Keldysh contour

There is no Hartree term for lesser function, because the times t and t2 are always at the different branches of the Keldysh contour, and the <5-function <5(ti — T2) is zero. 

Note, that the expressions for the retarded and lesser functions, described above, can be obtained in a more formal way by the EOM method formulated on the Keldysh contour. 

The ordering operator Tc places the operators in the Taylor expansion of the Tc exponent to the left with a later-in-time variable on the contour y. The operators are taken in the interaction representation on the Keldysh contour. 

Figure 5.1 The Keldysh contour in the complex plane with t on the forward branch and t on the backward branch. The contour begins at t = 0 on the upper branch and ends at t = — if) on the vertical branch. Here denotes the forward branch, whereas +" represents the backward branch.

Before defining nonequilibrium Green s functions, we introduce field operators (e.g., a fermion field) y/ lr) and anticommutator relation t/r(r), t/rt(r )J = 5(r — r ). The Green s function on a Keldysh contour is defined as... 

To account quantum correlations of the function between real and imaginary times on the Keldysh contour, we additionally introduce the following... 

The function with time arguments on both real and imaginary time branches of the Keldysh contour can be presented in the following manner ... 

To understand the diagrammatic approach, which is introduced below, we first determine the nonequilibrium Green s functions for uncorrelated electrons in the leads and the bridge on a Keldysh contour employing a diagrammatic expansion rather than the equation of motion [45,52,53]. For such a system, the Hamiltonian is given by ... 

The nonequilibrium zeroth Green s functions are determined by the Dyson equations (62) and (63) on the Keldysh contour. The standard way to solve these equations is to perform a Fourier transform and then solve the algebraic matrix equations for the Green s functions. For the Keldysh functions, this procedure cannot be implemented in a straightforward way because of two time branches. Thus, we should find the Fourier transform for each Keldysh function after applying the Langreth s mapping procedure described in Section 2 [41, 45]. In particular for — t ), the Dyson... 

The detailed diagrammatic analysis results in the following Dyson integral equation for the mixed Keldysh function on the Keldysh contour [99] ... 

The derivation on a Dyson equation on the Keldysh contour is similar to the derivation presented in Section 4 [54]. The difference is in the interactions. In the present derivation, we also include the interaction with irradiation. For any order diagram with respect to the Coulomb interaction, the semiconductor-quantum dot and quantum dot-light interactions retains the topological structure of a graph in the same manner as for noninteracting case where the ordinary zeroth Green s functions are substituted by the renormalized zeroth Green s functions described by Eqs. (121) and (122). 

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