D.2 Non-Interacting Bosons

The HAMILTONian for non-interacting phonons (Bosons) in momentum representation is

$\displaystyle H_{0} \ = \ \sum_{{\bf q},\lambda} \hbar\omega_{{\bf q},\lambda} \left(b^\dagger_{{\bf q},\lambda} b_{{\bf q},\lambda}\ + \frac{1}{2}\right) \ ,$ (D.11)

where $ \hbar\omega_{{\bf q},\lambda}$ is the energy of mode $ {\bf q}$ with the polarization $ \lambda$, $ b_{{\bf q},\lambda}$, and $ b^\dagger_{{\bf q},\lambda}$ are the Bosons annihilation and creation operators. The time-evolution of the annihilation operator in the HEISENBERG picture is

$\displaystyle b_{{\bf q},\lambda}(t) \ = \ e^{iH_{0}t/\hbar}\ b_{{\bf q},\lambda} \ e^{-iH_{0}t/\hbar} \ ,$ (D.12)

so the operator obeys the equation

$\displaystyle i\hbar \partial_t b_{{\bf q},\lambda}(t) \ = \ [b_{{\bf q},\lambda}(t),H_{0}] \ = \ \hbar\omega_{{\bf q},\lambda} b_{{\bf q},\lambda}(t)\ ,$ (D.13)

which has the solution

$\displaystyle b_{{\bf q},\lambda}(t) \ = \ e^{-i\omega_{{\bf q},\lambda}t}\ b_{{\bf q},\lambda} \ .$ (D.14)

The creation operator for Bosons is the just the HERMITian conjugate of $ b_{{\bf q}}$, i.e.

$\displaystyle b^\dagger_{{\bf q},\lambda}(t) \ = \ e^{+i\omega_{{\bf q},\lambda}t}\ b^\dagger_{{\bf q},\lambda} \ .$ (D.15)

The non-interacting real-time GREEN's functions for Bosons in momentum representation are now given by

\begin{displaymath}\begin{array}{ll}\displaystyle D_{\lambda_0}^{<}({\bf q},t;{\...
... \ e^{-i\omega_{{\bf q},\lambda}(t-t')} \right] \ , \end{array}\end{displaymath} (D.16)

where $ \hat{A}_{{\bf q},\lambda}(t)=b_{{\bf q},\lambda}(t)+b^\dagger_{{\bf -q},\lambda}(t)$, $ \hat{A}^\dagger_{{\bf q},\lambda}(t)=\hat{A}^\dagger_{{-\bf q},\lambda}(t)$, $ \omega_{{\bf
-q},\lambda}=\omega_{{\bf q},\lambda}$, and $ n_{{\bf q},\lambda}=\langle
b^\dagger_{{\bf q},\lambda} b_{{\bf q},\lambda}\rangle$ is the occupation number of the state $ ({\bf q},\lambda)$, where under thermal equilibrium one obtains $ n_\mathrm{{\bf q},\lambda} = n_\mathrm{B}(\hbar
\omega_{{\bf q},\lambda})$, with $ n_\mathrm{B}$ denoting the Bose-EINSTEIN distribution function (Appendix C.2). The GREEN's functions depend only on time differences. One usually Fourier transforms the time difference coordinate, $ t-t'$, to energy

\begin{displaymath}\begin{array}{ll}\displaystyle D_{\lambda_0}^\mathrm{<}({\bf ...
...\ \frac{1}{E+\hbar\omega_{{\bf q},\lambda}-i\eta} , \end{array}\end{displaymath} (D.17)

M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors