to describe the quantum theory of Laser 23,24 . In the quantum theory of Laser, the atoms are subject to bosonic reservoirs, while in our quantum transport case, the quantum dot is "pumped" by a fermionic reservoir (the left lead), and output to another fermionic reservoir (the right lead).
In contrast to the non-interacting case [see Eq. (35)], the noise operators couple to the system projection operators in Eqs. (60) and (61). The correlations between noise operators and projection operators, such as L † s (t)σ i j (t) , are calculated according to the generalized causality relation Eq. (28). Taking L † ↑ (t)σ v↑ (t) for example, whereσ v↑ (t) = σ v↑ (t)e iω ↑ t is the slow-varying amplitude of projection operator, andL † With these correlations, ensemble average of Eqs. (60) and (61) gives the "rate equations" for the diagonal elements: and for off-diagonal elements: where Eq. (66) shows that the coherence between energy levels vanish after a long time, i.e.
This indicates the two different spin channels are incoherent, which physically arises from the fact that noise operators with different spins are uncorrelated, i.e. L † ↑ (t)L ↓ (t ′ ) = 0. The rate equations (65) describe the population transfer between each energy levels, and the steady state populations are The average current is The current vanishes if γ R↑ = 0 or γ R↓ = 0. This is because turning off a certain spin channel, say the spin-up channel, i.e. γ R↑ = 0, will induce the accumulation of the spin-up electron on the quantum dot. Then the electron tunneling of both spin channels is blockade due to the strong Coulomb interaction. When the tunneling rates are spin-independent, i.e. γ L↑ = γ L↓ = γ L and γ R↑ = γ R↓ = γ R , the average current in Eq. (69) becomes Î R = 2γ L γ R /(2γ L + γ R ), which accords with the results obtained by other methods. 11,16,17

B. Current noise
Now we turn to the current noise. Similar to the noninteracting case, the two-time correlations of the following form should be calculated In the interacting case, the system projection operators cannot be expressed in terms of the simple integration of the noise operators as in the non-interacting case, see Eq. (37). The causality relations introduced in Sec II C provide us a convenient way to convert the noise-system correlation to the system-system correlation. Thus, in the white-noise limit, as a powerful tool, the quantum regression theorem is applied to calculate the two-time system correlations. Noticing that the noise operator R s (t) plays the role of "annihilation operator", the correlations between noise and system operators can be calculated following the spirit of the Wick's theorem(see Appendix A).
Taking the spin-up component for example, the correlation The second line of Eq. (71) is simplified by noticing the fact [R ↑ (t) , R † ↑ (t ′ )] + = γ R δ(t − t ′ ), and the third line vanishes since [a † ↑ (t) , R † ↑ (t + τ)] + = 0. This white-noise correlation provides a constant current noise background. Due to the δ(τ) function, only equal-time correlation (τ = 0) is relevant. By noticing Eq. (69), this correlation is written as For the correlation a † , it can be translated into the correlations between the system operators using the causality relations The first term cancels out the contribution of Eq. (70a). As a result, the spin-up current-current correlation is The current-current correlations of different spin components are calculated similarly, and in general, they can expressed in terms of the correlations of system operators as Here we have shown an analogous form of the current-current correlation to the second order optical coherence function. 24 The last term of Eq. (75) can be calculated from the quantum regression theorem. By this theorem, the current-current correlation function is determined by the rate equations (65), and in the Coulomb blockade case, it does not show the effect of the quantum coherence terms in Eq. (66). The total current correlation function is Its Fourier transformation gives the current noise spectrum S (ω). In the spin independent tunneling rate case, i.e. γ L↑ = γ L↓ = γ L and γ R↑ = γ R↓ = γ R , the noise spectrum is This result deviates from the single-level case [see Eq. (56)], due to the presence of the Coulomb interaction. Typical current noise spectra for the spin dependent tunneling rate case are shown in Fig. (3a). The Fano factor is as a function of the imbalance between spin-resolved tunneling rates P L and P R , which are defined as P L = γ L↑ /(γ L↑ + γ L↓ ) and P R = γ R↑ /(γ R↑ + γ R↓ ), for given total tunneling rates γ L↑ + γ L↓ = γ R↑ + γ R↓ = 1. The white thick lines are the boundary between sub-Poisson and super-Poisson regime, i.e. F = 1.
It is found that super-Poissonian noise arises when the tunneling is spin dependent, which can be realized, e.g., by using magnetized barriers between the leads and the quantum dot. Super-Poissonian noise appears when the numerator of the second term becomes negative. The Fano factor as a function of the tunneling rate imbalance is shown in Fig. (3b).
Physically, the super-Poissonian noise is the consequence of the dynamical channel blockade effect. 25,26 The tunneling rate imbalance induces different average currents for the two spin channels. Thus, in additional to the noises of each channels themselves, the shot noise between the two channels gives rise to the low frequency noise enhancement. Such kind of shot noise is absent when P L = 1 − P R since the two spin channels have the same current.

