D.2 Non-Interacting Bosons

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

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

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

so the operator obeys the equation

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

which has the solution

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

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

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

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

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

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

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