Partition-free theory of time-dependent current correlations in nanojunctions in response to an arbitrary time-dependent bias

# Partition-free theory of time-dependent current correlations in nanojunctions in response to an arbitrary time-dependent bias

Michael Ridley, Angus MacKinnon, Lev Kantorovich
###### Abstract

Working within the Nonequilibrium Green’s Function (NEGF) formalism, a formula for the two-time current correlation function is derived for the case of transport through a nanojunction in response to an arbitrary time-dependent bias. The one-particle Hamiltonian and the Wide Band Limit Approximation (WBLA) are assumed, enabling us to extract all necessary Green’s functions and self energies for the system, extending the analytic work presented previously [Ridley et al. Phys. Rev. B (2015)]. We show that our new expression for the two-time correlation function generalises the Büttiker theory of shot and thermal noise on the current through a nanojunction to the time-dependent bias case including the transient regime following the switch-on. Transient terms in the correlation function arise from an initial state that does not assume (as is usually done) that the system is initially uncoupled, i.e. our approach is partition-free. We show that when the bias loses its time-dependence, the long time-limit of the current correlation function depends on the time difference only, as in this case an ideal steady state is reached. This enables derivation of known results for the single frequency power spectrum and for the zero frequency limit of this power spectrum. In addition, we present a technique which for the first time facilitates fast calculations of the transient quantum noise, valid for arbitrary temperature, time and voltage scales. We apply this to the quantum dot and molecular wire systems for both DC and AC biases, and find a novel signature of the traversal time for electrons crossing the wire in the time-dependent cross-lead current correlations.

Department of Physics, The Blackett Laboratory, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom
Department of Physics, King’s College London, Strand, London, WC2R 2LS, United Kingdom

## 1 Introduction

Electronic circuit components with nanoscale dimensions can now be fabricated and tuned to form active circuit components [1]. In addition to the speed-up in processing power that arises from sub-micrometre size [2], molecular junctions also enable a massive speedup in device operation due to THz intramolecular transport processes and fast electron traversal time [3]. Subsequent to the initial proposal of molecular rectification in 1974 [4], chemical fabrication techniques have led to the realization of many interesting devices, including molecular wires [5, 6], single-electron transistors [7], frequency doublers and detectors [8, 9] and switches for fast memory storage [10, 11]. In addition, conductance properties of nanostructures subjected to strong time-dependent external fields have been the subject of intense experimental research. This research includes work on photon assisted tunneling [12, 13] and transport through AC biased carbon-based nanostructures in the GHz-THz regime [14, 15, 16, 17].

In contrast to classical electronics, the time-dependent current in molecular structures may undergo fluctuations that have a comparable magnitude to the current signal itself, so that a theory of time-dependent fluctuations is essential for the design and control of these devices [18]. Moreover, time-dependent current-current correlations and their associated frequency-dependent noise spectra contain information which is not present in the first moment of the current [19]. This includes deviation from classical behaviour in the Fano factor due to Pauli repulsion [20, 21], detection of fractional charges for quantum Hall quasiparticles [22] and the determination of transmission probabilities [23]. When the external field driving the transport process depends upon time, the transient current correlations provide information on intramolecular ‘circular’ currents that cannot be studied using the current alone [24]. Recent measurements of shot noise in graphene irradiated by THz fields showed an enhancement of the shot noise due to the excitation of electron-hole pairs in the sample [25].

In general, nanoelectronic devices possess noise spectra which are nonlinear functions of frequency. When in equilibrium, there are two regimes, namely the low regime, in which Johnson-Nyquist noise is evident [26, 27], and the high scenario in which zero-point fluctuations dominate [28]. When a bias is applied to the system, one observes in addition the shot noise, which results from the discreteness of electronic charge and the Pauli exclusion principle. At high frequencies, it was shown that the correct noise spectra is an asymmetric function of the frequency due to the dominance of zero-point photon fluctuations there [28, 29]. Distinct negative and positive frequency components of the current noise due to quasiparticle tunneling across a Josephson junction have been measured experimentally [30], and may be physically interpreted in terms of the transfer of energy quanta during the coresponding absorption and emission processes [31]. In the theoretical literature, both symmetric [32, 33, 34, 35] and asymmetric [36, 37, 38] noise spectra have been classified and studied.

The Landauer-Büttiker (LB) theory of shot and thermal noise represents a significant milestone in the development of the theory of current fluctuations in nanoscale systems [39, 33, 34, 40, 19]. Originally, it was developed within a scattering matrix approach to coherent quantum transport, wherein one typically considers a molecular junction as a subsystem coupled to macroscopic leads, which act as heat and particle reservoirs. Electrons in the leads are treated as independent plane waves, populated according to the Fermi distribution function, and propagated onto the molecule, where they scatter. Experiments have demonstrated a good agreement between experiment and the noise spectra obtained from the scattering theory for both the low-frequency noise [41, 42] and for power spectra that depend upon the frequency of the measurement device [43, 28, 44]. In these studies the scattering potential is chosen to be static, but time-dependent scattering formalisms have been developed which enable the calculation of current and current noise in response to an AC potential in the leads [45, 46, 47], in both the adiabatic [45] and non-adiabatic [48, 49] regimes. These approaches make use of the Floquet theorem, as do master equation approaches, which expand scattering states into a harmonic series [50, 35], generating functional approaches to the full counting statistics (FCS) [51] and reduced density matrix methods that make a perturbative expansion in the lead-molecule coupling [52]. The noise response to an AC field has been shown to carry information on the production of electron-hole pairs that does not appear in the noise response to a DC bias [53]. Moreover these electron-hole pairs are correlated and able to propagate through the molecular junction into separate terminals [54, 55]. In a generating functional approach to the full counting statistics of an AC-driven system, it was proven that a periodic Lorentzian voltage signal with quantized flux minimized the noise, i.e. it was reduced to the DC level [56, 57]. In recent experiments, these quantized voltage pulses, known as levitons, have been experimentally realized [58] and approximated by a biharmonic driving field [59]. Even given the restriction of periodic time-dependence, one can study a rich range of phenomena, such as photon-assisted tunneling (PAT) [60, 54, 35, 61], quantum pumping [62, 63], and the interplay of an external driving field parameters with Fabry-Pérot conductance oscillations in graphene nanoribbon (GNR) and carbon nanotube (CNT) systems [64].

The Nonequilibrium Green’s Function (NEGF) or Keldysh method for the calculation of dynamical quantum statistical averages can be used to re-express time-dependent transmission functions, currents and particle populations in terms of products of self-energies and Green’s Functions [65, 66, 67, 68]. The equivalence of this picture to the Landauer-Büttiker theory in the noninteracting case is well known [69, 70], but it can also be extended to perturbative calculations of noise in systems with a Coulombic interaction [71, 63]. Crucially for the present work, it involves the propagation of Green’s Function along a complex time contour that means the effects of the equilibrium preparation of the system are automatically taken into account in the dynamics resulting from the switch-on of a bias in the leads [68].

Many calculations of the time-dependent response of a nanojunction to the switch-on of a bias across the junction make use of the partitioned approach, in which the leads and molecule are completely decoupled prior to the switch on time , and suddenly coupled simultaneously with the addition of a time-dependent bias to the leads at [72, 73, 74, 75]. Partitioned approaches often involve relegation of to the distant past, because in noninteracting systems the Memory-loss Theorem [76] guarantees that the initial condition does not affect the long-time dynamics. However, transient dynamics was also studied within a partitioned approach following an artificial quench that instantaneously couples the molecule to the leads, as was recently done for phononic transport [77] (assuming that such an experiment can be done in practice). In the partition-free framework, one includes a coupling between the leads and molecule in the equilibrium Hamiltonian which describes the preparation of the system prior to the switch-on. Partition-free approaches to quantum transport have been implemented within NEGF [78, 76] and master equation [79] approaches. Recent calculations of transient noise characteristics have made use of the partitioned approach [80, 63, 81] and there are currently no published calculations of the transient current noise arising from a partition-free switch on process.

In recent years, partition-free generalizations of the LB formula for the current and particle number response to the switch-on of a static bias have been derived [82, 83, 68, 84]. This formalism makes use of the wide-band limit approximation (WBLA), and enables fast calculation of the transport characteristics of realistic systems at very low computational cost compared with other time-dependent schemes [84, 85, 86]. It was then extended by the present authors to the current response to an arbitrary time-dependent bias [87], and a practical scheme for implementation of this formula based upon the replacement of all frequency integrals with special functions was then developed [88, 89]. In the static bias partition-free switch-on approach pioneered in Refs. [83, 68, 84], an analytic result for the equal time lesser Green’s function was derived, from which the particle number in the molecular region and current in the leads can be derived. However, to calculate current-current correlations one needs an expression for the lesser Green’s function in the two-time plane, , and the formalism presented in Refs. [87, 88, 90] does this for the arbitrarily time-dependent bias. The ability to deal with arbitrary time-dependence enables us to study a wider class of switch-on problems, including those in which the bias is stochastic in time [90]. In the present work, we will extend our NEGF method further in order to develop an exact formalism enabling the study of transient current correlations resulting from an arbitrary time dependent bias in the leads. This method does not involve any assumption of adiabaticity or weak lead-molecule coupling, and neither is there any limitation on the kind of time-dependence which can be studied. This will be useful within the field of fast noise calculations for real molecular junctions driven by ultrafast pulses [3, 91, 24], and to new physics arising from the time-resolved nanoelectronic response to these pulses that includes the effects of the initial coupling.

The paper is organized as follows. In Section 2 we introduce the partition-free time-dependent NEGF formalism developed in Refs. [87, 88, 90], and show how to obtain generic formulas for the two-time current correlation function within the WBLA. In Section 3 expressions are derived for the long-time and static bias approximations in the frequency domain, thereby confirming that our formalism agrees with other published work. In Section 4 we present the results of numerical calculations of the two-time current-correlations in a two-terminal nanojunction, based upon a fast algorithm that is based on an expansion of the Fermi function with subsequent analytic removal of all frequency integrals. We calculate the time-dependent cross-correlations for single-site quantum dots and extended molecular wires of different sizes. We identify finite-size effects in the transient current cross-correlations which cannot be observed in single level systems. In particular, by studying the competition between wire length, end-site coupling and internal coupling on the molecule, we show that a resonant signature of the time taken for electronic information to cross the system can be seen in both the transient and steady state cross-lead correlations.

## 2 Partition-free Correlation Function

### 2.1 Time-dependent NEGF

In quantum transport processes, one is typically concerned with the time-dependent electronic response through a junction at measurement time to the switch-on of a bias at some initial time , which drives the system away from equilibrium. The equations of motion for quantum statistical averages are evolved along a complex time contour, consisting of an upper branch running from to , then along a lower branch running back from to , and finally along the imaginary time branch from to , where (it is adopted that in the following). Real times on the horizontal branches correspond to the nonequilibrium system, whereas on the vertical branch the equilibrium system is represented. The Hamiltonian we will use is formally identical to the one studied in Ref. [90] and is parametrized by the variable which denotes the contour ‘time’ variable specifying positions on the Konstantinov-Perel’ contour :

 ^H(t)=∑kαεkα(z)^d†kα^dkα+∑mnHmn(z)^d†m^dn+∑m,kα[Tmkα(z)^d†m^dkα+Tkαm(z)^d†kα^dm] (2.1)

Here, , and , are annihilation and creation operators of leads and central system electronic states, where for simplicity spin degrees of freedom are neglected. The first term is a Hamiltonian of the lead states belonging to each lead , the second is the Hamiltonian of the molecule sandwiched between the leads, describing hopping within the molecular structure, and the third term describes the coupling of the molecule to the leads. We collect elements of this Hamiltonian into a matrix consisting of ‘blocks’ corresponding to each of the physical subsystems it describes. For example the ‘block’ is the matrix with elements :

 h(z)=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝h11(z)0⋯h1C(z)0h22(z)⋯h2C(z)⋮⋮⋱⋮hC1(z)hC2(z)⋯hCC(z)⎞⎟ ⎟ ⎟ ⎟ ⎟⎠ (2.2)

In the molecular basis, we also define the -th component of the one-particle Green’s function on the Konstantinov-Perel’ contour:

 Gij(z1,z2)=−iTr[e−β^HM^Tγ[^di,H(z1)^d†j,H(z2)]]Tr[e−β^HM] (2.3)

The elements of the Green’s function form a matrix defined on the whole space of orbitals of all leads and the central region; correspondingly, one can introduce diagonal, and , as well as non-diagonal, , and , blocks of this matrix:

 (2.4)

The Green’s function for the central region is obtained by projecting the general equation of motion onto the matrix block:

 [iddz1−hCC(z1)]GCC(z1,z2)=1CCδ(z1,z2)+∫γd¯zΣCC(z1,¯z)GCC(¯z,z2) (2.5)

where is the unit matrix in the subspace, and

 ΣCC(z1,z2)=∑αhCα(z1)gαα(z1,z2)hαC(z2) (2.6)

is the matrix of the embedding self-energy, where is the isolated lead Green’s function, whose evolution is governed solely by the block of the Hamiltonian matrix, Eq. (2.2). The non-diagonal matrix blocks of the Green’s function are given by Eqs. (A.1) and (A.2) in Appendix (A). The blocks in Eq. (2.4) can then be further subdivided into subspaces defined by regions of the complex time plane. For example, the ‘left’ Green’s function is obtained by choosing and , and one can obtain its equation of motion using the Langreth rules [92, 93]:

 G┌CC(τ1,t2)⎡⎣−i←−−ddt2−hCC(t2)⎤⎦=[G┌CC.ΣaCC+GMCC⋆Σ┌CC](τ1,t2) (2.7)

where the differential operator in the left hand side acts on the left. One also defines the ‘right’ Green’s function by choosing and , the ‘lesser’ and ‘greater’ Green’s functions with, e.g., and , respectively, and the Matsubara Green’s function with . In addition, ‘retarded’ and ‘advanced’ Green’s functions are stipulated with a definite real-time ordering:

 Gr(t1,t2)=θ(t1,t2)[G>(t1,t2)−G<(t1,t2)] (2.8)
 Ga(t1,t2)=−θ(t2,t1)[G>(t1,t2)−G<(t1,t2)] (2.9)

The equations obtained by projecting Eq. (2.5) and its complex conjugate onto these subregions of the complex time plane are known as the Kadanoff-Baym equations, see, e.g., Ref. [68].

### 2.2 Generalized Expression from Wick’s Theorem

The current in lead can be obtained as the thermal average of the time derivative of the average charge in that lead, (where the spin-degenerate particle number is ). In all numerical calculations that follow, the electron charge will be set to . Given the noninteracting Hamiltonian in Eq. (2.1), the current operator has the form:

 ^Iα(t) = 2iq∑k,m[Tmkα^d†m(t)^dkα(t)−T∗mkα^d†kα(t)^dm(t)] (2.10)

We define the current deviation operator with a mean value of zero:

 △^Iα(t) = (2.11)

The two-time current correlator between leads and is defined as:

 Cαβ(t1,t2)≡⟨△^Iα(t1)△^Iβ(t2)⟩ (2.12)

This correlator obviously satisfies the symmetry property:

 Cαβ(t1,t2)∗=Cβα(t2,t1) (2.13)

Since and do not commute in general, is not guaranteed to be real and so in several studies the symmetrized correlation function is preferred [33, 34]:

 Pαβ(t1,t2)≡12⟨△^Iα(t1)△^Iβ(t2)+△^Iβ(t2)△^Iα(t1)⟩=Re[Cαβ(t1,t2)] (2.14)

Since is just the real part of , knowledge of the latter object is sufficient for a full characterization of the symmetric noise properties of the junction. The lack of two-particle interactions in the Hamiltonian (2.1) means we can simplify the non-symmetrized correlator using Wick’s Theorem, which is valid for a noninteracting Hamiltonian with arbitrary time-dependence [68]:

 −Tmkα(t1)T∗m′k′β(t2)⟨^d†m(t1)^dm′(t2)⟩⟨^dkα(t1)^d†k′β(t2)⟩
 −T∗mkα(t1)Tm′k′β(t2)⟨^d†kα(t1)^dk′β(t2)⟩⟨^dm(t1)^d†m′(t2)⟩
 (2.15)

One identifies the following Green’s functions in this expression:

 (2.16)
 (2.17)

where and correspond to either the lead or central molecule regions. It is then possible to rewrite Eq. (2.14) in the compact analytic form:

 Cαβ(t1,t2)=−4q2TrC[hCα(t1)G>αC(t1,t2)hCβ(t2)G<βC(t2,t1)−hCα(t1)G>αβ(t1,t2)hβC(t2)G
 −G>CC(t1,t2)hCβ(t2)G<βα(t2,t1)hαC(t1)+G>Cβ(t1,t2)hβC(t2)G

The expression (2.18) is structurally identical to current correlation functions in Refs. [81, 24], but we emphasize that here the two time Green’s functions appearing in Eq. (2.18) evolve in response to the switch-on of an arbitrary time-dependent bias in the partition-free approach, i.e. they contain convolution integrals taken along the vertical part of the Konstantinov-Perel’ contour as well. Notice that, in addition to correlation functions describing particle hopping events between the leads and the molecule, Eq. (2.18) also contains information on lead-lead hopping events and on ‘circular’ [24] currents involving electronic transport processes within the molecular structure. In some work on the time-dependent noise, the two-time correlator was given as a function of a single time [80], but we emphasize that we need to solve the Kadanoff-Baym equations for all Green’s functions ‘blocks’ in Eq. (2.4) in the two-time plane for a complete picture of current fluctuations. We present the main steps of this derivation in Appendix A, and the derived Green’s functions are inserted into Eq. (2.18), resulting in a sum of terms involving only self-energy components and components of the region Green’s function:

 +Σrα⋅(GrCC.Σ>β+G┐CC⋆Σ┌β))(t+1,t−2))⋅G
 +G>CC(t1,t2)(Σ<α(t2,t1)δαβ+((Σ<β⋅GaCC+Σrβ⋅G
 +Σrβ⋅(GrCC⋅Σ<α+G┐CC⋆Σ┌α))(t−2,t+1))
 −(Σ>α⋅GaCC+Σrα⋅G>CC+Σ┐α⋆G┌CC)(t+1,t−2)(Σ<β⋅GaCC+Σrβ⋅G
 (2.19)

Here the sign superscripts indicate the contour position of each time variable. So far, no assumptions have been made on the system Hamiltonian, i.e. we have not yet stated which regions are subject to a time-dependent perturbation, and neither have we made assumptions about the nature of the lead-molecule coupling. Up to this point, the derivation is completely algebraic, and so for noninteracting systems Eq. (2.19) is completely general.

### 2.3 Time-dependent Model and the WBLA

In this section we make assumptions on the model that enable us to solve the Kadanoff-Baym equations analytically. We assume that, prior to , the Hamiltonian is given by Eq. (2.1) with time-independent energies and molecular site and hopping integrals . The lead-molecule couplings are assumed to be present in equilibrium in the partition-free approach and unchanged by the switch-on process. As all subsystems are coupled during their equilibration, they all possess the same initial temperature and chemical potential , which means the system is initially described by the density operator (where is the partition function and is the number operator for the entire coupled system). Following Ref. [90], we add an arbitrary spatially homogeneous time-dependent shift to the lead energies as their bias. To the molecular Hamiltonian, we add a static correction [84], and a time-dependent shift that scales the particle number operator [90]:

 εkα(z∈C∓) = εkα+Vα(t) (2.20) Hmn(z∈C∓) = hmn+umn+δmnVC(t) (2.21)

Now we assume that the leads satisfy the WBLA, i.e. we neglect the energy dependence of the lead-molecule coupling. As described in Ref. [87], this assumption enables us to write down all components of the effective embedding self-energy in terms of the level-width matrix , defined as:

 Γα,mn=2π∑kTm,kαTkα,nδ(εFα−εkα) (2.22)

where is the equilibrium Fermi energy of lead . The self-energy components for this problem are collected together in Eqs. (B.5)-(B.10) of Appendix (B), where the time-dependence of the lead states is contained in phase factors of the form:

 ψα(t1,t2)≡t1∫t2dτVα(τ) (2.23)

Within the WBLA, the KB equations [66] for the different components of are linearized in terms of the effective Hamiltonian of the central region, where . The derivation of these components was published in Refs. [87, 90], and leads to the following compact formula for the greater and lesser Green’s functions:

 G≷CC(t1,t2)=∓i∫dω2πf(∓(ω−μ))∑γSγ(t1,t0;ω)ΓγS†γ(t2,t0;ω) (2.24)

where we introduce the matrix

 Sα(t,t0;ω)≡e−i~heffCC(t−t0)e−iφC(t,t0)⎡⎢⎣GrCC(ω)−it∫t0d¯te−i(ω1−˜heffCC)(¯t−t0)ei(φC−ψα)(¯t,t0)⎤⎥⎦ (2.25)

defined in terms of (i.e. defined without the tilde on the effective Hamiltonian), and the phase factor associated with the molecular time-dependence:

 φC(t1,t2)≡t1∫t2dτVC(τ) (2.26)

All other components of the GF can be explicitly calculated in the time domain [87, 90], and are listed in Appendix (B). The quantum statistical expectation value of the current operator (2.10) can also be reformulated as a sum of convolution integrals on the Konstantinov-Perel’ contour, which may be evaluated exactly within the WBLA. Setting the electronic charge , the current may be expressed in terms of the as [88]:

 Iα(t)=1π∫dωf(ω−μ)TrC[2Re[iΓαeiω(t−t0)eiψα(t,t0)Sα(t,t0;ω)]−Γα∑γSγ(t,t0;ω)ΓγS†γ(t,t0;ω)] (2.27)

The WBLA enables us to derive a closed form for the current correlation function. We substitute the equations (B.5) and (B.6) for the retarded/advanced self energies into Eq. (2.19), which then reduces to a rather compact form:

 Cαβ(t1,t2) = 4q2TrC[δαβ(Σ>α(t1,t2)GCC(t1,t2)Σ<α(t2,t1))
 +ΓαG>CC(t1,t2)ΓβG
 +i[Λ−α(t1,t2)Γβ+Γα(Λ−β)†(t2,t1)]G
 −Λ+β(t2,t1)Λ−α(t1,t2)−(Λ+α)†(t1,t2)(Λ−β)†(t2,t1)] (2.28)

Here we have collected convolution integrals on the Konstantinov-Perel’ contour into the objects :

 Λ+β(t2,t1) ≡ (Σ<β.GaCC+Σ┐β⋆G┌CC)(t−2,t+1) (2.29) (Λ+α)†(t1,t2) ≡ −(GrCC.Σ<α+G┐CC⋆Σ┌α)(t−2,t+1) (2.30) Λ−α(t1,t2) ≡ (Σ>α.GaCC+Σ┐α⋆G┌CC)(t+1,t−2) (2.31) (Λ−β)†(t2,t1) ≡ −(GrCC.Σ>β+G┐CC⋆Σ┌β)(t+1,t−2) (2.32)

We may now perform the convolution integrals in Eqs. (2.29)-(2.32) using the formulas obtained for the self energies and GFs of the region in Appendix B. The convolution integrals in are evaluated using the methods of Refs. [87, 90], where the transformation from Matsubara summations to frequency integrals [68] is done taking account of the ordering of time-variables on the contour. This guarantees the linearity of each term in the fermion/hole distribution function , and results in the following pair of functional identities:

 Λ+β(t2,t1) = ie−iψβ(t2,t0)∫dω2πf(ω−μ)e−iω(t2−t0)ΓβS†β(t1,t0;ω) (2.33)
 Λ−α(t1,t2)=−ie−iψα(t1,t0)∫dω2π(1−f(ω−μ))e−iω(t1−t0)ΓαS†α(t2,t0;ω) (2.34)

Here we have expressed in terms of the matrix defined in Eq. (2.25). Notice on the first line of Eq. (2.28) the presence of the object . In the single level case, all the objects inside the trace are replaced by scalars, and this object is equal to , where the particle number on the molecular region is defined by . The time-dependence of this object is thus entirely due to the internal dynamics of electron and hole populations on sites of the molecule. The lead dependent matrices and correspond physically to electrons propagating from lead and positively charged holes propagating from lead , respectively. We therefore interpret the two terms appearing on the second line of Eq. (2.28) as describing processes in which electrons in the leads interfere with holes in the molecular region, or holes in the leads interefere with electrons in the molecule. The terms on the third line of Eq. (2.28) are interpreted as cross-lead particle-hole interference terms.

In [87, 90], the greater and lesser Green’s functions were expressed in terms of the matrices following a line-integral of the Kadanoff Baym equations in the two-time plane, and these are given in Eq. (2.24). We thus have explicit formulas for all terms which appear in the two-time correlation function, which may be evaluated numerically in the plane as follows:

 (2.35)
 +i∑γ(ΓαSγ(t1,t0;ω)ΓγS†γ(t2,t0;ω)Γβ(e−iψβ(t2,t0)e−iω′(t2−t0)S†β(t1,t0;ω′)−eiψα(t1,t0)eiω′(t1−t0)Sα(t2,t0;ω′)))
 −(e−iψβ(t2,t0)e−iω(t2−t0)ΓβS†β(t1,t0;ω)e−iψα(t1,t0)e−iω′(t1−t0)ΓαS†α(t2,t0;ω′)
 +eiψα(t1,t0)eiω(t1−t0)Sα(t2,t0;ω)Γαeiψβ(t2,t0)eiω′(t2−t0)Sβ(t1,t0;ω′)Γβ)}

This expression contains a great deal of information, and it is the central result of this paper. It is the two-time correlation function for a molecular junction connected to an arbitrary number of leads, through which time-dependent voltages are passed. It contains transient parts which decay as , while remains finite. It automatically enables evaluation of cross-correlation functions between different leads when , and the correlation between currents through the same lead when . It describes the noise on the current signal due to non-zero temperatures (the thermal noise), and due to a non-zero bias (the shot noise), as will be made clearer in the next section. The leads are assumed to satisfy the WBLA, and the additive contribution of the voltage to the lead state energies is assumed, but the approach is otherwise exact for electrons interacting at the mean field level [38]. Under close inspection, using the definition (2.25), we find that the explicit time-dependence enters into (2.35) only within structures of the form , so that the noise does not distinguish between external fields that bias all leads identically (, for all ) or a gate voltage which moves energies in the negative energy direction (). This is also true for the current [90]. The expression (2.35) will be used for the proof of analytic identities in Section (3), but it is not entirely convenient for numerical evaluation. Instead, we describe in Section (4) and Appendix (D) how to evaluate Eq. (2.28) directly.

## 3 Recovery of Known Results For a Static Bias

To parameterize our system with experimentally relevant variables, we work in the relative time coordinate system so that and , where is the relative time that we wish to take a Fourier transform with respect to . Note that, to make the mapping to the Fourier space associated with , one needs to take on negative values. However, since both and must be times greater than , this means that is restricted to lie in the range , as was done in Ref. [71]. We define the Fourier-transform of the correlation with respect to the relative time , as a function of a single frequency and the measurement time :

 Pαβ(Ω,t)≡t−t0∫−t+t0dτeiΩτPαβ(t+τ,t)=12(Cαβ(Ω,t)+C∗αβ(−Ω,t)) (3.1)

where is the Fourier transform of with respect to . Note that the relation

 P∗αβ(Ω,t) = Pαβ(−Ω,t) (3.2)

immediately follows. In Section 2.2 we remarked that it is sufficient for knowledge of to know the non-symmetrized function .

In addition to the power spectrum, one can calculate several other useful quantities in terms of the . For instance, in a two-lead junction, one may focus on the net current,

 ^I(−)LR(t)=12(^IL(t)−^IR(t)) (3.3)

or on the sum of currents, which by the continuity equation is proportional to the rate of change of charge in the molecule [88]:

 ^I(+)LR(t)=12(^IL(t)+^IR(t)) (3.4)

The time-dependent noise spectra of these objects can be written:

 C(−)(Ω,t)=∫dτeiΩτ⟨△^I(−)LR(t+τ)△^I(−)LR(t)⟩=12(C(auto)(Ω,t)−C(×)(Ω,t)) (3.5)
 C(+)(Ω,t)=∫dτeiΩτ⟨△^I(+)LR(t+τ)△^I(+)LR(t)⟩=12(C(auto)(Ω,t)+C(×)(Ω,t)) (3.6)

where we have defined Fourier transforms of the average autocorrelation and cross-correlations:

 C(auto)(t+τ,t)≡12(CLL(t+τ,t)+CRR(t+τ,t)) (3.7)
 (3.8)

In general, and are complex quantities and so cannot be observed. However, due to the symmetry property (2.13), they are both real at the equal observation time point . This fact was exploited in Ref. [80], where the equal time autocorrelation in the left lead, , was studied in the time domain. Using the identity (2.13), one can show that the real parts of these functions are always symmetric in the line:

 Re[C(auto/×)(t+τ,t)]=Re[C(auto/×)(t,t+τ)] (3.9)

 Im[C(auto/×)(t+τ,t)]=−Im[C(auto/×)(t,t+τ)] (3.10)

To check the validity of our theory, we must confirm that it reduces to known expressions in the long time and static bias limits, as was already demonstrated for the current in Ref. [87]. We shall assume that the bias is applied only to the leads (), that the equilibrium and nonequilibrium effective molecular Hamiltonians are identical (), and that is constant in time (). In this case the defined in Eq. (2.25) can be evaluated explicitly, and in the limit we obtain

 Sγ(t1,t0;ω)ΓγS†γ(t2,t0;ω)t0→−∞⟶e−i(ω+Vγ)(t1−t2)Aγ(ω+Vγ) (3.11)

where . Other expressions appearing in the generalized two-time correlation function can be worked out in a similar way, for instance:

 e−iψβ(t2,t0)e−iω(t2−t0)S†β(t1,t0;ω)⟶t0→−∞GaCC(ω+Vβ)ei(ω+Vβ)τ (3.12)

The matrices enter into the general expression (2.35) only in the form of structures like (3.11) and (3.12), so we easily conclude that the correlation function depends only on the time difference , the power spectrum does not depend on time . Hence the current becomes a stationary stochastic process under the conditions that the bias is static and that the switch-on time is relegated to the distant past. This is implied by the fact that the current itself is simply the steady-state LB formula in this case, as it was proven in [87] that all terms arising from the initial conditions (vertical contour convolutions) vanish in the long-time limit. With the exception of the initial condition term, every vanishing term includes a convolution with a left or right self-energy. In the partitioned approach to the transport problem, these quantities vanish, as one can see from the definition (2.6) and the fact that for all . It also enables us to write down a generic formula for the lesser/greater Green’s functions and matrices that can be chosen either partitioned or partition-free, for all times, by premultiplying all terms arising from a vertical contour convolution integral by the partitioning parameter

 ξp={1,partition-free0,partitioned (3.13)

We include this parameter in the formulas for numerical implementation in Appendix D, which will enable us to directly compare the noise resulting from a partitioned and partition-free switch-on in Section 4 of this paper. In studies of high-frequency shot noise, the interesting physical observable is usually the static non-symmetrized power spectrum [28, 31], which is the regular Fourier transform (denoted via hereafter) of :

 Cαβ(Ω)≡limt0→−∞Cαβ(Ω,t)=∞∫−∞dτeiΩτCαβ(τ)≡F[Cαβ(τ);Ω] (3.14)

Note that infinite limits are possible here as . The above quantity satisfies the relation [94]. For those experiments which do distinguish between absorption and emission processes, the quantity of interest is most often , which in general satisfies the inequality . can therefore be used to describe measurements in which a quanta of energy is transferred from the measuring device to the system. By contrast, the symmetrized spectrum obeys , i.e. it does not distinguish between emission and absorption processes. Moreover, in recently published work [81], a master equation formalism was used to derive an exact formula for the frequency-dependent autocorrelation and cross-lead current correlations in a nanojunction composed of a quantum dot coupled to two leads, which were treated within the WBLA. In Appendix C, we derive an explicit formula for . Here we simply note that, if the discussion is restricted to a molecule coupled to left (L) and right (R) leads, we find that the non-symmetrized autocorrelation associated with a single lead is given by

 Cαα(Ω)=4q2∫dω2πTrC[(1−fα(ω+Ω−μ))f¯α(ω−μ)T(α¯α)CC(ω)T†(α¯α)CC(ω)
 +(1−f¯α(ω+Ω−μ))