A. Relation to Landauer-Büttiker formula
Here, we show that the Landauer-Büttiker formula can be reproduced by the quantum Langevin approach. For simplicity, let us consider the single energy level transport example.
According to Eq. (29b) and the boundary relations Eq. (23), the current operators can be expressed solely by the input and output noise operators. For example, Thus, it is clear that the average current is divided into the input current proportional to R † in (t)R in (t) and the output current proportional to R † out (t)R out (t) . Furthermore, defining the scattering matrix S, the Fourier transformation of output noise operators is expressed in terms of the input operators as and where the functions T i← j (ω) and R i← j (ω) can be regarded as the energy dependent transmission and reflection coefficients from lead j to lead i. The Fourier transformation of the average current is Noticing the relation Eq. (80) and the correlations between the noise operators we obtain the Landauer-Büttiker-like formula of the average current with the transmission spectrum (86)

B. Relation to non-equilibrium Green's function theory
Here we discuss the relation between the quantum Langevin approach and the non-equilibrium Green's function theory for the quantum transport problems. The retarded Green's function is defined as 27 for s =↑ or ↓, from which the local density of states (LDOS) D s (ω) is given by whereG s (ω) is the Fourier transformation of G s (τ). The LDOS contains the essential information about the system relevant in quantum transport. In the following, we take the Coulomb blockade example, and give the retarded Green's function and the LDOS using the quantum noise approach.
Noticing that the definition of the retarded Green's function Eq. (87) only involves the system operators a s (t) and a † s (0), we apply the quantum regression theorem to calculate their correlations. The retarded Green's function can be expressed in terms of the two-time correlations between the projection operators. Consider the spin-up component for example, The equations of motion for these projection operators are given in Eq. (66). By the quantum regression theorem, the two-time correlations are determined by d dτ with the initial condition for τ = 0 where the coefficient matrix M is defined as Here, γ L↑ = γ L↓ = γ R↑ = γ R↓ = γ is assumed for simplicity. The other correlations involved in Eq. (89) can be similarly calculated.
Thus the retarded Green's function is with the renormalized frequencies and the weight factors The Fourier transformation of the Green's function gives the LDOS (see Fig. 4). It is obvious that, for the large U case considered in this paper, the LDOS consists of two Lorentz shape peaks, centered around ω ↑ and ω ↑ + U. The two peaks separate from each other by U, which is a signature of the Coulomb blockade. 9

VI. CONCLUSIONS AND OUTLOOKS
In this paper, we have developed a quantum noise approach to treat the quantum transport through a nanostructure such as a quantum dot. We formulate the average current and the current noise in terms of the correlations between the noise operators. The quantum noise approach is applied to a paradigmatic example, namely, transport through a single quantum dot under large biases and both the non-interacting and Coulomb blockade cases are investigated. With the Markovian approximation for the tunneling processes, the electron-electron interaction in the quantum dot can be exactly treated.
The quantum noise approach provides a bridge between quantum optics and quantum transport. Thus notions and methods in the quantum optics could be adopted to study quantum transport through nanostructures. Although we show the application of the quantum noise approach by a single quantum dot example, the theory is not limited to this simple case. On one hand, the system could be generalized to more complicated ones, such as coupled quantum dots, multi-end nano-circuits, or systems with spin interaction. On the other hand, the reservoirs of other kinds, such as phonon baths or spin baths, could be included to explore how such reservoirs would affect the current and current noise, providing a method of studying the bath dynamics via current noises. The Markovian approximation may also be released with colored noise correlation functions of the reservoir used in lieu of the whitenoise model adopted in this paper.

APPENDIX A: PROPERTIES OF THE NOISE OPERATORS
In this appendix, we give the correlations between noise operators. We consider the single-level case here. The physical quantities of interest are determined by the noise correlations such as L † (t)L (t ′ ) . According to the definition of the noise operators, where D(ω k ) is the density of states in the leads, and is the thermal occupation number of the lead in quasiequilibrium. The Markovian approximation requires two assumptions. First assumed is the "flat band" condition that the relative change of the effective density of states around the resonant ω 0 over a range of the characteristic damping rate γ L is much less than unity, i.e., whereD(ω k ) ≡ ξ 2 (ω k )D(ω k ). Under this condition,D(ω k ) can be replaced by its value at ω 0 , and the correlation becomes Here, the zero temperature case has been considered for simplicity. Second, under the large bias condition, the resonant level ω 0 is far away from the fermi energy and the conduction band bottom (chosen as the energy origin), i.e.
In this case, the integration over ω k is extended to ±∞, and finally results in the white-noise correlation where γ L = 2πξ 2 (ω 0 ) D (ω 0 ). Similarly, for the right lead, Here, we have use the fact that the thermal occupation number n (R) th ω j = 0 for the right lead around the resonant level ω 0 . In the same way, one can show that other noise correlations vanish, i.e. L (t)L † t ′ = R † (t)R t ′ = 0.
(A8) Note that Eq. (A8) implies that the noise operatorsL † (t) andR (t) play the role of "annihilation operators", since they always give zero correlations when they stand on the rightmost position. With this observation, the normal-ordered product of noise operators can be defined by placingL † (t) andR (t) to the rightmost position, and the expectation value of the normal-ordered product vanishes identically. Thus, the Wick's theorem is generalized to the noise operators and the current and current noise can be exactly calculated in the white-noise limit.