Computational Nanophotonics

6.5.2.3.3  Laser Mode Density Matrices

In order to trigger quantum processes in the transmit node for generating photon state, the lasers $L_{01}^{\left(0\right)}$ and $L_{01}^{\left(1\right)}$ are simultaneously turned on, and afterward the remaining two lasers $L_{12}^{\left(0\right)}$ and $L_{12}^{\left(1\right)}$ are also turned on. Therefore, initially the two lasers $L_{01}^{\left(0\right)}$ and $L_{01}^{\left(1\right)}$ are prepared with N₀ photons each where N₀ ≫ 0, and the other two lasers $L_{12}^{\left(0\right)}$ and $L_{12}^{\left(1\right)}$ with zero photon. Basis vectors of these laser modes in the corresponding Hilbert spaces were defined in Equation 6.76. Based on these basis vectors and the said initial conditions, the initial density matrices for the lasers $L_{01}^{\left(0\right)}$ and $L_{01}^{\left(1\right)}$ can finally be written as

${\text{ρ}}_T^{\left(L01-0\right)}|{t=0}_{}=\left[\begin{array}{ccccccc}0& 0& \cdots & 0& 0& \cdots & 0\\ 0& 0& \cdots & 0& 0& \cdots & 0\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 0& 0& \cdots & 1& 0& \cdots & 0\\ 0& 0& \cdots & 0& 0& \cdots & 0\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 0& 0& \cdots & 0& 0& \cdots & 0\end{array}\right]$

(6.111) ...