We employ the standard device for obtaining a non-equilibrium state. At time
, prior to which the system is assumed to be in thermodynamic equilibrium
with a reservoir, the system is exposed to
a disturbance represented by the contribution
HAMILTONian. The external perturbation can for instance be a time varying
electric field, a light excitation pulse, and so forth. The total HAMILTONian is thus
for . One is not restricted to using the
statistical equilibrium state at times prior to as the initial
condition. As shown by , a non-equilibrium situation can be
maintained through contact with a reservoir. A discussion of the coupling of a
system to a reservoir has been studied in .
statistical mechanics is concerned with calculating average values
of physical observables for times
. Given the density operator
, the average of any operator is
then defined as (C.8)
is an operator in the HEISENBERG picture.
The non-equilibrium GREEN's function can be defined as
is the field operator in the HEISENBERG picture
evolving with the HAMILTONian
in (3.16) and the bracket
is the statistical
average with the density operator defined in (3.17).
One can evaluate GREEN's functions by using WICK's theorem, which enables
us to decompose many-particle GREEN's functions into sums and products
of single-particle GREEN's functions (see Section 3.4.1).
The restriction of the WICK theorem necessitates that the field operators and the
density operator have to be represented in the interaction
picture, or equivalently, their time evolution is governed by the
non-interacting HAMILTONian . The contour-ordered GREEN's function,
which is introduced next, provides a suitable framework for this purpose.
M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors